Thursday, 21 September 2017

A Dialogue on Mathematical Propositions

I wrote the following dialogue as an antidote to the dogmatism I felt myself falling into when trying to write a paper about a priori propositions. The characters A and B are present-day analytic philosophers. Roughly, A represents the part of me which wanted to write the paper I was working on, and B represents the part which made trouble for the project.

A: I've got a view about a priori propositions I'd like to discuss with you. I don't think you're going to like it.

B: Intriguing! I'll try to put up a good fight.

A: Good. Still, you won't just defend the opposite view no matter what, will you? I'm certainly going into this ready to modify my view, if not to completely relinquish it.

B: Sure. No, I won't just set myself up as an opponent debater. Let's try to give each other as much ground as our philosophical consciences allow, and see if we can agree on some things.

A: OK, great. So, here's the view: what is special about a priori propositions, which enables them to be known independently of experience, is that they have their truth values essentially. They do not reach outside themselves to get their truth values, but carry them within as part of their nature.

B: OK. Interesting use of the notion of essence. I'm used to associating views which tie a priori propositions' truth or falsity closely to meaning with more deflationary attitudes, not with philosophers who make positive use of metaphysical notions like that of essence.

A: Exactly. That's one of the exciting things about my view, I think. It brings out the fact that that sort of tight connection between meaning and truth value can be posited without embracing any problematic conventionalist or deflationary attitudes about essence or meaning.


B: I think you have a point there. A meaning-based view of a priori truth doesn't need to be deflationary or conventionalist. Still, I think it's wrong. Your view overlooks the fact that a priori propositions, or many of them at least, are about something, and we often have to inquire into that something to know them. When mathematicians discover new truths, they don't sit and try to get insight into the essences of the propositions they are wondering about. They try to get insight into the things that the propositions are about, like numbers, or sets, or graphs.

A: That is true, but does not affect what I am saying. Look, the a priori truths of mathematics either have their truth essentially, or accidentally. And if they really had to reach outside themselves for their truth, then they would only be true accidentally. And in that case it should be possible to depict those very propositions reaching out but getting the opposite truth value. But you can't even begin to imagine a situation where someone has expressed what is actually an a priori truth, but which in that situation is a false proposition. And it's not like the case of propositions whose instantiation vouchsafes their truth, like 'Language exists'. Instead, their truth is of their very essence. Now, we all agree that an a priori truth can have its actual truth value, but what would it look like for it to have the other one? The onus is on you to flesh out an answer here, and it seems to me that nothing you could say on this point would satisfy.

B: I do not dispute that I couldn't really flesh out a description of a situation where the same a priori proposition gets the opposite truth value, but I don't think I have to be able to. I can still maintain that these a priori truths do not have their truth off their own bat, due to meaning alone. The source of their truth lies in what they are about. However, unlike with empirical truths, what they are about is rigid and unmoving - necessarily the way it is. So it is no real objection that I cannot depict a situation in which their source of truth or falsity yields them a different truth value, since that is just because their source is necessarily the way it is. That doesn't make their source any less of a source.

A: So you are saying that the meanings of these a priori propositions are out there in a rigid, unmoving space of possible meanings, and that they get their truth or falsity from an equally rigid, unmoving space of mathematical objects. But since all this stuff is rigid, unmoving, and necessarily the way it is, it seems to me that your talk of sourcing is just empty talk. The very idea of sourcing seems dubious here. Granted, you may seem to have an advantage in the fact that our knowledge of these truths must have some source. But the sourcing you are talking about is all going on in Plato's Heaven. It does nothing to explain how we get the knowledge. So you might as well not posit it.

B: You are trying to cast aspersions on my talk of sourcing, but I want to suggest that what you are saying is, on examination, more dubious than what I am saying. You are no nominalist, no denier of the independent existence of mathematical objects. Right?

A: Sure. I mean, I think when people object to claims like 'Mathematical objects exist independently', they are perhaps bothered by something that really should bother them. But I do think that understood properly, such claims do make a sound and correct point.

B: OK, fine. And so, it seems to me that if you are saying that a priori truths about these objects have their truth essentially and off their own bat, you are positing a kind of harmony between the meanings and what they carry inside them on the one hand, and the mathematical objects on the other. But this harmony seems dubious. It cries out for explanation. Why should it exist? Coming around to the proper view, that the propositions are about the mathematical objects, and therefore the mathematical objects' being the way they are is the source of these propositions' truth values, the difficulty disappears.

A: I don't see how the harmony you complain about is particularly strange or objectionable. Don't parts of mathematics mirror and reflect each other in weird and wonderful ways? Since we accept that, it seems that it's not particularly costly to acknowledge that the meanings of mathematical truths are also part of this crystalline structure. Crucially, it seems less dubious than your sourcing talk - more of a piece with things we already acknowledge. And it seems to me that your view overdoes the analogy between mathematical and empirical truths, leading to confusion.

B: Do you see any positive value in your view? Or is it all about stopping that over-assimilation?

A: Well, perhaps my view helps with the problem of how we get mathematical knowledge. It seems to me an easier problem to say how we get in touch with meanings, than to say how we get in touch with things like numbers and sets. Our talk and thought instantiates meanings, I want to say, even if the meanings themselves are abstract, like numbers and sets.

B: But there are also "instantiation relationships", arguably more straightforward, between, say, numbers and piles of apples.

A: Hmm. Well, I don't know, I'll have to think more about that - but perhaps stopping the over-assimilation is enough. What value do you see in your view, anyway?

B: When I think about what is fundamentally wrong with your view, apart from my complaints about it being mysterious and ill-motivated, it seems to me that, in your effort to block the over-assimilation of mathematical and empirical propositions, you bring about another over-assimilation. Namely, between mathematical propositions which can be hard to discover the truth about, and what you might call paradigmatically analytic propositions - propositions where it really does seem that the way to know the truth about them is just to have insight into their meanings. Those propositions may perhaps be said to have their truth values essentially, since they don't seem to say anything substantial about anything, whether their subject matter be empirical or mathematical. And your view wrongly depicts substantial mathematical propositions as being like them. My view has the virtue of avoiding that over-assimilation. It may be that the over-assimilation you worry about is also a problem, but it should be combated in a different way.

A: Well, I am - or at least have been, up to having this conversation - inclined to think the corresponding thing about the over-assimilation that you are worried about. Positing a mysterious sourcing relationship between mathematical propositions and mathematical objects seems like a crude expedient. But I must acknowledge that the over-assimilation that bothers you is also a problem.

B: OK. So, it seems we can both agree that our respective views may have some power to prevent a certain over-assimilation, a different one in each case. And perhaps we can also agree that each of our respective views, when adopted, may increase the danger of falling into the over-assimilation targeted by the opposite view.

A: Hmm. I suppose we can both agree about that.

B: Now, isn't this worrying? I mean, where does it leave us? We have a question: Do mathematical propositions have their truth values essentially, intrinsically, inherently, off their own bat - or do they not? And it seems like our opposing answers have opposing strengths and opposing weaknesses. I feel the weakness of your view much more acutely, but I can't deny that your feeling that my view might be a somewhat crude expedient makes some sense as well.

A: I'm glad you're staying true to your intention of not just defending your view tooth and nail. Now it's starting to look like both our views have some merit, but that these merits crowd each other out. I am beginning to think that perhaps both our views can be said to suffer from crudeness on that score. We are both inclined to use a certain picture to ward off the over-assimilation which has most bothered us. And the pictures conflict, or at least seem to. Now, could it be that if our views were made clearer, these pictures could be seen to apply in different ways, so that there is no inconsistency in using one in its way, and the other in its way? The task then would be to clarify the difference between these two ways of using what appear to be conflicting pictures.

B: That is sounding more and more reasonable to me as a diagnosis of what's going on in this case. How Wittgensteinian! And to be honest, the Wittgensteinian-ness of this view worries me a bit, since this sort of approach, to this sort of problem, seems like it will turn many people off right away. If we are to try to resolve our difficulties this way, and if we expect the resolution to be given a fair hearing, I suppose we will also have to be careful to defend our resolution from objections which lump it together with features of Wittgenstein's views which people don't like.

A: I agree that is a worry. And it may be even worse than you are suggesting. What if the things people don't like and have turned their back on include this very power to resolve our difficulties!

B: Well, I see what you're saying. People are invested in a certain way of doing things, and in defending views of a certain type. And those ways of doing things may come naturally, at least to people with a certain background (including us), so that one slides back into them. But I think we may just have to try to give the naysayers about this method plenty of credit, and allow that there are serious problems with the sort of resolution we're talking about now. After all, why wouldn't there be? It could be that it's very promising, and still ultimately our best hope, but that there are serious difficulties with it which, in our desire to resolve our present issue, we aren't currently alive to.

A: I suppose I'm on board with what you're saying. As exciting and powerful as this approach may seem now, we must beware of coming off as if we think there's a silver bullet, a simple solution we've already got here. And I think that comes out more clearly when we come back from talking about pictures and consider the question, framed in terms of 'essence' or 'intrinsic' or what have you. Something about the idea of pictures makes us quite willing to allow different applications. Ambiguities, if you like. But it seems as though people, ourselves included, may be inclined to take a certain attitude to words like 'essence' and 'intrinsic', such that the word analogue of the move where we say 'These pictures appear to conflict, but if you look at their application, you see it's only an apparent conflict' seems less appealing. There is a feeling that with such words that for each there is a big, important, single job that they should be doing.

B: I think you're right. But again, I think you may be overplaying people's resistance. Yes, there will be people who just get turned off at the suggestion that such words should be understood as having various quite important roles to play. But probably, with many of the sort of people you have in mind, you must admit that they are willing to countenance such things as long as you keep things relatively clear and definite. I mean, if you start banging on about how complex and multifaceted it all is with these words, then yes, that will turn people off, because it sounds defeatist. It sounds like shirking hard and maybe very interesting work. But these sorts of people - and let's face it we're among them a lot of the time when we aren't just talking but trying to write papers - are quite willing to distinguish certain senses of weighty-seeming words, using little subscripts for example. So we shouldn't be too discouraged.

A: Yes, I suppose that's right. So, we should be ready to float the idea that our different pictures each having a role to play, but that just giving the picture and saying 'That's how things are' is a bit crude until we clarify and distinguish the application of the picture in each case. And we should be ready to try to take exactly this approach when it comes to our difficulties as posed in philosophical jargon, but be on guard against defeatist or wishy-washy sounding attitudes. I confess I'm worried about the extent to which this is possible. I mean, maybe once we try, we will find that the distinctions we might want to make by putting little subscripts on words like 'essence' tend to fall apart in our hands, or that possibilities multiply very quickly. But on the other hand, I must admit we haven't seriously tried yet. And maybe there is some progress to be made in that way, even if it does give out and get confusing again in a way similar to our original disagreement. So we should keep working on this.

B: Agreed.

A: I think I'm pretty worn out for now, though. And I suspect there are further problems with your view that I haven't brought out.

B: Same here, on both counts.

A: I hope we can find what it takes to continue soon.

B: So do I.

6 comments:

  1. Hi Tristan,

    Very interesting post, as usual. I wish I could dedicate more time to think about these matters. But for what's worth, I have a couple of thoughts, in light of B's over-assimilation charge.

    a. It's very easy to tell that 3 is a prime number. It's not at all easy to tell that 2^(274,207,281) − 1 is a prime number. In fact, it's way beyond unaided human cognitive abilities. But the fact that one of those is easy and the other is very difficult depends on the computing power of the mind assessing whether it's difficult. It does not seem to me like there should be a difference in terms of analyticity, or (more generally, if it's more general) about whether the truth-value is had essentially.
    But isn't the situation of other mathematical truths difficult to figure out (i.e., other than whether a number is prime?) similar?
    A possible agent who understands mathematical concepts and has a lot more computing power available (in its brain, however it works) might find mathematical truths that humans find very difficult to figure, actually very easy, even immediate if it's smart enough.

    b. A similar situation seems to happen not only when it comes to mathematical truths, but also to sufficiently complex logical truths, and theorems of the first-order predicate calculus. And wouldn't the same apply to the axioms as well? After all, it seems that there is no single way of picking the axioms. One can pick X as an axiom and Y will be a theorem, or pick Y as a axiom and X will be an theorem. Does B have a stance on this? Do we need to know more than the meaning of the symbols to know that the axioms of a first order predicate calculus are true? Are complex theorems of the predicate calculus relevantly different from axioms?

    It seems to me it's very difficult to find a difference between axioms and theorems, especially given that we can pick different axioms. Also, entities with much greater cognitive power may well be able to pick much more complex "self-evident" (to them, at least) formulas as axioms. It might be argued that even for the axioms (or axiom-schemes), one needs more than just knowing the meaning of the words in order to know them. But then again, if that is so, wouldn't that also be so of paradigmatic analytic propositions?

    Granted, one might distinguish between mathematical truths and truths of the first order predicate calculus. But then again, suppose that instead of trying to figure out whether 2^(274,207,281) − 1 is a prime number, someone writes that in first-order symbols (say "Pr(2^(274,207,281) − 1)" represents that), also writes down the axioms of formal number theory (say, FNM), and tries to figure out whether FNM->Pr(2^(274,207,281) − 1) is a theorem of the first-order calculus?
    Sure, one can insist and point out that the formalization of mathematics is in a number of ways incomplete. Still, given that:

    I. Mathematical truths aren't more difficult to figure out than logical truths in general (it depends on the truth), and
    II: Whether it's difficult depends on the agent's cognitive abilities,

    it seems to me that B's charge of over assimilation is at least debatable.

    I guess one might say that the truth of some propositions is clear to us from the meaning of the words alone, and those are also the propositions that have their truth off their own bat, due to meaning alone, while in the case of others, we need the meaning + deductive reasoning to know them. But if meaning + deductive reasoning are enough to tell that a proposition is true, wouldn't it be the case that those truths have their truth off their own bat? It definitely doesn't seem to require any rigid and unmoving stuff (though I'm not sure how B's theory about that works, and how non-deflated it is).

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    1. Thanks for the characteristically thoughtful comment.

      I agree that it is hard to maintain that an 'easy' mathematical proposition like '1 + 1 = 2' is analytic but that more complex ones aren't (although I think there may be a difference between hard-to-calculate but in principle straightforward mathematical questions and ones which call, so to speak, for conceptual innovation, and perhaps a wedge could be driven in there. I.e. I think there may be grounds for a negative answer to the question at the end of the second paragraph of your 'a'.

      But I can also imagine a philosopher, and I think there have been some, who want to say that even a very easy mathematical proposition like '1 + 1 = 2' is not analytic like 'All bachelors are male'. Rather, it states a substantive fact about numbers, just a very basic one, one even that everyone who understands the proposition knows to be true. B could insist on that viewpoint.

      '...wouldn't it be the case that those truths have their truth off their own bat? It definitely doesn't seem to require any rigid and unmoving stuff (though I'm not sure how B's theory about that works, and how non-deflated it is).' - I'm sympathetic to the idea that stuff that can be known on the basis of meaning and deductive reasoning needn't be construed as getting its truth value from rigid and unmoving stuff, but the non-essentialist about the truth values of mathematical propositions could, I think, reasonably deny that mathematical truths do follow deductively from essentially true propositions, or perhaps allow that many do but invoke Goedel's theorem as a reason for thinking that not all are (although of course that would be debatable).

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    2. Thanks for the thoughtful reply as well.
      Regarding conceptual innovation, when a mathematician offers a proof of something, the proof is based on already known results, and in order to be valid, the result has to follow logically from them. While they might introduce new concepts in order to facilitate the proof, and also while they usually don't write proofs in purely symbolic language (way too complicated), the key point is that the hard-to-prove result follows logically from other results. Some of those other results also were hard to prove, but then, those too followed logically from easy results.

      So, B's charge of overassimilation seems to me at least improperly motivated, and the distinction that he's making between what's substantive and what isn't, also seems very problematic in my view. I would argue as follows:

      If discovering that C logically follows from A1&A2&A_n does not involve the discovery of any substantive truths, then nearly all of mathematics does not involve the discovery of any substantive truths. The only substantive discoveries would be axioms, but those do not involve proofs. Mathematical proofs would never give us anything substantively new.
      Granted, B could point out that it follows from his position that C is a substantive truth, because A1, A2, etc., are themselves substantive truths. This is true, but the problem for B's view is that A1&A2&A_n->C does not appear to be a substantive truth if we go by B's concept of substantive truths (it does result from the meaning of the words alone, +logic ; its denial would be a contradiction). So, a really hard proof that resulted from perhaps centuries of work by some of the best mathematicians (some proving some previous results, etc., not necessarily aiming at C), is in fact a proof that A1&A2&A_n->C (or equivalent; they begin with A1, etc., and they reach C), and so mathematics, no matter how complicated, nearly always involves no substantial discoveries.
      So, B's distinction "between mathematical propositions which can be hard to discover the truth about, and what you might call paradigmatically analytic propositions - propositions where it really does seem that the way to know the truth about them is just to have insight into their meanings." has the problem that at least nearly always, mathematical propositions which can be hard to discover the truth about are such that the "hard to discover" part is a matter of whether some proposition logically follows from some other, known propositions.

      I think Gödel's theorem is not going to help B, because in practice, at least nearly always, mathematicians discover truths that are hard to know by proving them from other truths we know.
      Granted, B could still say that that doesn't tell us that the axioms are not substantive. But what is the advantage of this view?
      B claims is that it avoids some overassimilation. But then again, it wouldn't. After all, the actual proofs would still be assimilated to analytical truths, as they do not add anything of substance. Moreover, hard to discover proofs in first-order logic would also be assimilated to analytical propositions, on B's account.

      Now, B might say that on his view, complex logical proofs also add knowledge that can be considered substantive, but in that case, B's charge against A's position is immediately defeated, since A may reply: "Okay, then, I hold that mathematical proofs allow us to make substantive discoveries, even though I hold that mathematical truths are analytical."

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  2. I've been pondering your last comment. It brings up some deep issues. I'm not very satisfied with what I have to say, but anyway...

    'Regarding conceptual innovation, when a mathematician offers a proof of something, the proof is based on already known results, and in order to be valid, the result has to follow logically from them. While they might introduce new concepts in order to facilitate the proof, and also while they usually don't write proofs in purely symbolic language (way too complicated), the key point is that the hard-to-prove result follows logically from other results. Some of those other results also were hard to prove, but then, those too followed logically from easy results.'

    I think it can be argued that this embodies a conception of mathematical knowledge which, while it may contain a lot of truth, may also fail to emphasise an important part of the picture. This way of looking at things seems quite common. For instance, in the just-revised SEP entry on Philosophy of Mathematics, in the introductory part introducing the central problems of the area, we find:

    (...) the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way: by deduction from basic principles.

    And I think this passage from later in your comment is especially representative of the attitude I think B could question:

    'the "hard to discover" part is a matter of whether some proposition logically follows from some other, known propositions'

    B may protest here that it can be very hard, or a very non-trivial achievement, to form new mathematical concepts. (And along with them come "easy", basic propositions involving those concepts. They're easy and basic *once you have the concepts*.)

    I think B can perhaps quite justly complain that your arguments are neglecting this. Someone who is opposed to the idea that mathematical propositions have their truth-values essentially, i.e. do not reach outside themselves for their truth-values, if they admit that we indeed have some mathematical knowledge, might complain that this deduction-centric epistemological attitude to mathematics is one-sided, and leads to a wrong metaphysical picture.

    On B's conception, deduction is only part of the process of acquiring mathematical knowledge. There is also concept formation and insight - or just "sight" even, i.e. seeing the truth of something. With this in the background, I can see it coming naturally to B to picture a body of mathematical propositions ultimately getting their truth from outside themselves. We, somehow - with mathematical insight - get to know bits of this truth and then we can use deductive methods to work out other truths. (I am certainly not saying that this conception poses no problems - after all, A's conception isn't going to be a desert either when it comes to philosophical problems - but just trying to give it its due.)

    Also, I'm interested in what you think of the more meta issue of what's going on in the disagreement between A and B. Whether perhaps there is an issue of subtly clashing applications of a picture or a form of words. Is part of your motivation for exploring these ways of supporting A against B a feeling that, no, this isn't one of those cases of clashing applications, but rather one where the one view, perhaps A's, is the true, or at any rate a better, view about some matter?

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    1. Good point regarding conceptual innovation. I grant that it can be very hard to come up with new concepts, and that I left that important part of the picture aside in my previous comments. But I don't think this helps B's charge, for at least the following reasons:

      First, coming up with new concepts, is in most cases done as I described: a mathematician gives a definition to make the proof of something else easier. I grant it is not always like that. Sometimes they are working with known concepts, and they think of a generalization. Sometimes, they might come up with concepts by observing some concrete stuff, and making some model of it. But it's mostly like that, at least based on what I've seen (then again, the most new concepts aren't "big" new concepts, but small ones, so to speak).

      Second, at the very least, much (even if not nearly all as I said earlier; but I'd say most) of mathematics is like that, i.e., proving some very difficult things from easier ones (or from difficult ones already proven from easier ones, etc.). B's view on what is substantive would make all of that mathematical work (which is sometimes at least as difficult as any other mathematical work, if not more so) something that adds nothing substantive. But then, why would B think that other mathematical work does add something of substance?

      Third, even if B has good answers from his position, that does not seem to be what B's over-assimilation charge was about - or if it was, then the charge misses the mark. On that note:

      "B may protest here that it can be very hard, or a very non-trivial achievement, to form new mathematical concepts. (And along with them come "easy", basic propositions involving those concepts. They're easy and basic *once you have the concepts*.)".

      That can be very hard, but A needn't worry. In fact, A should hold that it's sometimes hard to come up with new concepts - because there is good empirical evidence of that. But there is nothing in the hypothesis of analyticity of mathematical statements that says that it's easy to come up with new mathematical concepts.

      B's charge of holds that said over-assimilation is "between mathematical propositions which can be hard to discover the truth about, and what you might call paradigmatically analytic propositions - propositions where it really does seem that the way to know the truth about them is just to have insight into their meanings. Those propositions may perhaps be said to have their truth values essentially, since they don't seem to say anything substantial about anything, whether their subject matter be empirical or mathematical. And your view wrongly depicts substantial mathematical propositions as being like them."

      As I understood it, B was talking about mathematical propositions, not about the activity of coming up with new mathematical concepts.
      However, if I didn't interpret that right and the charge was partly about that activity, A may properly reject that part by pointing out his position doesn't commit him to saying otherwise.


      As for my my motivation, I have to admit I just read your post, and I found the matter interesting.
      That said, I am sympathetic to A's view, at least to some extent. I think A's view is simpler, and doesn't have some of the problems that are usually attributed to it. I don't know whether there are other problems, though.
      Also, I do think the view of some rigid an unmoving stuff would be at least unwarranted, and I suspect that at least part of what's going on is that B is coming up with a theory to fix something B thinks needs fixing, but I think doesn't.

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