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You are not logged in. #1 20060709 22:16:36
Even and odd numbersYesterday I read David E. Joyce's Innocent Investigation into the the foundations of numbers.. I enjoyed it, so thought some of you might aswell. Go read it #2 20060713 00:38:21
Re: Even and odd numbersI gave it a read. Very interesting. I learned some notational symbols. If I were to rewrite the article myself though, it would be much longer because I would try to set up more basic ideas that say what can be done in a proof. What is math and what is obvious and what is an axiom. How certain axioms can be used for further theorms. igloo myrtilles fourmis #3 20060713 01:39:22
Re: Even and odd numbersI guess the "common sense" you're talking about is that we take even/odd numbers as a fact without thinking it over first? #4 20060713 02:19:21
Re: Even and odd numbersI only skimmed over it, but here's what I think. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #5 20060713 02:59:11
Re: Even and odd numbersThe problem is you only skimmed it over.. The question is not if x is an even or odd number, but if we(he) can prove that the numbers will continue to be even/odd and follow the pattern even odd even odd even odd... #6 20060713 04:18:13
Re: Even and odd numbersPatrick, the first two pages focus solely on the definition of even and odd. This is what I was referring to. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #7 20060713 05:26:51
Re: Even and odd numbersWell, you have to be sure that your assumptions are correct before you use them to prove something, don't you? #8 20060713 07:29:51
Re: Even and odd numbers
Well what I'm getting at is first of all maybe I've missed the courses in the subject matter I am referring to, but is there some subject matter that provides an environment to manipulate ideas in a proof. Sort of, an analogy would be a computer program, you have to stick to its syntax, but ofcourse this environment would be more human probably and allow for different types of logic and inferences, and they could be categorized. So perhaps 20 or 30 or 50 different types of thinking could be outlined with examples for starters to try to figure out if we can even agree on what thinking is, and what can be taken as simply given or common sense, and what cannot. igloo myrtilles fourmis #9 20060713 07:52:30
Re: Even and odd numbersso you wonna read more on proofs in general? I'm not sure I understand you #10 20060713 08:04:21
Re: Even and odd numbersI'm not entirely what you mean John, but I think you're talking about the following. There are different ways to prove things: "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #11 20060713 10:15:07
Re: Even and odd numbersNice one, Ricky. I wonder if there could be an example that works for all. Such as "there will be always be a winner at cointoss" (maybe not a good example). "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #12 20060714 00:20:23
Re: Even and odd numbers
Yes! That is what I am talking about! And also something called "proof theory", which is way over my head I just started reading about. igloo myrtilles fourmis #13 20060714 00:37:01
Re: Even and odd numbersInduction proof is just like a zip. you set the mover right at the begining, and the mover slip one by one(two by two), you should anticipate that anywhere will be covered and closed finally. X'(yXβ)=0 #14 20060714 02:11:48
Re: Even and odd numbersGreat analogy, George. I really like it. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #15 20060714 14:26:39
Re: Even and odd numbersGreat that you like it. It's not always easy to find such an approprate metaphor because a structure similar stuff does not always exist. X'(yXβ)=0 