math_remarks - Some remarks about math and proofs Math 8...

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Unformatted text preview: Some remarks about math and proofs Math 8 2009, Chris Lyons While I can’t speak about international standards, students at the average American high school receive little exposure to mathematical proofs other than the sort covered in a Euclidean geometry course. So freshman who arrive at Caltech are often shocked and confused to find themselves in lectures where the professor has an “unusually” high emphasis on proving mathematical statements rigorously. Then these feelings give way to some degree of horror, as they discover they’ll be required to prove mathematical statements on their homework and exams! Moreover, at a place like Caltech, there’s always a small but very conspicuous group of students who • completely understand why this is taking place, • have no trouble following the detailed definitions and arguments written on the blackboard, and • are enjoying every minute of it. And upon seeing these people, many of the other students are suddenly thinking to themselves: • “Why don’t I get this stuff? Is this all supposed to be clear at first sight?” • “Am I just supposed to come up with the kinds of clever and detailed arguments on my own that I see in lecture and the book?” • “If this is what ‘real math’ is like, then I guess I hate real math!” It’s incredibly disappointing when I see students who loved math in high school suddenly become turned against the subject due to initial impressions that they may get from a college math lecture. The point I want to make here is that your feelings about math shouldn’t be based on the shock experience you get in Ma 1a, because Ma 1a represents only a certain portion of how math is practiced and developed. After I try to convince you not to hate math, I’ll then make some general remarks on proofs, including how you should regard them generally and some vague strategies that can help if you have to do one. Ma 1a 6 = the whole of mathematics! If you love Ma 1a, great! There’s no need to read this. But given the reputation that Ma 1a has among Caltech students, I know there’s a lot of people who definitely don’t feel this way. The things I say in this section are an attempt to convince you that if you used to like math, but Ma 1a is making you rethink that, then you shouldn’t get disillusioned just yet. No one expects you to be a genius... but they do know you’re smart and hard-working! The origins of calculus, that is, the “seeds” of calculus in human thought, took place over hundreds of years. And even when the subject was finally invented by Newton and Leibniz, the actual rigorous foundations took a couple hundred years more to be established. Yet when you sit in Ma 1a, you see the subject rigorously unfold in front of you in a matter of months!...
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.

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math_remarks - Some remarks about math and proofs Math 8...

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