The Difference between Science and Mathematics
Date: 09/18/2006 at 21:58:46 From: Carlee Subject: How is a math project different from a science fair project? I need to do a science fair project (I'm in 11th grade, pre-calculus) and I was thinking about doing a math project, but my friend told me that math projects involve a "completely different process" from the normal scientific process. I asked her what she meant, but she wasn't really sure, so I was wondering if you could answer me that?
Date: 09/19/2006 at 10:25:49 From: Doctor Ian Subject: Re: How is a math project different from a science fair project? Hi Carlee, She's right. In math, we start with some definitions, and we see what follows from those definitions. For example, we define numbers, and operations, and then we find that we can define a category of numbers called "prime" numbers, and then we can ask questions about them and try to answer the questions. What's important here is that we KNOW what the "rules" are, because we CHOSE them. If we want to use different rules, we're free to do that. For example, if we start with the rules of Euclidean geometry, we find out that in any triangle we construct, the sum of the interior angles is 180 degrees. That's a consequence of choosing those rules. If we choose a different set of rules, e.g., the rules of spherical geometry, we find that this consequence is no longer true, e.g., we can construct triangles where the interior angles add up to 196 degrees, or 258 degrees or a lot of other values that aren't reachable in Euclidean geometry. It's very important to note that none of that has anything to do with what's going on in the world! It's just making up games and playing them. So the concepts of "true" and "false" become matters of internal consistency. Can the angles of a triangle add up to more than 180 degrees? The answer can be "yes" or "no", depending on which game you're playing. But it's all "math". In science, it works in the other direction. We don't know what the rules are, so we try to set up situations that can help us guess what they are. Or, more precisely, we guess what the rules are, expressing our guesses in terms of mathematics. Then we use math to generate some predictions. And we test those predictions against the world. So a mathematical question would be something like: Given [some set of definitions], and noticing that [some pattern that seems to hold], can we prove that the pattern must hold, or that it does not hold for at least one case? A specific example of this would be Given the rules of standard arithmetic and the standard definition of "prime number", and noticing that it seems to be possible to express every even integer greater than 2 as the sum of two primes, can we prove that this must always be the case, or find a counter-example? (This particular example is called Goldbach's Conjecture, and it's an open question, i.e., no one has been able to prove it either way.) A scientific question, on the other hand, would be something like: If [some theory is true], then in [some situation], we should see [some result], and if we see a different result, then the theory isn't true. So, do we see that result? A specific example of this would be If Einstein's theory of general relativity is true, then as light from a distant star passes close to the sun on its way to earth, the apparent distance of the star should be shifted by such-and-such an amount. So, does that happen? To be even more precise, when asking scientific questions, we're _really_ trying to differentiate between two competing theories. So that would look like [Theory A] predicts that in [some situation], we will see [Result A]. [Theory B] predicts that in the same situation, we will see [Result B]. Which prediction is closer to what we actually observe? and Einstein's theory of general relativity predicts that as light from a distant star passes close to the sun on its way to earth, the apparent distance of the star should be shifted by such-and-such an amount. Newton's theory of gravitation predicts that there will be no shift. Which prediction is closer to what we actually observe? Note that in the case of science, it may be the case that NEITHER theory actually captures the rules that govern what's going on. We're not asking which theory is TRUE, but which theory is MORE ACCURATE. But that's the best we can do--keep guessing, and checking our guesses. Because even if we get to the point where our best guess seems to predict the answer to every question we can think up, that might just mean we haven't been smart enough to think of the questions that would show us that the theory is wrong. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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