Wednesday, March 01, 2006

Chad on Zeno and Black

I agree with Black’s position that mathematics alone cannot break the argument put forth by Zeno. Moreover, his argument examines the point that trying to count an infinite number of steps in a finite amount of time as impossible. In nature, it is true that the faster runner will overtake the slower runner. In my opinion the answer does not lie in trying to prove that it is impossible to count an infinite number of something in finite amount of time. The answer lies in the whole definition of infinity.
To my knowledge no one has ever proved infinity can exists. It is just simply an approximation used to put at an end of what the human mind is unable to conceive of. Human beings have been approximating nature since the beginning of time. As the centuries have passed since the time of Zeno our approximations have become better and better.
A good example is from the time of Isaac Newton. He invented calculus and applied it to celestial and earth bound mechanics. This new formulation explained everything from the distance a baseball could be thrown in a finite amount of time to the motion of planetary objects. Except, one thing was unable to be explained by Newton’s mechanics. The orbit of Mercury did not fit the calculations or any other explainable reason for its peculiar orbit. However, Newton’s approximations were close on nearly everything else in nature, but they were still approximations none the less.
Three centuries later Albert Einstein formulated his own approximations and clearly explained the orbit of Mercury. More in particular he defined that nature does have finite values for what was previously thought to be infinite. For example, Newton believed that the speed of light and gravity are instantaneous or at a speed of infinity. Einstein discovered that the speed of light in not instantaneous. Therefore, light travels a finite distance in a finite amount of time.
Two of the greatest minds in all of history understood that mathematics is invented as just approximations. It is the best tool we have and the approximations are used to explain how we believe nature works. Therefore, their arguments, including Zeno’s, still hold until a better approximation can explain them otherwise.


note: This post is Chad's despite the fact that it is not posted by Chad.

1 Comments:

Blogger linford86 said...

You do not need to prove that infinity exists any more so than you need to prove that triangles, lines, or rectangles exist. Infinity may not even exist in the practical sense - but it most definitely does exist in the abstract, mathematical form, just as any mathematical entity does. Note that Euclid never felt that he needed to prove that circles, lines, triangles, etc. exist, despite the fact that he proved several theorems about them (using pure logic.) It may be that it is impossible to have a line in practicality (although we can approximate one) but a line does exist as a mathematical entity. In fact, the idea of infinity was extroardinarily important in the history of set theory, especially in the work of Cantor. In Set Theory, it was found that there exist mathematical entities which must be infinite. For more information on that one, see http://www.mathacademy.com/pr/minitext/infinity/index.asp

In a practical sense, there are phenomena in physics and engineering that are taken to be infinite because they are so large. You are correct that these are only approximations. For instance, in optics and photography, one can talk about focusing on "infinity" (focusing your lens on a distance that is significantly far away.) Newton and others in his time period believed that light travelled infinitely fast through space because they lacked the experimental apparatus to prove that it didn't (i.e. light moves far too fast for them to have measured its speed.) Also, it was believed that gravity "moved" infinitely fast because it wasn't looked at as being a disturbance which propogates from place to place. Gravity was seen as something that acts instantaneously, without having to move through the intervening space. In this way, gravity didn't really move at all - so to talk about the speed at which gravity propogates under that context is absolutely meaningless (namely because in Newtonian physics gravity doesn't propogate.)

But please also note the fact that although infinity is used as an approximation for these particular instances, that does not mean that infinity doesn't exist in general. Suppose we have the following argument:

1. We approximate the top of a table as being flat.
2. The top of a table isn't actually flat.
3. If (1) and (2) then flat surfaces don't exist.
4. Therefore, flat surfaces don't exist.

That argument may be valid, but its soundness is questionable. This is because premise (3) doesn't appear to be true. A similar problem occurs with your argument.

11:53 AM  

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