Agreed if does away with Euclid.

Post #225,298 by a6l6e6x 9/18/05 1:38:03 PM Reply	Agreed if does away with Euclid. Studying Geometry is the only contact most people have with the notion of a proof. Alex The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt. -- Bertrand Russell
Post #225,394 by ben_tilly 9/18/05 10:57:41 PM Reply	This is a good thing? My memories of having been taught "proofs" in geometry are not good. Actually they are very bad. Besides, people get the notion that proofs are a big, fancy and difficult thing. I'd rather that people think of proofs as straightforward. A highschool teacher did something fun with complex numbers. He introduced an abstract set of operations with pairs of numbers. Proved that algebra worked. And only later mentioned that (x, y) was normally written x + iy. This worked very well for me. Cheers, Ben PS Note: part of my dislike for geometric "proofs" may be that I fall in the 40% of the population that is not visually oriented. I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,403 by static 9/18/05 11:29:23 PM Reply	I also learnt complex numbers via an unconventional route. This strongly suggests that the "standard" ways of teaching mathematics are highly flawed. I believe you already hold this opinion. :-) Wade. d-_-b
Post #225,466 by mmoffitt 9/19/05 1:22:23 PM Reply	Re: think of proofs as straightforward. I would prefer that, too. The problem is that very few people I've met have the reasoning abilities required to follow a modus ponens argument. bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell
Post #225,525 by ben_tilly 9/19/05 6:35:17 PM Reply	I'll have to disagree with you on that I've met a lot of people with the ability but without the patience. I've met far fewer who could not learn it. In any case, real mathematicians almost never write out formal proofs. The exercise of doing so strikes me as nearly useless. Knowing that you can is useful, but doing it is not. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,598 by mmoffitt 9/20/05 11:18:42 AM Reply	Nit. In any case, real mathematicians almost never write out formal proofs. Except, of course, those who must publish to keep their positions. They do it at least once in a while. At any rate, I'm certainly glad Andrew Wiles did. ;0) bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell
Post #225,652 by ben_tilly 9/20/05 7:18:19 PM Reply	Nit denied The proofs that you see published are closer to vague outlines of how a competent person might construct a formal proof than they are to formal proofs. And yes, that includes Andrew Wiles. Whose first "outline" turned out to have a hole that a competent person could not be expected to be able to fill in, though he managed to come up with a workaround. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,741 by mmoffitt 9/21/05 12:37:40 PM 9/21/05 12:39:10 PM Reply	What's your point? That professors of mathematics never write formal proofs? Or that professors are not "professionals"? Because, CLEARLY, professors of mathematics (at least at Purdue and North Carolina) do write formal proofs and submit them for publication. [Edit]: BTW, where'd you get your M.S. in Mathematics? Presumeably, at a place where none of your professors wrote formal proofs, right? bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell Edited by mmoffitt Sept. 21, 2005, 12:39:10 PM EDT Expand All History
Post #225,760 by ben_tilly 9/21/05 2:49:18 PM Reply	My point is that... a formal proof is a very specific thing, and very few proofs that you'll find in the mathematical literature are formal proofs. If you find this strange, then you either have no familiarity with what real math papers look like, or you don't really know what a formal proof looks like. This applies as much at Dartmouth College, where I got my masters, as it does at Purdue and North Carolina. Furthermore if you talked to a logician from any of those three universities, I guarantee that they would agree with me. All of this is well-known within the profession. It even has shown up on the wikipedia, for instance see [link\|http://en.wikipedia.org/wiki/Mathematical_proof\|http://en.wikipedia....athematical_proof]. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,769 by mmoffitt 9/21/05 3:26:10 PM Reply	Ah. Dartmouth. That explains the arrogance. :-p bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell
Post #225,473 by a6l6e6x 9/19/05 1:59:58 PM Reply	It was for me. I ate 'em up. Therefore, QED! :) Alex The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt. -- Bertrand Russell

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