On 7/17/2011 2:35 PM, Jason Resch wrote:
On Sun, Jul 17, 2011 at 3:37 PM, meekerdb <[email protected]
<mailto:[email protected]>> wrote:
On 7/17/2011 1:18 PM, Jason Resch wrote:
On Sun, Jul 17, 2011 at 2:54 PM, meekerdb <[email protected]
<mailto:[email protected]>> 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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