Discover or Invent ?

One of the quesions that comes up is whether mathematics is discovred or invented. Like many mathematicians, I am firmly on the side of it being discovered. There are, of course, cases where it's fair to use the language of invention – for example novel cryptographic algorithms – so it is certainly fair to say that in some cases mathematics gets invented; but even here there is a strong sense in which the invetor is just discovering a mathematical form for which she can see a practical use. To be fair, of course, I must also point out that, at some level, much mathematics is invented, as a model to use for describing something; so it is perhaps better to say that the question is silly, inviting the respondent to make a distinction that's not really applicable. In some sense, for mathematics, the distinction between discovering and inventing is meaningless.

This question tends to come up, also, in the context of the remarkable efficacy of mathematics at describing the physical universe. At the simplest level this is so astonishing it might be counted miraculous. Then again, to anyone familiar with even the beginings of the vast range of mathematical structures it is possible to construct, it is immediately clear that the range of universes that could be described by at least some mathematical model is, in fact, so huge that it is quite unsurprising that there do exist mathematical models that describe our universe well.

In answer to that, though, it may fairly be objected that we have, in fact, remarkably simple (in suitable senses) mathematics that describes the universe we in fact experience; furthermore, there is a tolerably clean progression from the simplest mathematical models that described straightforward aspects of our reality well. For example, Euclidean geometry describes the relationships between property boundaires to far better precision than actual farmers need; it served as a viable foundation for Newton's mechanics, which in turn could be extended to encompass a respectably good approximation to how electromagnetism works – all of it beyond Euclid's wildest imaginings, yet tolerably well fitted by it. There is a natural sense in which the theory of smooth manifolds grows out of Euclidean geometry (just scrap his fifth postulate); and this theory suffices to give us Einstein's General Relativity, which delivers a considerably better description of both electromagnetism and gravity. Likewise, Euclidean geometry (via a different extension, to vector spaces over the complex number) supplies the theory of Quantum Mechanics, which serves to describe large parts of what Einstein's theory fails to reach. So it's not just that mathematics does supply good theories to describe the universe: it furthermore provides a theory that's easily understood from a very straightforward understanding of something very simple in our world, which does serve as a good foundation for significantly better models of far more of how the universe behaves than is countenanced by what could be countenanced by those developing that initial understanding.

This may fairly be considered somewhat remarkable: the promiscuous diversity of conceivable mathematical models, that would suffice to describe a vast range of universes utterly different from ours, suffices to say that some mathematical model could be developed to describe the immediate phenomena that denizens of such universes would experience; but it offers no firm assurance that the mathematics that works well for those phenomena would even vaguely resemble the mathematics needed to describe the system in its finest detail; nor, even, that some mathematics based on what worked for primitive phenomena would suffice to get even vaguely close to a full and rich characterisation of the aspects of such a universe other than those specific phenomena. The problem here is that, in the vast range of possible mathematics, most systems are so weird and complex that a denizen of a universe described by such a system would never see any chance of discovering that it was so, since the primitive phenomena they could analyze would all be well tolerably described by other, quite unrelated, mathematical models, that would give no clue to where to look to find a fuller theory.

The one exception to this would be if the universe at large were, in fact, governed by remarkably simple rules (as, indeed, ours is), so that its specialization – to the phenomena that would first present themselves to any intelligence capable of beginning the journey towards discovering even a tolerably good description of the whole – would necessarily itself be at worst a simplification of those simple rules. This arises because a simple system's specializations are mostly (although by no means exclusively) simple. So we come back to the remarkable simplicity of the mathematical models we have found, that suffice to describe our universe remarkably well.

Even so, the range of possible mathematical models is really vast. For any given abstruse and complicated model that describes a universe, in which the simpler sub-systems are better described by models entirely unrelated to those of the abstruse and complicated one that's the full description, it is entirely possible that there is some model, that would seem straightforward and simple to the denizens of that universe, that is either isomorphic to the abstruse and complicated one (and hence a simpler but perfect description) or models it to a greater a degree of accuracy than our present physical observations can discern. As long as some such theory does do a respectably good job of describing whatever universe one inhabits, there shall be more straightforward (as perceived by denizens of that universe) aspects of such a universe that (again, because many specializations of a straightforward theory are necessarily straightforward) would be adequately well described by models that would indeed serve as good start points form which to develop the more precise (even if not perfect) theory of which they are specializations. So there may well be a significant subset – potentially even almost all in the measure-theoretic sense – of universes mathematics would be cabable of describing, in which there are simple or straightforward (from a native perspective) models as good as we have for our universe.

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