>>14415034Yes, there are myriad applications, in much the same way as there are real-world applications for philosophy. The applications of mathematical proofs lie in their stimulation of understanding and inspiration, which then yield ideas with practical applications. Proofs and theorizing are to practice as planning is to doing.
The act of devising the mathematical proof can, by finding a set of fundamental conditions that could found the system or proposition that was known to begin with, reveal even more general systems or alternative systems. Consider that many of the propositions (i.e., the "theorems" before they were proven) of geometry, such as Pythagoras' theorem, were known to the ancients albeit without proof. Euclid then devised a deductive system that gave these theorems an axiomatic foundation, but which also revealed the parallel postulate to be an unproven assumption. By relaxing that assumption, we reveal non-Euclidean geometries, which are at the core of general relativity. This is to say, the act of proving that a geometric theorem such as Pythagoras' should be true revealed viable systems in which it may not be true.
Another instance in which proofs can lead to practical applications is how the purely theoretical tool or concept that is conceived of in order to complete the proof can itself sometimes precipitate a corresponding real-world object. An example is that, in order to solve Hilbert's Entscheidungsproblem, that is, disprove the proposition given by Hilbert, Turing conceived of a theoretical tool, a Turing machine, to carry out algorithms. But the Turing machine precipitated a corresponding real-world tool, the computer. The goal of completing the proof was in effect an incentive to create an abstract tool that provided a useful real-world correspondent.
They also form an educational tool and can reveal errors and incorrect assumptions in a theoretical landscape.
