Do there exist true statements with only one proof?

Question

The Pythagorean Theorem, for instance, has many, many proofs which are fundamentally different and use what appear to be very different ideas. These range from static geometric constructions to similarity factor arguments.

My question is: are there any provable statements which do not have more than one way to prove them (up to proofs being "effectively the same" in some way)? If so, what is this notion of "the same", and how do we know there is only one way to prove something, or that no such provable statements exist?

Equivalently, can everything which is provable be proved in more than one way? We know some true statements cannot be proven at all by Godel, but do some have just one proof? (or finitely many?)

EDIT: The notion of "effectively the same" and what that means is designed to be a part of the question; asking whether such a notion formally exists and, if so, what that notion is.

Derik Elkins does a very good job of answering with similar level of detail to what was asked. The answer was that, yes, there are some statements which only have one "normal form", and thus all their proofs are equivalent.

Answer 1

As saulspatz stated, this is a rather vague question if you don't pin down a proof system. Once you do pin down a (formal) proof system, then there are still questions about what "effectively the same" means, and there are multiple answers to that question. At that point though, the problem falls into the field of proof theory.

For many systems, especially constructive/intuitionistic systems, we have a good notion of "normal form" for proofs which means one way of defining "effectively the same" is when two proofs can be reduced to the same normal form. We talk about "normal proofs" in natural deduction systems and "cut-free proofs" in sequent calculi.

One particular form of the Curry-Howard correspondence links terms in typed lambda calculi to proofs in various logics. The archetypal example being the simply typed lambda calculus and intuitionistic propositional logic. From this perspective, your question is equivalent (using the same notion of "effectively the same" as the previous paragraph) to asking if there are types that have only one normal form inhabitant. The answer to this is definitely "yes" though it can depend on which logic you're using and some other details. A widely used example is the type $\forall A.A\to A$ in the polymorphic lambda calculus has only one (normal form) inhabitant, namely the polymorphic identity function (formally: $\Lambda\tau.\lambda x\!:\!\tau.x$). The polymorphic lambda calculus corresponds to intuitionistic second-order propositional (not predicate) logic. For contrast, $\forall A.A\to(A\to A)$ has two distinct normal form proofs corresponding to the terms $\Lambda\tau.\lambda x\!:\!\tau.\lambda y\!:\!\tau.x$ and $\Lambda\tau.\lambda x\!:\!\tau.\lambda y\!:\!\tau.y$.

This notion of "effectively the same" is somewhat reasonable but probably not perfect. On the one hand, it's not so coarse as to make all provable propositions equivalent, and the proof reductions it's based on generally do look "bureaucratic". On the other hand, there can be an exponential increase in the size of a proof when reducing to its normal form, and relatively minor details can still make proofs distinct even if they use the same "ideas". It's unlikely there is a (clear) formal analogue to the intuitive, informal notion of two proofs being "effectively the same". This likely depends on aspects of human psychology. Different intuitive perspectives can easily lead to the same proofs. To be very reductionistic about it, intuition is usually a guide to proof search (especially when you are working in a fixed formal system).