Why Study Math?
>> Sunday, January 23, 2011
"The continuum is that which is divisible into indivisibles that are infinitely divisible."- AristotleThe idea that there are myriad infinities boggles the mind and makes us wonder what we really do know. As the quote implies, we can take the real number line and divide it into such small pieces, literally indivisibles, and then be left with something that is yet infinitely divisible. Wow! Maybe we should stick with more mundane stuff like how to use the latest application on our smartphones; but then, we would not have half as much fun. Take the ride with me.
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Maybe I am unusual in that I find this idea absolutely fascinating, so much so that I light up like the July 4th night sky during fireworks when I think of such wonders. That the real number line, or the continuum, as mathematicians call it, can be divided ad infinitum and yet still be infinitely divisible, or that there should exist a whole hierarchy of distinct infinities, provable by mathematical theory---indeed provable without such exotic mathematics as might be necessary so as to prove Fermat's Last Theorem, let us say---is nothing short of miraculous. When the concept of the atom was proposed by the English scientist John Dalton to explain the behavior of chemical elements, he was ushering in the age of atomic theory. To wit, the word atom comes from the Greek and means "uncuttable," or "indivisible." It would seem that any subcomponent should be made of something, much as any living cell has subcomponents which we call organelles. These subcomponents are made of subcomponents and one could extend this argument ad infinitum. The question becomes, "When does it stop? Well enter the real number line.
The continuum, as we have heard it called, exhibits the same uncanny quality of being indivisible yet infinitely divisible. Case in point: no matter how small we shrink an interval, we can yet infinitely shrink it further. To make this point a bit more clear, let us relate this to the size of atomic particles. An Angstrom unit is 10^(-10)m which is one ten-billionth of a meter. Although the exact size of the electron is not known, it is known that its size is less than 10^(-13)cm. This is five powers of ten less than the angstrom. Indeed the electron is very, very small.
If we project this size onto the continuum then we can think of it as the length of the interval from 0 to 10^(-13)cm over to the right. A very small interval without question! It is easily provable that between this interval there are infinitely many real numbers. Mind blowing! Thus we can slice the real number line however close to 0 we want and set that as our new interval. Perhaps this is the interval from 0 to 10^(-25)cm. Yet in this interval we can still find infinitely many distinct real numbers. In fact, one of the results proved, by extension of previous results, by the famous mathematician Georg Cantor, the founder of modern set theory, was that the number of distinct reals in any interval, however small, was more numerous than the infinite set of natural numbers, that is the set {1, 2, 3,...}!
Now if you are not blown away by the facts laid out in the previous paragraph then one of the following must apply: 1) you are dead; or 2) you have become so jaded by living that you really should be dead. Hopefully none of these conditions applies to you. Now whether we can keep subdividing particles this way remains as arcana only known to the privileged of an unearthly domain. Yet I believe that the outlined construct gives us serious fodder to fuel speculation about not only mysteries of the universe but also such ponderous contemplations amenable to mathematical thought.
As till now it seems that the electron is a point particle and therefore not further divisible. Thus the electron would be comparable to a point real number which makes up the continuum. Yet it does have some size, however small. We have just seen that we can slice up ad infinitum any interval, however small. The question becomes, what occupies the space in those intervals of the electron? Pure empty space or something else? Intense! Maybe we should go back to that smartphone application after all.
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