Re: Math and religion - is the very topic crankish?
- From: "T.H. Ray" <thray123@xxxxxxx>
- Date: Thu, 09 Nov 2006 14:07:32 EST
Greetings
Obviously, mathematics is not a religion, which is
why I didn't post
this as a response to the "Math as Religion" thread
as I had originally
intended. I teach at a Catholic university, and when
students here want
to pray, they go to the chapel and not to the Math/CS
department. I
really wouldn't know what to do if any student turned
to me for
religious advice - it's not my area of expertise to
say the least.
But - There is no denying that a religious undertone
can be detected in
many of the (non-crankish) discussions of
foundational issues involving
Platonism, constructivism or finitism of various
sorts and infinity.
For example,Cantor himself was interested in how his
philosophy of the
infinite related to the notion of infinity in
Scholastic philosophy
and he carried out correspondence with German
Thomistic philosophers
(as is described, for example, in Joseph Dauben's
"George Cantor: His
Mathematics and Philosophy of the Infinite". Also,
see Dauben's
description of Cantor's views on God and infinity on
pages 228-232 for
a clear example of the religious nature of some of
Cantor's
philosophy).
Cantor had to overcome a lot of resistance to his
views, and it is no
wonder that he adopted a shotgun approach of
combining mathematical,
philosophical and theological arguments. It is
significant that by the
early 20th century, theological discussion faded from
the scene as far
as mathematical set theory is concerned and with the
formulation of
precise axioms like ZFC it became just another branch
of mathematics
(albeit one with foundational implications). Arguing
that set theory is
religious because Cantor had religious views of
infinity is like
arguing that physics is religious since Newton had
religious views of
how God ran the universe. One should not commit the
genetic fallacy.
Nevertheless, Cantor's case shows that it is not
intrinsically
irrational to think that there is some link between
the notion of
infinity and the notion of God, and that this link
might have some role
to play in the philosophy of mathematics.
A more recent example of explicitly theological
themes arrising in
mathematics is Errett Bishop's Constructivist
philosophy. Particularly
interesting is a paper he wrote called "Schizophrenia
in Contemporary
Mathematics" (Contemporary Mathematics, Volume 39,
1985):
"Our [constructivist] point of view is to describe
the mathematical
operations that can be carried out by finite beings,
man's mathematics
for short. In contrast, classical mathematics
concerns itself with
operations that can be carried out by God. For
instance, the above
number n_0 [0 if Riemann Hypothesis is true, 1
otherwise] is
classically a well-defined integer because God can
perform the infinite
search that will determine whether the Riemann
hypothesis is true...
You may think that I am making a joke, or attempting
to put down
classical mathematics, by bringing God into the
discussion. This is not
true. I am doing my best to develop a secure
philosophical foundation,
based on meaning rather than formalistics, for
current classical
practice. The most solid foundation available at
present seems to me to
involve consideration of a being with non-finite
powers -- call him God
or whatever you will -- in addition to the powers
possessed by finite
beings." (page 9). Chapter 1 of Bishop's "Foundations
of Constructive
Analysis" contains a similar argument. Ironically,
Paul Halmos in his
autobiography says something to the effect that
Bishop was a brilliant
mathematician "until he got religion
[constructivism]." (or words to
that effect - I don't have Halmos' book before me).
Another example of a theological theme in philosophy
of mathematics is
Edward Nelson's radical finitism. In his "Predicative
Arithmetic" he
writes:
"We are creatures (Kronecker had it backwards), not
too much older than
an infant in a crib, and we still feel the urge to
count on something
when we count. The infant counts on its fingers, the
mathematician on w
[omega] - but the infant at least nows its fingers to
exist. The
mathematician's attitude towards w has in practice
been one of faith,
and faith in a hypothetical entity of our own
devising, to which are
ascribed attribtes of necessary existence and
infinite magnitude, is
idolatry" (pg 80)
See "Ad Infinititum -The Ghost in Turing's Machine:
Taking God out of
Mathematics and Putting the Body Back In" by Brian
Rotman for similar
thoughts (although, Rotman's work strikes me as
crankish since it is
based on "semiotics" - a philosophical type of
linguisitics. He quotes
Derrida at the very beginning of chapter 1 - which
does not bode well
for the rigor of the subsequent argument).
As far as Platonsim goes - many thinkers have felt
that a commitment
to naturalism (the negation of theism) renders
Platonism at the very
least implausible, which is perhaps one of the
reasons why formalism
has been (officially) such a popular viewpoint in the
philosophy of
mathematics. Penelope Maddy in "Realism in
Mathematics" has a good
discussion of the problem, as well as what seems to
me an adequate
defense of Platonism within the context of
naturalism. Nevertheless,
even though she doesn't explicitly talk about God,
the effort she must
undergo to find a naturalistic basis of Platonism is
tacit admission
that it is not easy to completely disentangle
Platonism from theism.
I don't have any real point beyond the observation
that asking if there
is a relationship between religion and mathematics
does not
*automatically* render you a crank. On the other hand
- there is little
reason to doubt that most of those who do so on
sci.math are cranks.
So, cranks, take note: the mathematicians like Bishop
and Nelson I
quoted above have *earned* the right to raise such
questions. They have
made major contributions to mathematics both before
and after their
conversion to constructivism/finitism. Also - they
are both willing and
able to rigorously spell out their assumptions and to
see what follows.
Once you have even 1% of the accomplishments of
someone like Nelson,
you might find someone taking you seriously as well.
-John Coleman
I think it was John Barrow who interpreted Godel as
saying that if mathematics is a religion, it is the
only religion that can prove itself to be a religion.
If mathematics contains,to some perceptions,
the characteristics of religion -- such as faith in
mythology and assignment of value to personal
beliefs -- such observation is not in itself a "crankish
topic." In fact, the constructivist philosophy you
mention is quite concerned with eliminating mythology
from mathematics. Insofar as true religion necessitates
both mythology and priesthood,however, mathematics --
lacking a priesthood -- does not qualify as religion.
Tom
.
- References:
- Math and religion - is the very topic crankish?
- From: John Coleman
- Math and religion - is the very topic crankish?
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