On Sun, Jul 17, 2011 at 7:55 PM, meekerdb <meeke...@verizon.net> wrote:

> **
> On 7/17/2011 2:35 PM, Jason Resch wrote:
>
>
>
> On Sun, Jul 17, 2011 at 3:37 PM, meekerdb <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> 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".
>


Perhaps having infinite time and resources we could come to know everything
about 5, but if you admit the possibility that this universe does not afford
us the infinite time and resources necessary to know the infinite set of
relations concerning number 5, then the number 5 cannot be fully known (at
least by us).  What state of existence should we ascribe to these
undiscovered, perhaps undiscoverable, properties of 5?  If we don't know
everything about 5 is it truly our invention or are we just discovering
things about it piece by piece?  If there is more to know about the number
five than there is to know about the observable universe then to what does 5
owe its reality?  Five would, in a sense, be larger than the universe,
larger than us.  It seems arrogant then to believe we are its inventor.



>
>
>
>
>
>>
>>
>>>
>>>
>>> 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?
>
>
No, it has been proven, but the equation would not be true, (the left hand
side would not evaluate to 1) if the correct values for e and Pi were not
used.  These constants have objective (non human-invented, and non
human-known) values.  We know only a few terabytes of the infinite amount of
information required to describe the magnitude of Pi.


>
>
> 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.
>
>
What are the axioms of this meta-theory?  Are they things like x = x, and ~p
& p = false?  You might have to give up logic and rationality to coherently
deny the truth contained in this 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.
>

(I think you meant to say "about objects that don't exist physically")


> For example, "Sherlock Holmes smoked cigars." is false and "Sherlock Holmes
> lived on Baker Street." is true.
>

Sherlock Holmes is a real fictional character, not a real person who lived
in the history of our world.  Likewise, 5 is a real number, not a physical
construction.


> I don't think statements are properties of objects anyway - whether true or
> false or indeterminate.
>


If 5 is odd, and prime, and less than 6 and everyone agrees these are true
statements, then what are these statements talking about if not some
object?  Do these properties not need to be ascribed to something?


>
>
>  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,
>

According to this definition, the past is unreal, that beyond the
cosmological horizon is unreal, and other branches of the wave function are
unreal.


> 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.
>

Instead, I think what is required is a word to distinguish things like
tables and chairs from the past, beyond the horizon, other wave function
branches, and mathematical objects.  Perhaps "reciprocally changeable" or
"mutually affective" captures this ability to change and by changed by such
objects.  Whereas something like the past, or mathematical truth can effect
us, it does not seem like we can affect it.  The scarcity of 5's factors
might cause a mathematician to conclude 5 is prime, but a mathematician
can't cause there to be more factors.



>
>
>
>
>
>>
>>
>>
>> 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.
>
>
The expressions for deriving Pi all require an infinite number of steps to
determine its value.  If you are saying the expression is all that is
needed, and we don't have to actually execute the infinite number of steps
on a Turing machine, then you should also thank Bruno for creating the
universe when he wrote the Universal Dovetailing software.

If you think Pi exists independently of our derivation of it, then why not
other infinite and recursive mathematical definitions such as the UD?


>
>  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
>


Which axioms or rules of inference do you disagree with?  If you throw away
logic, reason, and math as we know them, what would become of our physical
theories which you hold in such high regard?  If you want to create your own
axiomatic system where 1 + 1 <> 2, you are welcome to it, but I doubt you
will convince anyone else to use it.



> or that "=" only means "the same for the first 10^10^100 digits".
>

This sounds like an axiom of an ultra-finitist theory.  It also would
quickly lead to inconsistent statements, such as 1 = 0.  For example:
A = 1, B = 0
Then if we add a sufficiently large number to both A and B, they will become
equal.  This would likely conflict with any useful axiom of addition.

I think given the statements you have made throughout this thread, you are
not a finitist.  You accept huge numbers and objective values of Pi.  The
only thing we disagree on is the conventional meaning of the word "exists".
You associate the term with something that you can in principle both affect
and be affected by.  I use a slightly looser definition, which is anything
which has objective properties.  Although you accept the existence of an
objective value of Pi, or numbers larger than can be represented or accessed
in this universe, you seem reluctant to call yourself a Platonist.  Is this
because mathematical statements can trace their truth to axiomatic systems?
What about the mathematical statements that are true but not provable in the
axiomatic system?  To me, this suggests that axiomatic systems represent
human attempts to formalize mathematical truth (they should not be
considered the source of mathematical truth).


Jason

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