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This question is vague and might be closed, but I also feel like its incredibly important to math and how we approach problems. Particularly difficult problems. Namely the concept of information.

One of the main principles in math is "there is no free lunch"; if a proof claims to show something, but didn't use crucial information/steps, then there is a mistake somewhere.

Because of this I use to believe although different proofs of a result might look different, they were all essentially the same on some level: they all found the crucial parts and simply combined it differently.

However more and more it feels like different proofs are fundamentally different. One might use Axioms A and B of ZFC, and another axioms B and C of ZFC, whereas A doesn't follow from B and C.

Is there any way this concept can be made more rigorous and is true or false in that encapsulating sense?

If proofs are the same in some informational sense, then we always begin the same way by collecting needed parts until it boils over. But if they aren't, this justifies much more disregarding the work of others as possibly unnecessary.

Stephen Harrison
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    "if a proof claims to show something, but didn't use crucial information/steps, then there is a mistake somewhere" Not necessarily. Consider the theorem "a quadrilateral with all angles and all sides equal is a rectangle". A reasonable proof would completely skip over the side information. So is there a mistake? No. It's just that not all restrictions on the quadrilateral were needed for the result. So what do we do? We trim away that restriction. Now we have "a quadrilateral with all angles equal is a rectangle". In known results this trimming is usually already done, but not in new research. – Arthur Feb 13 '22 at 23:35
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    I think of interest to you is Reverse mathematics. You restrict your base system and then check which sentences imply which or not. I think this covers all the queries your formulate. – Nikolaj-K Feb 13 '22 at 23:36
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    @Arthur It’s interesting that in new research this “trimming” likely hasn’t been done because it’s not obvious yet that it can be done, or not clear how to do it. I wonder if it could be possible to precisely formulate and prove statements such as “the statement of this theorem can be trimmed no further.” – littleO Feb 13 '22 at 23:41
  • In the example in the question, of a result in ZFC which follows from either axioms A and B, or from axioms B and C, I think that theorem is just a special case of two different theorems. You could make one theorem which says, "Under axioms B and C, such-and-such", and a different theorem which says, "Under axioms A and B, such-and-such". If you are only interested in ZFC, these theorems have the same consequences, but they are different theorems. – 1Rock Feb 13 '22 at 23:47
  • I don't think it makes sense to say, "The statement of this theorem can't be trimmed further", because you can often come up with generalisations of a result, and I don't think there's a real difference between generalisations from trimmed statements. E.g. just because a proof uses (say) commutativity, you might be able to prove something more complicated in a non-commutative setting, where the complicated terms disappear in the commutative case and the non-commutative result implies your result. – 1Rock Feb 13 '22 at 23:51
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    @1Rock But intuitively it often seems that there is some right amount of generalization beyond which further generalization no longer provides any new insights or clarification, or is simply not possible. When we generalize, we strip away the irrelevant details so all that is left is the "essential information" that needs to be used in order to prove a given result. – littleO Feb 14 '22 at 00:06
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    What do you mean by " this justifies much more disregarding the work of others as possibly unnecessary"? – ckefa Feb 14 '22 at 00:21
  • @Arthur what if someone claimed to prove it, but didn't use that all sides are equal? For ckefa's question I mean there could be ways to travel across the country in a day without, say, gas so you don't restrict your search to methods that use gas like others did. – Stephen Harrison Feb 14 '22 at 00:38
  • @littleO I don't share that intuition - I have a competing intuition that most settings are special cases of more general settings, and a result which you completely understand in one setting is often more complicated, but still useful in the general setting. For example, rings are mathematical objects where "primes" make sense, and it would be interesting to know under what conditions an infinite ring has infinitely many primes. – 1Rock Feb 15 '22 at 22:30

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Take a look at the opening pages of Martin Aigner and Günter Ziegler's justly famous Proofs from The Book.

The very first example they give of a cluster of elegant proofs is of SIX different proofs of the infinity of primes.

In this case, it is plain we can't say that though the six different proofs of the result might look different, they are all essentially the same on some level. Rather we have different proof-ideas, different conceptual apparatus, which are brought to bear in the different proofs. And in this way, the different proofs make different connections with the surrounding landscape. And those sort of connections generate the kind of enriched understanding that we want from an array of good proofs.

Peter Smith
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