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... and now I got a fire lit under me, to write a bit about the topic I dodged the other day in my Philosophy post for the @ifc competition run by @apolymask.

Platonism

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The idea of Plato's cave is a simple philosophy to begin thinking about whether or not abstract ideas are "real" or just a manifestation of our consciousness, a game that we play. It was most famously forwarded by Plato (surprise...), however the ideas had been bubbling around in Greek philosophy long before his time.

Plato's cave and it's relation to Mathematics

Think of a cave which is lit by a fire in the centre of the cave. Look to the walls of the cave and you can see the shadows of objects that lie between the cave wall and the fire. So, the idea is that if you were a person that was to forever face the cave wall, you would only see the shadows of the "real" objects and never the objects themselves.

This relates to Mathematics in that (to use Plato's example), we never see perfect geometric objects (circles and shapes). We only see their imperfect analogues in our reality. However, from the "shadow" approximations we see around us, we can imagine the "real" objects that have pure abstract forms and pure mathematical constructions.

Thus, leading to the question, do these pure abstract forms exist in some (not our current) reality? Or are they just figments of our imagination?

This question can be applied to all of the fundamental ideas of Mathematics, such as what is the "reality" of numbers? For instance, the concept of 3 is something that is quite abstract, whilst the idea of 3 people is concrete (you can point to a group of 3 people).

If these questions exist for the purest form of expression (Mathematics), then what hope is there for lesser fields!

Resolution

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_{So, first a disclaimer, I'm not a Philosopher and I've not studied Philosophy, so please correct me in comments if I've gone off track. Likewise, I won't be using formal labels for the ideas below, as I can't be sure that I will label them correctly!}

In the end does it matter to Mathematicians and the endeavour of doing Mathematics? No, not really. This question of "fundamental truths and reality" of non-concrete (abstract) forms is not really relevant. People can and have ignored the question to focus on doing "useful" (haha...) Mathematics.

However, for those concerned by these questions it perhaps is a question with an answer that will remain forever elusive. If these "abstract" forms are "concrete" in some reality, we would never know, being that we are bound to this reality with no way to cross over. Conversely, if they are just figments of our imagination, then by definition, we will never be cured of this hallucination.

Further information

If you are interested, the attempts to reconcile the fundamental notions of Mathematics can be found by beginning some length reading these starting points:

Formalism, this was the idea that Mathematics could be formalised into a finite set of rules (axioms). From my point of view, it appears that this school of thought (which used to be the dominant school) was dealt a death-blow with the idea (from Godel) that such rule-sets could not be consistent.

Intuitionism, the idea that Mathematics is all a construction of a the mind. Thus, if there were no minds to conceive of numbers and other mathematical ideas, then the ideas would NOT exist in an otherwise uncaring Universe. After all, the Universe doesn't need to know of numbers to continue to exist and evolve.

Logic, this is the idea that all of Mathematics derives from logic and logical statements. The modern set theory of numbers is the attempt to make concrete the otherwise abstract concepts of numbers. Again, it is possible that Godel's Incompleteness Theorem undermines this idea of a rules based description of Mathematics.

These are the major schools of thought into this problem, there are many finer variations and splinterings of these schools, but it is way beyond the scope and ability of this writer!

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