Tuesday, 22 July 2014
On the Nature of Mathematics
Hands up, I'm way outside my sphere of expertise here: I'm no mathematician, physicist or epistemologist. I know enough to be dangerous, but not enough to make any ground-breaking contributions. So why read any further? Well, I have something to say on this which has been worming around in my mind for years, sometimes peeping into my consciousness for a fleeting moment, before vanishing back to obscurity. But it's now well enough formed to describe as a starting point, if not with any degree of eloquence. And even if I haven't made any breakthrough, let me perhaps lay a cobble along a road which may be interesting to walk. Please be patient...
I have long thought about the nature of mathematics in relation to physics. The two disciplines are closely linked, particularly at the limits of today's advanced cosmology and particle physics. Many physicists and mathematicians have marvelled at the predictive power of mathematics, at the way that theories can be synthesized mathematically into structures which suggest discoveries to be made experimentally, if only to realise some perceived 'beauty' by completing an elegant mathematical structure. When lo and behold the discovery is made, theorists marvel at mathematics and its pre-eminence among the intellectual disciplines.
Some go further still: Roger Penrose holds mathematics to be the reality of nature, and if that wasn't enough, that mathematical concepts have a metaphysical existence in the universe independent of mathematicians, as if Pythagoras's Theorem was floating in the ether for the Greeks to discover and document.
I have long held mathematics to be a human construct which represents the world around us, and not some disembodied mystical entity. I have found every other metaphysical construct to evaporate under the harsh light of critical examination, and I have no patience to entertain disembodied equations emerging from the Big Bang! But I see a problem: my view of mathematics as a construct just doesn't fit with the predictive power mathematics has proved to wield. The discovery of the Higgs boson was a triumph of the predictive power of mathematics, and one which has not sufficiently been heralded in my view. So there must be something more to mathematics than just a toolbag of strategies for solving practical problems.
There is something transcendent about mathematics. If you know what a materialistic skeptic I am, you'll appreciate the enormity of that statement. Solving equations, as I do from time to time as an engineer, feels like refining truth - cancelling terms feels like spooning off the dross from the ever more pure and precious metal sought. The formal proofs of theorems are eternal - once proven they are never broken, and reveal their truth for eternity. There is some kind of magic in mathematics, but I just can't follow Penrose down his metaphysical road. That way lies madness!
I also have bags of humbug for the ancient Greek philosophers. Hemlock wasn't Socrates's only herbal vice: just what was he on when he came up with the Allegory of the Cave? So I'm more than slightly embarrassed that my resolution to the problem of mathematics has certain similarities to his shadows on a cave wall.
While listening to back issue podcasts of The Infinite Monkey Cage a few days ago, with Brian Cox perhaps stating as final that mathematics is truth, while Robin Ince teases him on multiple levels simultaneously, which you only realise are much more clever than at first appears some time later, a thought popped into my consciousness, and decided to hang around.
The thought was: "there is a structure of underlying truth to the universe which we hairless apes are not adapted to comprehend, but parts of which are projected onto our limited consciousness, and the shadows formed are what we call mathematics".
Sitting there, like a mischevous imp at the corner of my mind, that thought cast off other thoughts. I thought of the schematic map of the London Underground. When laid out geographically, the tube network is fiendishly complex. But the schematic representation just shows what we need to know to plan a route from A to B, and where to change lines. It's a functional representation of London, but it's not actually London. So if the underlying truth is like London, but we only have parts of a schematic tube map, there are limited things we can know about London, (er, I mean truth). We know schematically that the Jubilee line crosses the Circle line twice, and if we know that in real London it crosses at one point, (we have solid experimental evidence for one physical law), then we can infer from the rules of topology that it must cross somewhere else (and make a prediction to test experimentally), even if we've never been to Baker St.
There are truths which are so obvious to us that they seem pointless to express: like the number 2 is half of 4, and sits neatly between 1 and 3. Perhaps if we were not adapted to life as apes, but as supreme logicians, Pythagoras's Theorem would be similarly trivial, and unworthy of a name. So perhaps there is no need for a disembodied metaphysical law of right triangles in the universe, right triangles just are the way they are. And it's not obvious to us because we don't have the right kind of minds to appreciate it, and have to construct formal proofs instead. These proofs seem so magical and powerful to us, that some of us think they have a special existence, but that's just an illusion born of our limited perception. And perhaps the behaviour of waves and particles, and spacetime, and the unity of forces, are all logically deducible, if only we could perceive the logic so clearly.
So we build pieces of a reality map through our reasoning and by our observations, and call these pieces laws and theorems. But these laws and theorems are our constructs, our inventions to account for the way the universe is, to steer our ape minds to conform for a moment to the truth of reality, while the universe just goes on being what it is without any need for such trivia.
On this view then, mathematics really is the projection of reality onto human consciousness. And as the contours of our consciousness change, so do the mathematical strategies we use. When I learned basic number theory as a child, I used abacuses to count-on and perform basic addition. My children were taught the number line, which is a different concept. So their mathematics will be different to mine, not because truth is different for us, but because their consciousness of number is different from mine.
What can this idea tell us we didn't know before? Well it does suggest that there may be limitations to what we can discover. In terms of the analogy, there may be areas of our consciousness which our cerebral topography keeps in mathematical shadow, corresponding to universal truths we can never comprehend. But who knows, if we can find where these conceptual gaps lie, perhaps mankind's perseverance at solving problems will find routes around these gaps, allowing us to solve theoretical and practical problems regardless. Quantum theory could be one of those gaps - we just do not have minds equipped to understand the world on such small scales, but we have mathematical strategies which allow us to skirt the edge of our blind spot and solve quantum mechanical problems anyway. We've done rather well for ourselves, don't you think?