Ali’s blog

Mostly quant stuff with occasional digressions

The Mathenauts

Posted by alifinmath on April 19, 2009

An interesting question for every mathematician is the ontological status of the various mathematical entities that he is obsessed with, that he plays with in his mind, that are the focus of his being. Most mathematicians, if pressed, will admit to being Platonists: their belief — if not conviction — that the mathematical entities have an objective existence of their own. Somewhere. And that this existence is not inferior to mundane day-to-day reality, which is often an unreal shadow world to many research mathematicians. Thus, a Riemann surface, a moduli space, a tensor category, or the monster group are just as real — if not considerably more so — than some material object or sentient being.

In 1964, Norman Kagan attempted to give flesh to these musings with an SF short story, “The Mathenauts.” It is unfortunately not available online. I read it about twenty years ago and it had quite an impact on me at the time. A mathematician looks at mathematical objects on their own terms, not looking for some raison d’etre in terms of applications; of course, if the ideas are any good and of enduring worth, they arise by abstracting some element or aspect of the “real world” — but once this is done, they develop a life and dynamic of their own, deriving their sustenance from the axioms that grant them immortal life. This is something physicists and engineers never understand, and their approach to mathematics is more mercenary, more slip-shod, more expedient. For them clarity, clear definitions, proved relationships, the architecture of a theory, the limits to what can (or cannot) be asserted are of no fundamental importance. They will not examine the mathematical universe and the objects therein on their own terms, as possessing an independent life of their own, and deserving of minute and careful study. They are Aristotelians, not Platonists.

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