On 7/17/2011 2:35 PM, Jason Resch wrote:


On Sun, Jul 17, 2011 at 3:37 PM, meekerdb <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:

    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.


That's fair.





            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.


My question was what makes 5 less real than those other concepts.

Also, I would disagree that we know everything there is to know about 5, there are an infinite number of facts about the number five and we do not know all of them. For example, there was a time when humans knew 5 was between 4 and 6, but did not know that 5 is an element of the smallest pythaogream triple.

Of course our present view, since Peano, is that the natural numbers are a structure and so within that context 5 has infinitely many relations. But when you know it is the successor of 4 you in principle know everything there is to know about it. Note that I wrote "can know", not "does know".






            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.



It sounds as though you are saying I can provide a finite description of how to compute Pi, and thus define it without having to actually execute its infinite steps on a Turing machine. Is this an accurate statement?


         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".


Are you saying numbers like 10^10^100 do not exist?  Are you a finitist?

I think if one is not a finitist, they must a platonist.



    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.


That is the conclusion if you believe 10^10^100 is real.





            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.



For the left hand side of the equation to equal 1 and not some other number, the exact values must be used. I don't see how to get around that.

Then are you claiming that Euler's identity has not been proven because nobody has calculated the numbers on the left side to infinitely many decimal places?


The equation doesn't require validation by a Turing machine to be true, any more than a turing machine has to validate 1 + 1 for it to equal 2.

      "True" is just a value that is preserved in the logical
    inference from axioms to theorem.  It's not the same as "real".


True is more than inference from axioms. For example, Godel's theorem is a statement about axiomatic systems, it is not derived from axioms.

Sure it is.  It's a logical inference in a meta-theory.


Objectively true or false statements are properties of objective objects.

No, I can make true and false statements about objects that don't exist. For example, "Sherlock Holmes smoked cigars." is false and "Sherlock Holmes lived on Baker Street." is true. I don't think statements are properties of objects anyway - whether true or false or indeterminate.

What leads you to label only some of these objects as "real"? What does this label add to any object's definition?

"Real" is not a word I generally use (except to indicate part of a complex number). It seems to be a kind of honorific that some people think is important to bestow: on numbers, on God, on atoms,... The conventional use is that it applies to things we can interact with, and since language is mostly convention that suits me. If you want to apply it to numbers then I suggest you come up with a different word so we can still distinguish numbers from tables and chairs.





    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.


Okay, we are in agreement on this.


    Do you believe humans know what that magnitude is?

    In mathematical contexts, yes.



I think we have only approximations to its magnitude.

Depends on what you mean by "have". We have simple, exact expressions for it, which we can use in inferences. I'm not sure what would be added by having a decimal (or binary) representation of it.

We don't know what the 10^10^100th digit of Pi is; it may be physically impossible to determine in this universe. Yet, it must have a specific value, which if it were different from, then e^(2*Pi*i) <> 1.

When you say "must" you really mean "It follows from the axioms and rules of inference we have assumed". If we assumed other axioms it might come out that e^(2*pi*i)=1 is indeterminate or that "=" only means "the same for the first 10^10^100 digits".

Brent


Jason
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