I'm not that interested in Philosophy, but I just thought I'd write this down to get it out of the way. The following stuff seems obvious to me, but I suspect that many people will not agree with it.
Timothy Gowers' latest blog post (http://gowers.wordpress.com/2014/01/11/introduction-to-cambridge-ia-analysis-i-2014/) discusses the fact that there is exactly one complete ordered field, namely the Real numbers. [You don't need to understand that stuff for the following discussion]. He points out two ways of describing a real number: (a) via an infinite decimal sequence; (b) via pairs of complementary sets of rational numbers (Dedekind cut). It will mystify non-mathematicians that these different things are said to be the same. The point is that they are exactly the same for all the practical purposes that we need the real line for, as is the description of a complete ordered field without even explicitly constructing the Reals. What we'll come back to is the question of whether the Real numbers really exist at all. First we move from Mathematics to Science.
The other day I was telling my grandson what a scientific theory is, namely it makes predictions which can, at least in principle, be falsified by observation. Two scientific theories are the same if they make the same predictions, even if the theories seem different. We have a recent example that illustrates this, in quantum theory. The original quantum theory was expressed in terms of infinite dimensional Complex vector spaces (yikes). The predictions then made met the most exacting standards of accuracy available, and continue to do so. Richard Feynman showed how one could instead draw collections of diagrams (the famous Feynman diagrams we often see in popular science) and use those to do calculations in a simpler and certainly more humanly accessible way. But these quickly get hard in more complex situations. Recently there has been a claim that we can portray quantum situations in a sophisticated mathematical context that allows calculations to be done much more efficiently. All these ways of describing reality make the same predictions, with varying degrees of computational efficiency and human accessibility. It is just not sensible, and certainly not Science, to say that one of them correctly describes reality while the others just happen to give the same answer. However human accessibility and ease of computation are key ways of evaluating theories, and what follows is an extreme example.
As we know, the Earth gives every indication of being 4.5 billion years old, and to have undergone massive changes during that time. Geology and related Sciences are all written as if that were the case. But suppose you lived in a society where you were likely to get burnt at the stake if you expressed the view that the world was more than 6.5 thousand years old. Then you might come up with an alternative theory: that God created the world 6.5 thousand years ago, exactly as if it had existed for 4.5 billion years and as if it had experienced many exciting events since that time. Maybe he ran a perfect simulation to see what it should be like. Now all scientific papers can still be written, but they have to have frequent weasel words so as not to suggest that the world really existed before 6.5 thousand years ago. The two theories are identical and lead to the same predictions and expectations. To claim that one theory is true and the other false is therefore not Science. However, like Feynman diagrams, the idea that the world really is 4.5 billion years old is the one that is more humanly accessible. Also the scientific community has a role in picking, from equivalent formulations, standard ways of describing scientific theories so that the scientific conversation can proceed smoothly. If the Geology literature was a mix of those describing a 4.5 billion year old world, and those describing a recent but equivalent creation, then the conversation would not proceed smoothly.
We can see that making specific choices from equivalent descriptions of scientific theories is very similar to specific representations of the Real numbers. It is relevant to look into the status of the Real numbers themselves. Our conception of the Real line plays an important role in our understanding of space, time and many other aspects of reality. However the Real numbers themselves have a rather tenuous connection to reality. We can easily ascribe a meaning to the Rational numbers (such as 2/3) and also to the computable numbers. A computable number is one where a computer program can (for example) give you as many decimal places as you request for that number. However there are only a countable number of computable numbers (so we could in principle give each one an integer index from 1 up), because there are only a countable number of programs (since programs are finite strings taken from a finite alphabet of symbols). And it is quite easy to prove that the Reals of the complete ordered field are not countable (http://en.wikipedia.org/wiki/Cantor's_diagonal_argument). What can we make of these multiferous uncomputable Reals? The key point is that all Mathematics is done with finite strings taken from a finite alphabet. There are no infinities there. Yet hypothetical infinities are a recurring theme. The infinite things, such as uncomputable Real numbers, are a human construct that gives meaning to the Mathematics, which in turn enables us to understand reality.
In the early days of calculus, the practitioners (such as Newton) liked to talk about infinitesimal numbers. These sure make it easier to think about calculus. But there was a backlash against this, since "infinitesimal numbers don't really exist", and the whole edifice was reconstructed less elegantly using just the Reals. It was subsequently shown (Model Theory) that you can use infinitesimals in a rigorous way. In fact the reality of Real numbers is just as dubious as infinitesimals. They both enable human intuition to make sense of mathematics and then of the world.
Well it turns out that mathematicians are trying to build mathematics up from core principles in a new way: that respects the fact that Maths is done with finite strings from a finite alphabet; that takes seriously the question of when two things are equal. This is Homotopy Type Theory. I'm trying to understand it. I can also say that the Wombat Programming language (wombatlang.blogspot.com.au) makes extensive use of the idea of multiple representations of the same value, and also has a flexible notion of equality. Maybe when I understand HoTT I can make Wombat even better.