Philosophy of the Truths of Mathematics

Date: 02/28/2001 at 20:02:27
From: Lauren 
Subject: Philosophy of the Truths of Mathematics

This is a philosophy question, but I figured the best opinions to get 
would be those of mathematicians. 

After reading about Rene Descartes and how he was able to "doubt" the 
truths of math, I am presented with a similar question: Do the truths 
of math hold in any conceivable world? 

I suppose the "conceivable" part is kind of jarring. Pretty much, in 
any world you can imagine (where cats rule, where people can fly, 
etc.), do the truths of math still hold?  

I know there is no real answer; I'm just looking for some educated 
opinions. Thanks!

Date: 02/28/2001 at 22:41:24
From: Doctor Achilles
Subject: Re: Philosophy of the Truths of Mathematics

Hi Lauren,

Thanks for writing to Dr. Math.

This is actually a very involved and thoughtful question. Let me go 
over a few separate issues relating to it.

What is math? This is something that is rarely addressed directly in 
math classes, especially before college. You can find a great 
introduction to to what math is in our archives:

  What is Mathematics?
  http://mathforum.org/dr.math/problems/erum.09.22.00.html   

Math is just a set of logical truths. It assumes a few things (such as 
Euclid's five postulates), defines terms, and sees what follows 
logically from those assumptions.

There are thus two ways that you can question math. First, you can 
question the truth of one or more of the assumptions. This sort of 
questioning has been very fruitful in the history of math. Questioning 
of Euclid's fifth postulate led to an entire new branch of math called 
non-Euclidean geometry.

The second way to question math is to challenge the rules of logical 
derivation. This is a less popular thing to do, because the legal 
steps in a mathematical derivation are typically very obviously truth-
preserving. Here's a simple example of a mathematical derivation:

  1) Assumption: Every natural number has a successor
                 that is itself a natural number.
  2) Assumption: Zero is a natural number.
  3) Definition: One is the successor of zero.
 -------------------------------------------
  4) Theorem: One is a natural number.

The only claim being made here is that if we accept the two 
assumptions and the definition above, then we must accept that one is 
a natural number.

Admittedly, mathematicians can come up with some pretty weird stuff, 
but the individual steps in each derivation are built entirely on 
processes like the one outlined above. Additionally, mathematical 
publications are very closely scrutinized by several mathematicians 
other than the author before being published, and are often criticized 
by even more after publication. All of this amounts to a good 
assurance that theorems do in fact follow from assumptions.

Having said all that, let me make a few brief comments about 
Descartes. Descartes is famous for being a skeptic. However, his 
philosophy was actually largely an argument against skepticism. He 
wanted to show that the skeptics of his time were wrong in saying that 
"NOTHING is certain." The way he did that was to set aside EVERY 
belief that he had ANY reason to doubt, and then see what was left.  
What he ended up with was his famous "I think therefore I am" proof of 
his own existence. From there he argued that several other things 
followed logically.

One could criticize Descartes on one level by saying that in addition 
to postulating his own existence, the very fact that he argued 
logically implies that he assumed the rules of logic. It is, however, 
hard to construct arguments (either for or against Descartes) if one 
doesn't accept the rules of logic. So it's not such a bad assumption 
on Descartes' part after all.

To see just how incredibly strange it is not to assume the rules of 
logic, look back at the derivation I showed above. If you want to say 
that the rules of logic cannot be assumed and must be argued for, this 
is what the derivation would have to look like:

  0) Assumption: If every x has a y and A is an x, then A has a y.
  1) Assumption: Every natural number has a successor
                 that is itself a natural number.
  2) Assumption: Zero is a natural number.
  3) Definition: One is the successor of zero.
 -------------------------------------------
  4) Theorem: One is a natural number.

However, this derivation is itself only valid if you accept the rules 
of logic. You would have to postulate something like: If (0) holds and 
(1) and (2) hold, then the successor of zero is a natural number (one 
is just a convenient name for the successor of zero). You could go on 
like this forever, proposing higher-level rules and setting them down 
as further assumptions, but no end is in sight. (This comes from a 
classic Lewis Carroll dialogue.)

So do the rules of math hold in every imaginable world? Well, I can 
imagine a lot of things: dogs that spit fire, pigs flying, life 
evolving on the surface of a star. I can even imagine a world where 
the successor of zero is not a natural number (but only if you don't 
assume (1) and (2) above).  However, I _cannot_ imagine a world where 
assumptions (1) and (2) are true and the statement "the successor of 
zero is a natural number" is false.

The question of what _is_ a possible world is one that I cannot hope 
to answer here. Probably, what I can imagine is not a very satisfying 
criterion, for me or anyone else.  The rules of logic are reasonable 
enough for me, so I try to go by them. But whether that is the right 
way to go is not something I can answer from outside the system of 
logic.

Hope all this helps. If you have any other questions about this or 
other math topics, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/