Are mathematical proofs incomparable with proofs in other disciplines?
There's an interesting dispute just started on Wikipedia concerning whether it is reasonable to see some commonality of meaning between the concept of proof in mathematics and in other areas such as rhetoric, law, philosophy, religion, science, etc. The dispute is at one or both of http://en.wikipedia.org/wiki/Talk:Proof_(informal)#Disambig_page (Editors keep changing the name of the article, which was Proof (truth) when I wrote it and others have replaced "truth" first by "logic" and then by "informal", neither of which are an improvement.) The origin of the article in dispute is as follows. Some months ago I went to Wikipedia to look up what it considered to be a proof and found only a dab (disambiguation) page listing ten articles that seemed to about proof as applied to propositions and about as many more to do with testing and quality control as in galley proof, proof spirit, etc. It seemed to me that the former kind were not so much different meanings of the notion of proof as the same meaning arising in different areas all depending on that meaning. So, still some months ago, I wrote an article on that common notion which began "A proof is sufficient evidence for the truth of a proposition," which as it happens is essentially the first entry in the definition at dictionary.com. The article enumerated the various notions of proof arising in different disciplines (all of which have their own Wikipedia articles with much more detail), and made a start on characterizing the scope of "evidence" (need not be verbal, and need not contain the asserted proposition) and "sufficient" (strict for formal proofs, less so elsewhere, to different degrees). The main dispute at the moment is Gandalf61's insistence that "Proof in mathematics is not based on 'sufficient evidence' - it is based on logical deductions from axioms. It is an entirely different concept from proof in rhetoric, law and philospohy." He backs this up with quotes from Krantz---"The unique feature that sets mathematics apart from other sciences, from philosophy, and indeed from all other forms of intellectual discourse, is the use of rigorous proof" and Bornat---"Mathematical truths, if they exist, aren't a matter of experience. Our only access to them is through reasoned argument." My position is that logical and mathematical proofs differ from proofs in other disciplines in the provenance of their evidence and the rigor of their arguments as parametrized by "sufficient." Whereas evidence in mathematics is drawn from the mathematical world, evidence in science is drawn from our experience of nature. And whereas formal logic sets the sufficiency bar very high, mathematics sets it lower and other disciplines lower still, at least according to the conventional wisdom. Whereas I find my position in complete accord with the quotes of both Krantz and Bornat when interpreted as in the preceding paragraph, Gandalf61 does not. My questions are 1. Is mathematical proof so different from say legal proof that the two notions should be listed on a disambiguation page as being unrelated meanings of the same word, or should they be treated as essentially the same notion modulo provenance of evidence and strictness of sufficiency, both falling under the definition "sufficient evidence of the truth of a proposition." 2. Gandalf61 evidently feels his sources, Krantz and Bornat, prove the notions are incomparable. Are there suitable sources for the opposite assertion, that they are comparable? 3. Someone with a very heavy hand has tagged practically every sentence with a "citation needed" tag. For those that genuinely do need a source, what would you recommend? Vaughan Pratt PS. I hope this sort of argument doesn't put anyone off volunteering to help out on Wikipedia. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Wed, 7 Jul 2010, Vaughan Pratt wrote:
My questions are
1. Is mathematical proof so different from say legal proof that the two notions should be listed on a disambiguation page as being unrelated meanings of the same word, or should they be treated as essentially the same notion modulo provenance of evidence and strictness of sufficiency, both falling under the definition "sufficient evidence of the truth of a proposition."
Let me begin my answer with an aphorism. I don't know who said it first, but I heard it from Charles Wells. In principle, there is no difference between principle and practice, but in practice... If mathematical proof were simply logical deductions, there would never be mistaken proofs published. On the other hand, if proofs weren't logical deductions, we could never find errors in proofs, only in their consequences. Nonetheless, the biggest difference between mathematical proof and, say, legal proof, is that the latter depends on real world evidence. Legal terms do not have definitions that can be understood without reference to the real world. we believe that, in principle, every proof we publish is a surrogate for a formal logical deduction, but I once tried that for a simple argument and gave up after I had filled a couple pages with indecipherable chicken scratchings (cf. Russell & Whitehead).
2. Gandalf61 evidently feels his sources, Krantz and Bornat, prove the notions are incomparable. Are there suitable sources for the opposite assertion, that they are comparable?
I agree, however, that they are incomparable. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Vaughan writes: 1. Is mathematical proof so different from say legal proof that the two
notions should be listed on a disambiguation page as being unrelated meanings of the same word, or should they be treated as essentially the same notion modulo provenance of evidence and strictness of sufficiency, both falling under the definition "sufficient evidence of the truth of a proposition."
Geoffrey Lloyd's book "Adversaries and Authorities: Investigations into Ancient Greek and Chinese Science" has a discussion of how the Greek legal system influenced Greek concepts of rationality. While I haven't read it in a long while, I think it makes a good case that the concepts of "proof" in Western law, medicine, science, and mathematics are deeply intertwined. It's very useful to compare Greece with China on these issues. You could get a taste of it here: http://books.google.com/books?id=3820gVEQu1AC&printsec=frontcover&dq=adversaries+and+authorities&source=bl&ots=Z1qCx04ANb&sig=MROclDNl8-wjDKFi77o4z8Cs1_s&hl=en&ei=pz03TJLUHobEsAPglo1S&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBcQ6AEwAA#v=onepage&q=proof&f=false but the whole book is fascinating. It would also make a good citation - and it contains many further references. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Vaughan, I agree with your definition: "A proof is sufficient evidence for the truth of a proposition," The article http://en.wikipedia.org/wiki/Proof does not discuss the idea (of Paul Lorenzen) that a mathematical proof is essentially a winning strategy in a formal game. I first learned the idea from Andreas Blass who introduced the game semantic of linear logic, http://arxiv.org/abs/math/9310211 A proof can be viewed as an argumentation to convince others of the validity of a statement. In mathematics, the argumentation must be solid enough to resist any conter-argumentation by an ideal opponent. It can be compared to a winning strategy in a game with two players, one defending a statement and the other attacking it. Lorenzen associates to every mathematical statement S a formal game with two players G(S), the defender and the attacker. The defender has a winning strategy iff the statement has a formal proof. The rules of the games for a proof in intuitinistic logic differ from the rules for a proof in classical logic. In other words, the rules of the games are determining the logic and vice versa. I believe that game semantic is putting some light on the origin of logic. I guess that logic was discovered by peoples debating in a democratic manner. All communities need to choose between different courses of actions. There are many answers to the question: how should this choice made? One was given by Plato who favored a government by the "philosopher king" who "loves the sight of truth": http://en.wikipedia.org/wiki/Plato#The_State Plato does not like Athenian democracy because it is imperfect. He observes that its political debates are manipulated by sophists. I agree with Plato that democracy is imperfect. But it should be improved, not condemned. Logic is anti-authoritarian since it wishes to convince, not to coerce. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 7/9/2010 12:55 PM, Joyal, André wrote:
I agree with your definition:
"A proof is sufficient evidence for the truth of a proposition,"
Meanwhile the existence of some who prefer adding "or argument" to "evidence" seems to have been established. Maybe one day they'll surrrender, but this is not my highest priority. Higher is to make the following title changes on Wikipedia: Proof (truth) ---> Proof Proof ---> Proof (disambiguation) The first is the article on proof, the second is the Wikipedia disambiguation or dab page that takes you to other meanings such as "alcoholic proof" (an obsolete term even in the US, having been replaced by "alcohol by volume" or ABV) and proofreading. The suggested change would make the article on proof the primary topic (a Wikipedia concept) having a so-called hatnote (note at the head of the article) pointing to a disambiguation page for the lesser meanings. So far no but myself and one or two people have spoken up for this; until they do nothing will change.
The article
http://en.wikipedia.org/wiki/Proof
does not discuss the idea (of Paul Lorenzen) that a mathematical proof is essentially a winning strategy in a formal game. I first learned the idea from Andreas Blass who introduced the game semantic of linear logic,
http://arxiv.org/abs/math/9310211
A proof can be viewed as an argumentation to convince others of the validity of a statement. In mathematics, the argumentation must be solid enough to resist any conter-argumentation by an ideal opponent.
I would divide proofs into two kinds, those where the intended audience can argue back, as in a courtroom or ordinary conversation, and those where they can't, for example the subscribers to a journal. What you describe does cover both, but for the latter the game is very short. In computation this distinction is that between alternating computation and nondeterministic computation. Alternation computation is an on-going game, in non-deterministic computation one player makes one choice and the game ends. Interestingly the notion of nondeterministic computation preceded that of alternating computation under that name by some 15 years or so. But Fraïssé's alternation preceded both by a decade, although it took a decade for Ehrenfeucht to cast Fraïssé's approach as a game (1961). Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
My suggestion is that mathematical proofs still require a mathematical landscape in which to find our way. Michael Barr points out the problems of finding a completely `logical' proof. I propose the analogy of giving directions to the station, such as `go out of the house, turn right, go straight on until ..., etc., etc.' We don't need to specify all the cracks in the pavement, but we may need to warn of holes due to roadworks! The aura of certainty in a mathematical proof is partly because the `conceptual landscape' has been worked up over centuries, in terms of convenience and usability, and tested by thousands. Hopefully, arguments come to be produced which seem inevitable, indeed aesthetic, rather than ad hoc. Are there computer theorem provers which can work at a `landscape level'? The assumption is also that there are no cracks in our mathematical universe. On the more general point, I did hear of a University Vice Chancellor who asked his staff for a series of lectures on `The notion of validity in my subject'. Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
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John Baez -
Joyal, André -
Michael Barr -
Ronnie Brown -
Vaughan Pratt