Date: Thu Sep 05 2002 - 13:41:26 EDT
On Tue, 03 Sep 2002 21:29:03 -0400 firstname.lastname@example.org writes:
> Just a few remarks again. Mathematics is often said the base of
> etc. However there were in my time already three very distinct
> of mathematics, discussed by Vollenhoven in 1918: Empirism,
> Formalism, and
> Intuitionism (later after WWII, maybe earlier by some called
> constructivism). All of them have to discuss "time", which is
> supposed to be without beginning and without end? That is in itself
> a difficult thing to grasp, but which has a lot to do with Creation,
> and the
> Coming back of our Lord. This would make at least a fourth
> philosophy of
> math. These questions are not trivial, but are still discussed more
> or less
> by students and their teachers. I have helped three students who
> were very
> much interested in the subject, and produced theses on it.
> As doing physics without mathematics is very difficult, these
> views on math have results in the philosophy of physics.
> many people in physics do not realize the background of modern
> thinking, and
> thus the background of their assumptions, which most of us (myself
> accept without any questioning. I am more familiar with math, and I
> that "new" math has results in other areas.
I trust your wife is doing well and that you are not overloaded. But I
fear you are confusing philosophy with other studies. I am familiar with
the three schools of mathematics. The doing of mathematics is the same
for formalists and "empiricists," but the Dutch school (intuitionists or
constructivists) demands a different logic, denying consequentia
mirabilis (p&~p-->q). This means that they provide a smaller set of
theorems. They originally claimed that their intuitionistic logic could
not be formalized. It has been. Apart from using or not using reductio ad
absurdum, a working mathematician does not worry about the philosophy
adopted. The theorems are the same whether one is committed to a view or
is totally agnostic.
I do not see how consideration of time makes for a different philosophy
of mathematics. Its inclusion in relativity theory merely requires a
4-dimensional geometry rather than the ones we can visualize. The same
holds for the 10 or 11 dimensions in the latest physical theories, which
go far beyond discussions of time. I do not espouse a new philosophy
every time a different geometry (of which there are potentially an
infinite number) is formulated. Nor do fractals and complexity theory
force new philosophies, just more sophistication in math and the areas
where it gets applied.
What gets ignored in the battle among mathematicians is that there are a
wider variety of logics. None of Aristotle's three can be formulated in
the standard symbolic logics. In my dissertation I used a logic in which
p-->p is excluded. This was a matter of utility, not philosophy. The same
holds for some of the multi-valued logics.
The users of math do not much care about the philosophy of the
mathematicians who provide the calculi, except probably they will use
more than the intuitionists provide. Further, it does not matter what the
philosophical or religious bent of the practitioner of a strict science
is: materialist, realist, idealist, pantheist, etc., the lab work and
theory is the same, except that the explanation for notions such as
"mind" may be reductionistic or non-reductionistic or non-material.
History is different. The dogmatic materialist has to find some way to
explain away miracles and resurrection. But that is not science.
As a Christian, I have very clear philosophical commitments. But as a
philosopher I also need to understand where non-Christians are coming
from. Indeed, I must recognize the different bases of different
Christians, for Descartes, Thomas and Augustine differ from Polkinhorne
and Peacocke, who differ from each other.
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