Comments for Synechism http://synechism.wordpress.com A mixture of philosophy and mathematics Thu, 11 Jun 2009 02:49:42 +0000 http://wordpress.com/ hourly 1 Comment on Zeno and additivity by cssp http://synechism.wordpress.com/2007/12/22/zeno-and-additivity/#comment-3 cssp Thu, 11 Jun 2009 02:49:42 +0000 http://synechism.wordpress.com/2007/12/22/zeno-and-additivity/#comment-3 All the results of standard analysis may be derived in nonstandard analysis as well. In fact, there are a number of nice nonstandard analysis texts, Robert Goldblatt's <em>Lectures on the Hyperreals</em> being one of my favorites. Your question is a good one, and one I have been giving some thought to recently. Namely, how much should one's philosophy of mathematics influence one's teaching of mathematics? In writing my analysis book, I was writing a book for use in the undergraduate beginning analysis course I taught last fall. Given that this was the first (and, for many, the last) course that these students would see in analysis, I felt it should be a standard epsilon-delta treatment of the subject. If nonstandard methods make more inroads into the mainstream curriculum, I could see myself changing my thoughts on this sometime in the future. All the results of standard analysis may be derived in nonstandard analysis as well. In fact, there are a number of nice nonstandard analysis texts, Robert Goldblatt’s Lectures on the Hyperreals being one of my favorites.

Your question is a good one, and one I have been giving some thought to recently. Namely, how much should one’s philosophy of mathematics influence one’s teaching of mathematics? In writing my analysis book, I was writing a book for use in the undergraduate beginning analysis course I taught last fall. Given that this was the first (and, for many, the last) course that these students would see in analysis, I felt it should be a standard epsilon-delta treatment of the subject. If nonstandard methods make more inroads into the mainstream curriculum, I could see myself changing my thoughts on this sometime in the future.

]]> Comment on Zeno and additivity by Arthur Nardiello http://synechism.wordpress.com/2007/12/22/zeno-and-additivity/#comment-2 Arthur Nardiello Thu, 11 Jun 2009 02:15:34 +0000 http://synechism.wordpress.com/2007/12/22/zeno-and-additivity/#comment-2 Professor Sloughter, Can you shed some light on why you have such a wonderful nonstandard calculus text but an analysis text that uses epsilon delta proofs, open and closed sets, etc.? Is there no way to extend nonstandard notions beyond basic calculus? Professor Sloughter,
Can you shed some light on why you have such a wonderful nonstandard calculus text but an analysis text that uses epsilon delta proofs, open and closed sets, etc.? Is there no way to extend nonstandard notions beyond basic calculus?

]]>