Doesn't that imply some sort of redundancy or point to some inner connection between the axioms? Or is it the case that when all theorems used in the proofs are broken down to their axioms that the two proofs must in fact be equivalent?
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Axioms are chosen to be independent. Can you show an example of a theorem proven in truly different ways from axioms ? – Feb 08 '21 at 18:12
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@YvesDaoust "Axioms are chosen to be independent." Not always: the usual axiom systems of (first-order) $\mathsf{PA}$ and $\mathsf{ZFC}$, for example, have lots of redundancy. In fact, the general existence of non-redundant axiomatizations is not trivial. – Noah Schweber Feb 08 '21 at 23:18
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@YvesDaoust: Gauss produced several proofs of the fundamental theorem of algebra (and there are many more). All of these proofs can be formalised using the axioms of Zermelo-Fraenkel set theory (maybe including the axiom of choice). To justify your claim, the onus is on you to show how these proofs are not "truly different". (FTA is just one example out of a gazillion others.) – Rob Arthan Feb 08 '21 at 23:42
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You might like to lookup the concept of "proof normalization", although it tends to only to simple things like propositional logic. – DanielV Feb 09 '21 at 00:53
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@RobArthan: I'll be glad to read the formal proofs that you supply. – Feb 09 '21 at 07:58
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Some proofs are of a very different style from others; geometrical vs algebraic vs formal vs generalized, etc. Technically everything known in mathematics can be proven from the axioms; that doesn't mean those theorems are all superfluous. One gauge of a mathematician's breadth of knowledge is how many totally different proofs of the Pythagorean theorem they know. (I still only know two). Mathematics is highly interconnected; we usually have several ways to say the same thing, and it is often much easier to prove something one way than another. That is the whole reason different coordinate systems exist, for example. We can write down the axioms in a small space; but the endless consequences of those axioms, and the chains of thought to get there, are interwoven and form a rich structure.
RobertTheTutor
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The pythagorean theorem though is known to be basically a single proof in many different disguises. – DanielV Feb 09 '21 at 00:52
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To keep things simple, I want to just consider proofs using the same limited set of axioms. – user1153980 Feb 09 '21 at 20:52
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Complementary to Robert's answer, you might like to be aware of the proofs-as-programs or propositions-as-types paradigm, also known as the Curry-Howard correspondence. Under this correspondence, a logical statement (such as $\alpha \to (\beta \to \alpha)$) corresponds to a program (or function) that represents a proof of that statement (such as $x \mapsto y \mapsto x$ or $\lambda x \cdot \lambda y\cdot x$, if you prefer $\lambda$-notation). Under the Curry-Howard correspondence, the existence of multiple proofs of the same statement has very interesting consequences. For example, the proofs of the statement $(\alpha \to \alpha) \to (\alpha \to \alpha)$ provide a model for the natural numbers (the Church numerals).
Rob Arthan
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Could you elaborate just a little on the nature of the consequences of having multiple proofs for the same statement. I was not aware of the Curry-Howard correspondence, but I know that the Metamathematics Web site uses a similar idea in verifying proofs. Everything is expressed as parameters passed into functions. – user1153980 Feb 09 '21 at 20:59
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The idea of representing proofs as simple function calls is part of what motivated the question. With proofs so explicitly represented it should be possible to do a comparison of two proofs of the same statement. – user1153980 Feb 09 '21 at 21:13