David Henderson's short paper, Proof as a Convincing Argument That Answers -- Why? addresses two key issues. The first concerns what constitutes a "proof", and in particular, what makes a proof convincing. The second issue concerns how human beings communicate with each other about mathematics.

These are related by the opening quote by the Dalai Lama: it is our deepest personal meanings, coming from our deepest personal experiences, which determine whether a proof is satisfying to us; these inner experiences and meaning, in turn, can be a source of differing mathematical viewpoints between people, and conversely, a source of a rich diversity of mathematical viewpoints, to the extent we learn to communicate, and most of all, listen across our different experiences.

In a way, Henderson is radically defining the notion of "proof" -- "Proof as a Convincing Argument That Answers -- Why?" takes proof beyond "Showing that a proposition P is true" to "Showing us convincingly why P is true".

To the extent that Henderson does not mean to suggest that the former, more limited type of proof is without any value, I agree with his emphasis on the latter, deeper definition of proof. Indeed we can draw some conclusions which Henderson does not (explicitly) draw in this paper.

He writes that: "...you have probably had the experience of reading a proof and following each step logically but still not being satisfied because the proof did not lead you to experience the answer to your why-question" and goes on to say "In fact most proofs in the literature are not written out in such a [logical step-by-step] way ... [and] even if they were so written, most proofs would be too long and complicated.. for a person to check each step".

But another statement about "most proofs in the literature" is omitted, despite having just spoken about why-questions: Most proofs in the literature do not answer it, and do not, in general, even try to answer the why-question, and furthermore, there is a cultural attitude in mathematics that that would not be the path of greatest "efficiency", or would even be a waste of time.

Why prove 10 theorems with very clear, satisfying explanations, background, and motivation, with a full, clear exposition, when your career would be hurt by this, relative your having proved 20 (or even 11) theorems in the narrow, unsatisfying sense?

One cannot make a sensible (or even rational) attempt to analyze the state of the mathematical culture without looking at the cost/benefit, or perhaps better, punishment/reward system under which most mathematicians live -- Thurson's "Theorems-credit"

If the system credits you with $1 for each theorem proved, and $1 additionally for each theorem (your own or someone else's) made very clear, expanded, answering why-questions, etc, we would see a very different mathematical literature. A system which gives a $1 credit for the former, and anywhere from a nickel, to a negative amount for the latter, on the other hand, would produce results very similar to what we see in today's literature.

Such a cost/benefit analysis would explain the (otherwise) "irrational" allocation of human hours: what could have taken the author of a proof an additional, say, 3 hours to explain more clearly, is saved (savings = 3), while the authors (say) N readers must therefore struggle, each, an extra 1 (or 3 or 10) hours more, than they would otherwise have had to do, to decipher the proof -- at a total human cost of N readers (say 100) times, say 2 hours extra each, of 200-3=197 more human-hours wasted as a result.

Under today's mathematical political climate, I would be inviting upon myself explanations such as "well, you're just dumb" were I not to cite a Respectable mathematician, so I will: it is while observing my advisor, who has been called by at least one source the strongest topologist at Cornell, that I noticed that I am not the only one who find themselves struggling for way longer than it would have otherwise been necessary, trying to decipher a proof which, it becomes clearer and clearer over time, would have taken far, far less time to decipher were a better job done in exposition, motivation, and other efforts towards conveying an intimate understanding expended, the kind of expenditure that the current cost/benefit reward system in our mathematical culture does not promote, and in fact the opposite.

At the same time, I feel that neither Henderson nor we, should, by omission, unintentionally suggest that there is no interest whatsoever in why the commutative law follows from Peano's axioms. Such additions to that proof as, an exposition of the motivations for mathematician's seeking, in the first place, such an axiom system, would go a long way towards answering important why-questions, leaving it for each reader to decide how much of the "dry" part of the proof they need, at present, to read and understand, given their present mathematical interests and motivations for reading the paper in question.

These comments about Henderson's paper would not be complete without noting that his "multiculturalism" argument is a powerful one. It also implies that women need to listen to men more closely, but I don't think that makes it less feminist. It also implies that blacks need to listen to whites more closely, but I don't think that makes it less powerfully a statement against a "prejudicial ear" (mathematical or even otherwise) towards ethnic minorities. It does, I think present a strong case, based on facts, experience, and well-argued reasoning, which does not replace, but supplements well the other important arguments -- based on non-racist non-sexist assumptions about human beings, basic decency, and so forth -- for a "multicultural mathematical culture"

Just as Henderson's paper started with a quote, we would like to conclude by quoting another important figure, on what might today be called a multicultural theme:

"It seems to be a universal fact that minorities -- especially when the individuals composing them are distinguished by physical peculiarities -- are treated by the majorities among whom the live as an inferior order of being.

"The tragedy of such a fate lies not merely in the unfair treatment to which these minorities are automatically subjected in social and economic matters, but also in the fact that under the suggestive influence of the majorities most of the victims themselves succumb to the same prejudice and regard their brethren as inferior beings.

"This second and greater part of the evil can be overcome by closer combination and by deliberate education of the minority, whose spiritual liberation can thus be accomplished. The deliberate efforts of the American negroes in this direction deserve approval and assistance."

--Albert Einstein (From: The World As I See It, 1934.)

(Emphasis added --HB).