27 July 2008
Mathematics and Reality
Today I'll give a three-part answer: no, and no, and maybe. The "maybe" is the interesting part.
1. I don't think there's any broad case to be made for a Platonic treatment of universals as such.
2. Nor do I think its helpful to morality to think of virtues as disembodied essences -- Justice, Courage, Temperance, the Good.
3. On the third hand, there IS a strong case to be made that numbers and the equations made from them ought to be considered as an independent world, which mathematicians discover rather than invent.
This case is based largely on the experiences reported to us by the greatest mathematicians since the days of Pythagoras.
Heinrich Hertz put it well. "One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them."
Who was Hertz that we should pay heed to his views on math? A physicist who obtained his Ph.D. from the University of Berlin in 1880, and whose work helped refine the mathematical theory of electromagnetism developed before him by Faraday and Maxwell. He earned his bones in terms of the relationship between physical reality and abstract formulae employed to describe it. If he believed (as have many others of equal eminence) that the formulae have an independent sort of existence, this isn't testimony that those of us who have shown less facility in their use than he should take lightly!
Knowledge is warranted belief -- it is the body of belief that we build up because, while living in this world, we've developed good reasons for believing it. What we know, then, is what works -- and it is, necessarily, what has worked for us, each of us individually, as a first approximation. For my other blog, on the struggles for control in the corporate suites, see www.proxypartisans.blogspot.com.