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From: Math1723 on 13 Jan 2010 14:47
On Jan 13, 2:28 pm, Leonid Lenov <leonidle...(a)gmail.com> wrote:
>
> While Intuitionist logic is mathematically correct it is just silly...
> Their "problem" is that they cannot perceive mathematics separated
> form the mathematician hence all the methods used, according to them,
> must be within human capability. Platonist has no such problems :).

Exactly. Just as a gorilla cannot fathom some of the deeper meanings
of Differential Calculus, certainly there are much higher order truths
of mathematics which cannot be comprehended by the human brain. But
that doesn't make them any less truth.

The Fundamental Theorem of Calculus was still true when the dinosaurs
roamed the Earth and no humans were around. Deeper truths may forever
elude us (but for me, the journey is the reward).

> A typical example of their logic is this:
>    a_2n=0 and a_{2n+1}=1 if Goldbach conjecture is true or a_{2n+1}=0
> if Goldbach conjecture is false
> They maintain that since we do not know if GC is true or false that we
> cannot say:
>    "a_n either converges or does not converge"
> Of course, most of us "know" that the previous sentence is true. What
> their "problem" is, is that they identify truthfulness with "proof-
> ness".

Yes, this is the most compelling Platonistic argument for me. Another
is: Let d = the googleplex-th digit of the decimal expansion of pi.
Now let the proposition P be "d = 2". Is P true or false? This is
trickier since there is a known procedure for calculating the nth
digit of pi, so it's not in the same "unknown" state that the Goldbach
Conjecture is. The problem is, calculating d would likely take much
longer than lifetime of the human species. So here we do in fact have
a decision procedure, but one which cannot be completed in practice.
Yet, I say with certitude that P is one or the other: true or false.
(I presume an intuitionist would say that it is neither, until the
proof is completed.)
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