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/
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