Monday, 4 February 2008

A Mathematical Universe

My first love in academic study was mathematics. At school I drank it up, and I studied it for my degree at university.

Mathematics, Realism And Theism

Mathematics is also a very interesting field if you have an interest in philosophy and questions about design in nature. Almost all mathematicians are in practice realists - they believe that as they make progress in their field they are involved in discovering and not in inventing. (See here for more on this distinction). That is, they act and research as if there is already a transcendent, pre-existing mathematical universe "out there" that is waiting for us to find and explore it. The opposite of that is behaving as if mathematics is our arbitrary toy, to be played with, deconstructed and rebuilt as we please. Shall we adopt the convention that 2+2 = 5 from now on and see where that takes us?

In my observation most "mathmos" haven't done much by the way of questioning themselves over this realism - they just accept it. If, though, you want to have a coherent and comprehensive view of reality, you have to start looking at this. In my view, the atheist materialists who have tried to explain their view of reality are in an exceptionally weak position when they seek to explain mathematics in non-transcendent terms. Mathematics resists, at multiple levels, any attempt to treat it as an arbitrary invention of the human mind. Almost at every turn it cries out "I was here before you, and I am bigger than you!". Maths is a very theistic subject!

Abstract or Concrete?

What I wanted to talk about here though was down a little bit of a different track. It is about the connection between mathematics and the physical universe that we live in.

Mathematics is, from one angle, an entirely abstract subject. When we do algebra, we are manipulating symbols on paper - but we are really talking about something that exists behind the symbols. Mathematics basically has its seat of existence in the mind, and not on the paper. I can of course always add two apples to two apples and will always get four apples (an inconvenient truth for the atheists who want to argue that mathematical truths are not transcendent!) - but as I do so I'm conscious that there is a notion of "two-ness" or "four-ness" that goes far beyond the tasty bits of fruit and is independent of them. If I add two oranges to two oranges I get four of them as well. The more complicated the mathematics gets, the more obvious this becomes. I can move from the simple adding of objects to a dimension up and do calculus to work out the area under a graph. I can then accelerate to five or six dimensional spaces and work out their corresponding concept of volume. I can work out the properties of completely theoretical objects. you get the idea. Mathematics speaks to us of an ideal reality which depends on the mind.

We've already said more than that, though. Whilst it depends on the mind, mathematics also seems to have an unbreakable link to the physical world. In the most simple example, there's something about those two oranges that has the notion of two-ness. The notion of two-ness is contained, but not exhausted, by them. I can create a two-dimensional shape that is approximately (but never exactly - because we live in a world of discrete atoms and molecules) equal to the one in the equation of the graph I was using. This is all simple enough. What is more breath-taking, though, is to understand that correspondences between abstract mathematics and the physical world have also been discovered in far more complicated cases. In some areas, mathematicians discovered new theorems in highly abstract areas that nobody thought would ever turn out to have a practical application - but in fact they actually perfectly described physical phenomena observed decades later. Do you get that? Away in his dusty study somewhere, the mathematician was working on a problem that was thought to be far too abstract to have any real application. Some time later, a physicist realised that this bit of mathematics was the key to something that he was observing. Quantum physics provides a number of illustrations of this.

Summing Up...

I hope you're still with me! The point here is this:

Observation one: Mathematics has its seat in minds. Observation two: We also now know that mathematics is also embedded at a fundamental and essential in physical reality. Inescapable conclusion: Physical reality is the product of a mind.

As a Christian I believe that all knowledge can only ultimately be rightly understood when we see its integration point in our Maker. Or in other words, unless we begin with God in everything, we'll eventually go wrong somewhere down the line. When I say things like this, a response I often get is "but lots of knowledge is neutral and has no religious implications. Take maths - 2+2 is 4 whether you are a Christian or an atheist!" Above I've explained just one reason why that answer is on a wrong track. The objectivity of mathematics and its fundamental connection to the physical world are two more places where God speaks clearly - and leaves the atheist without excuse. They cannot be accounted for on purely materialist assumptions. Mathematics is universal, transcendent, comes from a mind, and is embedded at the deepest level in the world we live in. Mathematics speaks clearly of the immense and wonderful mind of the one who is.


Mike Gilbart-Smith said...

David, as a fellow Reformed Baptist Mathmo pastor I'm delighted with this post. Do you know of others who have written on this at more length?

David Anderson said...

Hi Mike,

Thanks for your comment... here's a guy who's compiled a long list: