From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.ecf!utgpu!cs.utexas.edu!qt.cs.utexas.edu!zaphod.mps.ohio-state.edu!van-bc!ubc-cs!unixg.ubc.ca!ramsay Wed Apr 22 12:03:51 EDT 1992
Article 5129 of comp.ai.philosophy:
Newsgroups: comp.ai.philosophy
Path: newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.ecf!utgpu!cs.utexas.edu!qt.cs.utexas.edu!zaphod.mps.ohio-state.edu!van-bc!ubc-cs!unixg.ubc.ca!ramsay
>From: ramsay@unixg.ubc.ca (Keith Ramsay)
Subject: Re: The 'Big Bang' and the origin of 'mathematical  objects'
Message-ID: <1992Apr17.005509.5046@unixg.ubc.ca>
Sender: news@unixg.ubc.ca (Usenet News Maintenance)
Nntp-Posting-Host: chilko.ucs.ubc.ca
Organization: University of British Columbia, Vancouver, B.C., Canada
References: <kth7fnINNflu@exodus.Eng.Sun.COM> <1992Apr2.181440.11808@guinness.idbsu.edu> <1992Apr14.115539.26623@hellgate.utah.edu>
Date: Fri, 17 Apr 1992 00:55:09 GMT
Lines: 69

tolman%asylum.utah.edu@cs.utah.edu (Kenneth Tolman) writes:
|Ummmm...  Mathematical objects only exist within your head.  

Perhaps you could explain what you mean a bit more.

For example, when one defines "prime", one says it is a positive
integer p such that there do not exist integers m and n, 1<m,n<p, such
that mn=p. Isn't this a clear definition? Surely it is not enough that
no such m and n exist "in my head". There must actually not be such m
and n, whether "in my head" yet or not.

Consider the RSA cryptosystem, which uses prime numbers. If instead of
using primes in RSA, we just used numbers for which factors didn't
exist "in anyone's head", it wouldn't work. In the process of looking
for primes, it is entirely possible to find numbers which are
allegedly shown not to be prime, although they have no currently known
factor (no factor exists "in anyone's head"). How do you explain why
such a system works?

|Your computation
|and internal "gut" feeling are physical artificacts, embedded in the physicsl
|world. 

This is no less true for thoughts about any other subject, if it is
true at all.

|Let us consider any sort of mathematical concept:
|
|  It relies on axioms, which are built up into proofs.

I don't agree that all mathematical concepts "rely on" axioms. What
sort of "reliance" is involved?

|Where did these axioms come from?  Why did you choose them?  Euclid chose
|his axioms, and some of them seemed to be incorrect to Riemann, who
|developed the basics needed for relativity.  

The mathematical concepts pre-dated the axioms of Euclid's textbooks,
for example. The axioms were introduced to systematize what had gone
before.

The situation is clearer if we consider the natural numbers 1,2,3,....
The concept of natural number was around for a long time before the
axiomatic method, and yet we are using essentially the same concept
today as was available in ancient times (just better understood). We
have all learned about the natural numbers, and most of us learned to
deduce facts about them, without having any axioms. I don't see any
other manner in which the concept of "natural number" is "dependent on
axioms", either.

I doubt whether Riemann can really be said to have thought the
parallel axiom was "incorrect". Doing non-Euclidean geometry does not
mean that one thinks Euclidean geometry is "incorrect"

|Where the hell are the
|axioms coming from?  From your head!  They are physically instantiated.

Mathematics "comes from" many sources. All beliefs, in some sense,
"come from one's head".

|Asserting that some aspect of humanity is beyond time and space
|is merely arrogant nonsense.

Who's asserted that (here)?
-- 
Keith Ramsay            Even if a proof from given axioms can, in
ramsay@unixg.ubc.ca     principle, be found mechanically, is it not
                        often found in ways not all that different
                        from finding a new axiom? -Georg Kreisel


