The Access Problem for Holes

[Hello to anyone out there!

Project: exposure therapy for my fear of posting sloppy/confusing stuff to this blog continues. So uh... sorry. I hope to return to regularly posting polished content soon.]

One of the least popular aspects of my philosophy of mathematics  is what I call Moderate Quantifier Variance. So I thought I'd explain one reason why I'm such a big fan of this view (and why I think you should be too) which has nothing to do with philosophy of mathematics.


The Access Problem for Holes

Consider our knowledge of holes. As discussed in D and S Lewis' On Holes, it's appealing take holes to be distinct from things like the air inside them, portions of the matter that `hosts' (e.g. portions of cheese with some diameter aronud a hole in Swiss cheese).

We seem to know how deeply a piece of cheese must be indented in order for there to count as being a hole in that piece of cheese. But (to a crazy philosopher, for a second) this knowledge can seem rather mysterious. For our senses tell us how the cheese is distributed through space. But how do we know where to draw the line: re how shallow a hole can be? No sensory experience seems to point out a single metaphysically special place to draw the line. So how can one explain the match between human beliefs about how shallow a hole can be and objective matters of facts?

It's appealing to say: there's no mystery about human accuracy here because if we had used `hole' differently (by taking the minimum hole angle to be larger or smaller) then the meanings of our words would have been different so these alternative hole-identifying practices would have yeilded true utterances. But since variant practices of "hole" individuation can require change in sentences whcih don't even use the word hole  (e.g. a purely logical sentence like the Fregean paraphrase for `there are >3000 things' can go from true to false). So it seems plausible that allowing such changes in meaning requires allowing variant meanings for logical vocabulary like ``there is'' as well as in the meaning of  ``hole''.

So I think a great way to solve this `access problem for holes'' is to accept

Moderate Quantifier Variance: there are multiple existential quantifier like meanings which the words `there is' can take on in different ideolects, (say English_1910, English_2010 etc.), though within any particular ideolect `there is' is univocal.

• These meanings are `quantifier like' in the sense that they obey the all instances of standard first order logical inferential rules/axioms for `exists' (within the language they belong to). 
• If there is a single maximally natural quantifier sense (as e.g. Sider thinks there is corresponding to our talk about `what there is' takes on when doing ontology), these variant quantifier senses need not be mere quantifier restrictions of this fundamental sense. 

I like to use Moderate Quantifier Variance to explain/vindicate the freedom mathematicians take themselves to have to introduce new logically coherent structures for study, by saying that when mathematicians consistently extend the axioms of pure mathematics by adding axioms describing some new mathematical structures like the complex numbers, their words can both give meaning to expressions like `complex number' and also change the meaning of `there is', so that these axioms express truths.

But whatever you say about math, I think moderate quantifier variance is already useful for understanding our knowledge of swiss cheese!

How Not to Title your Dissertation

So I just had to look up my official dissertation title on my grad school's website and....well, here's a little piece of advice I would give to my past self. (The 'do' example comes from the excellent Johann Frick)

DO

"'Making People Happy, Not Making Happy People': A Defense of the Asymmetry Intuition in Population Ethics"  <-- conveys the core of the view being advocated, what's controversial about it, and why it's appealing, in a punchy and fairly concise way

DON'T

"The Marriage of Rationalism and Empiricism: A New Approach to the Access Problem." <-- draws attention to question of whether dissertation does anything new in a defensive "protesting too much" way that invites a negative answer, compares self to William Blake and/or God.

Access to reference magnets: a bitter pill you’ve already swallowed?

[This post proposes a defense of a famous defense of naive scientific realism against a famous antirealist challenge.  So, sorry that I'll have to speed through certain classics to get to the action in a timely fashion (and I'll try to add explanatory links later). 

Also, Ted Sider may have scooped me re: this proposal in Writing the Book of the World section 3.2 (I go back and forth about how to interpret him). But, regardless of priority, I'm shouting about it on the internet now because I'd like to see more uptake.]

Hillary Putnam raised a model theoretic challenge to the commonplace realist idea that `even in the ideal limit of scientific investigation' certain aspects of our best theory of the world could be wrong. David Lewis responded by invoking "reference magnets" (i.e., intrinsically eligible concepts/joints in nature) as a response to this challenge. The idea is that some concepts are more intrinsically eligible candidates for the meaning of words than others. So, it can be correct to interpret someone as meaining plus vs Kripke's quus, or electron vs. electron-that-an-ideal-observer-starting-on-earth-could-discover, even if this involves attributing them slightly more false beliefs.

Swinging back on behalf of Putnam, Tim Button and Jared Warren press a kind of access problem for fans of reference magnets. Suppose that there really are intrinsically eligible joints in nature as Lewis argues. How could creatures like us have come to recognize where these joints are, well enough to know when there is a single reference magnetic joint for our use of some word to defer to?

I think this challenge is worth taking seriously and may point out a bitter pill* which the realist/friend of reference magnetism must swallow. But I want to suggest that this bitter pill may already be part of the larger bitter pill nearly everyone has already swallowed in taking our intuitions about how to do scientific induction at face value. That is, accepting reference magnets doesn't land us with any more of an access problem than we already face in rejecting Humean skepticism about induction.

To see what I mean, consider Nelson Goodman's picture of scientific induction. When we do scientific induction, we don't treat all concepts equally. We currently take some predicates (and relations and functions etc) to be more projectable than others, e.g., green v.s grue. And (Goodman notes) we do a kind of induction about how to do induction, letting experience and reflection change our beliefs about which predicates are projectable. So (in effect) we dogmatically presume both that certain predicates are more projectable than others, and that that certain ways of letting experience change our beliefs about which predicates are projectable are reliable. And plausibly, taking scientific induction at face value requires doing something like this.

But maybe the friend of reference magnets can say that access to these facts about what's joint carving (in the sense of being specially friendly to induction) is all they need for access to reference magnets. If the reference-magnet-fan's doctrine identifies being an intrinsically eligibile concept in the sense of reference magnetism with being an intrinsically eligible concept for the purposes of scientific induction (as, e.g. Sider does and I essentially want to**), then it seems that accepting this doctrine create any extra intuitive access worries.



*[i.e., Perhaps they must embrace a rather depressing picture of the human condition, on which `justified’ reasoning (to the extent that we have any such thing) involves going along dogmatically assuming that certain methods for detecting intrinsically eligible concepts/reference magnets are tolerably accurate (and then being lucky enough to be right about this)]

**[I think one tiny refinement answering Hawthorne's problem about Europe and the Ural Mountains discussed on pg 39 of WTBOTW is needed, but that this makes no difference to the access problem stuff above. More on this in a later post]

Trivializing Benacerraf and Monstrous Moonshine

[note: Sorry that this post is a little wordy. I'm trying to get back into blogging, and forcing myself to post stuff I'm not quite happy with is part of that. Also see this paper for a way more detailed and less aggro take on this issue.]

Remember that the access problem (aka the Benacerraf problem) for realists goes something like this: If moral/mathematical/etc realism is true, how can human accuracy and reliability about moral/mathematical facts be anything but a miracle or a mystery?

People like Justin Clarke-Doane and perhaps David Enoch, (call them The Trivializers) have been suggesting that we can answer all legitimate access worries besetting for realists about necessary domains like mathematics morals etc. just by “stapling together” two things to explain why we couldn't have easily been wrong (and providing a similarly trivial explanation for the sensitivity of our I won't discuss here):

• an  explanation (e.g.historical or evolutionary) for why we reliably believe certain moral/mathematical/etc claim,
• the fact that these claims are necessary truths

So, for example, a classic platonist might try answer access worries by explaining human reliability about mathematics like so:

TRIV: Mathematicians reliably believe truths because they reliably believe only those mathematical claims which can be proved in a certain formal system (e.g. ZFC) and this formal system is (necessarily) truth preserving w.r.t. the platonic mathematical objects.

Such an explanation is not likely to satisfy anyone who feels an intuitive access worry. For TRIV just explains one intuitively mysterious match between human psychology and objective which intuitively “cries out for explanation” (our acceptance of theorems that match whats going on in Plato's heaven) by positing another such mysterious match (our acceptance of axioms that match what's going on in Plato's heaven).

But it does (in some sense) suffice to explain the safety of human beliefs by deriving our reliability about realist facts concerning the relevant domain (i.e., that in all close possible worlds, our beliefs about these domains match up with the truth) from more general premises which the realist (if not their deflationary opponents) accepts.

And I take Trivializers to be suggesting that our intuitions that certain regularities involving necessary truths "cry out for" explanation in a deeper/more unified sense, which mere deductions of reliability/safety/sensitivity like TRIV need not provide, is an illusion.

There are two reasons why I don't buy this.

First, conceptually analyzing anything (from tablehood to justice) is infamously hard, but there are plenty of good paradigms for how to think about ”crying out for explanation” which seem to apply equally well to necessary and contingent regularities (e.g., norms that say we should preferring theories that have fewer degrees of freedom and or Kitcher’s idea of scientific explanation as unification).

Much more importantly though, embracing the Trivializers' position seems to have deeply implausible revisionary consequences for mathematical practice, since mathematicians sure seem to think that some regularities involving necessary truth truths can cry out for (non-trivial) explanation.

For example, consider this quote from a popularizing article about the history of John Conway's `Monstrous Moonshine' conjecture.

``Strangely enough, [the j- function]’s first important coefficient is 196,884, which McKay instantly recognized as the sum of the monster’s first two special dimensions.
Most mathematicians dismissed the finding as a fluke, since there was no reason to expect the monster and the j-function to be even remotely related. However, the connection caught the attention of John Thompson, a Fields medalist now at the University of Florida in Gainesville, who made an additional discovery. The j-function’s second coefficient, 21,493,760, is the sum of the first three special dimensions of the monster: 1 + 196,883 + 21,296,876. It seemed as if the j-function was somehow controlling the structure of the elusive monster group.'’’

Note that mathematicians already had separate proofs of facts about the j-function and monster group (and hence an `explanation' of for the match between these facts of the kind which TRIV provides, i.e., a deduction, from more general premises, of the fact that this relationship holds in all close possible worlds). But (once this match between apparently unrelated domains proved striking enough) they expected to find some further/deeper unifying explanation and this expectation guided choices for further research.

So I'm prima facie pretty suspicious of the idea that felt intuitive demands for unifying/satisfying/non-trivial explanations of regularities involving necessary truths are generally misguided. But maybe I'm not being charitable to Clarke Doane and/or he can find some way of separating the "this regularity cries out for further explanation" intuitions he wants to dismiss as unreliable from those which obviously do good work in mathematics.

Posthumus Vindication and Newton's Concept of the Derivative

In a recent Mind paper, `Incomplete Understanding of Concepts: the Case of the Derivative', Sheldon Smith vividly sets up some classic questions about Newton's concept of the derivative, and how later mathematical work can be seen as vindicating Newton.

However I'm not entirely convinced by Smith's answers to these questions.

Historical Background:

[Smith tells us how] Newton and Leibnitz had certain limited beliefs about the derivative

• that it was "the local rate of change of a function given by the slope of the tangent" so the derivative of x^2 kinda should be 2x
• that it was the limit as i goes to 0 of (f(x+i)-f(x))/i, hence derivative of x^2 was [(x+i)^2-x^2]/i which they thought was =(2xi+i^2)/i=2x+i=2x

but they did not have a very solid justification for the later reasoning (particularly the presumption that one can divide by i in the claim above).

Since then, mathematicians have defined multiple derivative-like notions which all let one defend reasoning like the above more rigorously, but don't always agree:

• the usual: the derivative of f(x) is the function f'(x) such that for every epsilon there is an i such that |(f(x+i)-f(x))/i - f'(x)| < epsilon
• the symmetric derivative: [like the normal definition but with (f(x+i)-f(x-i))/2i in place of f(x+i)-f(x))/i] (note that when f(x)= |x|, the symmetric derivative is 0 whereas the standard definition is undefined).
• a definition using infinitesimals
• a definition which also can apply to generalized functions like the Dirac delta function

Furthermore there is a common intuition that, in providing some of the definitions above and proving things with them, mathematicians like Weierstrass "justified [Newton's and Leibnitz's] thoughts" and that Newton and Leibnitz would have felt "vindicated" by subsequent developments of the derivative.

The questions:

Now, Smith argues that Newton didn't seem to be using any particular one of these modern concepts of the derivative.

• Newton didn't (somehow) implicitly have any of these precise concepts in mind, and which definition of limit he would have preferred to adopt (if he had been told about all of them) might vary with which one he found out about first.
•  There's no single "best sharpening" of what Newton believed/had in mind which must be accepted in limit of ideal science. We just have separate notions of derivative, each of which is mathematically legitimate. Thus we can't say that Newton meant, say, the standard contemporary notion of the derivative because he was conceptually deferring to the results of ideal science.

So he asks:

1. How `` should [one] think about the derivative concepts with which Newton and Leibniz thought''? 
2.  How ``could [Weierstrass] have managed to justify their thoughts even if their thoughts did not involve the same derivative concept as Weierstrass’s''?

Smith's Answers:
I take Smith's answers to the above questions to be as follows:

Q1: What was Newton's concept of the derivative [specifically, how does it effect the truth conditions for sentences]?

A: Newton's concept of the derivative (call it derivative_N) "only has a definite referent" in cases where all acceptable sharpening definitions of his concept agree.  So, for example, if the symmetric derivative and the standard derivative were both acceptable sharpenings, then expressions like `the derivative_N of f(x)=|x|' would fail to refer [or, perhaps, would refer to function which is undefined at 0 so that 'the derivative_N of f at 0' would fail to refer].

Q2: How was Weierstrass able to vindicate Newton, given that his concept of the derivative was different from Newton's?

A: One can vindicate Newton by justifying particular claims Newton made (e.g., about the derivative of x^2). And one can do this giving a proof of the corresponding claim employing Weierstrass's definition, if it also happens to be the case that all other permissible sharpenings of Newton's notion of the derivative would agree on this claim.

A Small Objection: 

I'm not entirely convinced by Smith's account of Newton's concept (Q1) for various reasons. But even if Smith is right about Q1, I think his answer to the vindication question (Q2) is fairly unsatisfying.

For suppose (as Smith seems to presume) Weierstrass vindicated Newton by showing the truth of particular claims he made about calculous -- that, say, what he expressed by saying ``the derivative of x^2 is 2x'' was true. If (as Smith's account of Newton's concept seems to tell us) the truth of this claim requires that all acceptable precifications agree in making ``the derivative of x^2 is 2x'' come out true, how can one adequately justify Newton's claim merely by discovering *one* such precificiation and showing that *it* makes the above sentence come out true?

A fix?

Maybe Smith could solve this problem (while keeping his account of the concept and the, IMO, good idea that vindicating Newton doesn't require assessing all possible derivative-like notions) as follows.

Say that "vindicating Newton's thought"  in the sense we normally care about (the in the sense that seems to have happened, and that, plausibly, Newton and Weierstrass would have cared about) doesn't require showing that some of Newton's specific mathematical utterances expressed truths. Instead, one can do it just  by showing Newton was right to believe some more holistic meta claim like `There is some mathematical notion which makes [insert big collection of collection of core calculous claims and inference methods] all come out true/reliable'.

Three Projects Involving Dispensing With Mathematical Objects

One of the many unfortunate things about academic fashions is that when a popular project goes out of fashion, superficially similar-looking projects which don't face the same difficulties can be tarred with the same brush.

Many people (myself included) feel cautious pessimism about formulating a satisfying nominalist paraphrase of contemporary scientific theories [one major issue is how to make formulate something like probability claims without invoking abstract-seeming events or propositions]. But we wouldn't want to over generalize.

 In this post I'm going to suggest three different motives for seeking to systematically paraphrase our best scientific theories (and many other true and false ordinary claims) in a way that dispenses with quantification over mathematical objects, and note that the requirements for success in the first (ex-fashionable) project are notably laxer in some ways than the requirements for the others.

[Note: I don't mean to endorse any of the projects below, but I think that 3 and Burali-Forti based versions of 2 are at least interesting.]

Three Motivations for Paraphrasing Mathematical Objects Out of Physics

1. General rejection of abstracta: You deny the existence of mathematical objects because you think allowing any abstract objects are bad. (this is the classic motive)
2. Explaining special features of mathematical practice by rejection of mathematical objects: You deny the existence of mathematical objects because you think that not taking mathematical existence claims at face value is allows for the best account of certain special features of pure mathematical practice, (e.g., by mathematicians’ apparent freedom to choose what objects to talk in terms of/disinterest in mathematical questions that don’t effect interpretability strength, or by the Burali-Forti paradoxes in higher set theory) not to take apparent quantification over mathematical objects at face value. 
3. Grounding math in logic/bringing out a claimed special relationship between math and logic: You may allow the existence of mathematical objects, but you’re moved by the close relationship between an intuitive modal notion of coherence/semantic consistency/logical possibility and pure mathematics to seek some kind of shared grounding and think that the coherence/logical possibility notion looks to be the more fundamental. As a result, it seems promising to seek a kind of "factoring" story, which systematically grounds all pure mathematics in facts about logical possibility, and all applied mathematics in some combo of logical possibility and intuitively non-mathematical facts.

 Distinguishing these motivations/projects matters, because what you are trying to do influences what ingredients are kosher for use in your paraphrases: 

If you have the first motivation (defending general nominalism), you need to avoid quantifying over any other abstracta, including: platonic objects called masses or lengths which don’t come with any automatic relationship to numbers, corporations, marriages, sentences in natural languages, metaphysically possible worlds, and (perhaps) propensities. 

But if you have the second motivation (defending mathematical-nominalism-adopted-to-explain-special-features-of-mathematical-practice), then quantification over abstracta which don’t have the relevant special feature, (e.g., objects in domains where we don't think appear to have massive freedom to choose what objects to talk in terms of or objects which don't give rise to a version of the burali-forti paradox), is fine. 

And if you have the third motivation, then quantification over any objects which aren’t intuitively mathematical-- or aren't mathematical in whatever way you are claiming requires a special relationship to coherence/semantic consistency/logical possibility -- is OK.

Hello World (again)!

As you can see, I haven't posted to this blog for ages.

I've been busy a) enduring the horrors of the job market b) getting a sweet 5 year postdoc (I still can't express how luck and grateful I feel) c) finishing a stack of old papers and d) writing a zillion page monograph to answer a minor technical question about Potentialism and logical possibility which my advisers asked in grad school (and then rewriting all the proofs 3+ times because a grumpy mathematician friend didn't think the prose was clear or concise enough!).

But now that I have time to focus on new research, I'm thinking it might be fun start blogging again. I'm certainly touched by the number of lurkers who still turn up to check this blog out.

Let's see how things go!

Art and the Examined Life

1.
It's a common thought that works of art can (somehow) imaginatively suggest different ways of approaching and experiencing the objective world around us. For example, a character in Rebecca Goldsmith's curious novel The Mind Body Problem says

``The interesting thing about art is your being presented with another's point of view, looking out at the world from his perspective, seeing the dreaminess of Renoir's world the clarity of Vermeer's, the solemnity of Rembrandt's, the starkness of Wyreth's.''

Some philosophers and literary scholars have suggested that engaging with such works of art is valued/valuable because it helps us understand and sympathize with other possible points of view (whether these really were occupied by specific artists or not) -- and might thereby make us nicer to people. Art experiences might make us more inclined to try to be nice because we are more sympathetic to certain people, and they might make us better at actually doing nice things because we understand these people better.

But  (to my knowledge) there's no clear empirical or folk-psychological case that having deep artistic experiences does make people significantly nicer. And I'm inclined to be skeptical.  There's a funny line in C.S. Lewis somewhere about how any inclination to think that art makes people more virtuous will be dispelled by asking an English professor to think about their colleagues. (Lewis was an English professor at Oxford and Cambridge).

2.
I'm gonna suggest a different idea about why we might highly value art for its power to evoke a different way of looking at things (in addition to valuing it as a source of pleasure, a tool for distraction etc.).

Maybe art experiences are valued for expanding our knowledge of how it would be psychologically possible for us to approach the world (including what adopting such approaches would feel like from the inside). In doing so, they help us a) live an examined life and b) choose how to live by expanding our sense what the space of (psychologically accessible) options is like.

For great works of art seem to reveal the possibility of ways of approaching and experiencing life [noticing things, finding projects appealing, reacting emotionally] which it would have otherwise been very hard for us to first personally imagine (imagine Jane Austen reading Nietzsche, or Nietzsche reading Jane Austen). Like Hume's first taste of pineapple (and unlike his first experience of a missing shade of blue) such art experiences expand our knowledge of what kind of experiences it is possible to have.

This has two benefits.

First, art can help us adopt a life we want to live, in approximately the same way that travel or visiting different social scenes does -- by making us aware of regions within the space of possible approaches to life [i.e. the space of options which are at least sufficiently psychologically possible for us to take up that we can imaginatively simulate them/enter into them for a while] is -- hence potentially aware of attractive options which were hitherto overlooking.

Second, I suspect that merely getting this kind of knowledge of psychological/phenomenological possibility space (whether it ever gives us practical benefits or not) contributes more to `living an examined life' (in whatever rough intuitive sense that seems desirable) than many big pieces of awesome philosophical knowledge would (e.g. knowledge of the right solution to the liar paradox).

Kripke's Paderewski and Frege's Common Coin

Maybe this was already obvious to everyone but ...

Kripke's Paderewski example can be modified to refute both the attractive principle Frege's Common Coin, below, and contemporary weakenings of that principle which exempt indexical and demonstrative sentences, or require that all speakers be normally linguistically component.

Frege's Common Coin:
a) When two sincere speakers "disagree over" a sentence*, there is a single proposition expressed by this sentence in this context which one believes and the other does not believe,(and indeed believes the negation of).
b) When two sincere speakers "agree about" a sentence, there is a single proposition expressed by this sentence in this context which they both believe.

*[I realize this is an awkward locution, but I just mean the intuitive kind of disagreement which occurs when I say "snow is red" and you say "snow isn't red" but doesn't occur when I say "I'm tired" and you say "I'm not tired". It sounds far more natural to say `disagree over a proposition' but we will see that it is actually not clear whether these scenarios involve disagreement over a proposition.]

Consider the following drama involving Pierre, a man in a thought experiment of Kripkie's who knows the musical statesman Paderewski in two different ways, and doesn't realize that Paderewski the pianist is Paderewski the anarchist. Suppose all the following utterances are sincere,
Act 1 (musical evening)
pierre:"Paderewski is tall" p1
alice:"yes, Paderewski is tall" p2
Act 2 (on the street)
alice:"Paderewski is tall" p3
bob:"yes" p4
Act 3 (political rally)
bob:"Paderewski is tall" p5
pierre:"no, he's not!" let p6 be the proposition that Pierre *denies*

Frege's Common Coin tells us that there are propositions p1…pn which speakers express attitudes towards in all the different phases of our play, that p1=p2, p3=p4, p5=p6 and that Pierre believes p1 and does not believe p6.

But this is a very bad thing to say: By the fact that a person like Alice or Bob who has a single grip on Paderewski presumably says the same thing by asserting this sentence on the street vs. at a political rally or a musical evening p2=p3 and p4=p5.
By transitivity of identity p1=p6.
Thus Pierre believes p1 and does not believe p1. Contradiction.

[If you are worried about the fact that Pierre is unusually ignorant for his society, and hence may not count as "normally linguistically compent" substitute in the name "John". Now cases like the one above will turn out to be so ubiquitous that denying that Pierre, Alice and Bob know enough to have linguistic competency would imply that no one ever has linguistic competency for names.]

Possible Moral: If we want to take propositions to be the objects of belief then, contra Frege's Common Coin, each sentence must be associated with (something like) a class of different propositions which someone could sincerely assert that sentence in virtue of believing.

The Sheffer stroke, and the blandest antirealism ever

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"

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

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.

Hamming Questions for Philosophy

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 "Hamming 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 paraphrase 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

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?

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

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

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?

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

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

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.