Spoiler alert…the answer is yes!
The world of mathematics received a severe blow in 1931. In those years the Princeton mathematician and logician Kurt Gödel proved a fundamental theorem which says:
“there are mathematical statements of which no systematic procedure could determine the truth or the falsity”
This theorem left no way out, because it provided irrefutable proof that certain things, in mathematics, are really impossible. The fact that there are undecidable propositions in mathematics caused a great trauma because it seemed to undermine the rationale for the discipline. Consider, as a simple introduction to this subject, the puzzling sentence:
“This proposition is a lie”
If the proposition is true, then it is false; and if it is false, then it is true.
Rather than giving a strictly “technical” explanation, that might be boring, I will say what the “practical” and “philosophical” consequences of such theorems are. In practical terms, those theorems introduced a fracture in mathematics: they proved that there are statements that are true but for which there can be no proof of their truth. The mathematicians of the nineteenth century, David Hilbert in the lead, were absolutely convinced that any statement of a mathematical nature could be proved to be true or false, and it was only a matter of time. Godel has broken this certainty by showing that the set of true statements does not coincide with the demonstrable ones.
Powerful, isn’t it?
From a philosophical point of view, the consequences of Godel’s theorems say something about the human mind: since mathematical theories are products of the human mind and are essentially all similar to each other, the split between true and demonstrable means that
the human mind has LIMITS
It cannot in any way reach all the truths contained in a construct (mathematics) that it has created itself and therefore of which it should be able to know everything.
In conclusion, we can prove that something is unprovable…but isn’t this itself a proof?
Can we prove a proof of unprovability of something?
Thank you for the attention.
