Math : Art or Science? (Part 3)

 This is part 3 of the series called Math : Art or Science?

 Read part 2 here 

So in my last post I talked about Platonism. For people seeing this concept for the first time, it was perhaps a little too extreme. I said mathematical objects reside in a realm of its own which is not affected by the physical universe. But where is this realm? It seems like the ideas of an overimaginative person.

And you would not be the first person thinking that way. Platonism is not universally accepted as the philosophy of Mathematics. It is just one of many.

Possibly the toughest opposing philosophy would be that of Formalism.  

What is Formalism? 

In the (late)19th and 20th centuries, Mathematicians, or rather a particular group of them, started to feel that Math is just a game. Just like chess, it has some players (mathematicians), it has some pieces with which the game is played (numbers, equations, geometric objects etc.) and most importantly it has some fixed predetermined rules of the game (axioms of mathematics). We all agree on some fixed set of rules and then we play. A move is only allowed if the preset rules allow it. Anything we can arrive at after a certain number of "legal moves" is called a theorem! And that is pretty much all.

And as such, Mathematics isn't really about anything. It is not talking about anything, there is no inner meaning, there is no interpretation to be made. It is just the outcomes of the legal moves of the set of axioms. And this is what is called Formalism. 

Arguably the greatest  Formalist was David Hilbert. 

 David Hilbert

Hilbert wanted to write or rather re-write the whole of Mathematics as a formal system. He wanted to give a set of axioms and everything else that follow from those axioms would be all of Mathematics. Sadly, or maybe in some sense happily, Hilbert's attempts were killed by Kurt Gödel, the greatest Logician ever born. We will talk about the how and why later on. But from a purely philosophical perspective let's try to see why Formalism isn't as acceptable as it might seem.

So Formalism in short says that Math is just a game like chess. And there is no reason to prioritize one  set of axioms over another. A different set of axioms will produce a new "game". 

But let us pause for a while from our abstract thinking and look at the physical universe.  If Math was just a game like chess, then why is it that only one of them can be used to explain the whole universe, from rocket science to biology? Math might be a game, but it is criminal to suggest that it is just a game.

There is something suspicious going on here and it seems it is beyond human understanding for the time being. How is it that a game that human beings produced using a choice of axioms is being able to explain everything that there is to explain? The same numbers that historically first arose from the necessity of counting cows is being able to explain planetary motion! Is the universe organizing itself in a way so as to accommodate our "game"?!

This seems to be an absolutely outrageous suggestion. 

But for some reason, Math, which is "just a game" is being able to explain the universe while other games (say chess for example) cannot. This suggests that there is something far deeper in the designing of the game. And as such it seems to be an arrogant suggestion that human beings invented Math.   

Everything seems to point to one conclusion : Math is discovered and not invented! 

 It seems then that Math cannot be characterized as just Science or just Art. It is a bit of both, and neither of any. Mathematics is a thing in itself.