Computer Proofs Give A Priori Knowledge

30 06 2008

I just read a very interesting paper by Tyler Burge on computer proofs. (“Computer Proofs, Apriori Knowledge, and Other Minds”, Phil. Perspectives 12, 1998 ) I suspect that many mathematicians will find the paper quite interesting, as well as philosophers. His major claim is that, contrary to most assumptions, computer proofs can in fact give mathematicians a priori knowledge of their theorems.

An important question here is just what it means for knowledge to be a priori. Burge defines the notion as just stating that the knowledge doesn’t depend for its justification on any sensory experience – however, he allows that a priori knowledge may depend for its possibility on sensory experience. This account allows for the knowledge that red is a color to be a priori, even though having this knowledge requires having sensory experience of red in order to have the concepts required to even formulate the idea. Burge’s main goal here is to defend a sort of rationalism, which is the claim that a lot of a priori knowledge is possible, so it’s somewhat important that his account of the a priori is broader than one might initially think. One might worry that this waters down the notion of the a priori so much as to make this form of rationalism uninteresting, but I think it still gets at some interesting distinctions. For instance, his account will end up saying that almost all mathematical knowledge is a priori (or at least, can be) while very little knowledge in the other sciences can be. This may be very interesting for those wanting to claim a principled difference between mathematics and the other sciences. (For those of you familiar with the talk I’ve been giving recently about probabilistic proofs, I suspect that in addition to a priority, the notion I call “transferability” gives mathematics a lot of its specific character, but that’s a different issue.)

The biggest premise that Burge asks us to grant is that most ordinary mathematical knowledge that doesn’t rely on computer proofs is itself a priori. In order to grant this, we have to grant that testimony can preserve a priority, since very many mathematical proofs depend on theorems or lemmas that the author has never worked through, but believes just on the basis of testimony (having skimmed a proof, or even just read a result published in another paper). For an extreme instance, consider the classification of finite simple groups – no one has worked through all the steps in the proof, yet it seems at least plausible that our knowledge of the result is a priori in some interesting sense. The sense that Burge suggests this is is that although a mathematician may depend on testimony for her knowledge of the theorem, the actual steps in the justification of the result were a priori for whoever carried them out.

Burge needs to do a lot of work to suggest that this transfer of knowledge through testimony can preserve a priority – he ends up showing that we can often get a priori justification for the claim that there are other minds by the same means. He suggests that the very fact that some part of the world appears to express a propositional content gives us some defeasible a priori reason to believe the content of that claim, and also to believe that some mind somewhere is responsible for that content, even if at some remove. (That is, although books and computers can display things expressing propositional contents despite lacking minds themselves, the only reason they normally succeed in doing this is because some mind somewhere gave them this capacity, either by directly putting the symbols on them, or causing them to shuffle symbols in intelligible ways. Although you often get significant and meaningful patterns in nature, it’s exceedingly rare that something natural appears to express a specific proposition. See Grice for more on this.)

Once we’ve got this idea that testimony can preserve a priority, it becomes more plausible to think that computer proofs can generate a priori knowledge. However, he still has to go through a lot of work to argue that we have an undefeated warrant for believing the claims of the computer. Clearly, in many cases where someone utters a sentence, we have strong reason to doubt the truth of that sentence, unless we have positive reason to believe that the person is a very skilled mathematician. In the case of something like Fermat’s Last Theorem, it seems that even that is insufficient (after all, even Fermat’s word on the theorem really doesn’t seem like sufficient grounds for knowledge). Burge needs to do some fancy footwork to argue that the means by which we build up our trust in a source of utterances can itself be a priori, since it only depends on success of apparent utterances, and not on the fact that the source of the utterances really is as it appears to be. (It doesn’t matter to me whether Andrew Wiles (whom Burge embarrassingly refers to as Michael Wiles!) is a real person, a computer, or a simulation in the Matrix – he’s done enough to prove that he’s a reliable source of knowledge of complicated mathematical statements of certain forms.)

I think in the end, Burge spends a lot of time focusing on the wrong sorts of support for the reliability of a person or computer who claims to have proved something difficult. He mostly focuses on the fact that this source has gotten many other difficult statements correct before. I think in actual mathematical practice the more important criterion is that the outline of the strategy used looks much more promising than previous attempts at the problem, and the source has given good indication of being able to carry out particular sub-parts of the proof. Burge does deal with this justification, but in not as central a way as might be desired.

So I think Burge has come up with some interesting arguments that computer proof still preserves the fact that most mathematical knowledge is a priori, even though I think he makes some mathematical errors in focus and about particular claims of mathematical history. I think his defense of computer proof also still allows for the fact that other mathematical arguments (like DNA computing, for instance) really don’t preserve a priority. After all, in these cases, the computer isn’t going through something like a reasoning process, but rather is doing something more like an observation of an experiment. The way that most ordinary computers process data still shares abstract features of reasoning that are important for a notion of the a priori, in ways that DNA computing don’t. (If we didn’t grant this, then it might seem that there’s no such thing as any a priori knowledge, since we always in some sense rely on the physical reliability of our own brains – he gives some good arguments for why we should dismiss this worry.)

I think this sort of epistemology of mathematics is probably of much more practical interest for mathematicians than the standard questions of ontology and logic that more traditional philosophy of mathematics deals with.




7 responses

1 07 2008
Carrie Jenkins

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.

1 07 2008

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.

1 07 2008
Carrie Jenkins

“… 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.

1 07 2008

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.

7 07 2008
Chris Pincock

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?

1 08 2008
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2 12 2008

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–

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