Well that is an extremely interesting question - well, interesting to me anyway. The answer, to me, is not obvious and it’s non-trivial.
Now at first sight you would say that mathematics is something we make up to describe the world around us - it seems that we can rightly categorise it as a language to describe our experience of certain aspects of reality. That is the view that Lakoff and Núñez are seeking to put in a structured way in “Where Mathematics Comes From” that’s mentioned above. There is a lot to be said for that view: the very foundations of mathematics are simple arithmetic (counting objects, ie keeping track of our sheep), mensuration (eg measuring the weight of our corn or the size of our field), building and navigation (geometry eg the properties of triangles and squares). In this view, mathematics arises from the natural processes of human cognition to quantify and formalise aspects of the perceived world. This was a view that I once held (or tried to hold because it is rather more comfortable for me as an atheist than a Platonic view of mathematical concepts), but I’ve come to find it unsatisfactory.
The reasons that I find it unsatisfactory are based on some of the points I have been making in this thread. It seems to me that one can make a strong case for the Platonic nature of mathematical truths - ie the case that says that mathematical concepts (propositions within a formal axiomatic logical system) exist independently of human cognition and that in some sense doing mathematics is not so much the development of a language to describe the physical world, as the uncovering of concepts which are universally and eternally true. I simply don’t think that formal theorems in mathematics depend on the specific nature of our human cognition or that some other reflective cognitive or brain structure would reach different valid conclusions. So, for example, the theorem that SQRT(2) is irrational or that the primes are infinite can be proven with logic but without reference to the world and in a real sense it seems to me that these and all other formally proven mathematical theorems are universal and not particular to human perception and cognition. Even mathematical concepts which originally arose from their utility in the real world, when they are formalised and axiomatised, can be and are derived without reference to the physical world: so that the mathematics of natural numbers can be derived from the Peano axioms without reference to counting objects. Morever, much of mathematics was actually derived by the application of logic to axioms and to earlier proofs based on those axioms independently of any relationship to reality. There are many examples, some of which I have given in this thread.
So if I think that mathematical truths exist independently of physical reality and of the limitations and peculiarities of human cognition, then does that mean that I think that maths is built into nature? Well no, at least not in the sense that makes it some sort of Creator or origin for physical reality. I think that there is on the one hand physical reality, which is what it is, and which behaves as it does. Then there are mathematical truths. Neither domain is dependent on humans - we *discover and systematise *facts about the physical world by doing science and we *discover *mathematical facts by doing maths. The facts precede their discovery - they do not depend on it.
Now we come to the, for me, astonishing thing about the world. Not only is it at least partly intelligible to us, but its intelligibility is enhanced in many cases and made possible in other cases by the use of mathematical concepts as tools to describe its behaviour. In some cases those descriptions are exact but in most cases they provide a model of reality that more or less *approximates *to reality. So mathematical theorems absolutely and exactly state truths within a system, but when applied to the physical world do not (usually) absolutely and exactly describe it. Nevertheless a huge range of mathematical ideas have been found to be useful in physics, including many complex and subtle ideas developed over the last hundred years. What is more, is that the physical concepts that we are dealing with in modern physics have outstripped our ability to conceptualise in any way other than through the language of maths, so maths has become our only conduit to understanding.
It seems to me that human intelligence is limited and there probably exist aspects of physical reality that we can never explain and there will be mathematical truths of which we can never conceive (if only because the number of such truths is potentially unlimited) - and these things are henceforth inextricably linked: the limit to our physical understanding will be determined by the mathematical concepts we can generate and prove. (The limits, however, are likely very far from where we are now.)
The intelligibility of the world and the connection of mathematics (the product of logic) to reality, those things I discuss in this and other posts in this thread, are by far the biggest challenges to me in my atheistic worldview. And yet, I wouldn’t have it otherwise - to me, the idea that mathematics is merely a tool that reflects our experience in the physical world debases the profound beauty of the actual state of affairs.
A long answer to a very short question.
Alec
evolutionpages.com
