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If we assume there is no limit to the amount of mathematics we create/discover, is it possible that there are infinitely many proofs about any true mathematical statement? For example, we have Wiles' proof of Fermat's Last Theorem, but due to the deeply interconnected nature of mathematics surely there are many other proofs of it out there waiting to be discovered. Another example is the plethora of proofs regarding the Pythagorean Theorem which seems to be understood better the more we generalize it. In other words, is mathematics an infinitely large web that is connected at infinitely many points? (this is probably an inadequate analogy for capturing mathematics but it's the only one I could think of)

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    $\begingroup$ Do you mean formal proofs? It is always possible to tack on tautological statements to a valid proof. $\endgroup$
    – qwr
    Nov 6, 2021 at 4:18
  • $\begingroup$ @qwr Sorry I'm not well versed in logic, I only recently got into mathematics. Could you please explain it in simpler terms? $\endgroup$
    – fnalkj
    Nov 6, 2021 at 4:31
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    $\begingroup$ See discussion at this question: link . Depending on your definitions, the question is either trivial or very difficult. One difficulty is the question of what does it mean for two proofs to be the same? You could always add in some unnecessary tautologies, but the proof would still be "essentially" the same. $\endgroup$
    – spin
    Nov 6, 2021 at 4:33
  • $\begingroup$ @fnalkj your question can only be answered precisely if you define what a proof is and what constitutes two different proofs. The proofs you see in textbooks and by humans are usually "informal proofs", in that they are meant to convince a human of correctness. "Formal proofs" are proofs that are defined in a proof system using symbols and axioms and take a mechanical approach to proofs, for example in natural deduction where proofs are written as trees of expressions. $\endgroup$
    – qwr
    Nov 6, 2021 at 4:43
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    $\begingroup$ Well we've already told you the question is unanswerable because any human notion of completely different ideas in a proof is not really well defined without further qualifications $\endgroup$
    – qwr
    Nov 6, 2021 at 4:56

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Suppose I have a proof of $\varphi\implies\psi$. I will now construct infinitely many proofs of $\phi\implies\psi$, where $\varphi$ and $\psi$ are propositions. The first one goes like this:

Suppose $\varphi$

Thus, $\varphi\land\varphi$.

Thus, $\varphi$.

Thus, $\psi$.

Here's the second one.

Suppose $\varphi$.

Thus, $\varphi\land\varphi\land\varphi$.

Thus, $\varphi$.

Thus, $\psi$.

I can do this as many times as I like, and get a different proof of $\psi$ that way.

In case you're unfamiliar, $"\land"$ means "and."

The thing is that none of these proosf are functionally different. It's just a cheap way of creating a "technically" different proof. If this is what you mean by "are there infinitely many proofs of any statement?" then yes.

What you might mean, though, is whether there are infinitely many proofs that "use different ideas" or use some different kind of insight. This isn't really a precise definition, but it's all I've got to work with. The question isn't really answerable until you define what you mean by "different proofs."

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  • $\begingroup$ I think you should mention the distinction between "formal proof" and "informal proof" that is central to being able to answer the question. $\endgroup$
    – qwr
    Nov 6, 2021 at 4:48
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    $\begingroup$ Please read the final two paragraphs of my explanation @qwr $\endgroup$
    – Luna145
    Nov 6, 2021 at 4:55
  • $\begingroup$ In HoTT a theorem P is a type and its elements p:P are proofs. P then has the structure of a weak infinity groupoid, and a reasonable definition would be, that two proofs are different, if there is no witness $H: (p =_P q)$ that they are the same. ;) $\endgroup$
    – Merle
    Nov 6, 2021 at 8:06
  • $\begingroup$ Well beyond the scope of the question, but you are correct. It sounds reasonable. I'm sure HoTT was not the intended direction. @Nico $\endgroup$
    – Luna145
    Nov 6, 2021 at 10:26

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