I got my first tangible reward for my conscientious blogging yesterday: Dinesh had commented on my Simulators post, recommending a Greg Egan book called Luminous, and I bought it. An introduction to a new book: what could be more valuable? Thanks, Dinesh :-).
The book contains a short story of the same name. It’s only about 30 pages long, and by normal standards, I would have polished it off in a single sitting. However, a few pages into the story, the author makes such an incredibly audacious claim that I was completely shell-shocked, gob-smacked, and had to keep the book down and think for several hours before I can get back to it.
What Egan claims is: there is no such thing as pure mathematics.
This is as bold as it gets. Mathematics – isn’t its purity something that we were always brought up to believe in? Men may come and men may go, but maths goes on forever.
Even xkcd concurs:
And Egan disagrees. I love this guy!
According to Egan, even the most abstract theorems in Mathematics – Fermat’s Last, for example – have some physical basis. This may be a property of the real world in some cases (for example geometry modeling the real world). But even the so-called ‘abstract mathematics’ – even pure logic – has a physical basis; it exists as a sequence of thoughts in human brains, and therefore as a sequence of neurotransmitters, perhaps, that fire in a particular sequence in the brain. Alternatively, they exist as patterns of charge and current in a semiconductor wafer; but the point is, whatever the manifestation, there is a manifestation.
We are able to see the truth or falsity of mathematical statements without recourse to experiment because of a series of chemical actions and states in the brain.
So if our brains were wired differently, would we perceive mathematical truths differently? Likely, but it’s not that simple. Animals, for instance, could be said to perceive mathematical truths similar to us (though their deductive and inductive capabilities are much lower than ours).
So what would have to be different for us to perceive maths differently?
Egan asks the question a bit differently. Where, he asks, did mathematics come from? Did it exist before the big bang? Or, shortly after the event, did the early quarks and gluons ‘create’ mathematics? Shortly after, he says. And if physics were different, maths would be different.
I am enormously intrigued by Egan’s suggestion. I want to believe it is true – it appeals to my engineering-centric conceptualization of the universe! But I also have my own reasons for believing that pure mathematics is not so pure after all.
The crux of my opinions around this was developed around an investigation of the way we perceive infinity. We are able to assert (and mathematically substantiate) that a statement like the ‘infinite primes’ theorem (that there are infinitely many prime numbers) must be true. Certainly we do not have the physical experience of infinity in order to establish the validity of this theorem by experiment. How, then, are we able to prove it?
The analogy to explain this is that of a chess puzzle. Say we are given a fairly complex chess position, and are asked to prove something about it; for example, to prove that black can never win from that position. Prima facie, it may appear that such a proof is impossible. Since we do not have a limit on the number of moves that are allowed, even if we comprehensively list all possible moves starting from the current position, we cannot eventually exhaust the infinite length of time for which the moves may go on. What is to say that black will not check-mate in the trillion-trillionth move?
Cast this way, the statement cannot be proved. However, there is another way of establishing the proof. On a chessboard, with a finite number of chess pieces, there are only a finite (though large) number of positions possible. Since any valid chess move is one of these positions, a game of chess is then a way of moving from one chess position to another. For a finite number of positions, there are a finite number of interconnections between the positions. To prove something like the ‘black will not checkmate’ hypothesis, therefore, we do not need to reason infinitely; we need only check (and this can be done with a computer in a finite amount of time) whether it is possible to go from the node representing the current position, to one of the nodes representing a white check-mate. If such a path exists, black can win.
The cleverness here is that we have converted the infinity of chess moves into the ‘finity’ of chess positions. This is the same process that works at multiple meta levels in mathematics to allow us to make general statements about infinite sets of mathematical objects. Because underlying the infinity of mathematical operations, there is the equivalent of a ‘finite chessboard’, which is a combination of all the axioms and rules of inference that our minds accept as true, and which we manipulate to go from ‘truth’ to ‘truth’ in logical sequence.
The frightening question is: is this ‘chessboard’ a feature of neuroscience and physics?
Because if it is – and this is what Egan’s story seems to be about – then ‘pure’ mathematics has the same clay feet as the rest of human knowledge, held up by a complex scaffolding of experimental physics.
I’ll get back to finishing Luminous now…