Wednesday, April 14, 2010

Contrast w/ Tait "The Platonism of Mathematics"

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

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

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

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

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

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

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

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

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

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

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

Parsons and Intuitability

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

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

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

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

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

Parsons Mathematical Thought and its Objects CH1 summary

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

1. Abstract objects defined + generic worries about them

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

2-3 What is an object?

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

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

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

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

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

ii. intuitability

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

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

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

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

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

4. objecthood=having the logical role of an object

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

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

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

5-6 are about b and c respectively

7. Quasi-concrete objects

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

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

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

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

Friday, April 9, 2010

Field on Normativity and Logic

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

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

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

A. Classical Logic Version:

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

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

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

B. Non-Classical Logic Version:

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

Learning about numbers by thinking about sets

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

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

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

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

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

Sunday, April 4, 2010

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?