| Where does mathematics come from? Good question. How about a not so satisfactory answer - we like to make sense of the world around us and we also have the ability to map the world around into patterns. With that, we try to put what we observe into some possibly useful form. Now those attempts at making patterns can take a number of different forms, of which one is mathematics. In the case of mathematics, we create axioms as a basic framework on which we build the various mathematical edifices that we use. We have applicable rules that are used and from that people come up with all the various fields in mathematics that exist today. However, as part of a challenge that I accepted some time ago, I am finding that everything we use in mathematics is but a map of reality and not reality itself. No matter what mathematics one uses in whatever area one uses mathematics, it is only an approximation to the reality around us. When the rubber hits the road, our mathematics often fails us. Forty years ago, one of my engineering lecturers advised all of us students at the time that though mathematics could give precise answers, reality had a penchant for not being precise at all, as in, there is much variation in the physical mediums in which we would work. Always treat all mathematical results obtained as only being approximate and design for the worst case scenario. Every map in existence is but an inaccurate description of the observable universe. From the very small to the very large, from the very simple to the very complex, mathematics gives us a possible way of inadequately describing what we see. Though, in its inadequacy, mathematics provides a means of studying and manipulating the universe around us from which we can get useful results. So we have all sorts of interesting technology and science with which we manipulate our environments. The problem I see is that there are those who attribute an unwarranted "honour" (I suppose) to mathematics without stopping and seeing its limitations and that it is a tool which we have created or invented so that we can make sense of the universe around us. Studying the Metamath system is interesting for me in that by using some very simple rules we can build all the mathematics of today. What I find interesting in your last paragraph is that you, yourself, seem to make the distinction between reality and mathematics, which would indicate, at least to me, that you recognise that it is but a tool created for a use and not something that had a pre-existence that we could discover. However, I may be completely misunderstanding your argument at this point. |
That's the thing though: mathematics can be inspired by the world around us, but more often than not, it does not aim to describe it.
Mathematicians working, say, on exotic 4-manifolds, are not concerned in the least with how well these structure describe anything out there in the "real world" of yours. The short answer is: it doesn't. The longer answer is: 4-manifolds are the real world, or, rather a part of it that, perhaps, the only way you can observe and interact with that part of the real world is through mathematics.