## Imaginary friends…

There’s a new attempt to claim that the existence of God is rationally probable kicking around at the moment, divided into five “reasons”.

Therer’s a general problem about all of these, in that while they may point to there being something which we don’t yet fully understand underlying existence, the directions the author is going in would lead to a “God of the philosophers”, which (as I’ve complained regularly) looks nothing like the God of the Bible. In fact, it looks a lot more like Stephen Hawking’s “Theory of Everything”, and while I would be absolutely fascinated to see Hawking or some other brilliant mind come up with such a theory, and I would no doubt regard it as wonderful, awesome and similar words, I can’t see myself worshiping, loving or having allegiance to a theory.

I may come back to the other four reasons, but at this point I want to talk about the first, which has been called by others “the unreasonable effectiveness of mathematics”. The link I use there refers to a number of objections by Richard Hamming, but the list of names who have regarded this as a puzzle which requires answering includes some very great thinkers, and I don’t think it can be dismissed out of hand. As Max Tegmark suggests, perhaps at a fundamental level everything IS mathematical. It is definitely the case that mathematics comes up with concepts, and those concepts later find use when some theoretical scientist realises that that piece of maths describes (at least reasonably well) the mechanism which they are studying. The use of the Riemann mathematics in General Relativity, rather a lot of years later, is indeed a fine example.

I am not going to set out to dismiss the idea, but I do see a number of problems (apart from the fact that equating God with mathematics would negate virtually every religious or spiritual writing in history). Hamming mentions one, which I think has a lot of force – mathematics continues to produce a load of concepts, and not all of them by any manner of means manage to find a natural mechanism to describe. Some of them don’t describe the mechanism particularly well – I would argue, for instance, that string theory (which is an admirably complex piece of mathematical thinking) doesn’t actually describe the fundamental state of matter particularly well, given that to date it has failed to make any prediction which could be tested and that it keeps on being modified by legions of theoretical physicists in the hopes that one day it might.

He then develops that (it’s listed as a separate objection, but I think it flows from the above) to argue that we use the conceptual tools we have (which are, in science, largely provided by mathematics) to try to explain things. If we lack a mathematical concept for something, science doesn’t explain it, at least not yet.

What concerns me more, however, is the fact that mathematics throws up concepts which have no physical correspondent. Infinity is one such; we cannot observe an infinity; if we could, it would not be an infinity. It (together with a class of mathematical concepts which are quasi-infinities, called “transfinites”) is incapable of being experimentally verified; they just result from a contemplation of what would happen if an operation which you can perform a lot of times were continued indefinitely. I’ve written elsewhere of the problems faced by referring to attributes of God such as omnipotence and omniscience as infinite; I am deeply uncertain of the wisdom of this habit of saying “well, it looks as if it’s going there” without actually doing the experiment, as concepts have a habit of breaking down in limit conditions.

However, there’s another mathematical concept which cannot exist in the real world at all (it isn’t just not verifiable by experiment, it cannot exist) and that is the square root of -1, called “i”. The definition is i^{2} + 1 = 0. It is actually called an “imaginary number” for just that reason – it can have no real world equivalent. Mathematics therefore (arguably) axiomatically overspecifies what actually exists (axiomatically as opposed to the as-yet-unused mathematical concepts which may find an application some day).

I grant you, a very common use of imaginary numbers is in complex numbers of the form a + bi, where a is the “real” and bi the “imaginary” part; the imaginary part is then thought of as somewhere on an axis at right angles to the real axis. Any point on a two dimensional graph can therefore be represented as a single complex number.

The thing is, imaginary numbers are all over the place in some fields of mathematics, notably in areas like Rieman spaces (mentioned above) and anything to do with waves, including quantum physics. The mathematics for things which do exist therefore relies on concepts which don’t and can’t exist, despite the comments of the mathematicians talking with Melvyn Bragg in this BBC programme.

There are, I suppose, two ways of looking at this. The first is to say that mathematics clearly includes nonexistent things, and therefore cannot demonstrate the existence of God, because, well, God exists and they don’t.

The other is to say that if, just perhaps, there is something in the author’s argument that mathematics can tell us something about God, it is that “God exists” is at best a deceptive statement – because God includes some aspect which is, strictly speaking, imaginary…

So, my atheist friends, forgive me if I laugh at your comments about God as my “imaginary friend”. You’re reading this courtesy of techology which relies on imaginary numbers to exist.