Expert: Daniel Mazur - 4/19/2010

Question
Hi Daniel,

I just read about the Zeno's paradoxes. (here's the Wiki link : http://en.wikipedia.org/wiki/Zeno's_paradoxes )

My question is not about the paradox itself, but rather on something i realized after reading it.

After reading it what i've realized is that space and time, mathematically should be considered as made up of infinite discrete quantities, since mathematically, there are infinite no. of points between any two points and similarily we can think of infinite instances of time in between any two instances of time.

Now, if we take any two instances of time (or it could also be space), the only way we can connect these two instances is by imagining infinite no. of instances lying in between, but we can not connect them in a "continous" fashion.
Or stated differently, there is no meaning to "continuity" mathematically. Even continuity is made up of discrete quantities.

Am I right in my conclusions above??

Applying a similar argument to calculus, while calculating areas of curves using integral calculus, we divide the area into infinite small area elements.

(for eg, consider calculating the area enclosed by the parabola y^2=4ax and the x-axis in the first quadrant by constructing thing rectangular vertical strips having a base lenght dx and height y)

Some part of such vertical rectangular strips is always either lying outside or inside the parabola. Had dx been exactly 0, the upper part of the vertical rectangle would have coincided with the curve. But since dx is never exactly 0, the area calculated in such a way would always be either greater or less than the actual area of the curve.

So using such an approach, an error would be introduced. The error would be smaller, if we take smaller area elements but the error would never be exactly 0.

My question then is, that while using such an approach, how is it that we always get exact values of the areas of various curves?? How does the above error vanish away????


Thanks,
Shikhin

Answer
Hi Shikhin,

I have little time this week for answers, butI'll give you my view.

I think that continuity is an illusion of finely spaced points in space and time. In the real world (not just mathematical abstraction) there are not an infinite number of points between given two points, however. The number should be large and variable (!) but finite, limited by Heisenberg's Uncertainty Principle, speed of light and perhaps more. This is my view, I don't know if an absolutely correct answer exists.

The magic of differential calculus you mentioned lies in understanding the exact proofs of all relevant mathematical statements. The key word is "infinitesimal", understanding to this term is crucial. There is nothing really magical about it, one just has to do the painful task of learning to understand the statements, their proofs and their implications. I am not a mathematician, so I am hardly qualified to provide insight into this matter. It seems to me, though, that the full answer would be beyond the time anybody would volunteer. If you are looking for an "intuitive reasoning", why it should be so that the error vanishes, I can just say that if you add up an infinite number of elements of arbitrarily small areas, the sum must be exact. A computer of course has limited precision, the smallest resolved number is finite, so it will produce errors due to the non-compliance to the requirement of "arbitrarily small".

Take care,
Daniel