In a post I wrote many months ago I asked the question what type of mathematician are you ? I have been having various debates on the Fora about this, Including one with Nilo

The big divide is between those who think (like Nilo) that we discover maths and those like myself that we use maths in order to make sense of the world around us. On my view I have no problem with seeing maths essentially as a construction like a work of Art or music. It's relationship with the real world being an accident rather than some god given laws. Indeed when one applies mathematics to physics there will always be a gap. The Simple Harmonic Oscillator only works well when applied to small amplitudes. If one applies group theory to particle physics it only really works if we assume the particles have the same mass but of course they don't. Nevertheless it is still productive to see how far this can be taken.

Anyway an interesting article I came across in the Physics Arxiv concerns a debate between 3 physicists/mathematicans which delineates three views

1) The Fundamentalist view articulated by Max Tegmark who seems to be prominent on the TV more and more. This essentially on the extreme view that Max Tegmark asserts that to every mathematical concept there is a corresponding element of reality if not in this universe then in another universe. It has it's roots in Platonism and many mathematicians have been attracted to Platonism such as G H Hardy and in our own time Penrose and Marcus Du Satoy are prominent examples although they probably would not be as extreme as Tegmark. For them as I have commented before mathematics is about uncovering the deep nature of reality so that whenever a mathematical theorem is deduced no matter how trivial something fundamental is discovered about the nature of reality. On this view mathematics seems to be akin to a new form of mysticism and it is interesting that most TV programmes which talk about maths always seem to have people who express the Platonist view. Very rarely if at all are other views

2) The mystical view represented by Peter Hut which claims that in 500 years time the laws of physics and the principles of mathematics will have changed beyond recognition so that we are only getting glimpses of what is really the case. I'll leave you to read the article to find out more

3) Finally the secular view represented by Marcus Alford and which I would identify with. For us secularists mathematics is essentially a construction of the human mind which we impose on the world around us in order to make sense of it. There are only a limited subset of mathematical concepts which are applicable to the world around us and there will always be a gap between the real world and the mathematics that is used to understand it. For example (and this is the point I was making to Nilo) the symmetries of particle physics assume that all particles have the same mass which they don't. Nevertheless they are useful in classifying particles and making some order out of the apparent chaos. Again not all mathematics has to apply to the real world but one shouldn't dismiss out of hand their success when it occasionally works. This is not to denigrate maths at all but to see it as one amongst many of humanities greatest achievements.

Going back to Hume's distinction between relationships between ideas and matters of fact mathematics concerns itself with relationship between ideas and not necessarily matters of fact. On the one hand there is this pristine pefect world in the realm of ideas as exemplified by mathematics and on the other hand there is the messy world which aided by mathematics and our knowledge of the laws of physics we can begin to make a little sense of. But we have to acknowledge that there is no route to absolute truth even if humanity occasionally wins some battles.

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