mathematical explanation
May. 28th, 2008 07:22 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
guslI love this reply to Mancosu, by Thomas Forster:
I would say that mathematics has "doxastic counterfactuals" (because, thanks to the nonobviousness of math, we can imagine theorems being false), but that they are different for each person (maybe you can imagine the Pythagorean theorem being false while somebody else cannot). Garber has written something on logical counterfactuals. Gaifman has written on assigning subjective probabilities to logical statements.
I don't understand why they don't turn to cognitive science. The only person I know in this area of "mathematical explanation" is Jeremy Avigad, and he's not at all interested in approaching this from cognitive science. I don't get it.
<< Well, here's a position that has at least the beneifit of clarity and brevity:
No explanations without counterfactuals
No counterfactuals without contingency
All of Mathematics is necessary
------------------------------
No explanations in mathematics!
I'm not saying i believe it, but finding the right place to pick holes
might be a useful discipline. >>
I would say that mathematics has "doxastic counterfactuals" (because, thanks to the nonobviousness of math, we can imagine theorems being false), but that they are different for each person (maybe you can imagine the Pythagorean theorem being false while somebody else cannot). Garber has written something on logical counterfactuals. Gaifman has written on assigning subjective probabilities to logical statements.
I don't understand why they don't turn to cognitive science. The only person I know in this area of "mathematical explanation" is Jeremy Avigad, and he's not at all interested in approaching this from cognitive science. I don't get it.
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(no subject)
Date: 2008-05-29 02:08 am (UTC)(no subject)
Date: 2008-05-29 03:49 am (UTC)This is the idea of mathematical explanation:
<< Much mathematical activity is driven by factors other than justificatory aims such as establishing the truth of a mathematical fact. In many cases knowledge that something is the case will be considered unsatisfactory and this will lead mathematicians to probe the situation further to look for better explanations of the facts. This might take the form of, just to give a few examples, providing alternative proofs for known results, giving an account for surprising analogies, or recasting an entire area of mathematics on a new basis in the pursuit of a more satisfactory ‘explanatory’ account of the area. >>
(no subject)
Date: 2008-05-29 03:58 am (UTC)(no subject)
Date: 2008-05-29 04:31 am (UTC)David is also not interested in it, but one could potentially do a project with the both of them, but one would have to be awfully motivated.
(no subject)
Date: 2008-05-29 04:55 am (UTC)(no subject)
Date: 2008-05-29 04:59 am (UTC)(no subject)
Date: 2008-05-29 05:32 am (UTC)(no subject)
Date: 2008-05-29 02:25 pm (UTC)We could make cognitive models that generalize across tasks, each of which requiring different pieces of (declarative and procedural) knowledge. Students who understand will be able to transfer.
Once we've defined understanding, and shown how it could be measured experimentally, proceed empirically.
This way, we should be able to distinguish illuminating proofs from non-explanatory proofs: the former increases understanding.
(no subject)
Date: 2008-05-29 02:41 pm (UTC)I think that education researchers struggle with this.
(no subject)
Date: 2008-05-29 06:53 pm (UTC)Also, I'd definitely want to contest the first premise in the argument (at least, I don't get the intuition behind it), and I'd probably contest the second one as well, since there really do seem to be sensible counterpossibles. (I also think the third premise may be false, though I don't think it would be false in a relevant way.) There may be a way to shore up this argument if you replace "explanation" by "causation" and slightly restrict the other premises as well, but that's a very different conclusion. I'd use that conclusion as a reductio for the claim that explanations are always causal.
math + cognitive science
Date: 2008-05-31 01:19 am (UTC)Re: math + cognitive science
Date: 2008-05-31 01:26 am (UTC)Thanks, but I hope it's the only one too. I have many complaints about that book.
Re: math + cognitive science
Date: 2008-05-31 01:29 am (UTC)Re: math + cognitive science
Date: 2008-05-31 01:50 am (UTC)Your post suggests you think logic is the way to go. My PhD is in logic. I studied predicate calculus, modal logic, many-valued logic, meta-logic, model theory, computation and logic, temporal logic, and, especially, the logic of science: induction, abduction, discovery, confirmation, etc. I no longer believe that logic is an insightful way to approach cognition or explanation.
I hope you'll reconsider Where Mathematics Comes From. Maybe it will help to look at some of Lakoff's other books, such as Metaphors We Live By, if you haven't already. I know Lakoff can rub people the wrong way with his writing style. I disagree with many of the details, but I think that the central idea is right: math is based on metaphor and analogy.
Re: math + cognitive science
Date: 2008-05-31 01:56 am (UTC)Re: math + cognitive science
Date: 2008-05-31 01:59 am (UTC)Not at all.
<< I no longer believe that logic is an insightful way to approach cognition or explanation. >>
and I agree. This is why I stopped working with the Amsterdam logicians (where I got my MSc).
It may well be the case that I read the book with detail-oriented view. I can be literal-minded sometimes.
Re: math + cognitive science
Date: 2008-05-31 02:01 am (UTC)Re: math + cognitive science
Date: 2008-05-31 02:01 am (UTC)Re: math + cognitive science
Date: 2008-05-31 02:07 am (UTC)Re: math + cognitive science
Date: 2008-05-31 02:12 am (UTC)Re: math + cognitive science
Date: 2008-05-31 02:36 am (UTC)As a graduate student, I remember discussing with a graph theorist a particularly cryptic proof of a theorem. He replied with a metaphorical analysis of my difficulty in understanding the proof. He said that making a proof is like making a building. You need a lot of scaffolding during the construction. The scaffolding is intuitions, analogies, and metaphors. When the proof is complete, you take all the scaffolding down, and you only publish the final building. Then, when the other professional mathematicians read your new proof, the first thing they do is phone you up and ask you to explain; that is, to restore all the scaffolding that you removed from the final publication. His job, as a professor, was to give me that missing scaffolding. It seemed somewhat hypocritical to me. I asked him why journals don't allow mathematicians to include the scaffolding in the publication. I don't recall his answer; I'm sure it didn't satisfy me.
Re: math + cognitive science
Date: 2008-05-31 02:49 am (UTC)Re: math + cognitive science
Date: 2008-05-31 02:56 am (UTC)I sympathize. I think Zeilberger has a rant (or 10) about this.
I think journals do allow scaffolding, but the authors remove it to save space. Reviewers would probably appreciate the scaffolding too... but it might give them more power to reject too. As the cynical view goes: "the more obscure the content, the fewer reviewers dare complain".
<< There is a paper that shows how you can consistently and coherently allow division by zero >>
Is this about ComplexNumbers U {infty}? stereographic projection?
Btw, I think William is also a fan of Lakoff.
Re: math + cognitive science
Date: 2008-05-31 03:12 am (UTC)this Amazon review rings true to me
Date: 2008-05-31 05:07 pm (UTC)2.0 out of 5 stars
Good ideas, but self-aggrandising and full of blunders, November 23, 2001
By Idiosyncrat "casillas8" (California)
Lakoff and Nuñez have a very interesting general framework for approaching their topic: mathematics arises by the extension of innate human capacities (e.g. subitization) or basic universals of human experience (spatial and motor experience), and the means of extension is cognitive metaphors which preserve the basic inferential structure of the source domain. The first few chapters provide a plausible sounding, perhaps workable account of arithmetic, simple logic and set theory, but one that they should have developed in far more detail (e.g. their account of intersection in terms of container schemas is criminally underexplained).
The biggest problem is that they do not stop there. They proceed from the speculative but plausible ideas I just mentioned into butchering higher mathematics in ways that have been amply documented in reviews both here and elsewhere. (To add an example, they massacre the Compactness Theorem for logic, by absolutely failing to mention the fact that it only works for first-order logic. Their "account" of "a mathematician's understanding" of the Theorem is just wrong, since you can't really be said to understand it if you don't know why it fails for second-order and higher logic, much less if you don't know the fact in the first place).
More worrrying is their completely misinformed philosophical attack on mathematical realism. They manage the task of listing in their bibliography more than one fundamental collection on philosophy of math, yet completely ignoring their contents. They attack a "folk Platonism" that, no matter how popular it may be among actual mathematicians, is actually widely criticised in the philosophy of mathematics (the case of Brouwer, that somebody below mentions, is only one of many). They pass off as their own well-known arguments (e.g. a version of their argument against numbers as abstract objects, that we can't decide among the numerous candidates, was given by Benacerraf I believe in the late 60s, and it could well predate that). If they want to argue the philosophy of math they should read philosophers of math and engage them. Otherwise it is extremely hard to take them seriously.
In short, they have very interesting ideas, but the mass of technical vagueness and blunders, plus the big strawman that is their "philosophical" argument, suggests that they are more interested in passing off as intellectual revolutionaries among the pop-science book audience than in contributing to our understanding of the topic.
Re: this Amazon review rings true to me
Date: 2008-05-31 09:14 pm (UTC)Re: this Amazon review rings true to me
Date: 2008-06-01 06:59 am (UTC)Maybe what Lakoff should do is put a seal on his books that says: "hey, I'm being kind of a crackpot here, but it may inspire you!".
Re: this Amazon review rings true to me
Date: 2008-06-01 01:25 pm (UTC)I think your Lakoff seal sums things up nicely, although he might possibly take offense.
To be an inspiring crackpot is a noble ambition. ;-)