Philosophy in Progress

The Sheffer stroke, and the blandest antirealism ever

2011-12-09T09:22:00.000-08:00

In college my metaphysics prof said realism was the doctrine that there is a complete true description of the world, though it might take infinitely many sentences to express this description. Do the following incredibly bland considerations about different truthfunctional connectives commit me to "antirealism"?

(1) If you want thinking about propositions to play a role in psychology, then you need to individuate propositions narrowly enough that truthfunctionally equivalent sentences express different propositions.

(2) If you individuate propositions as narrowly as required by (1) then tautologies involving the sheffer stroke will be different propositions than any corresponding sentences which use more standard truthfunctional connectives.

(3) The same argument goes for all the other infinitely many n place truthfunctional connectives.

Conclusion: no language with finitely many basic connectives can express all true propositions. No finite language can be used to give a complete true description of the world. In particular if the a language L has only n place truthfunctional connectives then there will be tautologies using n+1 place truthfunctional connectives that cannot be expressed in L.

"Just what I mean by the term"

2011-12-09T09:15:00.000-08:00

Whether or not you think there are philosophically interesting facts about analyticity, normal people do respond to certain a) epistemic challenges and b) calls for scientific/philosophical explanation by saying 'that's just what I mean by the term'.

So I think any philosopher who doesn't want to accept massive error about justification and argument owes some account of what is going on here. What does this expression do?

Maybe one thing the claim 'x is just part of what I mean by the term' maybe does is diffuse the expectation that there is an interesting natural kind or scientific fact in the neighborhood. Here's what I mean...

Suppose I had a made a map with mountain ranges labeled and you said, how come all mountains have a slope of at least 1 percent (cf. wikipedia definition). If you said "how do you know that mountains all have a slope of less than 1percent?" or "why do all mountains have a slope of less than 1 percent" I might say `that's just what I mean by the term mountain'.

I *think* hearing this might helpfully lead us to rule out possibilities like the following: 99% of paradigm mountains are caused by a certain geological process, and this process always produces things that have a steep slope, and then reliably maintain this steep slope through the process of erosion. If there was some more natural kind in the neighborhood of "mountain" then there would be a less trivial answer to questions like 'why do all mountains have at least a 2% slope?' and 'how do you know that all mountains have at least a 2% slope.

Pictureability and Definiteness in Mathematics

2011-11-01T04:19:00.000-07:00

A fairly popular position in philosophy of mathematics is the following: there are definite right answers to questions about arithmetic, but not to questions about set theory (especially not to questions like the Continnum Hypothesis whose answers can be changed by forcing), because we have a clear notion of what the numbers are supposed to be like but not what the sets are supposed to be like.

This claim about clarity is sometimes supported by appeal to notions of mental picturing. The idea would be that we can picture all the numbers, but that we can't picture things like `all the subsets' of the numbers, or the hierarchy of sets as an item that contains, at each level, all subsets of the levels below. But I want to ask: In what sense can we picture all the numbers? In what sense CANT we picture the sets?

All the ways I can think of of picturing all the numbers involve some kind of "..." e.g. you might picture all the numbers by having a mental picture like the physical picture I have drawn below:

| || ||| |||| ....

When we use this picture to conceive of the structure of the numbers, we are employing a highly non-trivial method of picturing, which lets us represent the state of there being a countable infinity of numbers by considering some finite collection of drawings like the above.

And people certainly do *in some sense* draw pictures of the hierarchy of sets. At the beginning of a set theory class (and at the beginning of my college analysis class) the professor will frequently draw up a V on the board. In a way that is a picture of the heirarchy of sets. And, presumably, one could mentally picture the heirarchy of sets in the same.

For this reason I am (a little) skeptical about justifying the claim that we have a definite idea about the sets but not the numbers by appeal to considerations about what we can mentally picture. We can picture the numbers if we are willing to let "..." represent the idea of all finite successors to a stroke symbol. We can picture the hierarchy of sets if we are willing to let the increasing girth of the V represent the idea that each level of the hierarchy of sets contains all possible subsets of the level below. What (mathematical) objects one can picture depends on what methods of picturing we will accept, as allowing one to genuinely conceive of a situation by entertaining a mental picture.

At the very least, I think that if someone wants to argue that claims about numbers are definite and claims about sets are not on the basis of appeals to mental pictures, they owe us some kind of story or explanation for why one method of picturing is OK and the other is not.

Hammond Questions for Philosophy

2011-10-23T08:46:00.000-07:00

I was just taking a break by rereading this heartwarming little essay , which you should definitely check out. This made me think about what some "Hammond Questions" for philosophy are (i.e. big questions, which we nonetheless seem to be in a position to reasonably attack now)

1. Is there a meaningful distinction between statements that are "true by convention" and those that "state substantive claims about the world"? If not (I would guess there is not) what genuine distinction are people getting at when they say:
-it's a matter of convention that mountains have to be 100 feet
-the fact that I am a US citizen is a social/conventional truth (?)
-deliciousness and disgustingness are not something independently in the world but something that we project onto the world (???)
I suspect different things are at stake in each of these cases.

2. Why, despite our ability to quickly learn new words like `table', `policeman' and `gouche' has it proved to be so hard (impossible?) for philosophers to produce elegant necessary and sufficient conditions for the application of these terms using more primitive ones. Is it...
-that in learning new words we learn paradigm cases not definitions? (but cf. the known problems with e.g. trying to account for understanding "green apple" in terms of composing paradigm cases of "green" and "apple")
-that there is an adequate definition for being a policeman etc. to be found but they are ugly enough to be considered by linguists not philosophers (e.g. maybe they typically have many special clauses)?
-that external facts about natural kinds help determine the extension of terms like policeman, in a way that either gives these terms a very messy relationship to the extensions of other terms that we understand and/or makes it epistemically difficult for us to figure out what these relationships are?
-that there are clean necessary and sufficient conditions to be given, but `atoms' of these definitions turn out to be metaphysically wild and wooly notions like `purpose' `agent' `blameworthy' so that providing necessary and sufficient conditions for claims about policemen, tables, etc. in terms of primitives like these doesn't feel like (and maybe isn't) making philosophical progress.

2' Why has it proved so hard for philosophers to paraprhase away ceterus paribus clauses, despite their apparently unproblematic use in the sciences.

3. Do facts about forcing independence and large cardinals have any consequences for the trilemma about how to think about the height of the hierarchy of sets. If mathematical facts can't decide this issue, what can?

4. What are the truth or assertability conditions for claims about literary "function" e.g. x foreshadows y, x alludes to y, x raises questions about whether y? Provide a metaphysical story about what makes statements of the above kind true, plus corresponding "logic" of literary criticism e.g. a formal algorithm that captures many if not all truth/assertability preserving inferences about literary function? Does this
-bonus: add axioms and inference rules to your logic of literary function until you get something that captures all intuitively valid forms of argument, or prove that no recursively axiomatizable logic can do this.
-bonus: determine what if any relationship there is between claims about literary function and biological function (I guess Kant thought there was a connection but he clearly likes cute solutions too much to be trustworthy on a matter like this!)

5. In virtue of what does a piece of music count as expressive of sadness, excitement etc.? If these facts are relevant to some parameter like species or prior musical tradition state the relevant parameter. This is an old and daunting question but...
Currently fashionable psychological research into how pieces of music produce similar *judgements* about expressing sadness excitement etc. in different people may suggest promising new proposals with regard to the philosophical question of what features of music make it *count* as expressing sadness, excitement etc. What kinds of lower level features do people seem to be causally responding to when they say that a piece of music is sad, and what if any general-purpose causal reasoning is involved in these judgements?

Justifying Logic and the Normal Role of Proof in Justification

2011-03-29T06:22:00.001-07:00

Some philosophers aim to show how we can be justified in accepting certain basic logical truths by giving "rule circular" proofs of the soundness of these basic logical truths. They admit that most people will never have considered the proofs in question, and they admit that these people still count as justified in using logic. But, they say that they showing that such proofs can in principle be given makes sense of how we can be justified in believing the basic logical claims established by the proofs right now.

That idea seems prima face implausible. In general the fact that someone 100 years later will prove P from premises that I accept (like the ZF axioms) doesn't suffice to show that I am justified in believing that P now. So why should the case be any different for the proofs of logical principles?

[I would rather say that we are prima facie justified in believing these logical principles in a way that has nothing to do with the possibility of giving further argument; coming up with more or less circular ways of proving the soundness of our logical principles is (at best) a way of improving our justification]

are stipulative definitions a source basic knowledge?

2011-03-19T22:02:00.000-07:00

Random thought:

Whether or not its OK to make a certain stipulative definition can depend very messy questions - and not just mathematically messy questions like questions about harmony.
For example: it would seem that it's OK to stipulate that people are to count as "gleb" whereas bodies are not to count as "gleb" if and only if people are distinct from their bodies.

This suggests that knowledge by stipulative definition is not a source of basic knowledge. (basic knowledge= justified belief that doesn't depend on any other beliefs for justification) For, you can say 'of course people are gleb and bodies aren't, thats just what I mean by the term! remember when I stipulatively defined it...'. But (it would appear) the justificatory buck doesn't stop when you say this. If you are unjustified in thinking that bodies are distinct from people, this would seem to poison your justification for making and appealing to this stipulative definition.

However, perhaps we should say that only some stipulative definitions do have prima facie warrant, and the above stipulation about glep is just not one of the ones that does.

p.s. if we say that stiplative definitions aren't basic knowledge, we will probably want to say that analyticities aren't either.

Angst Over the Ordinals

2011-02-06T08:38:00.000-08:00

Ohhh, which of these three options is correct? Given my focus on philosophy of math it's mildly embarrassing not to have a fixed position on this, but I keep going back and forth...

1. Just say the hierarchy of sets goes "all the way up"

2. Say the hierarchy of sets goes "all the way up" in the sense that it contains ordinals corresponding to every distinct combinatorially possible way for some objects to be well ordered *except for the one that it, itself is an instance of*. (this would be appealing but i think it may be impossible to spell out in a consistent way)

3. Say that the hierarchy of sets goes up at least far enough to satisfy the axiom of infinity+ the rest of ZF, and leaves it vague what there is beyond the inacessables - much as our concept mountain leaves it vague how many really tiny mountains there are given that there is such and such a bit of lumpy terrain.

Five Reasons to be a Modal Carnapian

2011-02-06T08:00:00.000-08:00

I currently believe mathematics is best understood in terms of combinatorial possibility plus Carnapian framework stipulations. One reason for thinking this is, of course, that understanding mathematical objects in this way lets you tell a nice story about access to abstract mathematical objects, like the one I tell in my dissertation! But here are 4 other reasons.

1. Thinking in terms of combinatoiral possibility lets you solve the bad company objection for neo-carnapianism about objects.

You can say: It is OK to stipulate that there are sets but not to make the stipulations for tonk, or for Boolos parities because...

One can safely extend one's language by introducing object stipulations S if and only if it would be combinatorially possible for the objects one's language currently acknowledges at each possible world to be supplimented by new objects in such a way as to satisfy S.

2. Thinking about things in terms of combinatorial possibility + carnapian framework stipulations explains why certain mathematical questions are substantive questions while others are not (substantive ones involve disagreement over combinatorial possibility as well as over which from among the many objects which it would be combinatorially possible to stipulate you actually do stipulate)

For example:

There is no substantive mathematical question about whether there are sets or categories or both, because everyone agrees that it would be combinatorially for there to be objects satisfy Carnapian framework stipulations phrased in terms of combinatorial possibility for the sets, and for the categories.

There is a substantive mathematical question about whether the Goldbach conjecture is true, because it is combinatorially impossible that the Fs and the Gs should satisfy the informal (non-first order logical!) framework stipulations for sets could yield different answers to Goldbach.

3. "Mathematical Hypothesis"-based alternatives like structuralism, fictionalism and plenetudinous platonism already need *some* powerful modal notion or a sense of possible and impossible which can take into account vocabulary that goes beyond the connectives of first order logic, if they want to capture intuitive claims about all questions in arithmetic having right answers.

Intuitively there are right answers to at least all questions about arithmetic. Each first order logical hypothesis about the numbers necessitates right answers to all questions about arithmetic. Therefore, any mathematical hypothesis that claims to cash out the truth conditions for our claim that phi in terms of the claim that if H then phi must crucially use non-first order vocabulary. Therefore if you want to make some hypothetical view of math work, you must either bite the bullet that some simple questions of arithmetic have no answer, or invoke a more mathematically powerful notion of possibilty/consequence

4. The notion of combinatorial possibility is very crisp and simple, indeed it maybe well the crispest and simplest notion that yields enough mathematical power to reconstruct intuitive verdics about truthvalues in arithmetic as per (3).

For example, if you accept Tarski's definition of logical truth in terms of facts about how it would be possible to reinterpret all properties and relations involved (where possible doesn't mean that a person could give a rule for how to do it, or somehow list the new extensions) then you pretty much already accept the powerful notion of how it would be in principle possible for an arbitrary n-place relation to apply to some objects which I have in mind when I talk about combinatorial possibility. So if you're not too much of a finitist/intuitionist to accept the very broad sense of in principle possible reassignment of extensions to predicates used in the Tarski definition, I think you should be OK with my notion of combinatorial possibility.

Secondly, combinatorial possibility faces none of the quandires that make people skeptical about the notion of metaphysical possibility e.g. it is possible for something to be both red all over and green all over, or for there to be zombies? We avoid these problem because facts about combinatorial possibility ignore all metaphysical facts about particular properties and relations involved. In some cases one might perhaps argue that some natural language sentences are vague with regard to what pattern of relationships between objects they assert. But once you specify that you are asking whether e.g. combiposs( Ex Redallover(x)&Greenallover(x)), it is perfectly clear that this is combinatorially possible, since it would be combinatorially possible to choose extensions for Redallover() and Greenallover() that make this true.

What is Combinatorial Possibility?

2011-02-06T07:12:00.000-08:00

At the moment I think mathematics is best understood in terms of neo-carnapian/neo-logicist existence conditions for mathematical objects plus a kind of specifically mathematical modality (along the lines considered by Charles Parsons) which is looser than metaphysical possibility, and which I call "combinatorial possibility".

Here's a new way I thought of to explain what I mean by combinatorial possibility:

Combinatorial possibility is just like Tarski's notion of logical possibility, except a) we drop the assumption that you have to shanghai objects from the actual world to use in making a given claim true, allowing arbitrary choices of domain as well as arbitrary reassignments of extensions to properties and b) we allow more vocabulary to count as "logical vocabulary" in the sense that when reassigning extensions to predicates you can't change its meaning.

What more vocabulary?

In general anything like "finitely many" or "equinumerous" which functions like logical vocabulary in a sentence that its effect on the truthvalue of the whole sentence is systematically determined just by the domain and extension of relations in a set theoretic model is OK. Call these semi-logical expressions.

However I conjecture that we can cash out all of these all the semi-logical expressions, or at least all the ones that occur in modern mathematics and and its applications by merely using two things:
- an actuality opporator @F, so that you can say things about how it would be combinatorially possible for the things that are actually kittens, to be related by liking to the things that are actually baskets, e.g. there are enough different kittens and few enough baskets that it is combinatorially impossible that each kitten slept in a different basket last night.
- nesting, so that you can ask questions about the combinatorial possibility or impossibility of questions which are themselves described in terms of how it would be combinatorially possible to supplement them e.g. There are people at this party representing all combinatorially possible choices of whether to take sugar and/or milk in your tea = It would be combinatorially impossible to supplement the actual people with some additional person who differs from all actual people with respect to either whether they take coffee or whether they take tea = not combiposs(Ex person(x) & Ay [if @person(y) then ~x=y , and takessugar(x)iff~takessugar(y) and takesmilk(x) iff~takesmilk(y)) ] *

*yes, when you have multiple nesting of claims about combinatorial possibility, actuality opporators will need to be indexed to a particular instance "combiposs" so you will have combiposs_i and @F_i.

Three Arguments for A Priori Knowledge of (Very) Contingent Facts

2010-12-25T11:24:00.000-08:00

Can we have a priori knowledge of contingent facts? For example, consider the proposition below. Can we know truths like the following a priori? NOT PEA SOUP: 'It is not the case that everything outside of a 5 foot radius around me is made of pea soup, which stealthily forms up into suitable objects as I walk by' Here are three positive arguments (in ascending order of strength IMO) for the conclusion that we can know NOT PEA SOUP a priori.

1. Argument from Crude Reliablism

The belief-forming method of assuming that you aren't in a pea soup world is reliable. And even if we make things a little less crude by saying that good belief formation is belief formation that works via a chain of methods which are *individuated in a psychologically natural way* and are reliable, we will probably still get the same conclusion. For plausibly the most natural relevant psychological mechanism involved in generating that belief would be something like, 'believe not P when P is sufficiently gerrymandered'.

2. Argument from Probability and Conditionalization-Based Models of Good Inference

If you think that good reasoning is well modeled by the idea of assigning a certain probability measure to the space of possible worlds, and then ruling out worlds based on your observation, and asserting that P if and only if a sufficient fraction of the remaining probability is assigned to worlds in which P. There will be some propositions P that low enough prior probability to warrant asserting ~P before you have made any observations - and plausibly the pea soup hypothesis is one of them. Presumably in such cases your justification does not depend on experience. [I think Williamson has something like this in mind in one of his papers on skepticism, but his argument was more complicated]

3. Argument from Current Knowledge plus Inability to Cite Experiential Justification. The claim that NOT PEA SOUP is a priori follows from a claim about knowledge that only a skeptic would deny, plus a somewhat intuitive claim about the relationship between a priority and justification. The intuitive claim I have in mind is that if someone can count as knowing that P, without being able to point to any relevant experience (or memory of experience, or reason to believe that they had experience etc) as justification then they know that P a priori so P is a priori (i.e. a priori knowable). Everyone but the skeptic agrees that people know that they aren't in the pea-soup world. These people who know cannot point to any experience as justification. Hence, 'not-pea soup' must be knowable without appeal to experience for justification. You might try to defend the a posteriority of NOT PEA SOUP by saying that even if the man on the street can't make any argument from experience to NOT PEA SOUP, our intuition that people know that NOT PEA SOUP is based on the assumption that there exists some good argument from something about experience to NOT PEA SOUP, and philosophers just need to discover it. In this way, experience really is necessary to justify the belief that NOT PEA SOUP so the proposition is a posteriori.

However, this response threatens to generate the unattractive conclusion that people today do not know NOT PEA SOUP. For, in general, the mere existence of a good argument for some proposition that I believe does not suffice to make me justified in believing that proposition now, if I cannot (now) give that argument. If I believe some mathematical theorem T on a hunch or on the basis of tea leaf reading, the mere fact that there is a good argument for T on the basis of things that I accept, doesn't suffice to allow me to count as knowing that T. So even if there is some cunning philosophical argument yet to be discovered which justifies NOT PEA SOUP on the basis of experience, it would seem that this argument cannot suffice to justifies people now accepting that NOT PEA SOUP. If people now are justified that NOT PEA SOUP, and can give no argument from experience for this claim, it must be that the claim can be justifiably believed without appeal to experience.

Reliablism and the Value of Justification: The Angel's Offer

2010-12-19T09:39:00.000-08:00

A major objection to reliablism about justification is that it doesn't explain why we value having knowledge of a given proposition more than mere dogmatic true belief. For, believing a true proposition via a method that's reliable is just more likely to lead you to believe other true propositions; there's no obvious sense in which your relationship to true beliefs formed by reliable methods is thereby intrinsically any better or more valuable than you relation to mere true beliefs. If we don't like a particular cup of good expresso any better for it being the product of a machine that reliably makes good expresso, why should we like a particular state of believing a truth any better from the fact that it was produced by processes that reliably lead to believing the truth?

But maybe we DON'T value having the special relation we do to justified true beliefs over and above it's tendency to promote having stable true beliefs. Consider this thought experiment:

An angel convinces you that he knows the true laws of physics and maybe also that it can do super-tasks and thereby knows certain statements of number theory which cannot be proved from axioms which you currently accept. The angel offers to make it the case that you find these true principles feel obvious to you - the way that you now feel about 'I exist' or '2+2=4'. He will wipe your memory of this conversation so that you will not be able justify these feelings to yourself by appeal to the reliable way you got them - but of course you won't feel the need to justify them to yourself since they will just feel obvious and you will be inclined to immediately accept them. [Suppose also, if it matters, the angel will do the same to everyone in your community, that community members prefer to go along with whatever choice you make, that the angel is already going to blur your memories of not finding these claims obvious in the past etc.]
Would you accept the offer?

I personally would definitely take the offer. And I think many people would share this preference. If there were something intrinsically valuable about knowing verses merely dogmatically assuming a necessary truth, then this would be a strong reason not to take the angels offer. But if Plato is right (as thinking about the example tempts me to think that he is) to say that the only bad thing about dogmatically assuming truths rather than knowing them is that dogmatic assumptions don't stay tied down, then the angel's offer to make you and everyone else in your community find these truths indubitable fixes that problem - and you should take him up on his offer.

Dilemma re: (Platonist) Structuralism

2010-12-17T00:56:00.000-08:00

I've got to go reread my Shapiro. But before his smooth writing bewitches me, let me note down the very simple objection that I am currently unable to see how he would answer.

Structuralism is traditionally motivated by the desires to address a problem from Benacerraf: that there are multiple equally good ways of interpreting talk of numbers as referring to sets, so that either answer to "what set is the number 3" seems unprincipled. But now:

If you are not OK with plentiful abstract objects, you can't believe there are abstracta called structures.

If you are OK with plentiful abstract objects, then you can address this worry by just saying that the numbers and sets are different items. Certain mathematics textbooks find it useful to speak as though 3 were literally identical to some set, but this is just a kind of "abuse of notation" motivated by the fact that we can see in advance that any facts about the numbers will carry over in a suitable way to facts about the relevant collection of sets named in honor of those numbers. One might argue that analogous abuse of notation happens all the time in math e.g. writing a function that applies to Fs where you really mean the corresponding function that applies to equivalence classes of the Fs. This route seems like a much less radical move than claiming that basic laws about identity fail to apply to positions in a structure e.g. there is no fact of the matter about whether positions in two distinct structures (like the numbers and the sets) are identical.

Cog Sci Question

2010-12-14T02:25:00.000-08:00

[edited after helpful conversation with OE where I realize my original formulation of the worries was very unclear]

I was just looking at this cool MIT grad student's website, and thinking about the project of a) reducing other kinds of reasoning to Baysean inference and b) modeling what the brain does when we reason on other ways in terms of such conditionalization.

This sounds pretty good, but now I want to know:

a) What is a good model for how the brain might do the conditionalization? Specifically: how could it store all the information about the priors? If you picture this conditionalization in terms of a space of possible worlds, with prior probability spread over it like jelly to various depths, it is very hard to imagine how *that* could be translated to something realizable in the brain. It seems like there wouldn't be enough space in the brain to store separate assignments of prior probabilities for each maximally specific description of a state of the world (even assuming that there is a maximum "fineness of grain" to theories which we can consider, so that the number of such descriptions would be finite).

b) How do people square basing everything on Baysian conditionalization with psychological results about people being terrible at dealing with probabilities consciously?

Google turns up some very general results that look relevant but if any of you know something about this topic and can recommend a particular model/explain how it deals with these issues...

Putnam Indeterminacy Dilemma

2010-11-24T14:46:00.000-08:00

Putnam uses Skolem's theorem (every consistent first-order theory has a model whose domain is the integers or some subset thereof) to argue that the meanings of our sentences are indeterminate.

If considerations of elegance CAN make something a more natural candidate for the meaning of a given word (e.g. someone with behavior that doesn't distinguish between plus and quus means plus), then the mere existence of some (clumsly and arbitrary) Skolem model doesn't pose a problem for our meaning something definite - since the Skolem model's interpretation of expressions like "all possible subsets" will be much less elegant than the natural one.

If considerations of elegance CAN'T make something a more natural candidate for the meaning of a given word, then Putnam is wrong to assume that even the meanings of the first order logical connectives which his perverse Skolem model captures are pinned down. For why think that we mean 'or' rather than a quus like version of 'or' that starts behaving like `and' in sentences longer than a billion words long?

Old Evidence and Apologies

2010-11-21T20:02:00.000-08:00

If the problem of old evidence for Bayesian epistemology is just the following, then I don't think it's a problem:

Sometimes it seems like we should change our probabilities based on discovering logical consequences of a theory, but Bayesian updating only involves changing probabilities when you make a new observation.

For (it seems to me) this objection has the same ultimate structure as the following, surely bad, objection:

Sometimes it seems like we should apologize, but obeying so-and-so's moral theory involves never wronging anyone - and hence never apologizing.

If old evidence E is logically incompatible with hypothesis H, then Bayesianism says that you should *already* have ruled out all the worlds where H is true, and changed your probabilities accordingly, whenever you observed that E. So, I see no problem for the Bayesian epistemologist in saying that when you discover that you have failed to update in the way required by the theory (by not noticing a logical incompatibility), you should fix the mistake and change your probabilities accordingly.

[Compare this with the following popular intuition in ethics: you should promise to visit your grandmother and then visit her, but given that you aren't going to visit you shouldn't promise to visit her.]

Obvious vs. embarassing mistakes

2010-10-15T17:50:00.000-07:00

As you've probably noticed, this blog has been on a bit of a hiatus. I'm going on the jobmarket this year so things have been very busy. I do have a little time now though, to note something about the relationship between two phenomena that are ubiquitous in my life :)

Not all obvious mistakes are embarrassing mistakes. Any mistake you make while adding two numbers will be an obvious mistake, but nearly everyone doing calculations makes such mistakes some fair fraction of the time, and these errors are not (intuitively) embarrassing mistakes.

further questions:
-Is being obvious once pointed out a necessary condition for being an embarrassing mistake? [edit: appropriately enough, i had originally put "sufficient" :)]
-Is the mere fact that a mistake is made with high frequency in some community sufficient to prevent it from being an embarrassing mistake? (maybe inferring the consequent is made with high frequency yet also embarrassing).
-Will trying and failing to give principled necessary and sufficient conditions for a mistake being embarrassing make one feel less embarrassed by embarrassing mistakes?

[hat tip to E.M. for suggesting this would make a cute post]

On Moral Philosophers' Library Fines

2010-08-16T14:19:00.000-07:00

I just listened to this neat conversation, which summarizes some empirical research into the question of whether thinking about moral philosophy makes you any better at behaving morally. It turns out moral philosophers are actually slightly more likely to steal library books than philosophers in other areas, and political philosophers are no more likely to vote than people in other profession.

The speakers mention that these results are surprising, since they conflict with the hope that researching moral philosophy will have morally good effects.

Now I'm pretty skeptical about moral philosophy myself, for other reasons, but here's what the moral philosophers would/could say for themselves on this score:

"The phenomenon of weakness of the will, means that there are two components to doing what's good: the epistemic component of figuring out what's morally better/required in a given case, and then the practical component of actually doing that.

Moral philosophy only pretends to address the first component. Thinking hard about weird trolley cases, and abstract moral principles helps you figure out what you ought to do in cases where this is unclear. It doesn't address the second component of acting well - working up the will power to actually do what you ought to.

In this way, moral philosophers are like scientists who study fistfights not professonal boxers. They spend a long time studying the differences between different principles that only make a difference to what one should in principle do in certain rare cases. They don't spend this time practicing up their personal ability to implement the overall art of fighting well.

For this reason, testing whether moral philosophers are more virtuous in cases where it's *obvious*/uncontroversial what's virtuous (you should return library books, you should vote) exactly fail to capture the benefits that doing moral philosophy brings. Studying moral philosophy helps society make the world better, because the moral philosophers work out what we should do in novel, or controversial cases. This doesn't mean that it makes moral philosophers themselves substantially more virtuous. For, in most of the cases where ordinary people have a chance to act badly (adultery, embezzlement, falsifying data, refusing charitable aid) the limiting factor isn't *figuring out* what the right thing to do is, but rather summoning the willpower to sacrifice individual pleasure and benefit to do whats right."

Maybe this was obvious to everyone else

2010-08-05T05:22:00.000-07:00

If Fodor thinks that elements in the language of thought get their meaning from counterfactuals about assymetric dependence (HORSE means horse, not horse-or-cow-on-a-dark-night, because if tokenings of HORSE hadn't tracked horses they wouldn't have tended to track horses-or-cows-on-a-dark-night either), what does he say about Swampman?

Since Swampman is supposed to have come into being from random electrical activity, none of these counterfactuals about different response patterns which Swampman could have had seem well defined. Does Fodor say that Swampman wouldn't be thinking?

I guess Davidson (who came up with the example) bites this bullet. But it seems like the exact kind of intuitions that motivate accepting mental representation in the first place (you could have just the same phenomenology, if you were paralyzed so you had no dispositions to use any external language; surely this should suffice for you to count as having thoughts) rebel at the idea that Swampman wouldn't be thinking.

Invention, Discovery and Creativity in Mathematics

2010-07-24T18:37:00.000-07:00

Non-philosophers I meet sometimes ask: do I think mathematical facts are invented or discovered? IMO, This is a weird question - and not one that comes up much in the phil math literature- because the contrast between invention and discovery is not very well defined. For example, did Alexander Gram Bell *invent* the telephone, or did he *discover* that putting components together in a certain way would build a telephone? Intuitively, one might say both.

Maybe what people mean to be asking by this question is just this: do mathematicians bring new mathematical objects into existence, or do they discover already existing objects? For, paradigmatic cases of invention typically do involve creating a new physical object, while paradigmatic cases of discovery involves visiting an already existing physical object. So e.g. Columbus discovered America (he went to visit it) whereas Bell invented the telephone, by physically building the first telephone. However, the contrast between invention and discovery doesn't really capture the distinction between cases where a new object is made vs. not. This is because making a new thing isn't required for invention *or* discovery. Consider a thought experiment where Bell just thought up a plan for a telephone, and told someone else who physically constructed the first one years lated. Bell would still have invented at telephone, if he though up the plan and then worked out from known principles that the plan would work, but never made one.

While we are talking about invention and discovery, I think there's a third notion -artistic creation (e.g. what happens when someone composes a story or a poem)- which bears an interesting relationship to mathematical discovery. When a writer writes a story, they are putting down a sequences of sentences which already exists as an abstract object. I mean, suppose that the story teller composes a story today. If a linguist said yesterday 'no intelligible sequence of English sentences has property P', the and the sequence or sentence which the story teller writes down today has property P, then then the linguist's claim yesterday was false. The domain of potential counterexamples to linguistics claims today, already contains all sequences of English sentences which literary ingenuity could ever devise. Note also that to compose a story or poem doesn't require writing it down anywhere, (the person in the Borges story who has time stop so he can finish writing a poem before he gets shot, still counts as creating the poem). For this reason the task of literary "creation" doesn't really seem to involve creating anything, (neither a physical artifact, nor an abstract string of sentences), but rather directing your attention to an abstract object that already exists - carefully sorting out which string of sentences will combine certain varied and subtle properties in the right way.

Now, if I'm right about this- the creativity of a poet or novelist doesn't need to involve creating any new object, but rather amounts to discovering a pre-existing string of sentences which has a certain property - this suggests a potential confusion about the relationship between mathematical creativity and ontology. Arguably, mathematical creativity is much like literary creativity. But, if mathematical creativity is like literary creativity, it does not follow from this that the mathematician creates the mathematical objects he describes, or that he creates anything else. For (if the above is right) literary creativity isn't a matter of bringing new objects into being, but rather a matter of discovering, amid the combinatorial explosion of possible sequences of english sentences, one that has a certain special features.

Why Math and Morals Aren't Companions in Guilt

2010-07-24T17:19:00.001-07:00

Intuitively, many people feel that epistemic worries about moral facts (if there are moral facts, how to explain why our moral intuitions should be even even remotely correct about them?) are WAY more serious than epistemic worries about mathematical facts (if there are mathematical facts, how to explain why our mathematical intuitions should be even even remotely correct about them?). But is there really a difference here?

Well, here's one thing that I think does make a difference: mathematical claims about number theory have direct and specific consequences for stuff that we can check by logic and/or scientific observation.

-what will happens whenever a person or a computer to successfully applies a certain syntactic alogorithm
-how many apples-or-oranges do you have when you have n apples and m oranges (cf Frege for why this is a logical fact)

This matters because, plausibly, the need to get these concrete applications right likely prevents our beliefs about number theory from getting too off the wall - whereas, our moral intuitions have no such multitude of consequences which are directly checkable by logic and observation.

Epistemology verses Foundations in Philosophy of Math

2010-07-17T10:07:00.001-07:00

The epistemology of math task: Get a true theory of what under what circumstances a person counts as knowing something. Or, at least, square our beliefs about what people have or lack knowledge of what particular mathematical beliefs, with general beliefs about what’s required for knowledge (e.g. causal contact.

The foundations of math task: extend our mathematical knowledge.

I claim that making this distinction matters a lot, because:

Arguments that are helpful for foundations of math are (in themselves) useless for the epistemology task. Suppose we have a working derivation D of certain facts of arithmetic from logic. And suppose we have a perfectly adequate, intuition-matching story about what it takes to count as knowing the relevant logical facts.
This still does not allow us to account for current knowledge of arithmetic (i.e. reconcile our theory of knowledge with the intuition that people now know things about arithmetic). This is because - in general - it is not enough for S to know that P, for P to be true, S to believe that P, and P to be derive*able* from things which S knows. In general, the subject S needs to have some kind of access to the derivation. The mere fact that I believe that P, and P can be proved from other things that I know, hardly suffices to establish that what I have counts as knowledge. If a lawyer is asked to show that some contractor knew that a bridge was safe, it doesn’t suffice to show that one *could* derive from laws of physics and facts about the blueprint which the contractor knew that the bridge was safe - we also need to suppose that the contractor did derive it, or get testimony from someone who derived it or the like.

Hence, a foundational argument which derives (say) one body of mathematics from premises that are more certain is not directly relevant to the general epistemological project.

Conversely, an accurate epistemology of mathematics can be almost perfectly useless to the task of setting some shaky region of mathematical theory on firmer foundations. For example, one classic account of knowledge is reliablism. If we modify reliablism so as to apply non-trivially to mathematics (following suggestions by Linnebo and Field) we get the idea that someone has knowledge if they have a true belief which is reliable in the sense that: they accept a sentence which expresses p, and if that sentence had not expressed a truth, they would not have accepted it. This is a perfectly decent candidate for a general account of mathematical knowledge. But note that, even supposing that it is right, it does nothing to help satisfy foundational desires for, say, more secure foundation for the axiom of choice. If someone has foundational worries about the axiom of choice, they have worries about whether it is true. They might express these worries by saying ‘how do you know that the axiom of choice holds?’ but the emphasis here is on truth, not on knowledge. It would be silly to respond by saying that we know AC because AC is true, and we have reliable beliefs (as defined above) to that effect. What the foundation-seeker really wants is to know whether AC. They want to acquire knowledge about whether AC, not get a general theory of what it would take to count as knowing AC.

So, I have been trying to argue that it’s important to make a distinction between the epistemological project of trying to come up with a general theory of when someone knows something about math, and the foundational project of trying to make it the case that we know more things about math, by supplementing inadequate arguments with additional arguments that appeal to premises which are already known. The one focuses on the most bland an uncontroversial cases of mathematical knowledge, and tries to reconcile our other beliefs about the nature of knowledge with our particular judgments about this case. The other seeks out the most controversial regions of mathematical claims, and seeks to secure knowledge for us about these claims, by connecting them to claims that are more securely known. Enticing answers to one project can easily seem to frustratingly miss the point for someone who is interested in the other, as shown in the examples above. Hence it’s important to make the distinction.

However, this is not to say that there’s no relationship between the epistemological and foundational projects. Thinking about big picture issues about justification in general, can influence your judgments about particular cases. A kind of trivial example of this is intuitions about what you can take for granted, while still counting as being justified. Just off the top of one’s head, it can seem attractive to say that someone doesn’t count as knowing that P if all they can give is a circular justification for P, an infinite regress of justifications, or a justification that comes to a halt at a certain point. But when you consider these three options together and notice that they exhaust all the possibilities, you will likely be inclined to give up the principle that someone who can only give a justification of one of these kinds must thereby not count as having knowledge. So, if two realists about AC are attempting to provide and evaluate firmer foundations for AC, it may be helpful for them to general questions about what’s required for knowledge and justification – to make sure that their evaluation of the evidence in this case, doesn’t depend on assumptions about justification which turn out to be incoherent or conflict with what they take to be sufficient evidence more generally.

Are mathematical truths "substantive"?

2010-07-07T21:04:00.000-07:00

One thing that that has caused me great puzzlement (in the past few years), is the question of whether math tells us anything 'substantive'. I want to suggest that our intuitive notion of "substantiveness" combines two distinct notions, which come apart in this case.

- mathematical truths DONT rule out any physically or even metaphysically possible states of the world. (This is just another way of putting the truism that mathematical truths are necessary, hence compatible with every metaphysically possible world. I like putting things this way, because it doesn't suggest that necessary mathematical truths arise from something (mathematical objects?) causally blocking any person that tries to being both more than three feet long and less than two feet long)

- mathematical truths DO combine with our background beliefs to lead us to form expectations we wouldn't have formed otherwise (e,g. about the results of future counting procedures, about the programs)

Presumably you admit that these are at least nominally different properties. But you might still wonder *how* these two things could come apart. How could knowing any proposition be useful, if this proposition didn't rule out any possible states of the world? Here's what I think the answer to that is in a nutshell:

Some mathematical facts (i.e. facts which are derivable from math and logic alone) which are useful because they tell us that whenever one description of the world holds, then so does another (e.g. anything that accelerates from standstill at this rate for this amount of time travels that distance, anything that's less than two feet long isn't three feed long.)

And here's the answer in more detail.

FOL as the language for science

2010-06-28T04:23:00.000-07:00

Maybe I'm missing something here...

Quine suggests that we adopt first order logic as the language for science. But, first order logic can't capture the notion of 'finitely many Fs'. It can only express the claim that there are n Fs for some particular n. Yet, we do understand the notion of finite, and use it in reasoning (e.g. if there are finitely many people at Alice's party, there is one person such that no one is taller than him) and potentially in science. Hence, we should not adopt first order logic as the language for science.

[The standard way to try to get around this, is by talking about relations to abstract objects like the numbers (There are finitely many Fs if there's a 1-1 map from the set of things that are F to the some set theoretic surrogate for the numbers). This would give you the right extension, if your scientific hypothesis could say that something had the structure of the numbers. But first order logic can only state axioms, like PA which don't completely pin down the structure of the numbers. Any first order axioms which you use to characterize the numbers will have non-standard models. This is Putnam's point in his celebrated model theoretic argument against realism. So, if you take this strategy, rather than saying that there are finitely many people at Alice's party, you can only say that the number of people is equinumerous items that satisfy a certain collection of first order axioms. And this does not rule out non-standard models.]

Is math logic?

2010-06-28T04:20:00.000-07:00

Is mathematics just a branch of logic? This is the first question many people ask about philosophy of math (sometimes with a vague idea that a) it would solve some kind of metaphysical or epistemological problems if math were logic or b) it's been proved that math isn't logic). Well, unsurprisingly, the answer depends on what you mean by 'logic'. Here are some different senses of the word 'logic' that one might have in mind.

1. first order logic
2. fully general principles of good reasoning
3. a collection of fully general principles which a person could in principle learn all of, and apply
4. principles of good reasoning that aren't ontologically committal
5. principles of good reasoning that no sane person could doubt

The sense in which it has been proved that math isn't logic is (to put things as briefly as possible) this: You can't program a computer to spit out all and only the truths of number theory.

This fact directly tells us that the mathematical truths are not all logical truths, if we understand "logic" in sense #1 - since we *can* program a computer to list off all the truths of first order logic. And it also tells us that the mathematical truths aren't all logical truths in sense #3 or #5 either - if we are willing to make the plausible assumption that human reasoning can be well modeled in this respect by some computer program. For if all human reasoning can be captured by a program, then so can all human reasoning from some starting finite collection of humanly applicable principles, and so can the portion of human reasoning that no sane person could doubt (to the extent that this is well defined).

However, if by "logic" you just mean #2 -fully general principles of reasoning that would be generally valid (whether or not one could pack all of these principles into some finite human brian)- then we have no reason to think that math isn't logic. We expect the kinds of logical and inductive reasoning we use in number theory (e.g. mathematical induction) to work for other things (especially for things like time, which we take to have the same structure as the numbers). If Jim didn't have a bike on day 1, and if, for each subsequent day he could only get a bike if he had already had a bike on the previous day, then Jim never gets a bike. If there are finitely many people at Jane's party, there is one person such that no one is taller than them. The laws of addition are the same whether you are counting gingerbread men and lemon bars, or primes and composite numbers. And this doesn't just apply to principles of mathematical reasoning which we actually accept. We also expect any *unknown* truths about the numbers (as the smallest collection containing 0 and closed under a transitive, antisymmetric relation like successor) to be mirrored by corresponding truths about any other collection of objects which contain some other starter element and are as few as possible while being closed under a transitive, antisymmetric relation (be this a collection of infinitely many rocks, or a collection of some other abstracta like the range of possible strings containing only the letter "A"). Hence, it is plausible that every sentence about numbers is an instance of a generally valid sentence form containing only worlds like "smallest", "collection", "antisymmetric" "finite" etc - and every mathematical truth is a logical truth in this regard.

Finally, if by "logic" you mean #4- ontologically *committal* good reasoning, the answer depends on a deep question in meta-ontology. For, it is well known that standard mathematics can be reduced to set theory, which in turn can be reduced to second order logic. But what are the ontological commitments of second order logic?

People have very different intuitions about whether we should say that there really are objects (call them sets with ur-elements or classes) corresponding to "EX" statements in second order logic. Does the claim that "Some of the people Jane invited to her party admire only each other, so if all and only these people accept, she will have a very smug party" assert the existence of objects called collections? More generally: the quantification over classes in second-order logic ontologically committal? Statements like the one above certainly seem to be meaningful. And, it turns out not to be possible to paraphrase away the mention of something like a set or class, in the sentence above, using only the tools of standard first order logic. This reveals a sense in which we treat reasoning about abstracta like classes (or, equivalently for these purposes, sets with ur-elements), very similarly to ordinary objects in our logical reasoning about them. But is this enough to show that second order logical is ontologically committal (and hence not logic at all, according to meaning #4)?

I propose that the key issue here concerns how closely ontology is tied to inferential role. Both advocates and deniers of abstract objects will agree that many of the same syntactic patterns of inference that are good for sentences containing "donkey" and sentences containing "set". But what exactly does this tell us about ontology? If you think about ontological questions as being questions about what the logical role of an expression in a given language, this tells you something very decisive. On the other hand, if you think about ontology can swing somewhat free of the inferential roles of sentences in languages (so an expression can have an object-like inferential role without naming an object), it's open to you in principle to say that - however similar their logical role- second order quantifiers are not ontologically committal. On this view, claims about sets with ur-elements are just ways to make very sophisticated claims (generally claims that could not otherwise be finitely expressed) "about" the behavior and relationship between ur-elements, and true claims about pure sets (i.e. sets that can be built up just from the empty set) are true in a way that does not involve any particular relationship to any objects, but can illuminate the necessary relationships between different expressions about classes that do have ur-elements. [At the moment I prefer the former view, that quantification in second order logic is ontologically commital, but this is a subtle issue]

Thus, to summarize, it is fully possible to say - even after Godel- that math is the study of "logic" in the sense of generally valid patterns of reasoning. However, if you say this, you must then admit that "logic" is not finitely axiomatizable, and there are logical truths which are not provable from the obvious via obvious steps (indeed, plausibly ones which we can never know about). Note that to make this claim one need not give up on the idea that logical arguments proceed from the obvious via obvious steps. For, if you take this route you can (and probably will want to) distinguish the human practice of giving logical arguments, from the collection of logical truths. You can say: only some of the logical truths seem obvious to us, and only some of the logically-truth-preserving inferences seem obviously compelling to us. We make logical arguments by putting these inferences together to get new results which are also logical truths. But (what Incompleteness shows) is that not all logical truths can be gotten from the ones that we know about. You can even claim that mathematical truths are logical in the further sense of not being ontologically committal, if you allow (contrary to the usual close association between objecthood and logical role) that the set quantifiers in second order logic are not ontologically committal.

Knowledge and Cannonical Mechanisms

2010-06-18T07:39:00.000-07:00

In my first epistemology class in college, the prof encouraged us to look for adequate necessary and sufficient conditions for knowledge by making the following (imo appealing) argument. We expect that there's SOME nice relationship between facts about knowledge and descriptive facts not containing the word knowledge, since our brains seem to be able to go, somehow, from descriptions of a scenario (like the Gettier cases) to claims about whether the person in that scenario has knowledge. However, philosophical attempts to find a nice definition of knowledge in other terms seem to have systematically failed. This suggests that there may be a correct and informative definition of knowledge to be found, but this definition is just too long to be an elegant philosophical hypothesis, but not too long to correspond to what the brain actually does when judging these claims.

So here's what I propose that the true definition of knowledge might look like:

We describe messy physical processes by talking about symple mechanisms, and a notion of what these mechanisms tend to do "ceterus paribus". People agree surprizingly much on which mechanisms approximate what (e.g. how to go from facts about swans to claims about the swan lifestyle, how to divide up actual dispositions to behavior into "behaving normally" vs, "something special happening whereby the ceterus aren't paribus"). One thing that can be so approximated is human belief forming. We think about actual human belief formation by saying that it "ceterus paribus" it approximates combination of various belief forming mechanisms (e.g. logical deduction, looking etc). A reliable beleif forming mechanism is one whose ceterus paribus behavior yields true beliefs.

Certain belief forming mechanisms are popular, and remain popular with people even when they undergo lots of reflection. Some of these are cannonical, in the sense that we count them as potential conduits for knowledge. But, if we ever come to believe that some such mechanism is not reliable (jn the sense defined above) we will stop saying that beleifs formed via it count as knowledge. So here's what I think a correct definition of knowledge might look like.

We have, say, 300 cannonical reliable mechanisms for producing knowledge, 200 cannonical reliable mechanisms for raising doubt (100 optional and 100 obligitory), and 200 cannonical reliable mechanisms for assuaging doubt. Call these CRMs. Our definition starts by giving a finite list of all these CRMs.

You know P, if and only if your belief in P was generated by some combination of CRMs for producing knowledge, and you went through CRMs from assuaging doubt corresponding to a) all optional CRMS for doubt raising that you did engage in b) all non-optional CRMs for doubt assuaging that apply to your situation.

Even though this is just a claim about what the form of a correct definition of knowledge would look like, it already has some reasonably testable consequences:
1. That situations where it seems unclear of vague what mechanism best describes a person's behavior (should I think of the student as correctly applying this specific valid inference rule, or fallatiously applying a more general but invalid inference rule?) will also make us feel that it's unclear or vague whether the person in question has knowledge.
2. That we should seem unclear whether to attribute knowledge about when reliable but science fictiony and hence non-cannonized mechanisms are described. For example, most people would say it's OK to take delivarances of the normal 5 senses at face value, without checking them against something else. But what about creatures with a 6th sense that allowed them to reliably read minds, or form true beliefs about arbitrary pi 01 statements of arithmetic (imagine creatures living in a world with the weird physics that allows supertasks, and suppose that they have some gland that has no effect on conscious experience, but whose deliverances reliably check each case). Would they count as knowing if they form beliefs by using these?

New Uses for Conceptual Analysis

2010-05-19T16:18:00.000-07:00

Coming up with a systematic way to paraphrase sentences involving some wacky new term W in biology, sociology, psychology, or art criticism, into sentences that are just a logical product of claims about sets, mereological sums of other more commonplace objects, and preserves all our intuitive reasoning about W, is useful in three ways.

a) New Applications for Old Knowledge: Getting a method of paraphrase lets us bring our logical/set theoretic/substantive knowledge about the terms used in the analysis to bear on the new term in question. If the facts about the Ws parallel the facts about sets of such and such kind, set theory may have interesting implications for facts about the Ws.

b) Avoiding Adding Terms Which are "Incoherent" or "Have False Presuppositions": Getting a method of paraphrase may let us prove the consistency and conservativity of reasoning about the wacky new entities. For example if you analyze 'x is bachelor' as 'x is unmarried & x is a man', and then only accept informal reasoning about bachelors that can be reconstructed using this analysis, then it is clear that adding the term 'bachelor' and doing this informal reasoning will not allow you to derive contradiction, or any other new consequences. So, adding informal reasoning about bachelors will do no harm. Here the proof theory (any proof of P which uses the term "bachelor" could be turned into one that doesn't) is so obvious that it's easy not to notice. But the mathematical issues involved in showing consistency and/or conservativity of adopting some term (together with analytic feeling reasoning that goes with that term) can become more interesting when conceptual analysis only provides an *implicit* or recursive definition of the term.

This is valuable, to the extent that you are worried a purported new concept may be 'incoherent' (in the sense that intuitive, analytic feeling reasoning about it literally lets you prove contradiction) or may have bad 'presuppositions' (in the sense that that intuitive, analytic feeling, reasoning using the term allows one to derive new propositions not using that term, which are false)

c) Teaching: Obviously getting a method for paraphrase sentences involving new terminology in terms of old terminology provides a way of teaching the new terminology to people who already understand the old terminology.

Note that none of these purposes require that conceptual analyses be unique. Different analyses of claims about, say, the imaginary numbers, in terms of set theory can each serve this purpose equally well. Nor do these uses for conceptual analysis require that one make any claim about the metaphysical status of the objects in question. It's useful to know you can reconstruct all intuitively acceptable reasoning about the imaginary numbers in terms of intuitively acceptable reasoning about sets, even if you don't want to claim that the imaginary numbers ARE sets or anything like that. Nor, lastly, do they require that the analyses have some kind of psychological reality - that when you are thinking about imaginary numbers you are really somehow implicitly (subconsciously?) considering one or the other paraphrase in terms of sets.

[Hidden agenda: Even if it turns out that Occam's razor doesn't apply to positing special sciences objects like livers, species, trade deficits and languages, so there's no need to look for paraphrases which would allow us to *deny* that such "extra" objects exist, finding Quinean-style parapharases will still be illuminating and useful for other reasons. So we philosophers won't be talking ourselves out of a job :). Also, to the extent that you feel like something substantial is going on when one looks for Quinean methods of paraphrase, this may be because these paraphrases illuminate the structure of our intuitive reasoning about Ws, and let us relate the W facts to facts about objects we understand better - not because there is a serious question about whether the Ws really exist.]

A Depressing Theory of Ceterus Paribus Clauses

2010-05-18T14:46:00.000-07:00

We want to say "sugarcubes dissolve in water, ceterus paribus", but what does that mean? Philosophical analysis of the phrase ceterus paribus has proved surprisingly difficult. For example, the quoted sentence doesn't mean that all or most pieces of sugar that actually will be dropped into water will dissolve.

Here's a depressing proposal for how ceterus paribus clauses work. We have a substantive (implicit) theory of what "the normal cases" are like, which is based on human daily life and maybe some random traditions too. We use this when evaluating ceterus paribus sentences to choose which way of making the target sentence true to consider. So, for example, 'ceterus paribus' clauses get filled in so that "dropped eggs break, ceterus parbus" is true, because people tend to hang out in places near the surface of the earth, which don't have thick rugs, so it's part of our substantive theory of what's "normal" that when something is dropped there's a hard surface below it (as opposed to a thick rug, or the empty expanse of space).

Miniature Phil Math

2010-05-09T20:38:00.000-07:00

Almost everyone agrees that our mathematical talk is practically helpful. Unlike astrology, doing math helps us build bridges. But how is math practically helpful? And does the way in which talking about numbers is practically helpful give us any reason to think numbers actually exist?

In this tiny essay I will propose a theory of how the practice of talking as if there were numbers is helpful. Then, I will say that we can appeal to numbers to explain how this practice is helpful, though there are also other correct explanations for this phenomenon which do not commit themselves to numbers. I will conclude by turning to the question of whether there are numbers. On the basis of the previous section I will propose that we do not *need* to posit the existence of numbers to explain the practical usefulness of our mathematical talk. However, we have another reason to believe in numbers which is the following: We want to make statements like "the number of cupcakes doubles every day" true (under certain circumstances), and the pattern of inferences we make with this sentences is quantificational. But this (being describable by some true sentences associated with a existential pattern of inferences) is the only thing that the many different kinds of non-mathematical objects which intuitively exist have in common.

1. How talking about abstracta like numbers is helpful

Talking about abstract objects, like numbers, is helpful because it lets us economically hypothesize patterns 'in the world around us' as well as patterns that might be described as artifacts of language (patterns in which distinct descriptions are logically or otherwise necessarily equivalent). We can say one sentence (about numbers) that will cause people to be willing to infer infinitely many different sentences that aren't about numbers.

For example, suppose I say: "The number of cupcakes doubles every day" This is a claim that quantifies over numbers and days, in the sense that we might represent it as "Ad An if d is a day, and n is a number, then there are n cupcakes on d there are 2n cupcakes on the day after d. "
Hearing this single sentence will lead my listeners to accept many different statements that do not quantify over cupcakes:
"if Ex7 cupcakes today Ex14 cupcakes tomorrow."
"if Ex8 cupcakes today Ex16 cupcakes tomorrow."
"if Ex7 cupcakes tomorrow Ex14 cupcakes the day after tomorrow."

2. What role do abstract objects play in explaining why talk of abstract objects is helpful?
Now we can ask: what role do various objects play in explaining the success of this talk? We might explain the helpfulness of my statement by saying that it is helpful because it...
- lets us track and predict what cupcakes there are and will be
- lets us track *the pattern in* what cupcakes there are and will be
- lets us track and predict how *the doubling function* relates *numbers*, and then predict what cupcakes there will be when, by relating this to facts about the behavior of the doubling function.

It seems to me that all of these are intuitively decent explanations. I take it that what we have here is a typical phenomenon where the same phenomenon (a war) can be explained by accounts that quantify over various different objects (countries vs. people vs. atoms). However, not much would be lost if we just stuck to giving the first explanation, which does not involve any mention of abstract objects.

3. Are there numbers? A good and bad reason for believing in numbers.
If this story about how math is practically helpful is right, should we believe that there really are objects of the kind talked about in these explanations e.g. patterns in the provenance in cupcakes, or numbers and a doubling function?

I don't think there is an *inference to the best explanation* for the existence of patterns in the provenance of cupcakes, or numbers from the helpfulness of this talk. It's not the case that we *need* to posit abstract objects called "patterns in the provenance of cupcakes" or "numbers" to explain how saying the thing described above could help people cope with the cupcakes around them.

Instead, I think it's reasonable to believe in numbers because we have an intuitively true sentence ("the number of cupcakes doubles every day") which allows a existential pattern of inferences - and playing this logical role is all there is to being an object.

The idea here is that when we look at the variety of different "objects" in the world e.g. electrons, magnetic fields goats, holes, waves, contracts, countries, these different kinds of talk don't seem to have much in common with regard to their relation to the physical world. What they do have in common is the pattern of inferences we make between sentences between them. In each case we accept sentences, such that the inferences with these sentences in are elegantly captured (in first order logic) by something of the form "Ex Fx". Now it turns out that talking about numbers and the doubling function shares this same feature.

Contrast w/ Tait "The Platonism of Mathematics"

2010-04-14T18:07:00.000-07:00

Both my view (Lumpist Platonism) and Tait's might be considered unusual or quirky versions of platonism. Platonism (in phil math) is the view that mathematical objects exist.

I think that the world is fundamentally (something like) a space-time manifold [as opposed to a set of facts, or a set of objects and relations], and that all statements are true or false in virtue of how the manifold is. This includes statements about objects, and different statements about objects will correspond to very different claims about the state of the manifold (e.g. saying that there's a table vs that there's a whirlpool vs a trade deficit vs. a marriage contract vs. a number or string of symbols or a proposition). So facts obtain, and objects and relations exist, in virtue of how the physical stuff of the world is configured, not vice versa. Necessary truths (like all statements of pure math) correspond to the trivial claim about the state of the manifold (one that doesn't rule out any possible configurations).

Tait, as I understand him, thinks that mathematical sentences show that objects exist by constructing suitable objects. He writes "A proof is a presentation or construction of an object: A is true when there is an object of type A and we prove A by constructing such an object."

Both of these views contrast with what you might call a "two worlds" version of platonism. On this view: in addition to whatever objects exist in virtue of the physical stuff of the world comporting itself a certain way, there is also an "extra" component of reality. So far as I understand the force of the word "extra" here, the point of saying that there's an extra component of reality is this: An infinite and putatively exhaustive description of the world given purely in the language of microphysics e.g. (this point has that property, this point has that property etc.) would be missing out on the existence of sets, *in some stronger sense then the sense than in which it would be missing out on rabbits and trade deficits*.

Tait and I also agree that sentences are the right place to start when considering how semantics relates to metaphysics and ontology. For a sentence to be meaningful you just need the whole sentence to somehow make a claim about the world. Thinking about particular words in the sentence as having favored relations with particular chunks of matter will help in some cases but not others.

However, I disagree with Tait on some really important points:

Firstly, I don't really understand what he means by construction. The best sense I can make of the idea of constructing mathematical objects (how can you bring an abstract object into being?) is that it's something like the way I can create a) a marriage contract with another person by signing things the courthouse, or b) the set with ur-elements {Sharon's mullet} by giving myself an ill-judged haircut and thereby bringing a particular mullet-token into being, and hence it's corresponding singleton. But if this is what he had in mind, then...
a) it has the (at the very least) wildly counterintuitive to say that there wasn't a number between 3 and 5 before someone wrote down a proof inscription.
b) quantification in math works very weirdly and differently from quantification in general. For, since people have only written finitely many proofs there will be some number - say 347892-, such that no one has inscribed a proof of "3457892 has a sucessor". On the other hand, we certainly have inscribed proofs of "Ax if x is a natural number then x has a successor". So it would seem that the general statement is true. But the instance is (at the moment) false.

Secondly, Tait doesn't seem to allow that quantified statements of arithmetic (like, say, the Godel sentences for various formal systems) already have truth values now. He seems to think we are free to choose which kinds of proofs to construct (i.e. what formal system to adopt). And then he says that "the incompleteness of formal systems such as elementary number theory can be proved by induction, is best seen as an incompleteness with respect to what can be expressed in the system rather than with the rules of inference." And he points out that by extending the language (and adding suitable instances of the induction schema) you can prove the Godel sentence (and con) for this system.

But when I wonder about e.g. con(PA+X) [it's pretty hard to wonder about con(PA) imho] or con(ZFC), I'm not just wondering whether I could extend my formal system in such a way as to allow these sentences (or their negation) to be derived. Obviously, I could start making derivations (and hence constructing objects, for Tait) in any formal system I want. Nor am I pondering what kind of lifestyle choice to adopt in the future. Rather, I think that *right now*, I understand what it means to ask whether there's a proof of 0=1 from ZFC. And this is what I want to know. Is this sentence provable in that formal system or not? Is there such a proof or not?To the extent that we can ever be sure that we really understand something, and are asking a sharply meaningful question, this is it! [I think this may be why my advisor PK disagrees with Tait too]

Overall, I'm tempted to suspect that Tait is getting into bed with unattractive antirealism because he wants to avoid an epistemological problem. He sees how (/doesn't worry about how) you could know that something exists if you are able to bring it into existence (construct it). Such knowledge is sometimes called "maker's knowledge". And then he wants to say what mathematical knowledge is, in such a way that all mathematical knowledge turns out to be accessible in this way - which leads to weird consequences about large numbers, and unknown arithmetical facts.

In contrast, if you use the ...ahem... magic of Sharon's thesis, to provide a general naturalistic mechanism for how physical creatures could have gotten a faculty of reliable rational insight into abstract mathematical/logical truths :) , then you don't have to do any of this fancy (and potentially distorting) footwork.

Parsons and Intuitability

2010-04-14T18:02:00.000-07:00

I've just been summarizing CH1 of Charles Parsons' Mathematical Thought and it's Objects. It set me thinking that Parsons is oddly concerned with whether you can "see"/percieve/intuit mathematical objects. I say oddly, because IMO what matters for assuaging worries about the weirdness of mathematical objects or the weirdness of our knowing about them (which seems to be part of his aim) isn't whether we can strictly speaking *see*/perceive/intuit abstracta, but rather a) whether positing abstracta isn't a violation of Occam's razor and b) how there can be enough of a connection between mathematical facts and our dispositions to form beliefs about them, for what we have to count as knowledge.

I mean: even in the empirical case, questions about what we can see, as opposed to merely inferring from what we see are super murky. Who knows whether you can "see" that the light is on vs. that the electricity is back on vs. that Jones succeeded at his task etc. as opposed to inferring them or justifiably and reliably forming true beliefs about these subjects)? What matters (for the epistemology worry b) is just that there needs to be some suitable and clear reliable mechanism at work leading you to form true beliefs on these subjects - as there obviously is in the empirical case of the light. Once we see how this reliable mechanism could work, it's (in my opinion) a matter of indifference whether you want to describe this mechanism as seeing the light and then immediately and unconsciously but justifiably inferring that the electricity is back on vs. directly seeing that the electricity is back on.

And the same goes for knowledge of mathematical objects. What we'd like is something that was like perception in the sense that it provided an unproblematic mechanism whereby we could get the relevant kind.Once we have that in place, we can say whatever we like about whether someone staring at a piece of paper can see/percieve/intuit that there's a proof of SS0+S0=SSS0 in PA, or a palindrome containing the word 'adam' vs. merely reliably and justifiably infer these statements from the concrete object that they do see. The million dollar question is how we manage to do this putative seeing/inferring correctly.

Similarly, if someone thinks that construing math as stating truths about genuine abstract objects is a violation of Occam's razor, (as per objection a) they aren't going to be impressed by claims to "see" the abstract object (a string) in the concrete object (a series of inkmarks). When the Platonist stares at the sheet of paper and says they are seeing that there's a proof SS0+S0=SSS0, the Fictionalst will say that they are seeing that there would have to be a proof in the relevant mathematical fiction, and the modalist will say you are seeing that a certain proof is possible.

My point here is not to knock Parson's interest in the relationship between concrete things you can see and abstract mathematical objects. Hearing him talk about this connection was a decisive inspiration for my own view, and I think it's absolutely crucial to think about the concrete physical processes going on when we form and revise mathematical beliefs, if you want to understand how creatures like us could know about (or even think about) something as abstract as math. But I would claim that the key point about string inscriptions isn't what they represent/allow us to intuit (can you stare through the string inscription to the string itself?, can you at least see that a certain string exists?), but (as it were) what you take these inscriptions to represent, i.e. how you are willing to form and revise your beliefs about other things, like strings as abstract objects, in response to seeing them. This is what starts to give us traction in linking up our dispositions to form mathematical beliefs to mathematical facts, to answer challenge (b). (IMO answering challenge (a) requires something else entirely, namely Lumpism, but more about that in the next post)

Parsons Mathematical Thought and its Objects CH1 summary

2010-04-14T17:57:00.000-07:00

No one I've talked to is really sure what's going on. Especially me. But here's my current best guess. Maybe the magic powers of saying something wrong on the internet will help us work our way incrementally to a better interpretation.

1. Abstract objects defined + generic worries about them

Mathematical objects would be abstract objects = acausal, not located in space and time.
Worry: They aren't perceptable, if perceiving something requires locating it. Maybe this suggests there are no such things?
Response:
- electrons don't seem to be directly perceptable either, but they exist
- if we say that mathematical objects don't exist then we will have to explain why talking as if they did is so helpful for science.
- it's not clear whether we can avoid quantifying over abstract objects, hence (if we accept Quine's criterion) saying that they do exist.

2-3 What is an object?

It's hard to answer the question 'what is an object?' since unlike with gorillas we can't point out a contrast class of things that aren't objects.

the right answer: logical role
Philosophers usually ask 'what's an object?' in the context of trying to figure out how language can relate to the world - how we can talk about objects. For these purposes we can define being an object in terms of logical role: objects are what we talk about by using singular terms (e.g. 'Bob' in Bob is happy= Happy(Bob)) and quantification (e.g. 'Ex x happy').

other conceptions of objects/requirements philosophers have had for objects...

i. actuality/causal powers
Digression about Kant: general notion of object vs. "Wirklichkeit"
Kant invented the phrase 'concept of an object in general'. Kant's "categories" are concepts of an object in general. He is conflicted about whether these categories have to be perceivable by the senses [and hence whether "the concept of an object in general" would allow abstract objects?]
a) the categories are supposed to be derivable from logic and general considerations that don't take into account anything specific about the kind of object involved.
b) applying the categories is only supposed to generate knowledge when combined with stuff from the senses (namely: " the manifold given in sensory intuition")
Kant and Frege seem to have a notion of the actual = "wirklich" which only applies to objects you can causally interact with
Kant clearly accepts mathematical objects in some sense, but it's not clear whether he somehow thinks they are merely possible.

Idea: Many people find abstract objects spooky because they assume that they would have to be Wirklich, or something like it. The merely logical conception of object above doesn't require any such thing. So maybe mathematical objects exist in the logical sense i.e. we can state truths using singular terms for them and using quantifiers, but they are somehow not Wirklich.

ii. intuitability

Kant digression:
You use intuition to discover whether things could fall under it. [presumably round square would be an example of a putative concept that doesn't pass this test.]
geometric figures = forms of empirical objects
We can learn about them using intuition.

Perhaps it's an requirement that all objects are 'intuitable'?

defining intuitable
We will use intuition to mean a kind of perception that could apply to physical objects or abstract objects. We can distinguish
- having an intuition of an object, like perceiving an object (e.g. 'I intuit the equilateral triangle')
- having an intuition that some proposition about the object holds (e.g. 'I intuit that the interior angles of the equilateral triangle add up to 180')

Some issues:
-Should we require that one can have intuition *of* the object, rather than merely intuiting some suitable proposition about it? (call this strong intuitability) Or is it enough if you have an intuition of concrete objects that represent abstract objects, like the sequence of strokes Kant appeals to in his proof that 7+5=12? (call such a representation a quasi-concrete representation)
-On what sense does need to be possible to intuit something for that something to count as intuit*able*, and hence satisfy the requirement?

Idea cont. - Maybe mathematical objects are real in the logical sense, and intuitable, but not wirklich/causally effecations...

4. objecthood=having the logical role of an object

We will stick with Quine and Frege and say that the logical criterion (not wirklichkeit or intuitability) is all that's required for objecthood.

Some questions arise if you accept this definition of "object", about how to further spell out the view.

a) Which logic has the property that *its* singular terms and quantifiers correspond to objecthood? Maybe we should allow modal or other intentional notions, and if we do we will get different answers about what objects there are.
b) Maybe there are some entities which aren't objects? (i.e. maybe there's some important ontological category that's wider than objecthood - like some kind of meinogian being)
c) Maybe there are some objects which don't exist? (i.e. maybe there's some important ontological category that's narrower than objecthood - like fictional objects might be said to logically objects, but not really exist)

5-6 are about b and c respectively

7. Quasi-concrete objects

We will call abstract objects quasi concrete if they have a special relationship to certain concrete objects that 'represent' them e.g.
strings of letters --- inscriptions of strings of letters
sense qualities --- experiences of those sense qualities
shapes --- physical things that have that shape

We can look at the physical representatives, and keep in mind individuation criteria for the abstract objects. These individuation criteria say when two different concrete things 'represent' the same abstract one.

Some sets are quasi-concrete: sets with concrete ur-elements are represented by those ur-elements. But pure sets are not quasi concrete.

Overall Conclusion: mathematical objects exist in the logical sense, although they are not Wirklich, and although some of them are not intuitiable even in the weak sense allowed by looking at concrete objects that represent them.

Field on Normativity and Logic

2010-04-09T22:05:00.001-07:00

In "What is the Normative Role of Logic" Field argues that you can't understand logic descriptively as (eg. the project of studying necessarily truth preserving syntactic manipulations), and so are forced to a more normative conception of logic (logic is the study of how one ought to reason), by the following dilemma.
-classical logics can't state a general truth predicate (if they could, we could inductively argue for the soundness of logic, and hence a consistency proof for logic L in logic L, contra Godel 2)
-non-classical logics which can state a general truth predicate, sometimes fail to preserve truth, in some degenerate cases (in places where good reasoning wouldn't lead you to in the first place).

So (Field says) the only people who can *state* the descriptive criterion for being a logic, deny that logic has to have that property.

But I think there's a gap in this argument: why should you have to be able to state your criterion for what a good logical system is, *in the formal language of that logic*? In particular, why can't the anti-normativitst about logic reply like this:

A. Classical Logic Version:

Logic is the study of formal systems of syntactic manipulation which are truth preserving for various fragments of our language (e.g. english sans any truth predicate, english sans any repeated application of the truth predicate). Practically speaking, this is all we need for almost every purpose except philosophy of logic and truth. And the moral of Tarski-Godel considerations above is that this is all we can get.

Formal, exceptionless, rules for truth-preserving reasoning are great when you can get them (i.e. for limited fragments of our language) but what Field has shown, is that we can't get any such rules that apply to the informal notion of truth (as opposed to the notion of truth-of-a-sentence-in-L, for various restricted L)

Admittedly, taking this route involves giving up the traditional and somewhat attractive Fregean idea that logical principles are fully general, and hence would apply to all possible reasoning, but - at least- this seems way less revisionary than the normative relativism about logic where Field winds up.

B. Non-Classical Logic Version:

It was indeed wrong to say that logic studies patterns of inference that are always truth preserving. Field is right that Logic studies patterns of reasoning that are truth preserving "where it counts". But "where it counts" doesn't mean something normative like 'with regard to premises that one could be justified in believing', but rather, something descriptive like 'with regard to premises that people are likely to every actually accept'.

Learning about numbers by thinking about sets

2010-04-09T20:26:00.000-07:00

Maybe I just haven't done enough research yet, but I don't see why it's puzzling that we could learn new things about the numbers by learning things about the sets, and then applying them, given that we know perfectly well how facts about the numbers relate to facts about the sets (some people even identify the numbers with certain sets).

I mean: Is it puzzling that adding to a theory of shapes on a computer monitor (e.g. trangle, square etc) a theory of individual pixels that make up the shapes should let you derive new consequences about what shapes the monitor can display? I don't think this is puzzling - we see phenomena like this all the time e.g. new facts about chemistry can teach us new facts about how DNA will behave, hence about biology.

Or what about the way that reasoning about sets (with ur-elements) could teach you things about ordinary objects: If there's no non-empty subset S of the people you invited to the party such that each person is in that subset was formerly married to some other person in S, then if anyone shows up to the party (and only invited people come), there will be at least one person who fails to meet an ex-spouse there.

---
I am tempted to suspect that this whole thing is not a problem if you are as much of a realist about math as about computer displays or chemestry or biology or party-goes, and if you face problems about how we can *ever* know *anything* about mathematical facts, head on. (what my thesis claims to do). I mean, maybe if you thought that all mathematical knowledge was just a matter of stipulative definition it would puzzle you how we could learn things about the numbers from reasoning about the sets which was (presumably) not part of the stipulative definition of the numbers (or the sets?). But even then, the mere fact that we can *ever* know bridge laws relating the numbers to the sets should be puzzling, not the fact that these bridge laws are fruitful...

Does anyone have ideas for a more charitable understanding of the concern here?

McDowell on Rule-Following pg348

2010-04-04T22:44:00.000-07:00

In 'Wittgenstein on Following a Rule' McDowell's objection to the idea that language use just involves contingent agreement among speakers in their dispositions to go on in the same way, rather than some linguistic community in a richer McDowellian sense seems to be this. If the former view is right, we can never have more than "inductive" certainty that the rest of our community uses the word the same way. Hence, when we apply a certain term in a certain way, e.g. when we say "arthritis is inflamation of the joints" we can only be `inductively' certain that this expresses a truth - it's logically possible that everyone in our language community uses the word differently.

But why is this a problem? This supposedly bad consequence seems directly *true* in the arthritis case. Maybe it's worse to say that you can only be inductively certain that 2+2=4, since it's logically possible that your whole language community uses the word differently. But - come to think of it- don't we individuate language communities by common linguistic practice. So, arguably, if any community were to count as your linguistic community it would have to agree with you about many (most?) assertions that are really central to you, which you feel confident about. So the worry about the rest of our community using "2+2=4" differently enough for it to express a falsehood seems very very slender.

p.s. does anyone know if McD thinks he has a transcendental argument for the existence of other people, from the claim that we can have meaningful thoughts, and hence must belong to some non-private-language community?

Different Senses of the Quantifers?

2010-03-28T07:49:00.001-07:00

Carnapians want to say that different things can be truely said to exist when speaking in different language-frameworks. So the existential quantifier "Ex" will mean different things in these different frameworks. But can there really be multiple different meanings for these different uses of Ex, which would qualify as different kinds of e.g. existential quantification?

An argument that you can't is: The meaning of Ex is determined by its introduction and elimination rules. So any putative kind of existential quantifier would need to obey them. Hence different senses E1 and E2 from different frameworks would both have to obey the standard introduction and elimination rules for Ex. But if E1 and E2 obey these rules, then you can prove E1x from E2x and vice versa. Hence there is no room for ambiguity.

This argument can't be right though, if restricted quantification ('There is nothing in the fridge'. 'All the beers are in the fridge') - something that even the most ardent anti-Carnapians accept- counts as `a kind of' quantification. And intuitively it is. Hence in order to seem like a kind of quantification, a connective need not obey the full introduction rules. It suffices if there's a more limited range of instances of the introduction schema
P(x) --> Ex P(x) that speakers accept, together with all corresponding instances of the elimination schema Ex P(x). (A^B^C..^P(z) > F) ---> F (in cases where z does not occur free in A,B, C... or F). This is what we have for beers in the fridge.

Why can't the Carnapian claim that the same thing goes on with different linguistic frameworks? The different choices for when P(x) --> E2x P(x) is acceptable will each correspond to a different meaning for the existential quantifier. We can even formally represent these different possible senses for existential quantification formally, by saying a kind of existential quantification E_i corresponds to each subset S_i of the set of predicate-expressions (i.e. to each choice of what predicate-expressions the introduction and elimination schema are supposed to hold for).

You are probably worrying that this turns the Carnapian into a kind of maximalist (all the objects in question really exist, different frameworks just correspond to different framework restrictions) but I can't actually see any argument for that. So speak up if you can!

Seeing Sets

2010-03-18T09:40:00.000-07:00

I used to laugh about (early) Pen Maddy claiming that we could see sets. But now I think that's almost right- though not in the way that Maddy intended it.

1.
I can see that my program doesn't infinitely loop, or that the 1000th prime is 7919 by pressing enter, waiting a few seconds and then looking above the command line prompt on my computer. These are all claims about mathematical objects, yet (given suitable equipment and background knowledge, we would ordinarily say that I can see these things to be true).

This seems just as literally true as the claim that I can see that the electricity is on, when I look at the lit windows of the house next door.

In both cases I immediately form the belief, probably am justified, am depending on a lot of contingent assumptions about electronic wiring etc.

2.
But maybe we should distinguish seeing Xs from seeing that some fact about Xs obtains? Maybe there's something especially problematic about believing in objects which you can't see?

-If seeing x = seeing that x exists, then I can see that there is a 1000th prime in the above example (suppose I wrote the program but had never seen the proof that there are infinitely many primes)

-If we take a more intuitive approach to seeing xs (i.e. is it awkward to say I am now looking an X) then:
a) certainly it is awkward to say `I am now looking at a number'...hmm though we might say `I am now seeing the line of the program that causes the crash (and lines in programs are abstract objects, just like lines in poems),'.
b) it's also pretty awkward to say `I am now seeing a drought', or `I am now seeing North America' or 'I am now seeing a proton'.

3.
If you can see a drought when you look at a color map of precipitation, why can't you see a pair of twin primes by looking at a chart?

4.
Overall conclusion:

Seeing that P really means little more than having some visual experience which causes you to immediately believe that P, which you might cite as part of your justification for believing that p. So if you can know things (e.g. all the background mathematical beliefs involved in the program case) about numbers, then it's not too hard to arrange to see things about them.

Of course, the anti-platonist won't think that you can know things about numbers either - well that's where my thesis comes in. But if we can know some things about the numbers, its not hard to arrange things so that we can see further things about them ie rig up reliable methods for forming beliefs about them whose last step involves visual experience.

"Ideal" vs. "Ideal"

2010-03-16T09:19:00.000-07:00

Scientific explanations, which explain the behavior of an actual object by relating it to the behavior of an ideal object, don't usually involve a normative element. It's not as if we think that inclined planes should be frictionless, or planets should be perfectly spherical. These ideal models aren't somehow better then the actual objects in question, they are just easier to think about.

I wonder if psychological explanations of actual human behavior by relating it to rational human behavior ("the price rises because if everyone was a homo economicus with this set of beliefs and desires they would..." "actually, getting a beer is what a fully rational person with Jim's beliefs and desires would do right now..") are just instances of this. If they are, then the normativity makes no difference to the explanation. The idea that one ought to be rational (assuming there is such a fact) plays no more role in the success of the explanation than the claim that inclined planes ought to be frictionless plays in the success of the ordinary physical explanation.

Potter and the Loch Ness Monster

2010-03-16T08:37:00.000-07:00

M. Potter asks why some philosophers intuitively require so much less evidence for introducing abstracta than for concrete objects. How come the requirement not to "multiply entities beyond necessity" doesn't apply to these? Without an answer taking this relaxed attitude towards positing yuppie cliques and category theoretic arrows, while being very skeptical about the Loch Ness Monster looks a bit unprincipled. Well here's a sketch of an answer.

Start with a reliability based notion of justification: we evaluate a creature's justification by thinking of it as having certain faculties i.e. mechanisms that reliably produce true beliefs (e.g. infra-red vision, smell, first order logic). We say a belief is justified when it is the result of one of these reliable mechanisms 'working as intended'. Now in order for mechanisms that produce contingent beliefs to be reliable, they will typically have to be causally sensitive to facts about the outside world - so that e.g. they tend to only produce the belief "there's a llama" in situations where there is actually a llama. In contrast, you can build a faculty that reliably produces the right results about necessary aspects of the world, without using any such external input. And if there are necessary truths such as: whenever there are yuppies behaving in such and such a way there's a clique of yuppies, you can build in a reliable mechanism that makes this transition immediately, without requiring any further input from the environment. So it's not surprising that the reliable belief forming mechanisms we humans have should require less justification for introducing necessary abstracta, or ordinary objects whose existence is necessitated by already known facts about other objects vs. for introducing concrete objects (like the Loch Ness Monster) which lack either of these properties.

Now obviously, what I just said won't convince anyone who has some *other* other reason for rejecting abstract objects, and ordinary objects, to believe in them. But it does provide a unifying explanation, and hence (I think) a way for those who a) have the intuition that introducing abstract objects needs less justification and b) are inclined to take this intuition at face value to defeat Potter's challenge that their intuitions about justification are unprincipled. Quite to the contrary, this distinction falls out of a reliable-mechanisms theory of justification almost immediately!

Anti-Russell

2010-03-16T08:26:00.000-07:00

If (all) propositions intrinsically have a logical structure, then does an english speaker's utterance of "I will go to the store unless you already bought milk" typically express a proposition with the structure ~P>Q, or one with the structure PvQ?

Does it depend on the situation? Who bought milk last time? :)

It seems better to say that propositions expressed by natural language sentences only have a logical structures only relative to a choice of logic, and a method of translation.

Carnap Disenchantment

2010-03-14T10:30:00.000-07:00

[Sigh, I can never make up my mind about Carnap. I guess I'm feeling anti today]

I understand what it is to say that it's "merely a pragmatic choice" whether to use first order logic with the usual connectives vs. with the sheffer stroke. In both cases you will be expressing truths when you derive things in accordance with the logical laws, so the only harm you can do with choosing the sheffer stroke is make your proofs take longer.

But the "choice" of accepting a weaker vs. stronger and (say) inconsistent logical system, does not have this feature. In one case, you will be deriving truths. In the other case, you will now be deriving some falsehoods/crash your whole language so that none of your sentences are meaningful at all.

So I don't see what Carnap can mean by saying that adopting a system is merely pragmatic choice. Adopting a consistent system is a hard epistemic task! The only pragmatic choice is choosing which system - of a menu of systems of reasoning which are coherent enough to give their terms meaning and count as truth-preserving - to use.

Paradox of Analysis

2010-03-13T05:06:00.000-08:00

The paradox of analysis is roughly this: If a conceptual analysis of a term like justice was successful, then the two sides of the analysis should mean the same thing, so it should also be trivial.

The notions of cognitive triviality (analyticity?) and sameness of meaning are infamously hard to spell out, but I think we can get much of the intuitive puzzlement of the paradox of analysis by rephrasing it as follows:

If you know already what 'justice' means, how can it be useful to you to have a conceptual analysis that says an act is just if and only if it is ____?

If you accept this restatement of the problem, I propose the answer is this:

Your "knowledge of what `justice' means" consists in something like a disposition to accept some collection of methods of inference, which - under favorable conditions- tend lead to your beliefs about what's just correctly tracking the facts about what's just. Call the particular algorithm for making and revising judgements about what's just α. So your understanding of the word justice consists in the fact that your brain implements α.

The potential usefulness of conceptual analysis comes from the fact that your brain can implement α without:

a) your knowing what algorithm α is (e.g. some processes in your brain recognize grammatical english sentences, but you don't know what these processes are).
b) your knowing that the descriptions of actions which algorithm α ultimately gives a positive verdict on are exactly those which have property B. (this is useful when your usual methods of checking for B-hood are faster/easier to deploy than your usual methods of checking for justice)
c) your knowing that property C applies to most of the things which A would ultimately give a positive verdict on, but C is easier to apply, and all the purposes normally served by considering which actions are just would be served even better by thinking about which actions have property C. (the classic definitions of computability and limit are examples of this kind)

"Coherence" and Mathematical Existence

2010-03-11T06:59:00.000-08:00

When I say that "the more practically benign a system of proto-mathematics is, the more likely it is to count as expressing largely true claims about some domain of objects", I realize that this sounds horribly woolly. People naturally ask me: but how practically benign does a practice have to be to guarantee that it succeeds in talking about some object? (not to mention: how do you measure `largely true'?)

Why Be Woolly

Here's a classic example of a claim that isn't woolly:

H: Any logically consistent system of mathematical beliefs counts as expressing truths about some suitable domain of objects.

We can see H is false because it implies that if I believe ZFC+{X} and you believe ZFC+{~X} where X is some statement about number theory independent of ZFC, since we both have logically consistent systems of belief, we will both be right - just talking about different objects.

But what goes wrong?

Note the problem isn't that there aren't enough mathematical objects (if we just have sets every first order consistent theory has a model). Rather (I claim) it's because actual people will use words in the mathematical theory like 'finite' or 'smallest' or 'number' which have meaning that goes beyond their role in this first order logical stipulation.

When we both say that by the "numbers" we mean (among other things) the smallest collection containing 0 and closed under successor, smallest (intuitively) means the same thing for both of us, so it is NOT correct to then interpret each of us as talking about whatever larger non-standard model makes our claim true.

Hence our informal use of the words like "smallest" or "all possible collections" imposes constraints on interpreting us which go beyond the first order logical content of our mathematical statements.

If you buy this, here's why you should be wooly. In general there will be some vagueness with regard to how wrong you can be about Madagascar, Christmas or to use Quine's famous example, atoms, and still count as talking about these things. Once a theory is sufficiently wrong it can be a tossup whether to say that the objects in question are real, and the person is wrong about them, or that there are no such objects. But this is exactly what we face with regard to mathematical objects as well! We have an amorphous informal practice, and a norm that people count as referring to whatever the most natural object is that best satisfies their methods of reasoning about these putative objects, provided there is one that matches suitably well.

There's no bright line about how wrong your various formal and informal beliefs about some putative object can be while you still still count as referring - for the same reason in math as in physics or history.

Hence, I don't try to draw one, and that's why I'm woolly on this issue, and why you should be to!

All we can say generally is: Mathematicians can posit new objects, and the more logically consistent their reasoning about these objects is, and the less their intuitions about consequences of reasoning about these objects lead to false conclusions about other things, the more likely it is that they will count as expressing largely true claims about some suitable piece of the mathematical universe.

p.s.
The other non-woolly alternative is to give a list: a mathematician counts as referring if they have x beliefs (which are true of the integers), y beliefs (true of the reals), z which are true of imaginary numbers, w for quaterinians, v for sets, k for arrows ... and thats all the mathematical objects that it is metaphysically possible to think about! But surely this is insane.

trolly problems and literature examples

2010-03-11T06:42:00.000-08:00

I've heard it suggested that moral philosophers should consider examples from literature rather than simplified cases as in trolley problems. Here's a theory of why examples from literature might be particularly bad for moral philosophy purposes.

Kant says (as I understand him) that the experience of beauty happens when observing an object provokes the "free play" of the conceptual faculties, producing a harmonious volley between the intellect and the imagination. This works most naturally for novels and poems, where reading a line can set off a chain of thoughts which aren't logical deductions, but are still somehow naturally suggested by the line.

In contrast, in much moral philosophy you are looking for (relatively) general principles [it's an interesting question why this is], that different people might agree to and be guided by even when particular interest leads them in different directions. So you want something like "all actions of X kind are impermissable", For these purposes, you want to show that your general principle is acceptable even in, as it were, the worst case scenario, even in the most perverse instances. You also want to avoid features that would be distracting, from the question of whether the given action is permissible, and also unclarity about what the descriptive scenario is supposed to be.

Now, if we buy the kantian idea about beauty we get a quick explanation for why literature examples will tend to be bad for the purposes of moral philosophy. Beautiful cases will be ones that promote the free play of the intellect, considering all kinds of different aspects of what's being described, and reaching out into all sorts of other questions. Hence they are particularly likely to involve a) simulatious application of multiple apparent moral reasons for and against b) interesting factual questions about what the situation really is (do we really know that soldier is unpersuadable?) c) other different but related moral/philosophical issues - that one might easily confused with the issue of unpersuadability.

So literature examples may be good a suggesting questions, but there's some reason to think they are - so to speak- actively engineered to be distracting when consider as examples in debate about some particular moral principle.

Species and Couches

2010-02-21T12:25:00.001-08:00

A smart philosopher of biology I know claims to be researching "whether there are (really) species, as opposed to just individuals". So far as I can tell, he is investigating whether biological explanations that appeal to species are not really always better put in terms of individuals. That is, he's studying whether talking about species serves a certain kind of (ineliminable?) role in biological explanation.

That definitely seems worth worth investigating - especially since there are so many cases where the distinction between different species looks very unprincipled. (Because of ring species and ligers it won't do to just say that two things are the same species if they can produce fertile offspring.)

But it seems strange to me that he puts this in terms of `investigating whether species really exist'. This is because, presumably, he thinks couches really exist, and yuppies too, even though we could surely phrase an adequate biological and scientific theory in such a way as not to entail any sentences of the form Ex couch (x) or Ex yuppie(x).

What I THINK might be going on is that he thinks objects need to earn their keep, in a way that concepts don't. That is: it's fine to apply scientifically useless predicates like "...is a yuppie", but not to introduce scientifically useless *objects* like species. On this reading he would be fine with saying that dogs exist, or that two newts are consepecifics, but not with saying that there are (abstract) objects called species.

But I don't see quite how one would motivate this differential treatment. (Admittedly this may have something to do with my current adherence to the merely logical notion of objecthood). Also, the problems for the notion of species looking unprincipled seem to apply just as much to claims about being a dog or being two animals being conspecific.

*Obviously some scientifically useless objects are bad to introduce, like the flying spagetti monster but that's because their existence would entail false claims about the distribution of matter in space-time. In contrast, just proposing new ways to think of the same old distribution of matter etc. in space-time as constiuting objects e.g. (tables, vs. half tables, vs. complete livingroom sets vs. dearths of tables) in different ways, seems harmless.

[edit: there is something SLIPPERY about the way I am using the concept/object distinction here. must think more about this, and ask the philosopher of bio]

Explanation Puzzle

2010-02-21T10:05:00.000-08:00

On the one hand, we think that the fact that a theory T1 allows for "better explanation" of a certain phenomena than T2, gives us reason to believe T1 rather than T2 is correct. It's obvious (if not particularly explanatory) reason to prefer one theory to another that it "does a better job of explaining the data"!

On the other hand, we think that a better explanation can be one that better helps human beings "grock" patterns in the behavior of physical systems which may be mathematically very complex. Given human psychology, attention span etc. a simple ceterus paribus statement about struck matches tending to light can be a better explanation than an explanation that appeals to more specific details. To choose a more extreme example, even if there were a completely successful theory of microphysics, most people feel we would still have an explanatory task. We would still want elegant theories that told us about general high-level patterns in how the microphysical facts would evolve forward through time. (e.g. the ideal gas law, biology and maybe psychology and economics).

But now here's the problem: do we really think that the fact that a theory T allows for nice tractable/human-grockable explanations of high level phenomena makes it more likely to be true? For example:

I find "consider a spherical cow" style economics explanation, or "consider philosophers building up society from a state of nature" style early modern philosophy explanations way more attractive, satisfying, and easy to remember than explanations that cite lots of boring contingent historical facts. But this doesn't really make me feel that these explanations are more plausible, or getting at the heart of matters more.

I mean, I wouldn't be surprised if primate intelligence is optimized for avoiding getting double crossed by other monkeys, and making practical plans etc. so we like explanations better if they relate the explananda to these things (i.e, people with plans). Indeed don't we actually find this with explaining a phenomenon to people in different disciplines- that people familar with different areas find different explanations more satisfying?

We like explanations where lots of correct consequences "fall out immediately" from a tiny theory. But what seems to fall out immediately (vs. just be an ugly mathematical consequence) may well depend on how familiar you are with inferences of that kind. And folk (belief/desire) psychology is something we are *all* very familiar with from daily life. Hence, when someone says "this electron wants to escape the other electron" or "countries covet land", we have lots of immediate ideas about what behavior should follow from that, because we are experts at drawing consequences from belief desire psychology, and then we just convert these consequences back to the task at hand.

But surely allowing for nice parallels to common problems in monkey social climbing, is not a feature that has much to do with genuine theoretical elegance/ how likely a theory is to be correct.

On Rationalizing Explanations

2010-02-21T09:30:00.000-08:00

Philosophy papers (like David Lewis' Languages and Language) often seem to want to explain the fact that some X is actually the case (e.g. we all use the word "fire" in roughly the same ways) by showing why it would be rational for people to make X the case/preserve that state of affairs X. But this seems potentially problematic:

a) Historians wouldn't generally accept the idea that showing why declaring war was rational for a certain leader explains why he actually did declare war. If the presedent has the policy of always taking the first proposal suggested when he's tired and wants to go home, and P was the first proposal suggested, the fact that it would be rational to do P rather than Q is not the correct explanation for why the president actually did P rather than Q. Similarly if the president never even considered Q, the fact that P serves his interests better than Q seems like an incorrect explanation for his choosing Q.

b) More generally, rationality explanations seem to have exactly the issues that philosophers of biology make a huge deal about when considering evolutionary fitness explanations: the mere fact that some trait would be eliminated by natural selection isn't always the correct explanation for why we don't find it. For example, the fact that humans don't levitate isn't explained by natural selection, but rather by the fact that the total space of mutations available from the original organisms doesn't include that. (That is: plausibly, even if all creatures had had all the offspring they could, and lived as long as they could so there was no culling to generate natural selection, we would *still* find 0 creatures that levitate. )

Applied to the David Lewis case of conventions, this works out in the following way. It *might* be that we get linguistic conventions because once some people are using language a given way, each person works out that it would be rational for them to do the same (this is the nub of lewis' account). Or it might be that most possibilities for doing things differently don't even occur to them - people just brutely follow custom and habit and imitate those around them. (Apparently apes' tool use is like this: chimps in a given area all crack nuts using the same techniques, even though different techniques would work just as well and are used in different areas. Note here that there's no rational benefit to cracking nuts differently from your neighbor). Or it might be some combination of habit plus considerations of rationality. And surely its an empirical matter to find out which.

Given this, I think we can read Lewis either as making a bold empirical conjecture, or as explaining actual behavior by comparing it to the behavior of a simpler, but in some respects similar, model system (as often happens in biology).

bold empirical conjecture: In fact, adherence to linguistic conventions is always produced, not by custom and habit whereby doesn't occur to people to behave otherwise, but by speakers recognizing that it would be rational for them to continue with the convention.

simplified model system explanation: In ideal system S (where there are no limitations on computational power etc and people always behave in the way that best advances their aims), linguistic conventions arise and persist. The actual world of people talking is `relevantly similar enough' to S, for these facts about S to explain how actual linguistic conventions arise and persist in the actual world. [Slot in whatever notion of relevantly similar explains how facts about waves in infinitely deep oceans can explain facts about waves in actual finitely deep oceans].

this keeps coming up...

2010-02-13T08:43:00.000-08:00

You might think that a satisfactory account for mathematical knowledge would have to at least prevent us from ever winding up with/keeping contradictory mathematical beliefs.

But this is wrong for two reasons:
a) OLD REASON: "We" did have contradictory beliefs about mathematics at various periods in history (naive set theory, inconsistent reasoning about infinity).
b) NEW REASON: Many people have contradictory beliefs about baldness (as brought out by the sorites paradox), but this doesn't prevent them from knowing many things about baldness, who is bald etc.

de re beliefs about numbers?

2010-02-13T06:57:00.000-08:00

I just read some Azzuni which seemed to attribute the following argument to Burge:

"The difference between de re and de dicto thought, is that de dicto thought can have content that `goes beyond' your concepts and picks up info from the environment. So if I think de dicto "the nearest vase is green" the proposition which this expresses is purely determined by my concepts. Whereas, if I think de re "*that* vase is green", this picks up content from the context (in particular it claims something which would be false if that vase got painted white, but some other vase wound up getting put in front of me instead).

Now, (one might go on to think) , this suggests mathematical thought involves a de re component. Why? A de re component would explain how mathematical talk can pick up content from facts about matheamatical objects outside the head. Hence, it could explain why the truth conditions for mathematical facts go beyond my (probably recursively axiomatizable) inference dispositions."

The problem with this is that, as Burge himself is famous for pointing out, what someone means by concept-words like arthritis can also require one to `go to the context', (in a slightly broader sense) of how experts near the speaker use the word arthritis, to determine what proposition/truth conditions a sentence about ``arthritis'' has.

So, if the evidence is just that mathematical truths can depend on stuff that `goes beyond'*[Yuck, if there were some typographic convention stronger than academic shudder quotes I'd be using it here :)] our presumably recursively axiomatizable inference dispositions, then I see no evidence for the claim that our number talk is de re. Dependence on broader context could be achieved either by the the object-word "3" functioning as some kind of hidden de re ostension, or the concept word "is 3rd in a number-sequence" (or whatever other pseudo-definite description you would want to associate with three) having a meaning that isn't entirely determined by stuff in the head.

Many other things seem shady too, but I should really read more of Burge's own words before getting too dismissive :).
[edit: ok Burge himself does not seem to be making this argument in the relevant article which is here]

Funny

2010-01-22T21:12:00.000-08:00

Have you seen this article by Willfred Hodges about reading people's "refutations" of Cantor's diagonal argument?

It's pretty funny, and he even raises some interesting philosophical issues about logic, how to think about logical mistakes, etc.

Fallacious but psychologically attractive inferences

2010-01-22T20:56:00.000-08:00

I like to start with really simple theories and see where they go wrong. Recently this lead to an interesting combination of experiences

When I say:

People are justified in making those a priori inferences which are both necessarily truth preserving and psychologically compelling for normal humans.

People say: But reasoners do have initial justification for accepting certain attractive but ultimately fallacious arguments e.g. tricky arguments for the existence of God.

But when I take out the requirement of being genuinely truth-preserving and say...

People are justified in making those a priori inferences which are psychologically compelling for normal humans.

People say: But what about those bad but psychologically compelling inferences like inferring the consequent?

So which is it (do you think)?

When someone makes a psychologically compelling but invalid inference like the gambler's fallacy or inferences about naive set theory are they:
a) justified, (though presumably thinking about the right questions may later give them justification for changing their mind) or
b) unjustified
c) somehow there's a difference between the gambler's fallacy and naive set theory in this regard

I don't have a dog in this race, or think anything deep is going on here, but I'd really like to know which way normal language intuitions go.

Speculation re: "Faculty of Reflection"

2010-01-20T22:06:00.000-08:00

This may just be totally ad hominem speculation (though there's no particular person I have in mind), but...

Maybe the reason why some are inclined to think we should always be able (in principle) to reflect and formulate a system that captures any recursively reasoning we do about math and logic has to do with a certain view about reasoning + rule following.

Remember how Wittgenstein (in the Blue and Brown books) criticizes this theory that we manage to obey the command "bring me a red ball" by first imagining a red patch to get an idea of what color "red" is, and then picking the color that matches this imaginary patch. He says: do you first need to imagine another red patch, in order to know what color to make your sample patch? So, this theory leads to a kind of regress. [I'm always tempted to call it the smoke-two-joints-before-you-smoke-two-joints theory of understanding language, after Bob Marely's famous song about regress]

Presumably everyone will agree that in this case we need to just posit (and perhaps scientifically explain) a direct ability to pick red things, when commanded to. Invoking a further layer of person-level thought (where you pick the red balls by first doing something else like imagining a red patch) just leads to regress.

But, an analogous theory with regard to mathematical reasoning would be that when we are asked to answer some mathematical question, what we do is first consider certain rules for how to reason about mathematics, and then do what these rules say. Now, I think this is a very bad theory. But, if you accepted it, you might think there could well be a special process of reflection where you, in effect, remember these rules, or become consciously aware of the rules you were unconsciously appealing to all along. That is, you might think: any recursively enumerable portion of mathematical reasoning you accept, you should be able to formalize (and recognize to be correct) by making explicit all of the (presumably finite number of) rules that you implicitly consulted when doing that reasoning.

Reflection and Limits on Mathematical Knowledge

2010-01-20T21:03:00.000-08:00

It's a classic question whether there are mathematical truths which are unknowable by creatures like us. And, as Bill Clinton might have said, the answer to this question naturally depends on what you mean by "creatures like us".

A smart philosopher recently suggested that I should take the following possibility seriously: even if particular human's brains were well well approximated by a Turing machine, the faculties by which humans access mathematics include (once suitably idealized) a faculty of reflection, whereby one could transcend the possibilities of any system to which Incompleteness applies.

So, I'm taking that possibility seriously, here's why I reject it :)

If by `a faculty of reflection' you just mean something that lets you say, of any particular formal system which you believe to be sound, that it is consistent, then this is not enough to get around incompletness for familiar Putam vs. Penrose reasons.

If by `a faculty of reflection' you mean something which lets you produce a system which formalizes all your current reasoning about mathematics, and then recognize that the system does this (so that then then you can deduce this system is consistent and arrive at its con sentence) then I don't buy that humans can be plausibly idealized as having anything like this kind of faculty.

Behaving in a way that matches a given algorithm is one thing, coming to know that this is what you are doing is quite another! The issue here is essentially the same as with the Kant Puzzle I poster earlier. Certainly we can work out that particular examples of conclusions that the formal system proves, and check that yes we accept that conclusion. But to arrive at the con sentence you would need to know that everything the formal system proved was correct not just some finite number of instances.

Now, admittedly, after trying enough cases, (if the system was simple and elegant enough) you might be willing to accept that yes everything the formal system proved was something you accepted, and hence infer the con sentence.

But,
a) its somewhat controversial whether beliefs formed in this way would count as knowledge

b) this process might dead end at a point where all the reasoning you accepted could only be summarized by an alogorithm/formal system that looked ugly and gerrymandered to you, and hence was not a plausible candidate for induction.

c) if this is the sense in which we could always get access to con statements, no faculty of 'reflection' in particular would be involved, just a general ability to apply something like scientific induction to mathematics. The same kind of reasoning that gets you from 'the first million things proved in this system are ones I accept as true, so all of them are true (so the system is consistent)' would also get you from 'the first million numbers have property p, so all numbers have property p'. In both cases we're accepting a simple general principle on the basis of seeing that it holds true in finitely many cases.

Thus I think that if there's any sense in which idealized human mathematical reasoning transcends the limits imposed by incompleteness, it's not because we have a specific faculty of reflection.

Instead, it's because of something much less glamorous: because we're willing to apply not-always-truth-preserving methods like scientific induction to mathematics, and hence disposed to accept certain claims that are actually inconsistent.

Doubting Conceptual Truths

2010-01-20T20:42:00.000-08:00

Some things, some Kantians say about the justification for logic suggest the following superficially attractive idea:

a) Certain sentences have the property that anyone who can think about them must thereby be inclined to accept them (e.g. you can't even think thoughts involving `and' if you aren't willing to accept 'If it's raining and it's snowing then its raining')

b) We are justified in accepting such sentences, because we have no other option.

But actually, its plausible that we can reasonably doubt many sentences with this property. This is because sometimes you can turn out to have been working with an 'incoherent concept' like tonk, or bosh, the naive concept of set or perhaps various philosophical concepts. In such a case, you don't count as thinking with the concept unless you are willing to make certain (bad) inferences.

Now, you might argue that someone who was taken in by this kind of incoherent concept doesn't count as thinking anyway (e.g. there's no proposition which ``it's raining tonk its snowing" expresses). So maybe you only count as *thinking* in the good case, where it turns out that your concepts are coherent. But, given that we know that very smart and conscientious people can wind up with bad concepts, it intuitively seems reasonable to not completely dismiss the possibility that various new concepts you are learning are among the bad ones.

Hence*, it would seem that, we can rationally doubt claims which it would not be possible to deny. (what we're concerned about here is not the possibility that the claim is false - which we can't entertain- but that one of the concepts figuring in it is incoherent, so the claim is nonsense)

*if a) is true

Frege and Obviousness vs. Self-Evidence

2010-01-09T06:57:00.000-08:00

In a recent article Shapiro writes"There are obvious propositions, such as 2 + 3 = 5, that are not self-evident. Frege emphasized that to know sums like those, one need not invoke any intuition, Kantian or otherwise, but he insisted that one must reason one’s way to this knowledge."

But in what sense did anyone before Frege "reason their way" to the claim that 2+3=5? From what? The quoted claim seems to have the consequence that either:
a) no one knew any arithmetic before Frege's derivation of arithmetic from logic
b) people before Frege could count as knowing arithmetic because they were unconsciously going through Frege's derivation

Wacky Aesthetics

2010-01-02T12:29:00.000-08:00

Currently physics background for another project is devouring most of my mind, but I had this kindof wacky idea for a big-picture theory of aesthetics.

Suppose we say: Naive intuitions about beauty correspond to a folk theory whereby people naturally would all like the same things, except for certain deceptive conditions like:
a) only liking something because you were told it's a great work of art
b) not liking something which you would otherwise like because you've seen
it so often
c) liking something only through ignorance of some of its descriptive
properties
d) liking something which you would easily get sick of
etc.
These are platitudes about which causal influences on aesthetic judgement
are "misleading".

But now, pull a Kantian revolution, and say that it;s not that these traits are misleading because they lead us to fail to track some antecedently natural and interesting category of the beautiful, but rather that `beautiful' refers to (roughly) "the kind of thing that people would like if not for the conditions indicated in the platitudes." So far this sounds kind-of like response-dependence theories but...

There's one more wrinkle, since there might be enough natural variation between people for there to be no interesting class of objects which everyone is (ceterus paribus) disposed to like, when free from the deceptive conditions indicated by the platitudes above. It's a psychological matter whether getting more ideal in these respects would
lead to convergence or divergence in aesthetic judgements.

So here's the view. If there is a fairly definite class of things people (in our linguistic community) are inclined to like when free from the platiduinous bad influences then `beautiful' rigidly designates that class of objects. If there is no such class, `beautiful' is implicitly speaker relative. In the latter case, we can think of arguments about beauty as involving the implicit assumption between both participants in the argument that their dispositions to like or dislike that objects would be the same if free from platitudinous bad influences.

What the Indispensibility Argument Isn't

2009-12-15T14:41:00.000-08:00

When I first heard of Quine's indispensibility argument (We are committed to the existence of abstact objects, since we must quantify over them in order to state our best physical theory), years ago, I misunderstood it.

I thought Quine was trying to draw an analogy between nominalists and, say, people who deny that there are planets, but will admit all the usual observations through telescopes. We find the planet denier's position implausible. We want to say "If there aren't planets, how come -as you admit- everything we see through telescopes behaves just as it would if there were planets? If there are planets, this explains the order and regularity of what we see through telescopes. But if there aren't planets, how come - out of all the mindbogglingly many possible patterns of optical illusions - we happen to have ones that are just like ones that could be produced by seeing persisting objects in space?"

But, even aside from being not what Quine meant (as I was soon told), this is really not a good argument, as I shall now argue for the benefit of anyone else who is tempted by it.

The explanatory inadequacy argument above, crucially turns on our having a notion of the observations we would have if there were planets vs. various other patterns of observation which would *not* be consistent with there being planets. The planet denier's position is unattractive, because on their theory it looks like a miracle that we happened to get a coherent pattern of optical illusions that could have been produced by normal vision of real objects.

But, in the mathematical case, there is no such contrast. There is no pattern in the behavior of physical objects which suggests the existence of numbers. For what would the contrast class to exhibiting this pattern be? It's not like we think: actually cannonballs accelerate towards the earth at 9.8 m/s^2, but if there weren't numbers, they would probably just fly around crazily. On the platonist's own view the objects (numbers and functions) in calculus don't come down and beat on cannonballs to make them behave in ways that are describable by short differential equations. Nor do they prevent cars from going two meters per second for two seconds, in a given direction, but only traveling 1 meter in total (cars don't need to be prevented from doing the metaphysically impossible). So it's not the case that he would expect different behavior if there weren't any numbers (the way we would expect different behavior of telescopes if there weren't any planets). Thus, there's no argument to be made against the nominalist, along the lines of "If there aren't numbers, how can you explain the fact that things happen to look just like they would if there were numbers?"

Instead, the point of the Indispensibility Argument (or the only plausible version of it) is that the Nominalist cannot even *state* his theory of the physical objects he accepts, and how they behave, without quantifying over abstract objects, and hence contradicting himself. To summarize: Quine isn't saying we need numbers to explain observed patterns in the behavior of physical objects. He's saying we need numbers to even state the relevant patterns in the behavior of physical objects.

"No Fact of the Matter" Paradox?

2009-12-15T11:41:00.000-08:00

Here's a line of reasoning I just came up with, that seems paradoxical.

(1) Quine points out that there's a kind of Sorieties series of different theories posting "atoms", ranging from Democritus' theory where the whole point of something being an atom was that atoms are indivisible to the current theories on which atoms are in fact divisible. (Let's use "atom0" to express Democtitus' notion of atoms.)

(2) This suggests that when you are far enough away from having a correct overall theory some phenomemon, the truth value of your scientific words can be vague. For, if it is vauge whether someone intermediate scientist counted as meaning atom by "atom" rather than atom0 or some other notion, then it is vague whether their assertion "there are atoms" expressed a truth.

(3) Science progresses, and we clearly have more to learn about fundamental physics (e.g. how to reconcile QM and Relativity), so we are probably in the same boat with regard to some of our current theoretical terms, maybe "quark" or "superstring". Suppose (without loss of generality) this is true of "quark".

(4) If (3) is right, there's no fact of the matter about whether "there are quarks" (as said by me now) expresses a truth.

(5) But (assuming we can apply Tarksi's T schema to an ordinary looking case like this), "there are quarks" expresses a truth if and only if there are quarks.

(6) So, there's no fact of the matter about whether there are quarks. (!)

(Conclusion) Either there's no fact of the matter about whether there are quarks, or there's no fact of the matter about whether there are strings or etc. for some term with a similar role in phyiscs.

At least, if the conclusion is true, this would be very surprising since when someone says "there's no fact of the matter as to whether" X we usually take them to be suggesting that we dismiss the question, while, presumably, scientists studying whether there are quarks/strings is a paradigm of the kind of question we DO want to invest energy in discussing.

Does mathematics "need" new axioms?

2009-12-14T09:41:00.000-08:00

Here's something I'd like to figure out. When philosophers ask "Does mathematics need new axioms?", what is the is the intended task, such that they are asking whether we would need new axioms to accomplish it?

Here are some possibilities:

-to know all mathematical truths (well, we can't do that, with or without new axioms)

-to formally capture all our intuitive judgements about mathematics (there are familiar putnam vs. penrose reasons for thinking we can't do that either)

-to formalize some particular body of generally accepted mathematical reasoning, where everyone agrees on what's a good argument, but this can't be captured by logic plus the axioms we currently accept, and having a formalization would be practically helpful.

-to be in a state of believing all propositions which we are justified in believing.

It seems to me that, there's a great danger of launching into the debate about whether "math needs new axioms", and taking a position based on whether e.g. you like or dislike set theory, without having any clear sense of what you are claiming that we do/don't need new axioms for. Hence, I'd like to get clearer on different senses the question can have, and which one(s) are at stake in typical philosophical discussion.

Bookclub: 'Compositionality, Understanding, and Proofs'

2009-12-09T22:18:00.000-08:00

In the latest Mind, Peter Pagin argues that Dummett's proof theoretic semantics is incompatible with the compositionality - a popular view in philosophy of language.

Compositionalty is the view that the meaning of a sentence is completely determined by the meaning of its parts i.e. for every connective that might be used to build up a sentence, there's a composition function which takes the meanings of whatever components the connective is being applied to, to the meaning of the overall thing you get after applying the connective.

Proof theoretic semantics is the idea that: a) understanding a sentence consists in an ability to recognize (canonical) proofs of that sentence, and b) the meaning of a sentence is "the property of being a proof of that sentence".

Odd as I feel defending Dummett, on any subject, I think Pagin is wrong to say these two things are incompatible.

What compositionality (as stated in e.g. the stanford encylopedia, and "informally" by Pagin himself) requires is that, for each connective phi, there be a function Cphi which takes the < property of being a proof of p, the property of being a proof of q, the property of being a proof of r > to <the property of being a proof of phi(p, q, r))<. But if you accept compositionality at all, this has to be the case, because the property of being a proof of phi(x) can only be different from that of being a proof of phi(y) if x and y are different, and hence the property of being a proof of x is different from the property of being a proof of y. I don't think Pagin would deny this.

The problem is that Pagin seems to think compositionality + proof theoretic semantics requires something more. He writes:

"The combination of proof-theoretic semantics with the requirement of recognizability of proofs comes into conflict with compositionality. For assume that we have a semantic function phi for a language L. A generalized composition function {rho} for phi must then meet two conditions: (i) it must be possible to know the meaning of any complex expression in L by knowing {rho}, the modes of composition and the meaning of simple expressions; and (ii) the condition of being a canonical proof must, for every provable sentence A, be met by some proof that is recognizable by any speaker who understands A."

Note the switch here from the idea that compositionality says there must BE a function, to the claim that it must be possible to learn the meaning of words by KNOWING this function together with various other facts.

Firstly, the very idea of "knowing rho" (where rho is a function) makes me feel itchy and confused. I understand what it is to know *that something is the case* e.g. that a function f takes a certain value on a certain input. And I (kindof) understand what it is to know a person (e.g. I don't know Bill Gates, but I do know my advisor W.G.). But what's the equivalent of being on a first name basis with an abstract mathematical object? Does knowing a function mean being able to compute it? Being able to give a definite description that refers to it? Being able to give two distinct definitions definitions and knowing that they pick out the same function.

My best guess at what Pagin intends here, is that 'knowing rho' = knowing some proposition of the form:", is the composition function for whatever language L is in question'.

But now, note that Pagin's claim doesn't follow at all from the idea of compositionality - that the meaning of a composite sentence completely supervenes on the meanings of the pieces it is composed out of. The claim that a function with a certain property *exists* does not entail that it is possible to *know* such a function exists, or that this function is computable, or that it is possible to know which program computes it! So, compositionality doesn't imply that its even possible to have such knowledge, much less that it's possible to use this knowledge to learn the meaning of various composite expressions.

This distinction is especially crucial to remember in the context of discussing Godel's Thereom. For, remember from the Putnam-Penrose debate that all our reasoning about mathematics might well *be* recursively axiomatizable, it's just that we couldn't use mathematical reasoning to come to *know* what this recursive axiomatization was.

And, alas, Godel is exactly where Pagin is headed. For, his argument turns out to be that, if you could know some concrete specification of the composition function rho, you could mill out a recursive specification of the class C of acceptable proofs in number theory, then you could use this to construct an acceptable proof of the con sentence for C, which is itself a statement in number theory, but (by Godel I) cannot be proved in C. Contradiction.

Pagin's conclusion is that compositionality and proof-theoretic semantics are incomptatible. But, if this argument works, all it really shows is proof-theoretic semantics requires that one could not come to *know* a recursive specification of the composition function phi.

At this point, Pagin might say that the whole point of compositionality is to explain how we can know the meaning of complex sentences, by knowing their parts, so that accepting this point would be bad news for the proof-theoretic semanticist. But note that, we obviously don't understand composite sentences by explicitly breaking them don into parts. So the fact that we could never realize that something was a concrete specification of the composition function for our language, doesn't prevent compositionality from helping explain our linguistic abilities.

Justification vs. Truth Puzzle

2009-12-08T05:00:00.000-08:00

For the purposes of this post, I'm assuming something like the intuitive notion of justification makes sense.

Sometimes people say:

1. "You should believe what's true, and avoid believing what's false."

Other times they say:

2. "You should believe what's justified, and avoid believing what's unjustified."

But prima facie, these are incompatible demands, since there are many true propositions which I am not justified in believing, like statements of the form "Tommorrow's winning lottery number will be ....", and 1 seems to entail that I should believe these claims, while 2 seems to entails that I shouldn't.

Puzzle: Can these two claims be made compatible? What is the relationship between these them?

first pass- Maybe we want to widescope? e.g.
1 ='Should[(Ax) Believe(x) <--> Expresses-a-Truth(x)]
2 ='Should[(Ax) Believe(x) <--> You-are-justified-in-believing(x)]
Though this suggests the conclusion that you should bring it about (by some kind of superhuman feat of evidence gathering?) that you are justified in believing every truth. Which is, maybe, odd.

Anscombe + Descartes

2009-12-07T01:47:00.000-08:00

If Descartes' argument for dualism really is (as Anscombe seems to be suggesting in her essay on "The First Person"):

I know there's a thinking thing (namely myself).
I don't know whether there are any bodies.
Therefore: there's a thinking thing which is not a body.

then it seems to me, his argument is immediately fallacious. I mean, it's exactly like someone saying, after shaking a box and hearing a rattle:

I know there's a thing-inside-this-box (namely the one I heard rattle).
I don't know whether there are any marbles-inside-this-box.
Therefore: there's a thing in this box that isn't a marble.

Given that objects can have multiple different properties A and B, it's obviously possible to know that there's something with property A while being ignorant as to whether there's anything that has property B!

So there's no need for Anscombe to go to the lengths of denying that "I" refers to escape the force of such a weak argument as this, it seems to me.

Kant Puzzle

2009-12-07T01:15:00.000-08:00

Based on my Kant 101 level knowledge of the subject, it's tempting to think:
- Kant's problem of accounting for synthetic knowledge is to explain how we are able to make certain (non-analytic) judgments in advance of experience, which experience then bears out.
- Kant's answer is that our minds organize experience in such a way that whatever input comes in from the noumena we will always represent a scenario in which these propositions hold true. So, for example, I can know in advance of experience that there are no round squares because my mind organizes experience in such a way that it couldn't represent a round square.

BUT if this understanding is right, then Kant's answer doesn't seem to do a very good job of addressing the problem he poses. For, the mere fact that my mind couldn't represent a scenario in which ~P does nothing to ensure (or even make it likely in any obvious way) that I would *realize* that I couldn't have an experience as if of P.

I mean, think about what we find attractive. It might be a psychological fact about me that no physically possible configuration of matter would strike me as constituting a person who is both tall and attractive. The algorithms that produce my feelings of attraction and the ones that detect tallness might be such that no possible sensory input could set off both. But none of this entail that I *know* that I am incapable of finding tall people attractive. Perhaps all I know, at any given time, is that I have not seen or imagined an attractive tall person *yet*.

Thus even if Kant's claims that mathematics, principles of causation and so forth are somehow an artifact of the way the human mind organizes experience were true, this (it seems) would not yet constitute an explanation of how we can manage to know about these subjects.

So the puzzle is: what should Kant's reply be and/or where is the interpretive failure in the argument above?

Bookclub: Pedersen on Wright's Entitlement

2009-11-26T05:11:00.000-08:00

This latest installment is about Nikloj Pedersen's recent Synthese article on Crispin Wright. Pedersen criticizes (correctly in my opinion) certain possible motivation for Wright's idea that we are entitled to assume certain "cornerstone" propositions (like 'I'm not a brain in a vat') just because assuming these is requisite for getting any substantative theory of a given area off the ground. (You can't just point out that accepting ~BIV leads you to have many and no fewer true beliefs than the skeptic if BIV is true, and no fewer true beliefs if BIV is false. For, avoiding false beliefs is presumably also epistemically important, and assuming ~BIV imposes a risk of having many more false beliefs)

Instead he proposes that such cornerstone assumptions have "teleological value" insofar as they are aimed at something of value (namely, true belief), whether or not they actually succeed in producing such true beliefs. But this seems to immediately generalize to all beliefs - not just cornerstone ones.

For, what beliefs aren't aimed at the truth? It's just as true of the person who assumes the existence of a massive conspiracy as of the person who assumes the existence of the external world that they aim at having many true beliefs. Indeed, many people would say that it's a necessary truth, part of what it means for something to be a belief, that in believing that P one is trying to believe the truth.

With the possible exception of cases like the millionaire who bribes you to believe some proposition, all beliefs would seem to aim at truth. Hence it seems that all beliefs inherit teleological justification in Pedersen's sense.

One might be able to make this into an interesting view - all beliefs (not just cornerstone ones) are warranted until one gets active reason to doubt them. Such a position is remeniscent of conservitivism and coherentism. But, from the article Pedersen shows no sign of intending to say that all beliefs are default justified.

Crispin Wright and Rule Following

2009-11-25T02:10:00.000-08:00

In the paper on Rule Following here, Wright suggests that good reasoning proceeds in obedience to with concrete rules, rules which we can in principle give at least a rule-circular justification for. I claim that Wright's view is only tenable, IF humanlike reasoners count as `obeying' infinitely many rules in this sense.

For, suppose a good reasoner only obeys finitely many such inference rules. And suppose, as Wright wants to claim that reasoner can come (by reasoning) to provide a rule circular justification for each such rule, (i.e. show that applying this rule cannot lead from truth to falsehood). But then our reasoner can combine all these finitely many justifications, to arrive at the conclusion that anything arrived at by applying some combination of these rules must be true. Hence, he can derive that the combination of these rules doesn't allow one to prove "0=1".

But remember that the rules are supposed to be concretely described. So, our reasoner can syntactically characterize the system which combines all these rules (the one which, unbenounced to him capthers all his good reasoning), and state Con(some formal system which allows exactly the inferences allowed by these rules). But he knows the combination of rules is consistent, so he can derive the con sentence for this set of rules. But, by incompleteness II (on the assumption that good reasoner's reasoning extends Robinson's Q, so that the theorem applies) this is impossible.

Hence, if Wright's theory about obedience to rules is correct, any good reasoner who accepts principles of reasoning that include Q (like us) must be obeying infinitely many rules.

[This may be problematic if one wants the notion of obedience to a rule to have some kind of psychological reality]

[Note that this doesn't mean the good reasoner's behavior won't be describable in some more efficient way by some finite collection of rules, just that the reasoner doesn't have access to these rules, in Wright's sense of being able to prove that they are truth preserving]

Use, Meaning and Number Theory

2009-11-25T01:26:00.000-08:00

I like to joke that all philosophical questions in their clearest and most beautiful form philosophy of math. But I think this is actually true, in the case of questions about how much our use of a word has to "determine" the meaning of that word.

Consider the relationship between our use of the language of number theory, and the meaning of claims in this language.

I think the following two claims are about as uncontroversial as anything gets in philosophy:

a) The collection of sentences we are disposed to assert about the number theory (for any reasonable sense of the word disposition) is recursively enumerable.

b) The collection of truths of number theory is not. In particular there's a fact of the matter about all claims of the form "Every number has X recursively checkable property" e.g. whether fictionalism or platonism is the correct view about how to understand talk of the numbers, mathematicians are surely wondering about something when they ask "Are there infinitely many twin primes" (maybe something about what would have to be of any objects that had the structures which we take the numbers to have).

But what emerges from these two claims is a nice, and perhaps suprizing, picture of the relationship between use and meaning.

I use the words "all the numbers" in a way that (is r.e. and hence by Godel's theorem) only allows me to derive certain statements about the numbers. We can picture my reasoning about the numbers as what could be gotten by deriving things from a certain limited collection of axioms.

BUT in listing these limited collection of statements, I count as talking about some collection of objects/structure that objects could have. And, there are necessary truths about what those objects are like/what anything that has that structure must be like, which are not among the claims my use allows me to derive.

[If you're daunted by the mathematical example, here's another one inspired (oddly) by Wittgenstien on phil math. You use the words "bauhaus style" and "ornate" in a certain way, mostly to describe particular objects. This gives your words a meaning (though perhaps there is some vaguness). They would apply to some objects but not to others. Hence the question "Can any thing be both bauhaus style and ornate?" is either true false or perhaps indeterminate, if e.g. objects could be ornate in a way that makes it vague whether they are in the bauhaus style or not. But your use (e.g. your ability to say, when presented w/ one particular thing whether it is bauhaus style/ornate) does include anything which allows you to arrive at one answer to the question or another.]

So, there's a nice clear sense in which it strongly appears that: even if use determines meaning, facts about the meaning of our sentences can go beyond what our use of the words contained in them allows us to derive.

And, any philosophy that wishes to deny this claim will have to do quite alot to make itself more plausible than a) and b) above!

Plenetudinous Platonism, Boolos and Completeness

2009-11-21T05:44:00.000-08:00

Plenetuindous Platonism tries to resolve worries about access to mathematical objects by saying that there are mathematical objects corresponding to every "coherent theory".

The standard objection to this, based on a point by Boolos, is that if 'coherent' means first-order consistent, then this has to be false because there are first order consistent theories which are jointly inconsistent- but if 'coherent' doesn't mean first-order consistent, the notion is obscure.

I used to think this objection was pretty decisive, but I don't any more.

For, contrast the following two claims:
- TRUE: all consistent first-order theories have models in the universe of sets (completeness theorem)
- FALSE:all consistent first-order theories are true (Boolos point)

Which of these is relevant to the plenetudinous platonist?

What the plenetudinous platonist needs to say is that whichever kind of first-order consistent things we said about math, we would have expressed truths. But remember that quantifier restriction is totally ubiquitous in math and life (if someone says all the beers are in the fridge they don't mean all the beers in the universe, and if some past mathematician says there's no square root of -2, they may be best understood as not quantifying over a domain that includes the complex numbers).

So, what the plenetudinous platonist requires is that that every first order consistent theory comes out true for some suitable restriction of the domain of quantification, and interpretation of the non-logical primitives. And this is something the reductive platonist must agree with, because of the completeness theorem! The only difference is that the reductive platonist thinks there are models of these theories built out of sets, whereas the plenetudinous platonist thinks there's a structure of fundamental mathematical objects corresponding to each such theory.

Thus, plentudinous platonism's ontological commitments can be stated pretty crisply, as in the bold section above. And there's nothing inconsistent or about these commitments, unless normal set theory is inconsistent as well!

Rabbits

2009-11-21T05:26:00.000-08:00

Causal contact with rabbits seems to be involved in almost exactly the same way in the following two statements:

RH "There's a rabbit"
MP "The mereiological complement of rabbithood is perforated here" (Or, for short: "The Rabcomp is perf")

I mean, light bouncing off rabbits and hitting our eyes would seem to be what causes (assent to) both sentences.

Thus: if we try to say that RH refers to rabbits because assertions of it are typically caused by rabbits, we would (it seems!) also get the false result that MP refers to rabbits.

[Thus causal contact doesn't seem to be what does the work in resolving Quinean reference indetermenacy - which makes things look hopeful for the view that reference in mathematics can be as determinate as reference anywhere else.]

Speed Up Theorem and External Evidence

2009-11-20T14:04:00.000-08:00

It's been suggested (e.g. by Pen Maddy, Philip Kitcher, and possibly by my advisor PK) that we can get some 'external evidence' for the truth of mathematical statements which are independent of our axioms, by noticing that they allow us to prove things which we already know to be true (because we can prove them directly from our axioms) much more quickly.

However, Godel's Speed Up Theorem seems to show that ANY genuine strengthening of our axioms would have this property. I quote from a presentation by Peter Smith:

"If T is nice theory, and γ is some sentence such
that neither T
⊢ γ nor T ⊢ ¬γ. Then the theory T + γ got
by adding γ as a new axiom exhibits ultra speed-up over T"

"Nice" here means all the hypotheses needed for Godel's theorem to apply to a theory, and "ultra speed up" means that for any recursive function, putatively limiting how much adding γ can speed up a proof, there's some sentence x whose proof gets sped up by more than f(x) when you add γ to your theory T.

Smith just points out that we shouldn't be surprised by historical examples of proofs using complex numbers of set theory to prove things about arithmetic.

But doesn't this theorem also raise serious problems for taking observed instances of speed up to be evidence for the truth of a potential new axiom γ ?

More Davidson Obsession

2009-11-20T12:47:00.000-08:00

In their book on Davidson, Lepore and Ludwig suggest that when davidson says an expression E is a semantic primitive if "the 'rules which give the meaning for the sentences in which it does not appear, do not suffice to determine the meaning of sentences in which it does appear'", he means that:"someone who knows [these rules for how to use all sentences not containing E] is not thereby in a position to understand" sentences containing E.

Intuitively, I presume the idea is supposed to be something like this: "big cat" is not a semantic primitive, since you could learn its use just by hearing expressions like "big dog" and "orange cat" but "cat" is a primitive, since you wouldn't be able to understand this expression without previous exposure to sentences containing it.

However, I think this definition turns out to be rather problematic.

Firstly, by 'rules' Lepore and Ludwig later clarify that they don't mean consciously posited rules which we might have "propositional knowledge" of. So they don't mean something like "i before e, except after c". Rather, the relevant rules are supposed to be tacit, or unconscious.

So it seems like we can restate the criterion by saying something like:

E is a semantic primitive iff merely learning how to use expressions that don't contain E doesn't put one in a position to understand the use of E.

But now here's the problem.

-If "being in a position to understand" the use of E means being able to logically derive facts about the use of E then all words are semantic primitives. There's nothing logically impossible about a language in which there happens to be a special exception where, where by combine "big" and "cat" this means hyena rather than big cat.

- On the other hand, if "being in a position to understand" the use of E means being likely to use E correctly, this is a fact about about the relationship between a language and varying aspects of human psychology.

Here's what I mean:

Model someone learning a language as having a prior probability distribution over all possible functions pairing up sentences of a language they know with propositions, and then reacting to experience by ruling out certain interpretation functions, when they fail to square with the observed behavior of people who speak the relevant language. On this model, theories like Chomskian linguistics amount to saying that babies assign 0 prior probability to certain regions of the space of possible languages.

We can imagine a contiuum of logically possible distributions of prior probability, ranging from the foolhardy tourist who assumes that everyone is speaking English until given strong behavioral evidence against to the poet who feels sure he knows that a "fat sound" is the very first time he hears fat applied to things other than physical objects, to the anxious nerd who asks for examples of "fat" vs. "thin" sounds, to they hyperparainoid person who worries about the possibility that the combination of "fat" and "cat" might fail to mean a cat that's fat, just as the combination of "toy" and "soldier" fails to mean a soldier that's a toy.

Presumably actual (sane) people won't differ too much in their linguistic priors. [Though I wouldn't be surprized if babies and adults differed radically in this regard.]

But notice that being a semantic primitive turns out to have nearly nothing to do with the role of a word in a language. Rather it has to do with our cautious or uncautious tendency to extend examples of verbal behavior in one way rather than another. For the foolhardy tourist no English words are semantically primitive (on hearing a single word he comes to understand everything in one swoop) whereas all expressions are semantically primitive for the hyperparanoid person. Two people could learn the same language, and a word would be a semantic primitive for one of them, but not for the other.

Thus, so far as I can tell, the notion of 'semantic primitive' is incorrectly, or inadequately, defined for Davidson's purposes.

There's no limit to how complex a language a finite creature could "learn" on the basis of even a single observation. Whatever pattern of brainstates and behaviors suffice for counting as understanding the language, we can imagine a creature could just start out with a disposition to immediately form those, if it ever hears the sound "red". The only real limit on complexity of languages has nothing to do with learning, but rather with the complexity of the kind of behavior which competence with a given language would require. Our finite brains need to be able to produce behavior that suffices for the attribution of understanding of the relevant language.

Thus, I think, the claim that all human learnable languages have to have only finitely many 'semantic primitives' adds nothing but giant heaps of philosophical confusion and tortured metaphor to the (comparatively) clear and obvious claim that there have to be relatively short programs capable of passing the Turing test.

bookclub: Gareth Evans on Semantics + Tacit Knowledge I

2009-11-18T09:26:00.000-08:00

I just discovered Gareth Evans has a neat article (probably a classic) about the very issues of what a semantic theory is supposed to do, which I've been worrying about recently. I found it so interesting that I'll probably write a few posts about different issues in this article.

The article starts out by paraphrasing Crispin Wright, to the following effect:

If philosophers are trying to state what it takes for sentences in English to be true, there's a very simple schema '"S" is true iff S' (this is called Tarski's T schema) which immediately gives correct truth conditions for all English sentences.

But obviously, when philosophers try to give semantic theories they aren't satisfied with just doing this. So what is the task of formal semantics about?

I think this is a great question. When I first read it I thought:

Perhaps what we want to do is notice systematic relationships between the truth conditions for different sentences in English e.g. whenever "it is raining" is true "it is not the case that it is raining" is false. If you want to make this sound fancy, you could call it noticing which syntactic patterns (e.g. sentence A being the result of sticking "it is not the case that" on to the front of sentence B) echo interesting semantic properties (e.g. sentence A having the opposite truth value from sentence B).

However, I would call this endeavor the study of logic, rather than semantics. So far we have logical theories that help us spot patterns in how words like "and" and "there is" (and perhaps "necessarily") effect the truth conditions for sentences they figure in. There may be similar patterns to notice for other words as well (e.g. color attributions - something can be both red and scarlet but not both red and green) and one could develop a logic for each of these.

We aren't saying what "and" means (presumably if we are in a position to even try to give a logic for English expressions we already know that "and" means and), rather we are discovering systematic patterns in the truth conditions for different sentences containing "and".

So, rule one other thing off the list.

Instead, Wright suggests (and Evans seems to allow) that semantics to go beyond trivially stating the truth conditions for English sentences by "figuring in an explanation or the speaker's capacity to understand new sentences". (I am quoting from Evans, but both deplore the vaguness of this statement).

This sounds initially plausible to me, but it raises a question:

Once we have noticed that attributions of meaning don't require anything deeper than the kinds of systematic patterns of interactions with the world displayed by wittgenstein's builders (maybe with some requirement that these interactions be produced by something that doesn't look like ned block's giant look-up table), the question of how human beings actually manage to produces such behavior seems to be a purely scientific question.

There are just neuroscientific facts, about a) how the relevant alterations of behavior (corresponding e.g. to learning the word "slab") are produced when a baby's brain is exposed to a suitable combination of sensory inputs and b) what algorithm most elegantly describes/models this process.

So, what's the deal with philosopher trying to do semantics? And what does it take for an algorithm to model a brain process better or worse? I'll try to get more clear on these questions, and what Evans would say about them, in the next post.

Practical Helpfulness: Why Care?

2009-11-14T23:00:00.000-08:00

Readers of the last two posts may well be wondering why I'm going on so much about the "practical helpfulness" of mathematics.

One thing is, I wish I had a better name for it than "practical helpfulness", so maybe someone will suggest one :).

More seriously, I think the fact that our mathematical methods are (in effect) constantly making predictions about themselves, and other kinds of a priori reasoning - not to mention combining with our methods of observation to yield predictions that observation alone would not have yielded (see the computer example) has two important consequences.

Firstly, it shows that our reasoning about math is NOT the kind of thing you are likely to get just by making a series of arbitrary stipulations and sticking to them. All our different kinds of a priori reasoning (methods for counting abstract objects, logical inference, arithmetic, intuitive principles of number theory, set theoretic reasoning that has consequences for number theory) fit together in an incredibly intricate way. Each method of reasoning has myriad opportunities to yield consequences that would lead us to form false expectations about the results of applying some other method. And yet, this almost never happens!

Thus, there's a question about how we could have managed to get methods of armchair reasoning that fit together so beautifully. Some would posit a benevolent god, designing our minds to reason only in ways that are truth-preserving and hence coherent in this sense. But I think a process of free creativity to come up with new methods of a priori reasoning, plus Quinean/Millian revision when these new elements did raise false expectations, can do the job. This brings us to the second point.

Secondly, if we think about all these intended internal and external applications as forming part of our conception of which mathematical objects we mean when we talk about e.g. the numbers, then Qunian/Millian revision when applications go wrong will amount to a kind of reliable feedback mechanism, maintaining and improving the fit between what we say about "the numbers" and what's actually true of those-mathematical objects-whose-structure-mirrors-the-modal-facts-about-how-many-objects-there-are-when-there-are-n-Fs-and-m-(distinct)-Gs etc.

Examples

2009-11-14T19:10:00.000-08:00

In my last post, I proposed that that our methods of reasoning about math are "practically helpful", in (at least) the sense that they act as reliable shortcuts. Mathematical reasoning leads us to form correct expectations about (and hence potentially act on) the results of various processes of observation and/or inference, without going through these processes.

Now I'm going to give some more interesting examples of (our methods of reasoning about) mathematics being practically helpful to us in this way.

The general structure in all these is the same: Composing a process of mathematical reasoning M with some other reasoning processes A yields a result that's (nearly always) the same one you'd get by going through a different process B.

Examples:

1. Observe computer (wiring looks solid, seems to be running program p etc.), derive that program it's running doesn't halt, expect it to still be running after first 1/2hour <--> observe computer after 1/2 hour

2. Observe cannonballs, form general belief about trajectory of ball launched at various angles, observe angle of launch, derive where trajectory lands <---> measure where this ball does land.

3. Prove a general statement, expect 177 not to be a counterexample <---> (directly) check whether 177 is a counterexample.

4. Conclude that some system formalizes valid reasoning about some math truths, expect that you aren't looking at an inscription of a proof of ``0=1'' in that system <---> check what you have to see if it's an inscription a proof in the system, if it ends in ``0=1''.

5. Count male rhymes in poem, count female rhymes, then add <---> Count total rhymes

[Special Case Study: Number Theory

If we focus on the case of reasoning about the numbers, we can see that there's a nice structure of mathematics creating correct expectations about mathematics which creates correct expectations about mathematics, which creates correct expectations about the word.

- general reasoning about the numbers: Ax Ay Az ((x+y)+z) = (x+(y+z))
- calculations of particular sums: 22+23=45
- assertions of modal intuition: whenever there are 2 apples and 2 oranges the must are 4 fruit
- counting procedures: there are two ``e"s in ``there''

Note that each of the above procedures allows us to correctly anticipate certain results of applying the procedure below it. ]

How (Our Beliefs About) Math are Practically Helpful

2009-11-14T18:22:00.000-08:00

All philosophers of math will agree that people do something they call "math", and that this activity is practically helpful, in a certain sense. This is often put pretty loosely by saying `Math helps us build bridges that stand up'. But I think we can say something much clearer than that. Here goes:

Our grasp of math (such as it is) has at least three aspects:

- We can follow proofs. You will accept certain kinds of transitions from one mathematical sentence to another, (or between mathematical sentences and non-mathematical ones) when these are suggested to you.
- We can come up with proofs. You have a certain probability of coming up with chains of inference like this on your own.
- Proofs can create expectations in us. Accepting certain sentences makes you disposed to react with surprise and dismay should you come to accept other sentences. e.g. if you accept "n is prime" you will react with surprise and dismay to a situation where you are also inclined to accept "n has p, q, and r as factors".

Now, the sense in which our mathematical practices are helpful is this:

First, our reasoning about math fits into our overall web of beliefs in such a way as to create additional expectations. Here's what I have in mind: Fix a situation. People in that situation who realize their dispositions to make/accept mathematical inferences arrive in a state where they will be surprised by more things than those in the same situation who don't.

For example, plonk a bunch of people down in front of a bowl of red and yellow lentils. Make each person count the red lentils and the yellow lentils. Now give them some tasty sandwiches and half an hour. Some of the people will add the two numbers. Others will just eat their sandwitches. Now, note that the people ho did the math have formed extra expectations, in the following sense. If we now have our subjects count the lentils all together, the people who did the sum will be surprised if they get anything but one particular number, whereas those who didn't do the math will only be surprised if they get anything outside of a certain given range.

Secondly, the extra expectations raised by doing math are very very often correct. When doing mathematical reasoning about your situation puts you in a state where (now) you'd be surprised if a certain observation/reasoning yields anything but P, applying this process tends to yield P. (This is especially true if we weight the satisfaction/dissatisfaction of strong expectations more heavily). Thus, composing a process of mathematical reasoning M with some other reasoning processes A yields nearly always yields correct expectations about the result of going through a different process B, if it yields any expectations at all.

And finally, this is (potentially) helpful, because it means not only do we acquire the disposition to be surprised if B yields something different, but any further inferences/actions which would get triggered by doing B happen immediately after doing A and M without having to wait for B to take place. For example, in the case from the previous post: if we imagine that all of our sample population have inductively associated counting 1567 lentils in total with having enough to make soup, the people who did the addition after counting the lentils separately, start cooking earlier than those who did something else instead.

To summarize:

Doing math is practically helpful in the sense that spending time doing math raises extra expectations (relative to spending that time eating sandwiches) about the results of certain other processes, and these expectations are generally correct. Thus, mathematical reasoning constitutes a reliable shortcut, leading us to take whatever actions would be triggered by going through some other process B without actually going through B.

NOTE: I don't mean to suggest that this is all there is to math, or that math is somehow *merely* instrumental. I'm just trying to concretely state some data about the successful "applications" of math, which I think everyone will agree to.

Analyticity: No Free Lunch

2009-11-14T17:31:00.001-08:00

Consider the following pardigmatic examples of analytic and synthetic sentences:

(1) "Dogs once existed."
(2) "Prime numbers have two distinct divisors: themselves and one."

Both of these statements feel extremely obvious to us. And, if anything we're more likely to stop asserting (2) than (1) - if some perverse person wants to count 1 as a "prime" number, that's fine with me, so (if he's insistent enough) I'll adopt his usage and hence stop saying sentence 2 (and e.g. change how I state the fundamental theorem of algebra accordingly). So - we wonder, after reading Quine, what does the further claim that (2) is analytic amount to?

Here's an idea: If someone asked me to back up my assertion of (1), I'd be surprised, but there are things I would do to support this e.g. give an example of a dog. If (bizzarely) I couldn't state any other claims in support of (1), I'd be troubled. In contrast, if asked to justify (2) I wouldn't be able to give any kind of argument for it AND I wouldn't be troubled by this, or inclined to revise. (Note: this is exactly when claims about analyticity and meaning come up in ordinary contexts - people say 'that's just what I mean by the term' when faced with skepticism about certain things.)

S is basic analytic in P's idiolect iff: either P is happy to accept S without being able to provide any further justification

S is analytic in P's idiolect iff: S is basic analytic S is derivable via some combination of premises and inferences, each of which is basic analytic.

This seems to pick out a relatively sharp class of sentences, and accord with our intuitive judgments of analyticity (at least if we assume that experience can somehow be cited as a justification [or something more sophisticated], so that direct observations don't count as analytic for the observer).

Does this refute Quine? No. For, let's think about what epistemological siginificance (this notion) analyticity has. Do we have some kind of special access to analytic truths?

Making a bunch of new sentences analytic in your idiolect is just a matter of developing the inclined to say "that's just what I mean by the word" when pressed for a justification of these sentences. And this refusal to provide extra justification doesn't somehow ensure that the sentences you assert so boldly come to express truths.

For, what bucking up your insouciance like this does, is change the facts about your use of words so that (now), if the certain of your words are meaningful at all, these sentences will express truths. Thus, it makes these sentences/inferences function as a kind of implicit definition of your terms. But, as the famous case of Tonk shows, not all implicit definitions are coherent. Also, in changing the meanings of your words in this way, you run the risk of making other non-analytic sentences that you currently accept now express falsehoods.

Thus, saying that some sentence S is analytic isn't some kind of epistemic free pass for you to accept that sentence. All it does is semantically push all your chips into the center of the table with regard to S. Whereas before you ran the risk that S would express a falsehood, now there's a better chance that S will express a truth, but if it doesn't both S and a bunch of other sentences in your language will be totally meaningless.

So, here's my current position: the analytic-synthetic distinction is real, but it doesn't give the epistemological free lunch* which the logical positivists hoped it would.

*i.e. just saying that facts about something (like math) is analytic doesn't banish mysteries about how we came to know these facts.

Davidson Obsession

2009-11-07T22:29:00.000-08:00

Davidson proposes (in "Truth and Meaning") that for a language to be learnable by finite creatures like us there must be a finite collection of axioms which entails all true statements of the form '"Snow is white" is true if and only if snow is white'. Then he, and his followers, argue that people with various kinds of theories can't satisfy this constraint e.g. that nominalists can't get a theory that entails the right truth conditions for mathematical statements without using axioms that quantify over abstracta.

Something about this argument strikes me as fishy, and I've spent hours obsessing over it at various times, replacing one putative "refutation" with another. :( But I can't stop thinking about it, so here's my newest attempt.

First, grant that for someone to count as understanding some words they need to know all the relevant instances of Tarski's T schema. So they have to be disposed to assent to every such sentence. Now, as every sophomore seeing the Davidson for the first time points out, it's trivially easy to make a finite program that 'assents' to every query that's instance of the T schema in a given language, or enumerates all such instances. But Davidson requires more, there needs to be a finite collection of axioms, which logically entail all the instances. This is what gives Davidson's claim its potential bite. But now, we ask, why think this?

EITHER
Davidson thinks that to know the T schema you need to be able to consciously deduce them other things you antecedently know. In this case the requirement that each instance of the T schema must be deducible from a finite collection of axioms would be motivated.

But this can't be right because no one can consciously produce such an axiomatization for our language. If we learned the T schema by consciously deriving it from some axioms, we should be able to state the axioms. Therefore, conscious deduction does not happen, and cannot be required.

OR
Davidson allows that it suffices for each instance of the T schema to individually feel obvious to you, (and for you to be able to draw all the right logical consequences from it etc.)

But to explain the fact that each sentence of this form feels obvious when you contemplate it, we just need to imagine your brain is running the sophomore-objection program which checks every queried string for being an instance of the T schema and then causes you to find a queried sentence obvious if it is an instance. Once we are talking about subpersonal processes there is no reason to model them as making derivations in first order logic, so the requirement is unmotivated.

Perhaps Davidson might argue that the subpersonal processes doing the recognition are somehow doing something equivalent to quantifying over abstracta, so the nominalist, at least, would have a problem. But do subpersonal processes really count as quantifying over anything? And if they do, is there any reason we have to agree with their opinions about ontology?

Produce the Code!

2009-11-07T20:28:00.000-08:00

There's a three-way-debate going on between those who want to understand our ability to think in terms of manipulation of intrinsically meaningful items in the head (physical token sentences of a language of thought) vs. merely in terms of connections vs. behaviorists who think it doesn't matter how our brain produces suitable behavior.

Obviously, one would like to know a lot more about the neuroscience of language use. But, so far as I can tell, the philosophical aspects of this debate could be resolved right now, by producing toy blueprints/sample code. Then we look at the code, and consider thought experiments in which the brain actually turns out to work as indicated in the code...

Linguistic Behaviorism vs. Non-behaviorism:

If you think that stuff about how competent linguistic behavior is produced can be relevant to meaning, produce sample code A and B with the same behavioral outputs, such that we would intuitively judge a brain that worked in ways A vs. B would mean different things by the same words. [I think Ned Block has done this with his blockhead]

If you think stuff inside the head also establishes determinacy of reference, contra Quine, produce two pieces of sample code A and B for a program that e.g. outputs "Y"/"N" to the query "Gavagai?", such that we would intuitively say people whose brains worked like A meant rabbit and those that worked like B meant undetached rabbit part.

Language of Thought vs. Mere Conectionism:

If you are a LOT-er who thinks things the brain don't just co-vary with horses, but can actually mean `horse', produce sample code which generates verbal behavior, in response to sensory inputs, in such a way that we would intuitively judge pieces of the memory of a robot running that program to have meanings.

Then, produce sample code that works in a "merely conectionist way" and provide some argument that the brain is more likely to turn out to work in the former way.

[NOTE it does not suffice merely to give a program that derives truth conditions for sentences, unless you also want to posit a friendly homunculus who reads the sentences and works out what proper behavior would be. What your brain ultimately needs to do is produce the correct behavior! So, if you want to compare the efficiency of mere conectionist vs. LOT-like theories of how your brain does what it does, you need to write toy programs that evaluate evidence for snow being white, rocks being white, sand being white and respond appropriately- not just the trivial program that prints out an infinite list of sentences. "Snow is white" is true iff Snow is white. "Sand is white" is true iff sand is white... ]

Charitably, I think the LOT-ers want to say that the only feasible way of making something that passes the Turing test will be to use data structures of a certain kind. But until they can show some samples of what data structures would and wouldn't count, it's really hard to understand this claim. (I mean, the claim is that you will need data structures whose tokens count as being about something. But which are these?).

The Problem of Logical Omniscience and Inferential Role

2009-11-07T19:37:00.000-08:00

I just looked over a very old thing I wrote about the problem of logical omniscience. The problem of Logical Omniscience is: How can you count as believing one thing, while not believing (or even explicitly rejecting) something logically equivalent?

I suggested that propositions have certain preferred inferential roles, and that you count as believing that P to the extend that you are disposed to make enough of these preferred inferences, quickly and confidently enough.

So for example, someone can believe that a function is Turing computable but not that it's recursive, even though these two statements are provably equivalent, because they might be willing to make enough of the characteristic inferences associated with Turing computability, but not those for recursive-ness. (The characteristic inferences for "...is Turing computable" would be those that people call "immediate" from the definition of Turing computability, and ditto for the -different- characteristic inferences for recursive).

This is interesting because:
1. The characteristic inferences associated with a proposition/word will NOT supervene on the inferences which that proposition/word justifies. Since Turing computability and recursive-ness are probably equivalent, the very same inferences are JUSTIFIED for each one of them. But "This function is Turing computable" and "This function is recursive" need to have different characteristic inferences, to explain how you can know one but not the other.

2. Given (1), if you want to attach meanings to individual words, these meanings should not only include things like sense and reference which help build up the truth conditions for sentences involving that word, but also something like characteristic inferences, which helps you chose when to attribute someone a belief involving this word, rather than another which word would always contribute in exactly the same way to the truth conditions of any sentence.

2. It's commonly said that aliens would have the same math as us. If this means that they wouldn't disagree with us about math that sounds right. But if it means that they would (before contact with humans) believe literally the same propositions as we do, I don't think so.

For, think about all the many different notions we could define which would be equivalent to Turing computability, but have different characteristic inferences. If you buy the above, each of these notions corresponds to a slightly different thought. Thus for the aliens to believe the exact same mathematical claims as we do, they would have to have the same definitions/mathematical concepts. But it's much less clear whether aliens would have the same aesthetic sense guiding what definitions they made/mathematical concepts they came up with. For example, I'm much more convinced that aliens would accept topology than that they would have come up with it. I mean, just think about the different kinds of math developed just by humans in different eras and countries.

Freedom and Resentment in Epistemology

2009-11-07T18:14:00.000-08:00

Everyone likes to talk about Neurath's boat, but I think common discussion leaves out something critical. Not only do we all start with some beliefs, but we also start out accepting certain methods of revising those beliefs, in response to new experience or in the course of further reflection. This is crucial because it brings out a deep symmetry between all believers:

At a certain level of description, there's no difference between the atheist philosopher who finds it immediately plausible that bread won't nourish us for a while and then suddenly poison us, and the religious person who finds it immediately plausible that god exists, or the madman who finds it immediately plausible that he's the victim of a massive conspiracy. Everyone involved is (just) starting with whatever they feel is initially plausible, and revising this in whatever ways they find immediately compelling.

Thinking about things this way, can make one feel uncomfortable in deploying normative notions of justification. Being justified is supposed to be a matter of (something like) doing the best you can, epistemically, whether or not you are lucky enough to be right. But there's no difference in effort (or even, perhaps, in care) between the philosopher and the madman. It's just that the philosopher is lucky enough to find immediately compelling principles *that happen to be mostly true*, and inference methods *that happen to be mostly truth-preserving/reliable*. So how can we say that one of them is justified?

One reaction to this is to deny that there is such a thing as epistemic normativity. There are facts about which people have true beliefs, and which of them are on course to form more true beliefs, which belief forming mechanisms are reliable (in various senses) etc. But there are no epistemically normative facts e.g. facts about which reliably true propositions are OK to to assume, or which reliable inference methods are OK to employ without any external testing.

Another possible reaction is to say that even though "ultimately" there's no difference between believing finding it obvious that bread will nourish you if it always has in the past vs. believing you are the center of a conspiracy, there still are facts about justification. We can pick out certain broad methods of reasoning (logical, empirical, analytic(??), initially trusting the results of putative senses) which are both popular and generally truth preserving, and what it means to be justified is just to have arrived at a belief via one of those.

In either case, the result will give an answer to philosophical skepticism. The skeptic asks: "how can you be justified in believing that you have a hand, given that it depends on your just assuming without proof that you aren't a BIV?" Someone who has the first reaction can simply deny the contentious facts about justification. Someone who has the second reaction will be unimpressed by the point that they are "just assuming" that ~BIV. All possible belief is a matter of starting out "just assuming" some propositions and inference methods, and then applying the one to the other.

Funny Footnote

2009-11-05T03:50:00.000-08:00

Reading Mark Steiner's "Mathematics-Applications and Applicability", I noticed this footnote:

"Suppose we have a physical theory, like string theory, which postulates a 26 dimensional space. The number 26 happens to be the numerical value of the Tetragrammaton in Hebrew. Should this encourage us to try other of the Hebrew Names of God?"

[Note: in context, Steiner seems to think the answer to this question is yes]

Three jobs for logical structure:

2009-11-01T12:16:00.000-08:00

You might think the "logical structure" of a sentence is a way of cutting it up into parts [eg. "John is happy" becomes "is happy(john)"] that does three things:

1. gets used by the logical theory that best captures all the valid inferences.
2. matches the metaphysical structure of the world.
3. explains how we are able to understand that sentence, by breaking it down into these parts, and understanding them.

However, it's not obvious that the method of segmentation which does any one of these things best should also do the others. I don't mean that this idea is crazy, just that it is a bold and substantive claim that logic unites cognitive science with metaphysics in this way.

It's also not obvious that *any* method of segmentation can do 1 or 2.

Task 1 might be impossible to perform because there might not be a unique best logical theory. If we think that the job of logic is to capture necessarily truth-preserving inferences, then second-order logic is logic. Any recursive axiomatization of second order logic will be supplementable to produce a stronger one - since the truths of second order logic aren't recursively axiomatizable. (One might hope though, that all sufficiently strong logics that don't say anything wrong will segment sentences the same way)

Task 2 might be impossible because the world might not have a logical structure to reflect. What do I mean by the world "having a logical structure"? I think there are two versions of the claim:
a. The basic constituents of the world are divided between the various categories produced by the correct segmentation e.g. concepts and objects in Frege's case.
This is weird because "constituents of the world" sound like they should be all be objects. But presumably objects don't join together to produce a sentence, so the kind of expressions used in your chunking up can't all be objects.
Its also weird because it just seems immediately weird to think of the world as having this kind of propositional structure, rather than our just using different propositions with structure to describe the world.
b. The objects that really exist (as opposed to those that are merely a facon de parler), are exactly those which are quantified over by true statements when these are formalized in accordance with the best method of segmentation. To misquote Quine: "the correct logical theory is the one such that, to be, is to be the value of a bound variable in the formalization of some true sentence in accordance with that theory."
So, for example, if mathematical objects can't be paraphrased away in first order logic, but they can using modal logic, the question of whether mathematical objects exist will come down to which (if either) of these of logics has the correct segmentation.

Finally, Task 3 is ambiguous between something (imo) silly and something about neuroscience.
The silly thing is that a correct segmentation should reflect what components *you* break up the sentence "John is happy" into, when you hear and understand it (presumably none).
The neuroscience is `what components does *your brain* break up this sentence, when processing it to produce correct future behavior, give rise to suitable patterns of qualititative experience for you etc.?' This is obviously metaphorical, but I think it makes sense. It seems very likely that there will be some informative algorithm which we can use to describe what your brain does when processing sentences (it might or might not be the same algorithm for different people's brains). And, if so, it's likely that there will be some natural units which this algorithm uses.

Is there a logic that...

2009-11-01T08:32:00.000-08:00

Is there a logic that would capture inferences like:

-"John is very rich" --> "John is rich"
-"John is very very very very rich"--->"John is very rich"

Obviously it won't do to say "rich(John) ^ very (John)".

On Quantifying Over Everything

2009-10-31T10:11:00.000-07:00

Consider the following argument against quantifying over everything.

"It can't be possible to quantify over everything, because if you did, there would have to be a set, your domain of quantification, which contained all objects as elements. However, this set would have to have to contain all the sets. But there can be no set of all sets, by Russell's paradox argument."

I claim it's unsound for the following reason:

We presumably can quantify over all the sets (e.g. when stating the axioms of set theory). So, if (as this argument assumes) quantifying over some objects required the existence of a set containing all the objects quantified over, we would already have a set containing all the sets, hence Russell's paradox and contradiction.

Thus, meaningfully making an assertion about all objects of a certain kind does NOT require that there's a set containing exactly these objects.

---

BONUS RANT: Why would one even think that where there is quantification there must be a set that's the domain of quantification? Because of getting over-excited about model theory I bet. [warning: wildly programmatic + underdeveloped claims to follow]

Model theory is just a branch of mathematics which studies systematic patterns relating what mathematical objects exist and and what statements are always/never true. It's not some kind of Tractarian voo-doo that `explains how it's possible for us to make claims about the world'. Nor do sets (e.g. countermodels) somehow actively pitch in and prevent claims like "Every dog has a bone" from expressing necessary truths!

Is "Set" Vague?

2009-10-31T07:50:00.001-07:00

The (normal) intuitive conception of the hierarchy of sets is roughly this:

The hierarchy starts with the empty set, at the bottom. Then, above every collection of sets at one stage, there's a successor stage containing all possible collections made entirely from elements produced at that stage. And, above every such infinite run of successor stages, there's a limit stage, which has no predecessor, but contains all possible sub-collections whose elements have already been generated at some stage below.

But how far up does this hierarchy of sets go? Is there a fact of the matter, or does our conception not determine this?

The conception/our intuitions about sets doesn't directly tell us when to stop. For any stages we suppose we are looking at, it always seems to make sense to think of new collections that contain only sets generated by that point (e.g. the collection of all things so far generated). Of the sets generated by any collection of stages we can ask:
- Does the proposed next stage/limit stage of these stages really make sense? Are there really such collections?
- If so, are the collections generated at this stage still sets?

A textbook will tell you that at some point the things generated by the process above DO make sense, but DON'T count as sets. So, for example, there is a collection of all sets, but (on pain of paradox) this is not itself a set, but only a class.

However,
a) This just pushes the philosophical question back to classes: is there a point at which there stop being classes? Are there something else (classes2) which have the same relation to sets as sets do to classes? [One of my advisors calls this the "neopolitan" view of set theory]

b) We don't have any idea of WHEN the things generated by the process above are supposed to stop counting as sets.

Note that the issue with b) is not just that we don't know whether sets of a certain size exist. There are lots of things about math we don't know, and (imo) could never know. Rather, the uneasy feeling is that our conception doesn't "determine" an answer to this question in the following much stronger sense:

There could be two collections of mathematical objects with different structures, each of which equally well satisfies our intuitive conception of set.

For, consider the hierarchy of classes (note: all sets are classes). There might be two different ways of painting the hierarchy to say at what point the items in it stop counting as sets. Our intuitive conception just seems to generate the hierarchy of classes, not to say when things in it stop being sets!

In contrast, in the case of the numbers, I might not know whether there are infinitely many twin primes, but any two objects satisfying the intuitive, second order, characterization of the numbers would have to have the same structure (and hence make all the same statements of arithmetic true).

Thus, our intuitive conception of set seems to be hopelessly vague about where the sets end. Hence, even if you are a realist about mathematical objects, we seem forced to understand set theory as making claims about features shared by everything that satisfies the intuitive conception of set, rather than as making claims about a unique object.

Questions:

1. If you buy the reasoning in the main body of this post, does it give an advantage to modal fictionalism? e.g. the modal fictionalist might say: "You already need to agree with us that doing set theory is a matter of reasoning about what objects satisfying the intuitive conception of set would have to be like. What does incurring extra commitment to the actual existence of mathematical objects (as opposed to their mere possibility) do for you?".

2. An alternative would be to reject the textbook view, and say EVERYTHING generated by the process above is a set. Hence, you couldn't talk about a class of sets. Would this be a problem?

3. [look up] Is it possible that all initial segments of the hierarchy of classes that reach up to a certain point are isomorphic? (I mean, the mere existence of one-to-one, membership preserving, function that's into but not onto -the identity- doesn't immediately guarantee that some *other*, more clever, function that IS an isomorphism)
[maybe you can prove this is not poss by using the fact that one initial segment would have extra ordinals, and this iso could be used to define an iso between ordinals of different sizes which is impossible]

4. Is there some weirdness about the idea that collections in general (whether they be sets, or classes) eventually give out - so there's no collection of all collections.

We could say there are sets, classes, classes2, classes 3 and so forth. This lets us say there's a class of all sets, and a class2 of all classes etc. But as far as collections in general we must admit that there's no collection of all collections, on pain of contradiction via Russell's paradox.

Well, I don't personally find this that problematic. It's a surprising fact about collections maybe, but mathematics often yields surprising results.

Relations vs. Sets of Ordered Pairs

2009-10-31T07:20:00.000-07:00

(Normally in math) a relation is defined to be a sets of ordered pairs.

But the `elementhood' relation between sets can't, itself, be a set of ordered pairs - since there can't be a set which contains each ordered pair of sets such that x is an element of y. [From the existence of such a set you could use the axiom of collection in ZF to derive the existence of a set of all sets, and hence the Russell set and contradiction.]

Therefore, not all relations (in the ordinary sense) are sets of ordered pairs (i.e. relations in the mathematical sense).

Thin Realism #2

2009-10-30T17:07:00.000-07:00

Hmm on further reflection, `thin realism' is just Lumpism.

So see the essay above for why Lumpism is right and all that seductive stuff about the world having a logical structure/existence claims having a special epistemological status is wrong.

"Thin Realism" - what could it be?

2009-10-30T15:15:00.000-07:00

I always want to say "I think there are numbers - but I understand existence in a thin logical sense". But I feel kindof dishonest saying this. It's too much like the sleazy "Of course P - but I don't mean that in any deep philosophical way" which happens when Wittgensteinians get lazy.

So here are some actual concrete ways in which I differ from other platonists (i.e. other people who believe there are mathematical objects).

1. I don't think we need to posit numbers to explain how there can be unknowable mathematical facts.

2. I think fictionalism/if-then-sim is is perfectly coherent. We could have had a mathematical practice which was completely based around mathematical properties, and studying their relaitons to one another e.g.: `Insofar as anything heirarchy-of-sets-izes, it's mathematically necessary that it satisfies the contiuum hypothesis'.

And here's an attempt to say what having only "a thin logical notion of existence" means:

When we ask what objects exist, this is equivalent to asking what sentences with a given logical form (Ex) Fx are true. So far, this is just Quinean orthodoxy.

But now the question is: what makes a given sentence (say, of English) have a certain logical form?

Now, I think having existential form is just a matter of what inferences can be made with that sentence, and what other -contrasting- sentences are in the language. We cook up various logical categories in order to best represent, and exploit, patterns in which inferences are truth preserving. Furthermore, there's noting special about objects, and object expressions. Each component of a sentence (be it concept-word, object-word, connective or opporator) makes a systematic contribution to the truth conditions of the sentences it figures in (i.e. the class of possible situations where the sentence is true).

On this view, choices about the logical form of a sentence wind up not being very deep - the question is just what's the most elegant way to capture certain inference relations.

In contrast, (I propose) having a "thick" notion of objecthood and existence, means thinking that there IS something more than elegant summary of inference relations at stake when we decide how to cut sentences up into concepts and objects. For example, you might think

1. It's easy to learn statements which don't imply that any objects exist (all bachelors are unmarried), whereas learning statements that do imply the existence of at least one object (there are some bachelors) is harder.

2. The *world* has a logical structure too! - so the most elegant way of cutting up your sentences to capture inference relations might still be wrong, because it fails to respect the logical structure of the world.

[Oh yes, they are kindof seductive. More about why they are wrong later.]

Unknowable Truths without Objects

2009-10-30T14:33:00.000-07:00

I believe in mathematical objects, but I think the following appeal to them is dead wrong:

"The existence of mathematical objects is what allows there to be unknowable mathematical truths, whereas there are no unknowable logical or `conceptual' truths."

Corresponding to every unknowable AxFx statement in arithmetic, there's a purely modal statement, that's not ontologically commital, but would let you infer the arithmetical statement and hence must be equally unknowable, namely:

"It is impossible for there to be a machine on an infinite tape which a) acts in such and such such-and-such a physically specified way (here we have we list physical correlates of rules for some Turing machine program that checks every instance of the AxFx statement), and b) stops."

Why I am not Carrie Jenkins

2009-10-22T06:19:00.000-07:00

Carrie Jenkins' 2009 book Grounding Concepts: an Empirical Basis for Arithmetical Knowledge, proposes a theory that has a lot in common with my thesis project.

Both of us:
- want to give a naturalistic account of mathematical knowledge

- in particular, want to explain how humans can have managed get "good" combination of inference patterns that count as thinking true things about some domain of mathematical objects/having a coherent conception of what those objects must be like, rather than "bad", 'tonk' like patterns of reasoning.

-appeal to causal interactions with the world, to explain how we wind up with such combinations of inference dispositions.

BUT there are some important differences. Here's why (I claim) my view is better.

Jenkins' theory:

Jenkins winds up positing a whole bunch of controversial, and perhaps under-explained philosophical notions to account for how experience gives us good inference dispositions. She proposes that:

Experience has non-conceptual content which grounds our acquisition of concepts so as to help us form coherent ones. Then when we have a coherent concept of something like the numbers, we inspect it to see what what must be true of the numbers and reason correctly about them.

-The idea that there's non-conceptual content is a controversial point in philosophy of perception.
-The idea that experience can "ground" concept acquisition without playing a justificatory role in the conclusions drawn is not at all clear. What is this not-justificatory, but presumably not just causal relationship of grounding supposed to be? (Kant's notion of a posteriori concepts seems relevant, but that's none-too clear either).
-Finally, what is concept inspection, (presumably you don't literally visit the 3rd realm and see the concepts) and how is it supposed to work? Jenkins admits that this is an open question for further research.

My theory:

In contrast, my view gives a naturalistic account of mathematical knowledge that doesn't need any of this controversial philosophical machinery. I propose that:

People are disposed to go from seeing things, to saying things, to being surprised if we then see other things, in certain ways. When these inference dispositions lead us to be surprised, we tend to modify them.

Thus, it's not surprising that we should have wound up with the kind of combination of arithmetical inference dispositions + observational practices + ways of applying arithmetic to the actual world, which makes our expected applications of arithmetic work out.

For example: insofar as we had a conceptions of the numbers which included the expectation that facts about sums should mirror logical facts in a certain way, it's not surprising that we would up also believing the kinds of other claims about sums, which make the intended applications to logic work out (e.g. believing 2+2=4 not 2+2=5).

Note that we don't need to posit any mysterious faculty of concept-inspection, or any controversial non-conceptual experience. All I appeal to is perfectly ordinary processes. People go from one sentence to another in a way that feels natural them (whether or not they are so fortunate as to be working with coherent concepts like +, rather than doing reasoning like Frege did about extensions) And when this natural-feeling reasoning leads to a surprise, they revise.

[Well, perhaps I'm also committed to the view that innate stuff about the brain makes some ways of revising more likely than others, and certain initial inference-dispositions more likely than others, in a way that doesn't make us always prefer theories that are totally hopeless at matching future experience. But you already need something like this even to explain how rats can learn that pushing a lever releases food, so I don't think this is very controversial.]

Mathematical Concepts and Learning From Experience

2009-10-20T08:15:00.000-07:00

I've been reading Susan Carey's new book on the development of concepts, which features a lot of interesting stuff about the development of children's reasoning about number. The last two chapters are philosophical though, and bring up an important point, which it had not occurred to me needed to be stressed:

Learning from experience need not take the form of someone explicitly forming a hypothesis, and then letting experience falsify it/doing induction to conclude the hypothesis is true.

If this were all that experience could do, it would be hopeless to appeal to it to help explain how we could get mathematical knowledge. For, plausibly, you only count as having the concept of number, once you are willing to make certain kinds of applications of facts about the numbers, reason about the numbers largely correctly etc. So, by the time that experience could falsify hypotheses containing the mature concept of number, you would already have to have lots of mathematical knowledge.

Instead, experience helps us correct and hone our mathematical reasoning all through the process of "developing a concept". How can this be?

Well, firstly, think about the way students are normally introduced to the concept of set. No one makes a hypothesis that there are sets, nor do math profs attempt to define sets in other terms. Rather the professor just demonstrates various ways of reasoning about sets, ways of using these claims to solve other mathematical problems etc. and gets the students to practice. Given this, the student's usage and intuitions conform more and more to standard claims about the sets, and eventually they count as having the concept of set.

I propose (and I think Carey would agree) that the original development of many concepts in mathematics works similarly, only with trial and experience playing the role of the teacher.

You start out not having the concept, and try various usages. Here, however, rather than having a professor to imitate, you just have your general creativity/trial and error/analogical reasoning to suggest ways of reasoning about "the X"s and then an ability to check whatever kinds of consequences and applications you expect at a given time. Often this kind of creative trying and analogical reasoning will turn out to fail in some way, such as leading to contrdiction, or underspecifying something important. But then you can correct it. Inconsistent reasoning about limits in the 19th century and sets in the early 20th would be examples of the former. And the kind of process of refinment of the notion of polygon in Lakotosh's Proofs and Refutations would be an example of the latter.

We try out various patterns of reasoning about the world (e.g. calling certain things Xs, trying to apply the analogue of good reasoning about one domain to another) -with perhaps a nudge from brain structures subject to evolution effecting which patterns we are likely to try- and experience corrects these inference patterns until they cohere enough that we count as genuinely having some new concept. And note that no conscious scientific reasoning must be assumed to start this process, all we need some disposition to go from seeing things to making noises to doing things, together with a playful/random/creative inclination to try extending those dispositions in various ways!

p.s. I haven't emphasized this point the past, because I think questions like 'when exactly does someone start having the concept of X?', don't generally cut psychology or metaphysics at their joints. I mean: when exactly did people start having the modern conception of atom? The interesting facts are surely facts about when people started accepting this or that idea atoms "atoms", or reasoning about "atoms" in this or that way. Coming up with a decision about exactly what amount of agreement with us is necessary for people to count as having the same concept is a matter of arbitrary boundary setting.

But I realize now that ignoring the whole issue of concepts can be confusing. So let me just say:

When I say mathematical knowledge is a joint product of mathematically shapted problems in nature, correction by experience, the wideness of the realm of mathematical facts and the relationship between use and meaning, "Correction by experience" doesn't just mean what happens when hypotheses consciously proposed by people who already count as having all the right mathematical concepts get refuted. Rather, "correction by experience" includes what happens when you are inclined to reason some way, you get to an unexpected conclusion, and then subsequently become disposed to draw slightly different inferences/feel less confident when engaging in some of the processes that lead you there. You might or might not count as revising some hypothesis, phrased in terms of fully coherent concepts, when you do this.

p.p.s. The idea that experience helps us form coherent mathematical concepts, (while not figuring in the justification of our beliefs) is also a central theme in Carrie Jenkins' 2009 Grounding Concepts: an empirical basis for arithmetical knowledge.

Empirical adequacy and truth in mathematics

2009-10-16T18:58:00.000-07:00

The current weakest link in my thesis is this (IMO): how to connect merely having beliefs about mathematics that help us solve problems, and yield correct applications to concrete situations to having beliefs about mathematics that are reasonably reliable.

Couldn't totally false mathematical theories nonetheless be perfectly correct with regard to their concrete applications?

Also, even if our beliefs would indeed perfectly accurately describe some concrete objects, how can we count as refering to these objects, given that we have no causal contact with them?

My current best answer is this:

Think of human mathematicians as observing certain regularities (e.g. whenever there are 2 male rhymes and 2 female rhymes in a poem there are at least 4 rhymes all together), and then positing mathematical objects "the numbers" whose relationship to one another is supposed to echo these logical facts.

(This is a reasonable comparison because what we actually do is like this, in that we happily make inferences from a proof that "a+b=c" to the expectation that when there are a male rhymes and b female rhymes there are c rhymes all together. We behave as though we know there's this relationship between the numbers and logical facts, so it's not too much of a stretch to compare us to people who actually consciously posit that there is some collection of abstract objects whose features echo the relevant logical facts in this way.)

Now either there are abstract objects or not.

If there aren't abstracta (as the fictionalist thinks), the fact that mathematicians only care about structures makes it plausible to think of them as talking about the fiction in which there are such objects.
Thus, our abstract-object positing mathematicians will count as speaking about the fiction in which there are objects whose features echo the logical facts about addition in the intended way. They will also count as knowing lots of things about what's true in this fiction.

Also, note that insofar as these mathematicians propose new things that "intuitively must be true of the numbers" their intuitions will be disciplined and corrected by the fact that the relevant applications are expected, so there's a systematic force which will keep some degree of match between their claims about this fiction and what's actually true in this fiction.

If there are abstracta, then there are abstract objects with many different structures, in particular structures corresponding to every consistent first order theory (note this is even true if the only mathematical objects there are are sets! the completeness theorem guarantees that there are models of every such theory within the heirarchy sets). So there will be some collection of objects whose features match those expected by our positers (note that the positers only really care about structural features of "the numbers" not whether they are fundamental mathematical objects etc).

Now, how can our positers count as referring to some such objects? Well, as noted above, we have systematic mechanisms of belief revision which kick back and insure that their claims about the numbers must match with logical facts, and hence with the real facts about these collections of suitable abstracta. Just as looking at llamas helps ensure that certain kinds of false beliefs about llamas which you might form would be corrected, applying arithmetic insures that certain kinds of false general beliefs you might form about the numbers would be corrected (those which lead to false consequences about sums).

Thus, we have a situation where people not only have many beliefs that are true about the numbers, and the tendency to make many truth-preserving inferences, but also where these beliefs have a certain amount of modal stability (many kinds of false beliefs would tend to be corrected). Even Fodor thinks that making correct inferences with or is sufficient to allow or to make the right kind of contribution to the truth value of your sentences, so why should the same thing not apply to talk about numbers, given that we now have not only many good inferences but this kind of mechanism of correction which improves the fit between our beliefs about the numbers and the numbers?

You might still worry that there will be so many mathematical objects which have all the features which we expect the numbers to have - how can we count as referring to any one such structure, given that our use fits all of them equally well? And if we don't uniquely pick out a structure, how can our words count as refering and being meaningful? But note that to the extent that our use of the word "the numbers" is somehow ambiguous between e.g. different collections of sets, our use of the word "human bodies" would seem to be equally ambiguous between e.g. open vs. closed sets of spacetime points. So either meaningfully talking about objects is compatible with some amount of ambiguity, or the above kind of reasoning doesn't suffice to establish ambiguity.

Beliefs, natural kinds, and causation

2009-10-16T16:39:00.000-07:00

I think that having the belief that there's a rabbit in the yard is a matter of having some suitable combination of dispositions to action, dispositions to experience qualia, relations to the external world ect. (roughly: those that would make an omniscient Davidsonain charitable interpreter attribute you the belief that there's a rabbit in the yard)

But (I think) exactly which dispositions etc. are required is quite complicated, and in some respects arbitrary (e.g. verbal behavior that would equally well track the facts about rabbits and undetached rabbit parts counts as referring to rabbits).

Does this view that 'believing that there's a rabbit in the yard' may not pick out any supremely natural combination of mental states, prevent me from saying that beliefs can cause things?

No.

The facts about what physical combinations of stuff count as a baseball are equally complicated and arbitrary. But no one would deny that baseballs can figure in causal explanations e.g. the window broke because someone threw a baseball at it.

Just as the somewhat arbitrary fact that a regulation baseball has to have a diameter of between two and seven-eighths inches and three inches doesn't prevent talk of baseballs from figuring in causal claims, the somewhat arbitrary fact that it's easier to count as referring to/thinking about rabbits rather than undetached rabbit parts doesn't prevent talk of beliefs from figuring in causal claims.

Explaining vs. justifying beliefs

2009-10-16T15:55:00.000-07:00

Suppose I say that there's a fire in my room, and then you ask me why I believe there's a fire in my room. I could give a causal explanation for my belief (e.g. 'Some light bounced off a fire and this hit my eyes causing such-and-such brain changes in me) or I could try to justify the claim (e.g.'I seem to see a fire, and I don't tend to hallucinate').

These are two very different things! Thus, I think it's totally wrong to assume that the (potentially infinite series of) other beliefs I might express if asked to justify my claim that there's a fire in my room, somehow figured in causing the belief. If anything, these extra beliefs are probably simultanious results of a common cause, namely the fire.

Fire
-causes->
Light hits my retina
-simultaniously-causes->
I believe there's a fire.
I believe that I seem to see a fire.
I believe that I seem to seem to see a fire.
...

This is not to deny that beliefs CAN cause beliefs though, as in the case of conscious, Sherlock-Holmes-style chains of inference. Also the absence of certain beliefs might be necessary for the production of other beliefs (e.g. the absence of the belief that I have taken fire-hallucination causing drugs, might be required for causal stimulation by light from a fire to cause me to form the belief that there's a fire)

Conventionalim and Realism - are they incompatible?

2009-10-15T08:03:00.000-07:00

Conventionalism and Realism are often presented as alternatives (for example, I recently heard a talk about whether Frege should be understood as a realist or a conventionalist about number). But (at least on my own best understanding of what `conventionalism' might be) it's not at all clear that this is the case.

I'm tempted to understand realism and conventionalism as follows, in which case (I am going to argue) the two are perfectly compatible.

You are a realist about Xs iff you think there really are some Xs.
You are a conventionalist about Xs iff you think that we can reasonably address boundary disputes about just what is to count as an X, or what properties Xs are supposed to have by imposing arbitrary conventions.

Here's an example. I think there really are living things. But I don't think the distinction between living and non-living things is such an incredibly natural kind that much would be lost by stipulating some slight re-definition of "alive" that clearly entails viruses are/aren't "alive". Hence, (by the above definition) I'm both a realist and a conventionalist about living things.

Maybe compatibility between realism about Xs and conventionalism about certain facts about Xs only applies conventionalism with regard to tiny boundary disputes about the extension of the concept X? But here's another example where the extension of X will be completely different depending on what stipulation we make.

I'm a realist about human bodies, in that I think that there are indeed human bodies. But should human bodies be identified with *open* or *closed* sets of space time points? This issue, is (just like the viruses question above) one that it seems perfectly natural to settle by stipulation.

Thus, I don't buy the argument that Frege's willingness to allow some questions about what the numbers are to be determined by convention (assuming, as the speaker suggested, he was indeed so willing) shows that he's an anti-realist about about number in anything like the ordinary sense of the term.

[edit: To put the point another way - you can be a realist about the all items that potentially count as numbers but think it's vague which things exactly do count as numbers.

Taking the extension of a concept to be somewhat arbitrary/conventional doesn't require thinking that the objects which are candidates to fall under that concept are somehow unreal]

On Woodin on 'explaining' the consistency of large cardinal axioms

2009-10-15T07:33:00.000-07:00

One of the major highlights of MWPM was getting to hear eminent set theorist Hugh Woodin. He gave a great talk about his program of investigating large cardinal axioms, looking for a characterization of the sets that's as informative as our understanding of the numbers etc. I didn't quite buy his case for the truth of large cardinal axioms, for reasons which I present here with some hesitation (I mean, who is more likely to be wrong about explanation in set theory - me or Woodin?)

One of Woodin's main arguments seems to be that if you don't believe in large cardinals, you can't explain the fact that various large cardinal axioms turn out to be consistent. I'm not sure whether the explanation required here is mathematical (how come there's this pattern whereby all these different con sentences happen to be true?), or epistemic (how come thinking about large cardinals/the possibility of non-trivially mapping the universe into itself leaving a certain initial segment fixed reliably leads us to consistent theories, if it's not the case that in so thinking, we are seeing how the universe of sets actually is). But:

-If the explanation desired is mathematical, then it seems like there might be a purely number theoretic explanation for each of the con(ZF+{some large cardinal axiom}) statements. Why wouldn't this be explanation enough? (Indeed, I thought [?] each large cardinal axioms implied the existence of the smaller large cardinals, so giving a number theoretic explanation for some strongest axiom might simultaniously explain the others?)

-If the explanation desired is epistemic, you might think that people are reasoning about what's metaphysically/mathematically POSSIBLE - e.g. that a structure satisfying ZF and containing a large cardinal is metaphysically possible. We clearly do have mathematical/metaphysical intuitions about when a given body of claims are incoherent/couldn't possibly all be true. And, claims that are logically inconsistent are paradigmatic cases of claims that couldn't all be true. What's possible has to be *at least* logically consistent.

Thus, one might explain the fact we've got whole strings of large cardinal axioms A that are consistent by saying not that mathematicians saw that the sets really were A, but that they saw that objects *could possibly be* as required by A.

Is Realist Carnap Trivial?

2009-10-15T07:10:00.000-07:00

There were two neat talks about Carnap at the MWPM conference this weekend, which got me thinking. I like Carnap, I like realism (well, more like, I don't understand anti-realism) so I like to try to give a realist reading to all Carnap's stuff about the principle of tolerance. But my current best realist Carnap also seems kindof trivial.

Realist Carnap:

1. You can state truths in different languages, even languages which give different definitions/meanings to the same string of letters e.g. "atom".

2. Sometimes if you disagree with someone about "Xhood" (e.g. if you disagree about the question "viruses alive?") you can step back and use other facts that you agree on to characterize the situation, (e.g. viruses reproduce themselves in such and such a way, when they are dormant they don't do so and so, if we stipulate that something is alive iff it Xs then we will get the consequence that all physical things count as being alive) and decide what kind of stipulative definition of "alive" would be most useful to use in this context. Then you just go forward, using the word "alive" in this new sense, and not worrying about whether it was the same as what either of you originally meant by "alive".

Doing this lets you go on with biology without getting bogged down in likely unresolvable questions about whether viruses are alive.

BUT sometimes no stipulative definition given to a term will be as interesting as the one you started with (e.g. if you tried to re-stipulate the meaning of mathematical terms to avoid controversy about whether "there are infinitely many twin primes").

AND sometimes you disagree with your opponents so much about math/logic, that you can't agree with them about what the consequences of a given stipulation would be.

kindof a joke: an ad for my solution to the access problem

2009-10-09T07:47:00.000-07:00

After a really helpful but sad conversation with KY, I realized that I really haven't done enough to make clear to casual readers just what my thesis project (and paper on the access problem) are trying to do.

This lead to me making the following little advertisement.

addendum:

I just heard that Poincare thought that we evolve and/or prune our beliefs to believe what's advantagious, not what's true. In contrast, my thesis suggests that in evolving/pruning our beliefs to believe what's advantagious, we wind up believing (mostly) the truth about some suitable aspect of objective mathematical reality - but this doesn't make our mathematical beliefs a posteriori!

Skepticism and Normativity

2009-10-01T10:53:00.000-07:00

Thinking you have figured out how to solve age old philosophical problems, very quickly, is generally a bad sign.

Nonetheless, ever since TFing Intro Epistemology last semester, I find myself feeling more and more that worries about external world/other minds/memory skepticism involve an incoherent melange of a sharp proof theoretic question, together with a fuzzy normative question.

The proof theoretic question is something like:
- Can you prove the external worlds exist, starting from premises that contain only necessary truths?
- Can you prove memory is reliable starting from premises containing only necessary truths and true statements about current experience?
[Where "prove" can be cashed out in various formal ways - e.g. first order logic, or modal logic, or intuitionistic logic - to yield different variants of the question.]

And the normative question is:
When is it empistemically OK to assume premises in a given set X, given that I cannot prove them (in logic L) from premises in set Y?

Once we've made this distinction, and noted that some premises which one might assume are true, and others false, the normative question looses much of its interest (at least for me).

Furthermore, we can point out to the skeptic who e.g. believes in the reality of past experiences but not in the external world, that his position appears exactly analogous to our own. We can challenge the skeptic to provide any kind of distinction between what's OK to assume vs. not OK to assume that looks remotely principled enough to motivate our revising our judgments on the subject.

"In what sense," we can say to the skeptic, "do you know that e.g. there are infinitely many primes, or that it's impossible to know things about the external world, such that I don't also (by those very same standards) count as knowing that I have a hand?. In both cases, there are more radical skeptics whom we cannot persuade. Thus, in saying that you know, but I do not, you seem to be just stomping your foot and making the unmotivated value judgment that it's OK to assume what you assume and not OK to assume what I assume.

Why should I be more confident that you have correct moral beliefs about what it's OK to assume, than that I have correct descriptive beliefs about whether I have a hand?"

Stipulation and Easy Mathematical Knowledge

2009-10-01T10:28:00.000-07:00

As noted before, I think we get (mature, human) mathematical knowledge by benefiting from caual interactions with the world that lead us to find "coherent" combinations of mathematical statements obvious, and that our acceptance of these coherent stipulations helps determine the meaning of our words in such a way that these stipulations express truths in our language.

But this suggests a question. (Or at least, related views suggested a question to Shapiro and Ebert) Suppose someone accepts ZF and just guesses some elaborate provable truth T, and then stipulates {ZF+T}. Do they count as knowing that T? Doesnt my view commit me to thinking that they do?

The combination of ZF+T is indeed coherent, so I think that people who naturally found T just as obvious as people with mainstream mathematical intuitions find ZF would count as expressing mathematical truths, and indeed knowing that T. (see my paper the Doctoroids for more on this, though I wrote it before seeing the Shapiro).

But what about someone who feels uncertain about whether T, but tries to just stipulate it?

In general, I think, such a person won't count as having knowledge, because they are taking what is (relative to their current state of knowledge) an excessive epistemic risk - and hence they lack justification for their true beliefs. If their current mathematical faculties and other experiences do not give sufficient reason think that adding T to their beliefs would lead to a logically consistent system, they also lack sufficient reason to think that adding T would lead to a system of axioms that correctly describe some realm of mathematical reality. Thus, they are being epistemically irresponcible in adding this axiom.

However, if their current mathematical and other reasoning does suggest (though not prove) that adding T would be consistent, they can be justified in adding T as an axiom (although they may not be justified in assuming that once they have e.g. stipulated the axiom of choice to be true, they are talking about the same mathematical structure as they originally were).

bookclub: Wright and Hale's comeback

2009-10-01T10:17:00.000-07:00

Last time, I blogged about McFarlane's criticisms of Wright and Hale - here's what I think of their response, published in the same journal.

W+H say that the difference between stipulating the axioms of PA vs. stipulating Hume's principle is that the former stipulation fails their requirements by being "arrogant". So, what exactly is supposed to be arrogant about this stipulation:
A(x) x=x iff there is an object 0, and .... (the axioms of PA)
that doesn't also apply to all the instances of the schema below?
the Fs are equinumerous with the Gs iff the number of Fs = the number of Gs

I confess that I'm not sure exactly what W+H's answer to this, even after multiple readings of the paper. But here are some things I see in the paper that don't seem to work as explanations for why Hume's principle escapes arrogance:

1. Specification of truth conditions for all atomic sentences:
One criterion that comes up is that the latter collection of stipulations gives truth conditions to all atomic sentences in the language of numbers, in terms which someone who doesn't yet understand number talk can understand.

But then I don't see why the advocate of just stipulating the PA axioms couldn't break their stipulation down into a similar infinite series of stipulations, each of which is equated to a logical truth as follows and so on for all the countably many atomic sentences derivable from PA e.g.
A(x) x=x iff 0 is a number
A(x) x=x iff 1 is a sucessor of 0
..
You might worry that we can't intend to make any such stipulation, but note that both series of axioms will be recursively axiomatizable, (we aren't enumerating all the truths of arithmetic, just all the ATOMIC truths). So it's hard to see how we could be capable of intending all the instances of W+H's schema, but not the PA schema.

2. RHS can generate understanding of the new terms
Somehow W+H think that saying 'Let 'the numbers' name to some cannonical collection of objects which relate to each other in such a way that there's a "0 object" it stands in the successor relation to other objects etc' would not suffice to let someone who did not previously have the concept of number understand it, whereas hume's principle does.

But I don't buy the argument for this. All W+H say is that a) something else, namely the ramisfication of the PA stipulation, would have to be couched in second order logic and hence presuppose something like understanding of the numbers and b) the PA stipulation just adds to the ramsification by giving labels to the particular objects that stand in the relations stipulated by the ramsification.

They then seem to conclude that making the stipulation of the axioms of PA cannot suffice for understanding. This would be a plausible argument if they first showed that the ramsification doesn't suffice to express the concept of number, and then argued that the straight version doesn't add anything. But what they actually argued was that the ramsification would be incomprehensible to someone who didn't already have something more powerful than the concept of number. So, the fact that the straight stipulation of the PA axioms doesn't *add anything* to the ramsification, doesn't suffice to show that it couldn't be used to give someone understanding of the concept of number.
(which basically asserts that there are things satisfying the PA axioms using the language of second order logic).

W+H's argument seems to be exactly analogous to saying:
The stipulation 'a bachelor is a unmarried man' doesn't suffice to introduce someone to the concept of bachelorhood, because it doesn't add anything to 'a bachelor is an unmarried man who is either a happy bachelor or a sad bachelor' and that statement cannot be used to introduce someone to the concept of bachelorhood.
And this is surely not a good argument.

p.s. I don't think the notion of 'what it takes to gives someone a concept who doesn't previously have it' is well defined - like psycholgoically people can know related concepts, be susceptable to conditioning in various ways etc. Just giving some people one example of gouchery would suffice to make them understand the word, whereas you could say any number of things to a rock, and this wouldn't teach the rock the concept.

bookclub: McFarlane contra Wright and Hale

2009-09-29T09:03:00.000-07:00

I just finished reading a McFarlane's article in the newest Synthese, about neo-Logicism. He argues that the ideas which Wright and Hale put forward in favor of our license to adopt Hume's principle as a stipulation, would seem to equally support just directly laying down the axioms of PA.

This seems 100% right to me, and it occurs to me that my thesis proposal about how a priori knowledge is possible, can be seen as taking this strategy.

However, I wouldn't say that we literally do stipulate the axioms of PA (would anyone say this? I mean it seems like an obvious psychological/historical fact that people generally don't make any such stipulation). Rather, we come to find these statements (and others obvious) which then functions like a stipulation in that it helps determine the meaning of our words in such a way as to make it likely that these statements we find obvious will express truths.

Just saying this doesn't completely dispel worries about the epistemology of math though. For not all stipulations are OK - some stipulations, like those for "tonk" would not lead to a practice that counted as reliable reasoning. So there are two remaining questions.

1. Given that some stipulations are bad, how do we manage to make good stipulations so often?
2. Given that some stipulations are bad, how can we be justified in making any stipultion?

I think the answer to 1 is a combo of quinean/millian theory revision with a nudge from nature, together with facts about the relative profusion of abstract objects as targets for our stipulations.

I think the answer to 2 is that we can't help but start from what we find obvious and reason in ways that seem compelling, so we have prima facie warrant to do this - in the absence of any reason to think to doubt that these feelings of obviousness could be reliable in a given sphere.