NP Alabama Pi

[desert search for techno allah]

jporter writes:
>> Even believing that physics and pure math don't intersect (which I
>> think pretty much no one believes anyway), you're not forced into
>> metaphysical explanations. At least, not immediately - how much
>> the various solutions to this problem result in metaphysical
>> hooha is a matter of debate.
>
>If you believe in that intersection then the next question is: what is the
>product of that intersection?

I'm afraid you'll have to explain your question a bit more. Because
I have an obvious answer for you: open up a physics book, and look
down the pages for equations. These equations talk about physical
quantities and their measurements, etc., but stripped of their units
they are equations of mathematics, which probably require at base
real analysis (one of the two bases of most pure math done, along
with algebra) to make sense.

>> Wittgenstein's solution (that statements of mathematics are "rules of
>> syntax"), for one, appears to avoid metaphysical quandaries without
>> even caring whether or not there is a "pure" mathematics separate
>> from physics.
>> 
>
>Syntax are rules. It's the same wolf, naked. At some point there is no form
>independent of content.

Can you explain this some more? I don't get it.

>> There are definitions which don't rely on geometry.
>> 
>
>Whichever method you prefer for generating pi, and then decide is a "new"
>definition, especially as that process becomes more and more abstract and
>removed from the original conception of the ratio of the circumference to
>the diameter of an ideal circle, it only tends to make the point clearer
>that pi is an invention of human beings, not a discovery of a absolute
>perfect ideal, independent of human evolution and history.

This is a little bit disingenuous. Let me try to clarify what I mean.

I agree that pi, like all math, is not some eternal, unchanging ideal
which is untouched by human history. However, in most informal
presentations of the problem of certainty in mathematics, that
position (essentially, the Platonist one) is aligned with the
one that says truths of mathematics are always true. The position
you're taking, where pi is essentially mutable because it is only
a human construct, is usually aligned with the position that mathematical
truths are not necessarily always true.

I want to stake out a middle ground. Neither of these positions tells
the whole story. I won't admit any sort of platonic mathematical ideals,
or Kantian arithmetical and geometrical intuition. But that doesn't
make mathematical truths uncertain. It's the nature of mathematics
that its truths are certain - in the sense that we don't "allow"
math to be done with false statements. This is what I meant earlier
by "rules of syntax." Mathematical statements guide which further
mathematical statements are acceptable. In a very pointless sense
the digits of pi are not fixed _because mathematics is a human endeavor_,
but if we were to allow any changes in any of its digits, say we
passed a law or something, a la the original post in this thread,
then ENORMOUS portions of mathematics would suddenly be invalidated -
and it wouldn't be a simple little fixup either, you know, "OK,
start doing things this way."

>All the numerical techniques for generating more and more digits, are
>inconceivable without the original geometrical conception. You can
>hypothesize the possibility of a mathematics which invented a specific
>transcendental number out to 210 billion decimal places- for non-geometric
>reasons of its own- but that is speculation, and not the sequence of events
>which actually occurred. Such is not how the concept of pi entered into the
>culture.

What you're talking about is history of mathematics, though, and no
longer epistemology, which IIRC was the original thrust of this
discussion.



Josh

-- 
josh blog: http://www.public.iastate.edu/~kortbein/blog/
      tdr: http://www.public.iastate.edu/~kortbein/tdr/