Fred & all,

My goodness! I'd turn that question around: is there any proof (apart from an "indirect" proof, or "proof by contradiction") that one would *not* "consider as being explanatory in this sense?"

Speaking as a novice: yes, certainly. Isn't it a question of degree? Some proofs explain beautifully while others are clear as mud; most are between. Ideally a proof shouldn't depend upon natural language but most do. Striking sometimes how changing a few words of a sentence can make a concept obvious rather than nebulous. For example, I've proven some of the power laws for map objects. There should be a way to reduce the definition of a map object and the power laws to analogues in arithmetic. Still eludes me. My proofs have yet to help. So my understanding is incomplete and my power law proofs are poor. Best regards, ... Peter E. -- Telephone 1 360 450 2132. bcc: peasthope at shaw.ca Shop pages http://carnot.yi.org/ accessible as long as the old drives survive. Personal pages http://members.shaw.ca/peasthope/ . [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

My private reply to the original query from Jean-Pierre Marquis pointed to the style of combinatorial proof you refer to: they are called "bijective" or "combinatorial" proofs depending on the author, and rely on giving interpretations of "big ugly formula_1" and "big ugly formula_2" as enumerating the same thing by different means. For instance on can prove that n*2^{n-1} = \sum_{k=1}^n k*C(n,k) (writing C(n,k) for the binomial coefficient "n chose k") by differentiating the binomial theorem and evaluating at 1, but this hardly seems to explain it. Better is to observe that both sides count the number of ways to select a subset with a distinguished element from an n element set, the LHS by selecting the distinguished element, then the rest of the subset, the RHS by choosing a cardinality k for the subset, selecting the subset then selecting the distinguished element from the subset. David Y. On 22 Apr 2011, at 08:55, Graham White wrote:

And the folklore is (I haven't checked this in a proper history book) that Gauss proved quadratic reciprocity numerous times because he didn't consider the proofs sufficiently explanatory. It's certainly true that modern proofs (i.e. those using the methods of algebraic number theory) generalise it, and thereby explain, for example, what it is about the rationals, and the number two, that makes primes in the rationals obey quadratic reciprocity. I think one conclusion here is that, if you say "explanatory", I am entitled to answer "so what do you want explained?"

Another point is this: there are lots of combinatorial identities of the form

big ugly formula_1 = big ugly formula_2

which can be proved directly (for example, by induction and a lot of algebra), but where the proof is utterly unilluminating. And in many cases there are more conceptual proofs which people generally find more illuminating (depending on taste, of course).

Graham

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a friend told me that there was a conference where music critics and professors discussed the visual content of music. on the other side, there are learned essays about the deep links between music and architecture. (hegel in particular wrote about that.) the question whether a mathematical proof explains the theorem might be of a similar kind. while most proofs of the pythagoras theorem do explain why it is true, wyles' proof of the great fermat theorem (just a slightly different statement) does not seem to be explaining it to too many people. some proofs yield an explanation, some explanations lead to a proof - but there is a sense in which the *attitudes* leading to one and to the other are *opposite*: while explanations tend to increase the number of words in the world, many people prove things so that we can stop talking about them. maybe the situation resembles the discussion between zeno and parmenides: "as parmenides argued that the movement does not exist in the universe, zeno stood up and walked around". -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I cannot resist adding my grain of salt. Maybe we should distinguish between a human proof and a mechanical proof. The goal of a human proof is to convince other peoples of the truthfulness of a proposition. It is by nature explanatory and it can lead to new insights. A mechanical proof can be checked by computer but may not produce new insights. It is establishing a fact. Of course, it is better than none. Mathematics is above all a human activity. The value of a proof depends very much on its method. A new proof may suggest a new method. A method is a kind of toolbox for proving a large class of propositions. Commutative algebra is a method in geometry Category theory is a method in mathematics. proofs --->methods ----> proofs ----> methods ....... André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Jim, You are perfectly right! I am always amazed by the fact that a computation can yield a surprising result. It is as if the formal system knew more than me! Actually, I find a computation boring when the result is not surprising. Computing is probably the main vehicule by which we can move beyond a given body of intuitive knowledges. But after the initial surprise, we try hard to integrate the new result in a larger body, where it may become less surprising. It may even become obvious! The chain intuition--->computation---->intuition--->computation..... is probably more important than the chain proof--->method---->proof--->method..... André -------- Message d'origine-------- De: jim stasheff [mailto:jds@math.upenn.edu] Date: lun. 25/04/2011 20:52 À: Joyal, André Cc: Ronnie Brown; Graham White; categories@mta.ca Objet : Re: categories: Re: Explanations On 4/25/11 9:51 AM, Joyal, André wrote:

The goal of a human proof is to convince other peoples of the truthfulness of a proposition. It is by nature explanatory and it can lead to new insights. I disagree mildly.

convincing other people of the truthfulness of a proposition can involve staightforward computation which I don't find explanatory jim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Steve says: `Since mathematics is a formal system ...' is it? I gave a talk some years ago at the national college of the UK women's institute and the title I used was: Mathematics, a human activity. My point was that mathematics is done by mathematicians (amongst others). Until we find to the contrary, mathematicians are more often than not human (in the widest sense of the word!!!!!). The form and direction of mathematical investigation is determined by curiosity, and similar human emotions., (sometimes also by rivalry, hatred, envy , and other ones of less beauty). A (subjective view) good proof convinces the `reader' that the statement is true. The 'explanation' behind a proof by contradiction explains somewhere along the lines: the result is trapped, it cannot get away, therefore we have it. That is a human judgement and is sometimes accompanied by the sentiment of `but that argument leaves me dissatisfied as I do not see why'. (The level of belief in the use of contradiction is sometimes an issue but not always.) `Explanation' can be modelled by a worldview approach, but then you have the problem of the teaching situation where the teacher gives an explanation of some mathematical result, but has to say that the proof has to take a different route. In category theory, many proofs are transparent and of the form: what do we know about the situation, just one fact, so we have to use that.... it works. (I am thinking of classical Yoneda lemma type situations, since the only elements in hom-sets that we can be sure exist are the identities.) A thorough understanding of the proof does give an explanation of why the result holds. (The problem I have with the original request for examples is that explanation requires understanding of the situation so is dependent on the knowledge of the `codomain'/ reader!) Tim On 25/04/2011 14:17, ClemsonSteve wrote:

Quoted from Jean-Pierre Marquis email: "yes, of course, Salmon is certainly one of the important contributors to the field [scientific explanation]. In mathematics, Paolo Mancosu has been pushing the issue for the past 10 years or so, following the paths of Steiner, Resnik and a few others."

In science, the issue of explanation has been discussed for at least a century. Since mathematics is a formal system and not a physical system, we have to be more careful about what *explanation* means. This makes it a "worldview" problems? As a constructionist/computationalist I would say no constructive, computational proof then there is no explanation. Platonist have their own. Is their explanation useful to me? Don't know because if I can figure a constructive technique out from the plationic technique, I'm good.

steve stevenson clemson

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On 26/04/2011 06:55, Timothy Porter wrote: In category theory, many proofs are transparent and of the form: what do we know about the situation, just one fact, so we have to use that.... it works. (I am thinking of classical Yoneda lemma type situations, since the only elements in hom-sets that we can be sure exist are the identities.) [...]

The question is: "What facts and what objects do we have?", "What is given?". In general we need to work with what is directly given in order to arrive at some indirectly given thing that we are seeking (what we are seeking has to be given in some sense, or else the problem cannot be solved). In category-theoretic terms we can think of what is given as a subobject A of some larger object B in an allegory (where B represents things that "exist" but which we may or may not be able to refer to). A subobject of B is then given just in case it factors through A. (For this to work well the allegory we are working with should contain lots of different objects so that "subobject of X" and "part of X" become practically synonymous.) We may also view things in terms of symmetry and invariants. We are directly given certain subobjects A1, A2, ..., An of some object B in an allegory and we are indirectly given any subobject of B which stays invariant as we apply isomorphisms to B that fix A1, A2, ..., An. (The earlier object A would be the smallest subobject of B containing A1, A2, ..., An as parts.) Finally, we can view what we are given as a theory (axiom system), and the question is what can be defined/specified/referred to in that theory. Of course, mathematics inevitably involves axiom systems, but any theorem which starts "for all ..." can be thought of as involving an axiom system of its own. Mattias [For admin and other information see: http://www.mta.ca/~cat-dist/ ]