Mathematics and PhilosophyDate: 6/5/96 at 21:3:57 From: Anonymous Subject: Maths-phylosophy Hello, I have heard that maths is the universal language. I was wondering if mathematicians viewed maths as being 'transcendental' of the universe? The laws of physics apply specifically to this universe - does the same apply to mathematics? Is it really a universal language, or would an alien race have a completely different kind of logic? Thanks for your time. Date: 6/13/96 at 21:55:11 From: Doctor Tom Subject: Re: Maths-phylosophy It's difficult to know what an alien race might do or think. In fact, our understanding of mathematics and the foundations of mathematics has changed radically in just the last 60 or 70 years. There is a field called "metamathematics", which is the study of mathematics. It's "outside" of mathematics the same way "metaphysics" is outside of physics. But we can reason in metamathematics about the sorts of things that can and cannot be done with mathematics. For example, let's look at a "simple" problem. Given Peano's postulates that describe the natural numbers 0, 1, 2, ..., is it true that any theorem about them can be shown to be true or false given enough time, and given a smart enough mathematician? Up until 1930 or so, people thought that this was the case - that there might even be a mechanical way to decide (given enough time) whether any theorem about the natural numbers is true or false. But then Kurt Godel proved (using meta-mathematics) his so-called "incompleteness theorem". It states that there is a theorem about the natural numbers that can neither be proved true nor proved false. That you can add it or its negation as an axiom, and either will give a consistent theory of the natural numbers. What's more, even if you add this axiom, there will be another undecidable theorem, and no matter how many axioms you add, there will always be another. Some theories can be proven complete; others can't. The study of this is very deep, and very interesting. Mathematics is clearly more powerful (in a sense) than physics. For example, in physics in our universe, light always travels at the same speed. A mathematician, however, would be able to tell you what the physics would be like in a universe where light travels at a different speed, or where the electric field dropped off as 1/r^3, or where there were only protons and no electrons to balance the charges. I believe that mathematics is quite universal, but I don't think it's possible to "prove" that! -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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