This is probably going to be an inchorent post but what the heck. I want to lift up the corner of a debate which has raged since the Ancient Greeks about the meaning of mathematics. This is partly motivated having watched (finally as Nilo would say) the film Pi. In it the claim is made that everything can be reduced to numbers and their patterns with the implication at least according to the protagonists view that numbers are the foundation of the universe and everything is constructed out of them. For those unaware of the plot the protagonist becomes obssessed with trying to find a pattern in the digits of pi to the extent that he ignores everything else and shuts himself away . Despite some wise advice by his tutor to give up the pursuit which he himself tried he persists but to no avail and eventually drives himself insane and it's only by taking a drastic measure that he is able to free himself of his obsession. All pretty bleak and I suppose the film was trying to make the point that such obsessions can be destructive. However without such obsessiveness a lot of maths that we know just would not have happened. Maurice Wiles locked himself away for seven years with minimal contact with the outside world whilst he worked on Fermat's last theorem. And speaking personally I couldn't juggle the demands of a full time job with a family life and pursue my interests in mathematics. Not that I'll ever contribute much to the field.

So obviously if people are going to make such sacrifices there must be something about mathematics which drives them. A lot of mathematicians (and I suspect Number theorists are probably most prone to this) are attracted by Plato's ideas. In a very brief nutshell Plato had the idea that to everything that existed there was an ideal form of which any concrete realisation was a poor copy. Thus whilst we can draw a triangle it will never be an ideal triangle but just a pale imitation of it's real form. The extension of this idea to mathematics as a whole is encapsulated in the view that when a mathematician proves a theorem he is discovering something about the nature of reality. Or to put it another way to every mathematical concept there is a corresponding element of reality. For these people if mathematics wasn't doing this then it would be a waste of time. Current Platonists include Roger Penrose, Max Tegmark and G H Hardy who expounded this view in his book a mathematicians Apology. On this viewpoint mathematicians are the new priesthood entrusted with finding the fundamental structure of reality and every advance in Maths brings us closer to the ultimate truth. There is a quasi religious zeal about all of this hence the reason I suppose why some mathematicians become obsessed with their work. Ok I've probably over simplified this viewpoint so I apologise in advance for what follows if I've oversimplified.

I'm afraid I don't share this view at all, for me mathematics is a construction which we impose on the world around us which more or less fits. The first philosopher to expound this view was Kant (or at least he gave the most coherent statement of this view). Kant building on ideas from Hume argued that it was impossible to step outside the way we perceive the world to describe the world as it is initself. Indeed one cannot separate the world as it appears to us from the perceptual apparatus that we use to perceive it with. Very crudely we can only see with a limited range of wavelength's of light, it is true we can extend this range by using different wavelengths, hence all the various branches of astronomy from Infra red right up to gamma ray astronomy. However the way the world appears in these different branches is still a function of the wavelength used to investigate the phenomenon. Nevertheless Kant argued that there were certain concepts (which he called categories ) which were necessary to make sense of the way the world appeared to us.

(This is further elucidation added 25th March) Amongst these categories was the world of mathematics. We are forced to use mathematics as a key means of structuring the world of our experience. That is mathematics forms part of what Kant called the syntthetic a priori. Kant made a distinction between synthetic and analytic

concepts. Synthetic concepts are those which add some empirical information to the concept whereas analytical concepts are those which are contained in the definition. Thus Chris is a bachelor automatically implies that Chris is unmarried this is an example of an analytical statement. Whereas Chris is a short fat failed physicist tells you empirical information about Chris which is more than anything that could be deduced from the fact that Chris is a bachelor this is an example of a synthetic statement . Mathematics according to Kant involves concepts which are more than just empirical information but on the other hand are not just analytical concepts.

In the early part of the 20th century boosted by developments in logic due to Frege and Russell, the hope was that mathematics could be reduced to purely analytical concepts. This was dealt a major blow by Godel's theorems which if I understand it correctly has shown that any axiomatic system sufficiently complex to include the Peano axioms of arithmetic (Notably Russell's Principia Mathematica) is either incomplete or inconsistent. By incomplete is meant that there are known mathematical truths which are outside the given axiomatic system. By inconsistent is meant that the system is obviously inconsistent. The sting in the tail as far as I understand it is that if you try to make the system complete it becomes inconsistent. So much for trying to reduce arithmetic to logic. Interestingly enough geometry and abelian groups do not suffer from this defect. Russell's early book the Principles of Mathematics is full of hubris about how he has defeated the Kantian idea that mathematics is some form of construction used to make sense of the world as it appears to us. In the light of Godel's theorem such hubris seems unjustified. (When I do M381 in a couple of years time I'll be in a stronger position to assess the validity of the above formulation any experts reading this please let me know if I'm incorrect). So given that mathematics is not purely analytic there is some constructive element to mathematics which isn't done justice to on a purely Platonist account.

I'm not denying the effectiveness of mathematics in it's domain of applicability and the fact that many systems in the world can be described by linear mathematics is remarkable, but one mustn't be seduced into thinking that linear mathematics for example is the ultimate truth about reality. Try applying linear mathematics to biological systems and you will end up with a completely different ball game. Neither am I saying that because mathematics provides one way of looking at the world it's no different from astrology or voodoo. Thus I do not want to be accused of post modernist relativism which seems to be the fear of those who take something akin to the Platonist viewpoint.

For me there is a compromise which falls between the absolutist claims of the Platonists and the subjective relativism of the post modernists which is that if a theory purports to model some aspect of reality then provided it is empirically adequate that is all that matters. Progress in science or mathematics is found by finding the limits of the applicability of that theory but that doesn't mean the usefulness of that theory should be discarded if say for example another theory comes along which incorporates the successes of that theory but extends the range of applicability. For example despite all the advances of quantum mechanics engineers still design bridges using Newton's laws of motion and I still design antenna's using Maxwell's equations. Obvously we now know that a lot of the metaphysical baggage associated with Newton or Maxwell's theories such as absolute space and time and the aether are no longer tenable. But their equations still work in the domain they were intended to apply to. Thus they are empirically adequate even if their metaphysics is somewhat dubious. (I'll expound more in another post). Clearly Voodoo or astrology is not going to help you design bridges or antenna's thus they cannot be seen as empirically adequate for these particular problems but unlike voodoo or astrology we know the limitations of the applicability of Newtonian physics or classical electromagnetisn whereas by definition the limitations of voodoo and astrology are just not defined.

The implication is that it just is not true that theories such as Newtonian mechanics or classical electromagnetism have been falsified as someone like Popper would claim. What is true is that the metaphysical baggage associated with these theories has been falsified, but in their domain of applicablilty they provide perfectly adequate representations of the world as it appears to us and are perfectly adequate for all intents and purposes to make concrete predictions about phenomenon which they were constructed to explain. Any attempt to lessen their significance in explaining the world around us because they are seen as approximations to the ultimate truth misses the point. I do not know for example how to model rigid body motion or design an antenna using superstring theory or whatever is alleged to be the current ultimate theory of reality. It really is not helpful to see Newtonian physics or classical electromagnetism as having been superseded by quantum physics. Obviously it is helpful to have found their limitations clearly the metaphysical baggage is no longer valid and if that is the case then that places a severe question mark against the metaphysical baggage associated with more fundamental theories such as say superstrings or M theory or their claims to provide access to a hidden reality behind the world as it appears to us. So the concept of empirical adequacy provides a reasonable compromise between the idea that theories in mathematics and science are some how providing the absolute truth about reality and that there is no difference between Astrology or Voodoo and science or mathematics. It also does justice to how science and engineering is practiced today.

Ok so where does this leave us. Mathematics is essentially a game of rules and axioms some of which may apply to the real world and others may not. It should not need something like Platonism to justify its existence as a human activity. It is essentially a puzzle solving activity (those who are into Wittgenstein will know where I'm coming from) and I would argue that getting hung up on how mathematics relates to the world involves all sorts of unresolvable problems.

Welcome to the Pi club. ( Good to hear you had first view of Pi. ) More comments another time.

ReplyDeleteDarren Aronofsky's latest masterpiece, Black Swan, is definitely worth a view. This time it's about a ballerina slowly going insane. Natalie Portman got an Oscar for her part.

ReplyDeleteThank you I will try and take in Black Swan when it's out on DVD

ReplyDeleteI've attempted to make the above post a bit more coherent by expoinding a bit more on Kant's view of mathematics. Whether I've succeeded only you readers can decide.

ReplyDelete