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]

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'.

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.

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.

Putnam Indeterminacy Dilemma

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?

Maybe this was obvious to everyone else

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.

New Uses for Conceptual Analysis

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

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).

McDowell on Rule-Following pg348

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?

Anti-Russell

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.

Paradox of Analysis

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)

On Rationalizing Explanations

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].

de re beliefs about numbers?

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]

Doubting Conceptual Truths

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

"No Fact of the Matter" Paradox?

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.

Bookclub: 'Compositionality, Understanding, and Proofs'

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.

Rabbits

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.]

More Davidson Obsession

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.