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### Philosophy of the Truths of Mathematics

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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!
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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.
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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.

other math topics, please write back.

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