[FOM] The Gold Standard
Eray Ozkural
examachine at gmail.com
Thu Feb 23 05:42:11 EST 2006
Dear Professor Friedman,
Please consider my following comments.
On 2/23/06, Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> Already it can be perfectly well argued that the series of natural
> numbers is already Platonistic. In fact, one can already argue, if one
> wants, that the number 0 is Platonistic.
I do not see how that follows at all. The series of natural numbers
may be explained perfectly well by at least two other schools
of mathematics: formalism and intuitionism. I prefer psychologism
(cognitivism): The natural number is a psychological abstraction,
it is something that exists purely in your brain. Although in the
real world there are no such things as numbers, there are numbers
in one's brain, and they are quite useful, including the number 0 which
usually means the lack of objects in a discourse. (Also helpful
is the discussion about grammatical mirages, we often
mistake our abstract thoughts for things that do not exist in the
physical world.)
In many of the recent discussions, I have seen platonism
assumed and then platonism concluded. This is an expected
outcome. The main thrust of Weaver's philosophical analysis
cannot be dismissed by mere assumption of platonism. More
would have to be said. One such approach would be an
indispensability argument, e.g., we cannot even formulate physics
without being a real number realist. Unfortunately, that argument
has weak foundations, because for instance we already
know that all of quantum physics can be formulated in computable
mathematics. Likewise for several other physical theories.
In the failure of indispensability, one might try the approach of
uniqueness / independence, which is something that Godel
tried. However, that approach is also flawed, at least for the fact
that there are now many mathematical formulations of the concept
of "set", and unfortunately none of these formulations can be accepted
as a golden standard, or the "true" set theory from a philosophical
standpoint. They are merely competing theories, and there is no
winner. The formalist point of view is indifferent as to the reality of
each of these competing theories, and perhaps it is healthier this
way, since nobody has yet observed a set or a number in the
physical world, let alone an infinite set.
It may be also possible to work out Godel's explanation of
mathematical reality. According to him, mathematical facts
resided in a second order of reality. If I understood him correctly
he thought that the "first order", or sensory reality includes or
respects this second order of reality. He thought that mathematical
intuition can reach out to this second order of reality directly. He
implied also that this could be done by analysis of first order reality
(Godel scholars: correct me if I am wrong in interpretation of this
implication). This second way would be philosophically plausible. A
short explanation is order. Suppose that something like string theory
is a correct theory of everything. Such a theory may prescribe our
"universe bubble" as a solution to some uncanny equation. If that is
exactly true, then the mathematical facts inherent in the solution are
no less physical than a physical law (e.g., gravity). Then, "real"
mathematics is physics, and it is discovered. This point of view might
directly explain the "reality" of mathematics without succumbing to an
indispensability argument. The problem with this view is it does not seem
trivial to conclude that the sequence of natural numbers or the number zero
exists.
Finally, there is one other alternative to make sense of the "reality"
of set theory. Godel dismissed a view he termed "Aristotelian
Realism". According to this view, mathematical properties are
etched onto physical events. A simple example would be "the number
of electrons in X atom".
Regards,
--
Eray Ozkural (exa), PhD candidate. Comp. Sci. Dept., Bilkent University, Ankara
http://www.cs.bilkent.edu.tr/~erayo Malfunct: http://www.malfunct.com
ai-philosophy: http://groups.yahoo.com/group/ai-philosophy
Pardus: www.uludag.org.tr KDE Project: http://www.kde.org
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