2
$\begingroup$

Update: I edited the question as I saw it was closed. Let's see if with some improvements it can be considered worth reopening... (I already accepted an answer, but I'd like to see something more specific, if possible).


Premise: I think that a reasonable definition of what a "correct proof" is goes like: "one which is able to convince most mathematicians in the field that a formal proof can be written down, if one really wants". I don't remember where I read this first, but it seems sensible to me.

Now, suppose that, aimed at proving the true statement $A$, you write something like:

Since $B$ is true, then it clearly follows $C$, so that, as $D$ is always impossible in our assumptions, $A$ follows.

Suppose also that every single statement in this proof is correct, in the sense that $B$ and $C$ are true and $D$ is false in the proposed assumptions. Clearly, in a reasonable mathematical paper, the reader would be able to follow the reasoning by the author, so that the connection between $B,C,D$ and $A$ would be more or less clear. But in general, especially if the author is in a hurry (or if he's V.I. Arnold...), that connection can be highly non-obvious. It's also safe to assume that you cannot replace meaningfully $B,C,D$ by randomly chosen true/true/false statements in the previous proof. For instance, you can't write something like:

Since $57$ is not prime, then it clearly follows that every irrational rotation is ergodic, so that, as no locally compact Hausdorff space can be meagre, Gauss's Theorema Egregium follows

...and be taken seriously.

However, unless a formal proof is written down, there is always some degree of mathematical understanding which is left to the reader and it can be negligible or relevant, depending on the style of the writer. Take the following sentence:

Since $|M|>10^{34}$, then $|M|> 10^{50}$, so that...

Without context, this is simply nonsense. Even if you know that $M$ is a sporadic group, if you're in the 1950s the average reader would most probably frown upon this. However, today this makes sense. So it seems to me that the sociological aspects of the question about what a correct/wrong proof is cannot be easily walked around.

My question: has any well-known mathematician addressed the problem of how to meaningfully talk about "wrong proofs"? In particular, in cases in which the thesis is true (and provable) under the given assumptions, and all the statements in the proof are true as well?

$\endgroup$
18
  • 2
    $\begingroup$ @WilliamOliver have you read the question or just the title? $\endgroup$ Mar 28, 2023 at 14:53
  • 2
    $\begingroup$ I think the question hinges on the question of "difficulty", which is fundamentally a psychological/human question, and not one we have great answers about so far. Namely, a written proof is supposed to contain all the "hard" steps, with only "easy"/"routine" steps left to fill in (and the more detailed the reconstruction, the more only purely routine connectives are left missing) $\endgroup$ Mar 28, 2023 at 15:53
  • 5
    $\begingroup$ A valid proof is not just a sequence of statements. It is a sequence of statements connected by logical operations. For instance, I would not accept an argument of the form "Since $B$ is true then it clearly follows $C$" unless the implication "$B \implies C$" were clearly understood in the context. Even then I might, depending on the setting, insist on that implication being cited. $\endgroup$
    – Lee Mosher
    Mar 28, 2023 at 16:00
  • 2
    $\begingroup$ The definition I have heard is that an (attempted) proof is the sort of thing where, if it is flawed, then it is possible for a careful reader to find a specific piece of it that fails to convince them. (In a correct but insufficiently detailed proof, the reader should instead be able to point out a portion which they feel needs elaboration.) This definition appeals to me because it feels like the informal cousin of the complexity class NP. $\endgroup$
    – zeb
    Mar 28, 2023 at 22:51
  • 3
    $\begingroup$ Not so much in sociology of math, but in math education there is a lot of current research into the "social acceptability" of a proof. Here are just few random samples: * link.springer.com/article/10.1007/s11858-019-01039-7 * scholarship.claremont.edu/jhm/vol1/iss1/4 $\endgroup$ Mar 29, 2023 at 12:39

3 Answers 3

3
$\begingroup$

You might be interested in the paper What do mathematicians mean by proof? A comparative judgement study of students’ and mathematicians’ views by Davies, Jones and Alcock. (More generally, I consider the work of Lara Alcock to be interesting and insightful about a range of issues in university-level mathematical pedagogy.)

$\endgroup$
3
  • $\begingroup$ Thanks. It seems very interesting at first glance! $\endgroup$ Mar 29, 2023 at 18:56
  • $\begingroup$ @AlessandroDellaCorte I only skimmed the paper quickly, but it doesn't seem to address the "operational" part of your question, does it? Or maybe I don't understand what you mean by "operational"? $\endgroup$ Mar 29, 2023 at 20:06
  • $\begingroup$ @TimothyChow No, I think you're right, but it's interesting nonetheless. $\endgroup$ Mar 29, 2023 at 20:07
4
$\begingroup$

It sounds like you're asking for accounts of mathematical proof (other than the "traditional" definition of a formal proof) which say more than "proof is a social construct."

The closest thing I'm aware of is Andrew Aberdein's article, The informal logic of mathematical proof (in 18 Unconventional Essays on the Nature of Mathematics, which contains some other essays that may interest you). In particular, the concept of informal logic, which is distinct from formal logic, but which nevertheless has a certain non-arbitrary structure, might be close to what you're looking for.

There are some better-known authors who have rejected the notion of formal proof; Imre Lakatos (Proofs and Refutations) is perhaps the most famous, but Reuben Hersh (Experiencing Mathematics: What do we do, when we do mathematics?) has also said that "mathematics is ours, our tool and plaything, to use and enjoy as we see fit." Though neither Lakatos nor Hersh would say that "anything goes," I suspect that they are too "subjective" for you.

$\endgroup$
1
$\begingroup$

I don't quite understand what you want to ask. There is the notion of a formal proof (roughly speaking, a text with certain properties in some formal language). The existence or non-existence of such a text is a common natural science question. The decisive argument that a text can be constructed is a sketch of it, detailed enough so that all involved have great confidence in the existence of the text (say, $1 - 10^{-7}$). Of course, someone's single check does not give such a colossal degree of confidence for a complex text, it accumulates from all the checks carried out. Of course, the current average confidence level varies depending on how modern and complex the topic is. Usually the authors acquire a fairly high confidence (say, $0.8$) in the existence of the text long before this moment, from more shaky, but no less valuable arguments.

In this respect, mathematics is no different from any other field of knowledge: we collect evidence in favor of the correctness of certain statements about the real world. Only while most scientists are interested in objects of the world around us, we (to speak formally) are interested in texts in formal languages. Of course, this is only a formal aspect of what is happening, and languages are not chosen by chance. Our inspiration comes from physical phenomena, natural questions, philosophical/intellectual aesthetics. This is what most mathematicians really care about, as opposed to the formal notion of a proof :)

Perhaps the answer you want to hear: proofs that mathematicians write in their papers are people-oriented. Some readers may understand them, others may not. Someone writes more clearly and in detail, someone does not. It is the reviewer's job to maintain some standard of clarity in published works.

UPD: After editing the question (inserting "in the literature on sociology of mathematics"), I understand what you were asking. My answer is not relevant now and other good answers have been given. But I hope that some of the people watching this thread will still be interested in reading it.

$\endgroup$
0

Not the answer you're looking for? Browse other questions tagged or ask your own question.