Platonism, formalism, and Tonyism

What's the Context?

To begin, let's frame the question in which we are exploring these ways of thinking. What is the nature of mathematics? Is it something that exists regardless of human existence (discovered), or is it something which humans formulated (invented)? This is a question that has been tossed around for centuries (if you couldn't tell by the name of Platonism), and there still isn't one widely accepted answer. The goal here is to present two of the most prevalent fields of thought, my current thinking, and short-comings of all three.  

Allegory of the Cave

One of the more well known thought experiments is Plato's Allegory of the Cave. In this particular conceptualization, there is a group of prisoners which are restrained such that the only interaction they have with the outside world is to see a flat 2D shadow of the things that lie behind them. Plato considered mathematics to be no different than this. He believed there to be a "realm of mathematical forms" in which the perfect mathematical objects existed, and then humans have the ability to draw crude approximations of them. For example, were you to draw a circle right now, it wouldn't be a perfect circle, but rather an approximation of a perfect circle (which exists in the realm of forms). This concept then leads to the Platonism thinking regarding the nature of mathematics; it is something which exists in the realm of mathematical forms, and humans are discovering it. 

Math as a Game

Another way to look at the nature of mathematics is to consider it as a game; the axioms on which mathematics is build can be considered the rules of the game, and any resulting theorems, definitions, corollaries, etc., are simply strategies that different players employ while playing the game. To extend the metaphor further, consider taking one game and transforming it to another. If we have the game chess, disregard the limitations of movement imposed on different pieces (so that all pieces move in the same manner), and then impose a restriction on the starting position of pieces, we can get checkers. Each game has a different set of rules, however both can be played as a stand alone game. This is similar to the way that geometry can start with a set of axioms and build from there, while set theory can start with a different set of axioms, and build into it's own "thing". Formalism also believes that there exists a "mother of all games" which is structured such that all things mathematical can be achieved from one set of axioms.

Non-Euclidean Geometry

One major blow to the stance of Platonism was the development of non-euclidean geometry. Non-euclidean geometry came about from the finagling with the fifth axiom of euclidean geometry pertaining to a line and a separate point through which you can draw one parallel line. In toying with this axiom (including, excluding, or simply changing), it was discovered that other forms of geometry such as hyperbolic and spherical worked as well. Not only did they work, but they seemed to describe the physical world just as well as euclidean geometry. For a long time, euclidean geometry was considered the source of truth when it came to spatial reality, and therefore it seemed to be that humans were making discoveries about space. With the introduction of non-euclidean geometry it seemed more and more that this was not so - we could garner truths about spatial reality by simply setting up some rules to a game, and then playing it out. 

The Incompleteness Theorem

In the 20th century, one of the greatest logicians to date, Gödel, developed the incompleteness theorem; a theorem which rocked formalism back on its heels. The incompleteness theorem says, in layman's terms, that every axiomatic system contains statements which can neither be proven true nor false. Since formalism is completely based on the idea that you set up a system of axioms, and all of mathematics follows from there, Gödel's incompleteness theorem seems to have dealt a pretty major blow to this field of thought in proving that no single set of axioms can prove every statement. 

Tonyism

In an odd mosh pit of Mac's old advertising campaign (It just works) and formalism, my own thinking has fumbled to a halt. It seems to me that, while human intelligence would have to exist in order for it to be observed and explained, the way that nature is now would hold true whether humans were around to record it or not. A culmination of the manner in which humans have evolved to think, the way in which we use each other's work as stepping stones, and our ability to choose a type of mathematics suited to the problem at hand (fractions rather than topography or some other mathematical approach is used when discussing batting averages) seems to me that we are developing the language of mathematics in such a way that we can make discoveries about the natural world using it. 

My own particular vein of thinking is only recently formed, and also the "It just works" addition doesn't apply directly to mathematics (more so to nature), however I don't feel that an exact answer of "is mathematics invented or discovered" will ever be able to be explicitly answered.