Tuesday, November 12, 2013

Mathematical Realism and Anti-realism: all the way down to the Nature of the Universe itself

There's this age old question of where mathematics come from: is it a product of the human mind, or does it exist independently of it? Here's the labels:

Realism: A realist believes that math exists outside of our  minds. We discover mathematics, we don't invent it. Mathematics is real.

Anti-realism: An anti-realist believes that mathematics exists within our minds. We invent mathematics, we don't discover it. Mathematics is not physically real.

Realists argue that mathematicians make discoveries long before they become useful in physics, and that can't be a coincidence. When Einstein wanted to develop general relativity, non-Euclidean geometry had already been developed. Quantum mechanics is based on Hilbert spaces (functional analysis), which had also already been developed by mathematicians.

Anti-realists, on the other hand, argue that the truth of mathematical statements depend on assumptions, i.e. mathematics does not have any unconditional truths (apart from logical tautologies, such as "A implies A"). For example, let's reconsider geometry: there are various kinds of them, the Euclidean, and many non-Euclidean. For a long time the ancients saw Euclidean geometry as self evident, although we today know that this "self evident fact" is incorrect.

But let's take one question very seriously: why are the properties of the universe so mathematical?

It's almost as if the universe is made of mathematics, given that the laws of physics are mathematical relations (and moreover, they may all be computable). This hints towards a possible physical answer to the great debate between these two camps.

The universe may appear to be mathematical because it IS mathematical. The universe is a formal axiomatic system, and we exist as part of those axioms. The assertion may in fact be physically provable: if we discover a theory of everything (i.e. the ultimate laws of physics), then the universe is obviously mathematical, and realism turns out to be true: pi is not in the sky, but in the formalization of the laws of nature. As a special case, we it turns out that all physical processes of nature are computable, then the universe is not only mathematical, but a Turing-equivalent computer.

On the other hand, it may turn out that the universe is not mathematical, and we "discover" mathematical properties of the universe only because our mind creates them. This may sound crazy at first, but one may note that many concept are in fact purely man made (although indispensable to everyday life): concepts such as human, car, alive/dead, and temperature all rely on human interpretation of a huge amount of matter ("a macroscopic phenomena"). It is not fundamental to the universe: a car is just an arrangement of particles/energy ("a state"), no more special than a pile of junk. Put differently, the universe may be "blind" to our high level concepts, and that may actually include most of physics, possibly with the exception of quantum mechanics (although that is not certain either). From this perspective, the universe is not mathematical: we interpret it as having mathematical properties, and that approximate some things very well, but will never grasp the full extent of it (that is the case right now). In that case, anti-realists win: mathematics is only in the mind (yet very useful to our existence).