Wednesday, March 08, 2006

Sets, Infinity, Limits, and Functions

I hope I have substantiated my claim that one does not need facility with modern mathematics in order to grasp the philosophical problems posed by Zeno's paradoxes. Regardless, some of you have asked about the mathematics employed in addressing the mathematical problems posed by Zeno's paradozes. I have intentionally not focused on the mathematics in class. In case you are curious, I have posted two discussions by Wesley Salmon that deal with the mathematical concepts involved. The first discusses the concept of a limit, the sum of an infinite series, and mathematical functions. The second discusses sets and infinity. Both expositions assume no background in modern mathematics. If you read either and have questions about them, please comment to this post.

3 Comments:

Blogger linford86 said...

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6:32 PM  
Blogger linford86 said...

I don't really have a question, just a comment on something that I found interesting in my new found obsession with the mathematics of Zeno's Paradox, particularly with the mathematics relating to the Weyl Tile paradox/argument.

I found that the Weyl Tile paradox thing actually arises under a totally different name as the "Diagonal Paradox" in Quantum Electrodynamics (QED), where it has implications for how it is that physical quantities are calculated from the theory... For those who don't know anything at all about QED, don't sweat it - the fact that this "Diagonal Paradox" arises in the theory doesn't require you to understand the theory in order to understand the paradox... I think its interesting that I spent last night comming up results that, unknown to me, had already been discovered by mathematicians and/or physicists at some unstated earlier time - as I was quick to find upon surfing Steven Wolfram's web site (who, incidentally, invented Mathematica and wrote one of the most controversial mathematics/physics books in recent times...) Anyways, here's a link to an explanation of what physicists call the "diagonal paradox", which is actually identical to the Weyl tile paradox:

http://mathworld.wolfram.com/DiagonalParadox.html

In case any is interested, Richard Feynman gives a great explanation of QED (for which he won a Nobel Prize) in a series of video lectures for the general public which can be downloaded here:

http://www.vega.org.uk/video/subseries/8

6:42 PM  
Blogger linford86 said...

If anyone wants to see how physicists are currently working on the problem of discrete space-time, you can check out http://www.lns.cornell.edu/spr/2005-07/threads.html... It's a message board put up by Cornell University's Laboratory for Elementary-Particle Physics, and it contains a non-mathematical (but technical) disccussion of physics in a discrete space-time... I found it really interesting... Just go there, and scroll down to Re: A question of discrete space-time... I was unable to find the original post (Re: usually refers to a response to another post) but the discussion is interesting and relatively easy to follow (at least it was for me - I don't know if others have enough of a mathematical/physics background as I do in order to follow the discussion...) In any case, it's there if anyone's interested in the questions in physics raised by the possibility of a discrete space-time...

10:43 PM  

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