[FOM] Sharp mathematical distinction between potential and actual infinity?
Timothy Y. Chow
tchow at alum.mit.edu
Fri Sep 26 10:49:33 EDT 2003
Years ago, Stephen Simpson proposed a list of "the most basic mathematical
concepts," which included number, set, function, etc. I wonder if
"infinity" belongs on this list?
It seems to me that at minimum, "potential infinity" of some sort is basic
to mathematics. Consider the following caricature of a debate that has
been rehearsed innumerable times on the FOM list and elsewhere:
P: The concept of N, the set of all natural numbers, is crystal-clear.
F: No it isn't. For example, first-order PA has nonstandard models.
P: Huh? By appealing to mathematical logic, you betray your belief
that the concept of a *rule* is clear, and N is just as clear as
the concept of a rule.
F: Nonsense. A rule is finite, or at most potentially infinite, but
N is a completed infinity.
I confess that my sympathies lie with P here; I tend to think that
if we take skepticism towards N seriously, then we also need to take
"Kripkensteinian" skepticism towards rules seriously, and this leads
quickly to a wildly, and in my opinion unacceptably, ultraskeptical
view of virtually all mathematics and logic.
However, I might change my mind if someone could demonstrate a sharp
distinction between potential and actual infinity. The distinction
seems to have pretty much evaporated in modern mathematics, and it
seems that only the philosophers still talk about it. Or am I just
underinformed? Is there, for example, a way of drawing a clear
mathematical distinction between potential and actual infinity that
blocks the move from skepticism-towards-N to skepticism-towards-rules?
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