Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
fom
Posts:
1,035
Registered:
12/4/12
|
|
Re: Cantor's absurdity, once again, why not?
Posted:
Mar 14, 2013 2:38 PM
|
|
On 3/14/2013 5:43 AM, david petry wrote:
> On Thursday, March 14, 2013 3:17:06 AM UTC-7, fom wrote:
>
>> For example, Wittgenstein understood perfectly
>> well how to apply Cantor's argument and he
>> certainly is not thought of as believing in
>> a completed infinity.
>
>
> Here's an "acceptable" use of the diagonal argument: Given a provably well-defined list of provably well-defined real numbers (so that every digit of every number on the list can provably be computed), the diagonal argument gives us a new provably well-defined real number not on the list.
>
> Notice that that argument doesn't require the use of an actual infinite.
>
Right. I have a preprint of a text on enumerable sets written
long ago by Robert Soare. In the opening pages he discusses
the diagonalizability of total recursive functions. So, there
is an "acceptable" diagonal argument for mathematics restricted
to recursive function theory. Soare goes on to observe that
the total functions do not span every computable function
so that partial recursive functions are introduced.
>
>
>> However, he also did not attack the mathematicians
>> who conducted investigations along those lines.
>
> He most certainly did mock them.
>
Perhaps. I have a small book transcribing certain
lectures he gave to certain notable mathematicians.
It was civil. He gave wonderful explanations of how
natural language structure accommodates the typical
beliefs of mathematicians as they work with abstract
objects.
The problem is that "language game" theories do not
help to explain why a bridge of building does not
crash to the ground. Early pyramids do not have the
majesty of those built later at Gizeh. Once one begins
the process of using mathematics for real-world situations,
one begins referring to mathematical terms with the
same linguistic forms as one would with material objects
like chairs and tables.
Next, you have a 7 year old child asking "Why?"
Because physics has seemingly given up on explaining
"the observable universe" in favor of mathematical
abstractions, non-material physicalism is becoming
the only reasonable choice.
As for mathematics, I have no reason to support its
uses to discredit anyone's metaphysical beliefs. It
is perfectly plausible to treat mathematics along the
lines of Lesniewskian nominalism and regard its
essential value in the interderivability of its
facts. That is, because mathematics can be understood
in the sense of an Aristotelian demonstrative science,
there is no reason to confuse its facticity with
actuality.
More precisely, mathematical facticity relative to
the epistemic justification of a derivational system
is more important than mathematical actuality relative
to dialectical argument. Thanks to modern advances
beginning in the nineteenth century, it is possible
to divorce "essence" from "substance" in the
Aristotelian framework.
Modern logicians, however, try to distance themselves
from epistemology. Therefore, one has only the
Aristotelian distinction to guide this interpretation.
>
> FWIW, I was motivated to start this thread after reading an article you wrote about problems you had in your encounters with mathematicians who didn't accept your ideas. I'd like to know a little more about that, although I admit you really have no obligation to tell us more if you don't want to.
>
> The mathematics community has simply lost touch with reality.
>
>
>
I have no real issues along those lines except that
sometimes the personal frustration becomes somewhat
emotional. Professional mathematicians are busy
people. Most who might be interested in "pure
mathematics" have responsibilities that must be
met to justify the salaries they receive. On the
one hand, they are expected to teach. To the extent
that they receive additional time for their own
interests, they are expected to publish. That latter
expectation is demanded from the same people who
think quarterly profits are important to the security
of the endowments with which they are entrusted.
Because I had been unable to fulfill my own academic
goals, I can only hope that perhaps someone would
take an interest in my work. Given the reality of
the world, I probably have a better chance of winning
a lottery (which, of course, is less likely than being
hit by lightning walking in a grassy field during a
thunderstorm).
There is even more to consider. If common accounts of
Greek mathematics as the first mathematics that can
be considered "abstract" are reliable, then mathematics
is a 2500 year old subject. Any given mathematician
knows only a small part of what that entails and even
less of how mathematics is being practically applied.
Almost anyone who investigates a particular aspect
of mathematics that will require investigation into
the historical record will be surprised at what they
find.
In "A Mathematical History of the Golden Number,"
Roger Herz-Fischler writes:
"At first it seemed as if this
mathematical history would be fairly
short and straightforward. This
opinion was based on preliminary
reading not only of parts of the
Elements [Euclid], but also some of
the standard histories of Greek
mathematics. However, two things
soon became clear: the early Greek
aspect was not as clear-cut as it
was often made out to be and the
historical aspects that needed to be
considered neither started nor ended
with the early Greeks."
But, then if certain more informed accounts
are to be believed, I have mislead you.
In "Men of Mathematics" by E. T. Bell
we find the remark,
"[...] Finally we arrive at the first
great age of mathematics, about 2000 B.C,
in the Euphrates Valley.
"The descendants of the Sumerians in
Babylon appear to have been the first
"moderns" in mathematics; certainly
their attack on algebraic equations is
more in the spirit of the algebra we
know than anything done by the Greeks
in their Golden Age. More important
than the technical algebra of those
ancient Babylonians is their recognition --
as shown by their work -- of the necessity
for *proof* in mathematics. Until recently
it had been supposed that the Greeks were
the first to recognize that proof is
demanded for mathematical propositions.
This is one of the most important steps
ever taken by human beings. Unfortunately
it was taken so long ago that it led
nowhere in particular so far as our own
civilization is concerned -- unless the
Greeks followed consciously, which they
may well have done. They were not
particularly generous to their
predecessors."
How could anyone expect a handful of men
to be expert on the full array of
mathematical knowledge? And, how could
anyone expect men and women to be
different from the way men and women
have to be in order to live from day
to day and succeed in their personal
pursuits? I would be irrational if
I failed to recognize just how small
the small indignations I have experienced
really are.
Whatever difficulties I may have had
through the years, I was educated in
a respectable program. I respect the
people who study mathematics. I respect
mathematics as an academic discipline.
And, although one has to view one's own
interest in particular ways, I try to
respect other mathematician's interests
by trying to understand what they find
so interesting in what they study.
In view of my general isolation from
actively practicing mathematicians,
however, all of that effort is simply
personal effort to learn through
reading.
You may find some justifications
for your beliefs in comments you
might read that I have written.
And, I might sometimes be sharp
with my remarks to others. But,
you will not find me sympathetic to
your interpretations in this
regard.
|
|
|
|
