[FOM] RE: FOM Digest, Vol 5, Issue 15

[Axiomize@aol.com]

Matt Insall wrote 5/12/03:

> Lerma: In the realm of Mathematics there are no cabbages, nor apples, nor 
bricks, nor planets, nor anything "material" at all. 

Mathematics is the science that can be conducted without the use of our 5 
senses.  Thus it is "true of all (physical) worlds".

> But in Mathematics there are not "things" to start with.

The natural numbers divide everything into things.  (I think of them as being 
the delimiters between things, rather than being associated with the things 
themselves.  I feel this way from being a computer programmer who has for 
many years divided up files into fields with various delimiters, and then 
assigning the natural numbers to those fields.)

Set Theory can be built using only sets.  Recursion Theory (Kleene) is the 
science of computer programs whose input and output are only computer 
programs.

"There is a program that outputs only itself." = "There is a set that 
contains only itself."

(http://www.cs.nyu.edu/pipermail/fom/2002-September/005893.html is based on 
this analogy.)

> Lerma: The real problem of Foundations is how to build the entire building 
of Mathematics without any material bricks at all.  

I think that the above parallel between Set Theory and Recursion Theory can 
be exploited, especially since programs are more "concrete" than sets.

Charlie Volkstorf
Cambridge, MA