On 7/17/2011 1:18 PM, Jason Resch wrote:
On Sun, Jul 17, 2011 at 2:54 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 7/17/2011 11:50 AM, Jason Resch wrote:
For Euler's identity to hold, Pi must exist in its infinitely
precise form, but Pi does not exist in its infinitely precise
form anywhere in this universe.
You don't know that, since space may well be a continuum (c.f. the
recent paper by Feeney et al).
Pi is a number, that space may be a continuum doesn't make this number
appear anywhere in the universe. We can point to two electrons and
say that is an instance of the number 2, but where would we see a
physical instance of the number Pi?
I didn't say I knew where there was a physical instance - I said you
didn't know that there wasn't one.
Ben believes mathematical truth only exists in our minds, but
does Pi really exist in our minds, or only the notion that it
can be derived as the ratio between a plane circle and its
diameter?
But that's the characteristic of mathematics, its statements are
notions and notions are things in minds. So there is no
difference between the notion of pi existing in our minds and pi
"really" existing in our minds.
Is there no difference between the notion of the moon existing in our
minds and the moon "really" existing? We say the moon exists because
it has properties which are objectively observable. Mathematics, like
physics i a source of objective observations and therefore part of
reality. What makes the moon more real than the number 5? If you say
it is because the moon is some place we can go to or see with our
eyes, then what makes the number 5 less real than the past, or that
beyond the cosmological horizon, or other branches of the wave function?
One thing that makes them different is that you can know everything
there is to know about the number 5 (as a place in the structure of
integers), because it is a concept we invented.
Pi is so big that its digits contain all movies and all books
ever created, surely this is not present within our minds,
Expressing pi as a sequence of digits is a notion in our minds.
That Pi takes an infinite number of bits to describe, and an infinite
number of steps to converge upon, is more than a notion in our minds,
it is an incontrovertible fact.
But that fact is a finite notion. It's a consequence of a
non-constructive argument.
The sequence is no more in our minds than is 10^10^100.
Pi is not special, there are many numbers which exists that are beyond
the physics of this universe. I consider this further evidence of
mathematical realism.
So you simply have adopted a certain Platonic idea of "real".
If you say a Googolplex exists, then where is it? There are not a
Googolplex things in this universe to count. Therefore if you think a
Googleplex exists, then numbers exist independently of physical things
to count. Even if there was a universe with nothing in it at all, the
numbers would still exist.
So you say.
but it is exactly what must exist for e^(2*Pi*i) = 1.
I disagree. For Euler's identity to hold just means that if
follows logically from some axioms we entertain.
There are other ways to prove Euler's identity, but for that equation
to be true, those irrational numbers (e and Pi) must be used with
infinite precision.
Only to check the equation by computing the value on a Turing machine.
"True" is just a value that is preserved in the logical inference from
axioms to theorem. It's not the same as "real".
I have two questions for you:
Do you believe Pi has an objective magnitude?
Depends on what you mean by "objective". I think "objective" means
"eliciting intersubjective agreement"; in which case I would say yes.
Do you believe humans know what that magnitude is?
In mathematical contexts, yes.
Brent
Jason
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