In the first part of Roger Penrose’s book The Road to Reality (which I have not yet finished) Penrose argues that mathematics has an objective reality. It’s not entirely clear to me what he means by this, but I suspect that he is wrong. He points out that everybody who studies math comes to the same conclusions, and that seriously studying math has the feel of uncovering deeper and deeper truths. He also brings in Mandelbrot’s set, though I’m not sure why since it doesn’t seem to support his thesis.
It is undeniable that math is a deep subject, and that mathematics is an investigation shared by the human race across time. However, this does not mean that math has an objective reality beyond its existence in humanity’s shared knowledge. I think the easiest way to see this is to compare it to newer branch of knowledge, such as cellular automata.
The study of cellular automata is not mathematics. Cellular automata are abstract, but they are not based on numbers or geometry. Cellular automata are nevertheless a complex field of study, as can be seen by the extensive studies of just one instance, Conway’s Game of Life. There are undoubtedly general laws and insights waiting to be found in the study of cellular automatons.
However, I think it is silly to argue that cellular automatons have any sort of objective existence in a Platonic ideal space. The only way to support that is to become a thoroughgoing Platonist and believe that there exists an ideal object for all abstract concepts. So since I don’t see any essential difference between the study of mathematics and the study of cellular automata, I don’t see any reason to believe that mathematics has any objective reality.
There is a related issue, which is why reality is apparently describable in mathematical terms. Of course this is in part probably just an effect of how we observe things: we describe reality in terms of mathematics because those parts of reality which can not be described that way tend not to be described at all.
For example, consider the physics of gases. There are simple equations describing the relationship between the volume, temperature and pressure of an ideal gas. These equations do a good job of describing reality. However, we know that these equations are really just a way of describing the interactions of many many individual molecules. In this case, we are able to reduce the complexity of tracking individual molecules with simple mathematical formulas. Note that the gas laws are perfectly real: they correctly predict and describe the behaviour of gases. However, there is an underlying, more complex, system which produces the results the gas laws describe.
The weather on the Earth is also a complex set of interactions of many many individual molecules. However, in this case we have so far completely failed to describe the weather using mathematical formulas. Our best attempts to describe the weather involve complex models–in fact, a form of cellular automata. So when we say that reality can be described by mathematics, we evidently don’t mean the weather. Are we simply picking and choosing the things which we can describe? Also, when we find a simple mathematical formula in physics, I think we always have to ask: is this reality, or is it a formula which manages to capture underlying complex behaviour?
There is another way to consider the relationship between mathematics and reality, based on the weak anthropic principle. We are complex creatures, and our complexity was formed by evolutionary processes over a long period of time. Evolution is a very flexible technique, but it can only create complex objects when it operates in a relatively static reality. If the laws of physics changed unpredictably, it’s difficult to see how evolution would ever be able to build up complexity. Therefore, since we exist, we can conclude that the reality we inhabit must be relatively static. And mathematics is well suited to a description of many static systems. This does not prove that reality must be describable mathematically, but it suggests that we need not be surprised that it can.