Comments on: Computer Proofs Give A Priori Knowledge

By: 01010001

01010001 — Tue, 02 Dec 2008 18:22:15 +0000

The Burgean sort of justifications of a priori knowledge seem to follow in the Frege/Russellian tradition (if not platonic tradition), what one might term the “Pi doesn’t appear in nature” argument: the argument being essentially, empirical science cannot for logical/mathematical knowledge (at least at present); ergo, logical/mathematical knowledge is a priori, transcendent, universal, etc.

The constant “Pi” could be any sign, but Pi does of course denote something objective: the relationship of the ratio of the circumference of circle to its diameter. Is that relationship a priori as well? If it is, it’s not clear that’s what most logicians mean by a priori. It’s objective, and the relationship may hold independently of our minds–but the relationship is really still a part of nature. There may be few perfect circles in nature, but find a circle, measure the circumference of circle, and the diameter, and one would find something close to 3.1416. (assuming western numbering): the pythagoreans or whoever who established Pi did not merely tune into their immortal soul or “synthetic a priori”– It’s a relation which is perceivable at least.

That’s not to say it’s empirical ala Mill (tho Mill not as naive as some think)–but a certain rational constructivst view of mathematical knowledge seems rather feasible, instead of the usual vague platonic (and Kantian) reliance on the a priori. Humans may have the a priori skills to perceive and construct Pi (from experience of some sort), but the relationship itself is not mentally a priori …………

Given more time cognitivists will most likely map out the specific neural pathways of supposed “a priori” mathematical knowledge (they have already begun to do so): Pi will probably be shown to be cortically located on a sort of neurological hard drive–

By: Bookmarks about Fermat

Bookmarks about Fermat — Sat, 02 Aug 2008 04:16:35 +0000

[...] – bookmarked by 2 members originally found by glusberg on July 16, 2008 Computer Proofs Give A Priori Knowledge http://antimeta.wordpress.com/2008/06/30/computer-proofs-give-a-priori-knowledge/ – bookmarked by [...]

By: Chris Pincock

Chris Pincock — Mon, 07 Jul 2008 13:22:15 +0000

A crucial element of Burge's view is his notion of entitlement -- note how he defines "warrant" as "either a justification or an entitlement" (3). Very roughly, an entitlement to a belief is something an agent has even if they don't form the belief and even if they aren't aware that they have a reason to form the belief. So, being entitled to a belief is more like being in a position to justify and know something, rather than actually knowing it. (This fits with Carrie Jenkins' point.) The second big move at work in Burge is the claim that testimony and memory can figure in a priori entitlements because they preserve content, as opposed to being the source of new content. (Hence "Content Preservation".) I have never understood his argument here, and for testimony, at least, we now have the extended debates on the epistemology of testimony to show that Burge's view is not the only one. Still, I side with Kenny in finding Burge's view, especially as developed by Peacocke, very promising for mathematics. Whether it works, even for testimony, is another matter. Perhaps there is more than one way to ground a priori knowledge?

By: Kenny

Kenny — Tue, 01 Jul 2008 13:57:02 +0000

That sounds very plausible. I don’t have a strong commitment to any view of the a priori, but I definitely agree that the notion Burge is discussing is much broader than most notions of the a priori would be. However, for the purposes of the discussion of computer proof, it might be perfectly fine if he ends up talking about apriori* rather than apriori. He just needs to be able to argue that whatever status most ordinary mathematical knowledge has, knowledge by computer proof very often has the same status. I don’t think that point depends on whether that status is really a priority or not. Though I admit that this further question is the more interesting one to people working on the a priori, even if not to the mathematicians.

By: Carrie Jenkins

Carrie Jenkins — Tue, 01 Jul 2008 13:44:25 +0000

“… if we don’t grant this premise, then we have to say that most mathematics is not in fact known a priori.”

But that consequent sounds fine! It only sounds dodgy if you mistake it for “we have to say that most mathematics is not in fact knowable a priori”. I can’t see any other reason to worry about it, nor I suspect would most people working on the a priori be worried by it.

By: Kenny

Kenny — Tue, 01 Jul 2008 09:02:22 +0000

I’m not sure if that’s true. His big un-argued-for premise is that actual mathematical theorems that depend essentially on testimony often are known a priori, and not just knowable a priori. He does defend this point to a large extent by suggesting that we have a priori warrant to believe propositional contents that we seem to perceive. But I think the main force of the argument is supposed to be from the fact that if we don’t grant this premise, then we have to say that most mathematics is not in fact known a priori. He then suggests that we can do the same thing with computer proofs so that their results are known a priori. From the fact that each of these things is known a priori, we can also conclude that they’re knowable a priori, but that’s not his direct concern. So I’m not sure where the confusion is between known a priori and knowable a priori – I might have introduced it at some point in my hasty summary of his argument. Or do you think that the confusion still remains, and is important here? I don’t really know – I don’t have a particular axe to grind in this debate, though I suspect that you might.

By: Carrie Jenkins

Carrie Jenkins — Tue, 01 Jul 2008 08:37:04 +0000

Sounds to me like Burge is confusing what’s knowable a priori with what’s known a priori, especially in the argument that testimony preserves a prioricity of justification.