This vid says it all regarding math. Or does it? It is a great teaching aid for those not familiar with principles of math.
However, I dont see Alan Turings hole, affecting the principle that states, there can only exist five, regular/symmetrical and convex polyhedra of Universe, and only three of those have all equilateral triangular surfaces.
Hope others will find time to watch this vid, as I did. Best wishes, as is very informative, with clarity.
I watched it, it's OK at best, but the problem being discussed is fascinating.
First, it's not a hole in math so much as it is the case that math doesn't have a logical foundation to rest upon. The big unresolved question in mathematics is this: just what is the connection between mathematical truth and physical reality, do mathematical objects exist independently of the human mind or not?
There were really only three approaches to the Philosophical Foundations of Mathematics, Hilbert's Formalism, Frege's Logicism, and Brouwer's Intuitionism. Hilbertand Brouwer said mathematical objects are created by the mind, but on the otherside of this, you have Eugene Wigner writing “The Unreasonable Effectiveness ofMathematics in the Natural Sciences” which tells us that there’s got to be moreto mathematics than that.
In the end, and as the video shows, Hilbert’s Formalism fell to Godel’s proof that Formalism isn’t possible by showing that it is provable that you cannot prove the truth of all arithmetic statement within arithmetic, and Turing’s demonstration that you can’t have an effective decision procedure. Gottlob Frege’s Logicism fell to the self-referential paradox, even after Russell tweaked it with the Principia, the attempt to reduce mathematics to logic failed. And we were left dangling with Intuitionism which we intuitively feel cannot be true because the challenge to the law of contradiction left us with nothing to logically work with, and it’s foundational postulate that the truth of a mathematical statement is a subjective claim leaves us with an unacceptable vagueness of the intuitionistic notion of truth. It became clear that Brouwer's Intuitionism has no hope of explaining “The Unreasonable Effectiveness of Mathematics in the Physical Sciences”.
Godel showed us that the Formalist interpretation of mathematics is not possible, as the Logicist wasn’t, and Intuitionism didn’t stand up to scrutiny. There were only really three horses in the race and none of them won.
In the end we just have to accept that the greatest mathematical minds ever produced have pretty much collectively concluded that we do not understand the connection between mathematics and reality.
