In mathematics education, "proof" is widely used for many kinds of argumentation. For instance, one example could be called a proof, if it is paradigmatic in the following sense: The argument can immediately be transferred to the general case. (To be more concrete, assume you want to prove the sum of two even numbers is even again. Say we have $4+6$, then we see $2\cdot(2) + 2\cdot(3) = 2\cdot(2+3)$, not really depending on the numbers chosen.) The other extreme is given by formal proofs, where every(!) step needs an argument. This is sometimes practiced in courses on logic and it takes half a page to see $2+1=1+2$.

Most of the proofs in professional mathematics are far away from both poles; however, I think they will be closer to the formal pole.

Question: Do you know any literature (book, paper, website, blog, ...) which tries to define what a proof is for the working mathematician or students at (under)graduate level? I would like to restrict the answers to the literature and not personal opinions of which I have heard a lot. :-)

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    $\begingroup$ an unintentionally deep question taking into account automated theorem proving see eg adventures/commotions in automated thm proving $\endgroup$
    – vzn
    Mar 29 '14 at 20:33
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    $\begingroup$ I think Princeton Companion to Mathematics will be a valuable read in this context. $\endgroup$
    – user3319
    Sep 4 '15 at 3:16
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    $\begingroup$ I have seen at least one text take an operational view and define "proof" to mean "something that mathematicians would call a proof". $\endgroup$
    – user797
    Sep 7 '15 at 11:33

This collection of essays:

Reuben Hersh: Experiencing Mathematics: What do we do, when we do mathematics?, Amer. Math. Soc., 2014

contains, among other topics, also lots of excellent discussion of the problems you mention. I highly recommend it, some of the articles are even available online if you google for them.

  • $\begingroup$ Great collection of essays... $\endgroup$
    – Jon Bannon
    Nov 1 '15 at 21:15

From Bill Thurston:

When I started as a graduate student at Berkeley, I had trouble imagining how I could “prove” a new and interesting mathematical theorem. I didn’t really understand what a “proof” was. By going to seminars, reading papers, and talking to other graduate students, I gradually began to catch on.

Within any field, there are certain theorems and certain techniques that are generally known and generally accepted. When you write a paper, you refer to these without proof. You look at other papers in the field, and you see what facts they quote without proof, and what they cite in their bibliography. You learn from other people some idea of the proofs. Then you’re free to quote the same theorem and cite the same citations. You don’t necessarily have to read the full papers or books that are in your bibliography. Many of the things that are generally known are things for which there may be no known written source. As long as people in the field are comfortable that the idea works, it doesn’t need to have a formal written source.

At first I was highly suspicious of this process. I would doubt whether a certain idea was really established. But I found that I could ask people, and they could produce explanations and proofs, or else refer me to other people or to written sources that would give explanations and proofs. There were published theorems that were generally known to be false, or where the proofs were generally known to be incomplete. Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic. This knowledge was supported by written documents, but the written documents were not really primary.

I think this pattern varies quite a bit from field to field. I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think. In more algebraic or symbolic fields, this is not necessarily so, and I have the impression that in some areas documents are much closer to carrying the life of the field. But in any field, there is a strong social standard of validity and truth. Andrew Wiles’s proof of Fermat’s Last Theorem is a good illustration of this, in a field which is very algebraic. The experts quickly came to believe that his proof was basically correct on the basis of high-level ideas, long before details could be checked. This proof will receive a great deal of scrutiny and checking compared to most mathematical proofs; but no matter how the process of verification plays out, it helps illustrate how mathematics evolves by rather organic psychological and social processes.

So essentially, his view is that what a proof is is essentially based off of societal norms backed up by paper documents when necessary.

In a classroom, I feel this translates to 'a proof is an argument which convinces the grader that you have either written all the details or are able to provide them.

Formatting of the quote may be off.

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    $\begingroup$ See also New Directions in the Philosophy of Mathematics, ed. T. Tymoczko, which contains this piece on the question and others. $\endgroup$
    – user173
    Aug 30 '15 at 16:35

I'd go first for the literature on mathematical writing, e.g. Knuth, Larrabee and Roberts' "Mathematical writing" (MAA, 1989). Mathematics writing is not just "proofs," in the end, you want to convince your reader that what your reasoning is correct (and not bore them to tears in the process), and (hopefully) convince them that it is relevant. It very much depends on the audience, for starting undergraduates the proof may need to be much more detailed and explained than the brief sketch when talking to a coleague, and that one probably won't ever be accepted by a journal either.

For another view, refer to the journals devoted to completely formal proofs...


For a mathematics students: How to Read and Do Proofs, by D. Solow.

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    $\begingroup$ I know that book is a traditional favorite, and many of my colleagues endorse it, but somehow it seems to me to set the wrong tone. "Could be worse", yes, but I think it (perhaps inadvertently) falls too much into sympathy with mechanist/formalist philosophical approaches, in which "proof" is a different thing from "persuasion" or "explication of intuition", and so on. Maybe depending on the state of the reader, this could push their "gradient" in a good direction, but I perceive grad students as generally being too formal, not insufficiently so... $\endgroup$ Aug 30 '15 at 21:59

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