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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