Pointless diversion of the day ... Countably infinite.

3개월 전

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I broached this question a long time ago, and I've learned since then that, with appropriate expression, it is an open conjecture in number theory. It's basically this: The set of numbers that are individually specifiable by any mathematically symbolic means is countably infinite.

Or to put it in the negative: the set of numbers that man can ever specify by a finite number of symbolic means no matter how large must remain only countably infinite. That would imply an uncountably infinite number of numbers that must escape human specification.

Now, Platonists among us, what are we to think of the reality of such real numbers forever unspecified? Forever unspecified does not imply, necessarily, forever unspecifiable. But maybe that stronger condition also holds.

Pope Benedict has said that numbers are human constructs; I guess he is Aristotelian in that regard. Yet when mathematicians make discoveries, they speak and think as if they are discovering real properties of real though immaterial objects, and not just unfolding the implications of a man-invented system or game. And I think that it isn't just a stray issue for philosophers, since the natural world in large part "obeys" laws that can be mathematically expressed. But a law is ontologically prior to what obeys the law; unless we try to dodge the implication, that immaterial and immutable objects exist, by "folding" the law into what obeys the law, and saying -- not altogether plausibly -- that there is no difference between them. But that too appears to me to strike a dagger to the heart of materialism.

Staying up too late tonight ...

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