On 26 Jul 2010, at 18:22, Brent Meeker wrote:
On 7/26/2010 6:24 AM, Brian Tenneson wrote:
Does this mean that sets of numbers are inventions or just
particular numbers are inventions?
If the latter, then there must be a largest number which is, to me,
counterintuitive.
Numbers existed before 10,000 years ago when they were first
understood by humans to some extent. There was a specific number
of atoms in the universe one day before any numbers were understood
by humans, for example.
But there weren't an infinite number of atoms (or anything else).
But arithmetical realism does not ask for an infinite number. All
finite numbers is quite enough. You need only to believe that
statement like
s(s(0)) + s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) =
s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) (commonly written 2
+ 13 = 15)
are true independently of you, matter, universe, bibles, etc.
No theorem of math, even of intuitionist math makes any sense, without
such belief. You can threw Pythagorus theorem in the trash.
Only ultrafinitists are not arithmetical realist. But it is impossible
for them to say so, because the cognitive abilities you need to say
that you do NOT believe in arithmetical realism needs arithmetical
realism.
Bruno
http://iridia.ulb.ac.be/~marchal/
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