Mathematical Consensus

Question

Can anyone give me a reason why mathematics may require consensus to determine the quality of knowledge from the general mathematical community?

Also what would be the counter to such a claim, as in why wouldn't mathematics require consensus for determining the quality of knowledge?

My thought in general would be math requires consensus as i there wasn't, no one would be able to understand the quality of knowledge produced as there is no other ways to prove its correct. However I would also say mathematics does not require consensus as the work produced will not lose its quality regardless of whether people accept it or not as if its true it will always be true whether people choose to believe it or not.

I am studying mathematics in University, and I am actually quite interested in this topic. I know why consensus and disagreements would be beneficial in the natural sciences, but i am not sure in the case of mathematics. i would appreciate some in depth review or thought to my questions.

Answer 1

Consider the parallel to physics. Everything is either proved by experiment, or disproved (or unfalsifiable so not involved), right? Wrong. Popper in his consideration of the demarcation of science, pointed out that hypothesis creation is not in this schema.

It is easy for someone who doesn't see new physics getting done, or new mathematics, to not realise that it is a fundamentally creative process. If you could just crank the handle and churn out more, there would be no breakthroughs. That just isn't how discovery and innovation work (mostly!).

Physicists beyond the realm of the experimentally testable, have to hoard inconsistencies, puzzles, contradictions. They have to go hunting for reconciliations to these, and apply everything they think they know about good physics to do so.

Godel proved not all true statements within a system are provable. So what do we do in practice? As creative human minds, we step outside a given system, say out of number theory into set theory, out of algebra into lie groups. A conceptualisation of this is Hofstadter's strange loops. A naive view of mathematics, like say Hilbert's, will inevitable founder on the Munchausen trilemma: either reliance on unprovable axioms, circular reasoning, or infinite regress. We avoid this in practice through what Hofstadter calls 'tangled hierarchies'. Cross-referencing, step out and back in, starting in the middle and working out and back. Whatever yields interesting results, and can be reconciled with other interesting results, ideally without letting too many slip. But as the failure of most of Hilbert's program showed, sometimes discoveries are more interesting than the structures associated with cherished assumptions. We find a firmer basis. But don't ever think there is a basis so firm it cannot move, and yield tomsome deeper more universal insight. Beware of thinking like Lord Kelvin, who around 1900 allegedly said: "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement." A few tiny inconsistencies, yielded relativity and quantum mechanics.

So. In physics experiment is always the final arbiter. In maths, rigorous proofs. But once that is completed around a topic, that is no longer an active area of development. How should active areas of development be handled? Journals, reviewed by peers. Conferences. Research programs and projects. In other words, under the direction of the consensus of the community. What we called proven, or verified, we can expect to be valid as at least a special case of a new paradigm. But every time we thought we had Truth, it turns out to be just plain old truth, just one more stop on a journey into deeper understanding.

Answer 2

In principle, math does not require consensus. In practice, there are two reasons.

One is that people are fallible. Mathematicians have come up with innumerable proofs over the years that turned out to be flawed. This goes back to Euclid. It is not possible to prove Proposition 1 of Book 1 with the definitions, axioms, and postulates given. (The trick is that Euclid's definition of a circle applies also to arcs of circles, and the proof doesn't work if arcs are used instead of circles.) However, if a proof is valid, people who study it will continue to conclude that it's valid, and if a lot of mathematicians agree that a proof is valid it probably is.

Another is that not all valid theorems are created equal. It's possible to prove a lot of things that aren't interesting or useful. If enough mathematicians are interested in a proof, we can assume that it's interesting.

Answer 3

Simply put, mathematics is entirely built on consensus, because there's no other way for it to be practised (apart of course from purely formal/mechanical proofs)

Student: Then really what is a proof?

Ideal Mathematician: Well, it's an argument that convinces someone who knows the subject.

Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons. Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?

I.M.: No, no. There's nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you'll catch on.

Student: Are you sure?

I.M.: Well, it is possible that you won't, if you don't have any aptitude for it. That can happen, too.

Student: Then you decide what a proof is, and if I don't learn to decide in the same way, you decide I don't have any aptitude.

I.M.: If not me, then who?

Davis, P. J., & Hersh, R. (1998). The ideal mathematician. New directions in the philosophy of mathematics, 177-184.

[...] many great and important theorems don’t actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community. [...]

In any case, there is a web of semi-proved theorems throughout mathematics. Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation. [...]

We (the mathematical community) believe that the proofs are correct because a political consensus has developed in support of their correctness. [...]

How do we recognize mathematical truth? If a theorem has a short complete proof, we can check it. But if the proof is deep, difficult, and already fills 100 journal pages, if no one has the time and energy to fill in the details, if a “complete” proof would be 100,000 pages long, then we rely on the judgments of the bosses in the field. In mathematics, a theorem is true, or it’s not a theorem. But even in mathematics, truth can be political.

Melvyn B. Nathanson, Desperately seeking mathematical truth, Notices Amer. Math. Soc. 55:7 (2008), 773.

This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.

The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true. [...]

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. [...]

When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs. [...]

Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas.

Thurston, W. P. (1995). On proof and progress in mathematics. For the learning of mathematics, 15(1), 29-37.

As I’ve blogged about before, proof is a social construct: it does not constitute a proof if I’ve convinced only myself that something is true. It only constitutes a proof if I can readily convince my audience, i.e. other mathematicians, that something is true. Moreover, if I claim to have proved something, it is my responsibility to convince others I’ve done so; it’s not their responsibility to try to understand it (although it would be very nice of them to try). [...]

I recently described a proof to be a convincing argument of why you think something is true. I’ll stick to that definition in spite of a few commenters who want there to be axioms or postulates, because I really don’t think that’s what happens in real life (which is a good thing! It would be an incredibly boring life!). Since I’m a utilitarian, I only care about and only want to discuss what actually happens.

The above definition immediately begs the question, convincing to whom? Can a proof to someone be a non-proof to someone else? Absolutely, proofs are entirely context-driven. If I’m trying to prove something to you and you remain unconvinced, then it is no proof, even if I’ve used the same argument before successfully.

https://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-been-proved/ + https://mathbabe.org/2012/08/06/what-is-a-proof/