### AND, HOW SMALL CAN SMALL BE?

I don’t have the answers but I can’t help thinking about “infinity” and its reciprocal twin “infinitesimal”. This is dangerous to my mental health. The mathematician Cantor went mad while studying “infinity”. However, I don’t think the risk to me is THAT large – I will never know enough to endanger myself.

Ancient natural philosophers, as shown in the historical record, were asking how small can pieces of a “thing” be made by successive halving of pieces of the original “thing”. A little thought and one arrives at the conclusion pieces of the “thing” can never be reduced to a size smaller than the half of the piece last cut. Cut again and one still has half of the piece last cut. This is the line of reasoning which led to the first “atomic theory of matter”. The ancients concluded “matter”is composed of “atoms” that are indivisible. They had no idea what these “atoms” were. This discussion illustrates the line of reasoning used to ponder the world of big and small. For instance, one can get a sense of “infinity” by first imagining a sum of successive numbers, starting at zero, suming out to as far as you care to. Have we arrived at “infinity”? No! It is obvious I can add to your sum one more number and thus create a number larger than your “infinity”. It is equally obvious this game can continue forever without reaching a final value for “infinity”. The definition of “infinity” is ” be without bound”. As an aside, note the concept of “time going on forever” is now part of the discussion. Poor Dr. Cantor!

An ancient natural philosopher named Zeno advanced a motion paradox which is still giving fits to theorists today. It is useful to use this paradox as a way to gain a little insight into the concept of the “infinitesimal”. Also known as “the very small”. The Zeno paradox, loosely interpreted, states: (1) a body, set in motion at point A toward point B, will never arrive at point B, (2) experience demonstrates a body set in motion at point A will arrive at point B. (3) this is an obvious paradox. What is Zeno’s reasoning?

Zeno reasoned the moving body will reach half the distance to point B within a definite time. He further reasoned the body will reach half the remaining distance within a different definite time. Zeno continues this reasoning until the conclusion the body will never reach point B is reached. The body always has half the remaining distance to go. I just got back from filling my coffee cup and thus demonstrated point B will be reached in reality. Thus the paradox.

Now the insight part. As the time of the motion increases, the remaining distance traveled (half of the previous half, etc., etc.) will become smaller and smaller, ie. become infinitesimal. This “infinitesimal” is closely related to the “infinite” because the division of any number by a very small number is a very large result. In fact, if division by zero is attempted, the resulting quantity is defined to be without bound, ie, infinite. Mathematicians punted this problem long ago by declaring division by zero is not allowed. Issac Newton, the man who gave us “Newton’s Equations” of motion and the first explanation of how the Solar system uses Gravitational attraction to maintain orbital motion, was faced with the Zeno paradox and all that is associated with it. His solution to these difficulties was the invention of what is known today as ” The Calculus” mathematics. If you have studied “The Calculus” then you know about “taking the limit”, ie, division by a number arbitrarily close to zero, but, never allowed to be zero. ( Historians will point out that Leibnitz shares credit for the invention of “The Calculus”.)

A contemporary problem of long standing involves the most successful pair of thoeries yet devised to “explain” how matter works. The “General Theory Of Relativity” by Albert Einstein has been verified to high accuracy many times over using astronomical observations and very precise measurements of changes in time, length, and mass predicted to occur as the relative velocity between observers change. The second of these theories is the “Standard Model” of atomic particle physics. This theory has been verified to high accuracy using the results of many high energy particle collision experiments such as those now being conducted at CERN in Europe. At the energy levels of matter in our normal everyday existence, four fundamental forces are observed. The first is predicted by the “General Theory Of Relativity”; it is known as the Gravitational Force. The remaining three forces are the Weak force, the Strong Force, and the Electro-magnetic Force. The latter three forces have been shown to unify into a single force at very high energy levels. Many theorists expect further unification to occur at even higher energy levels.

The contemporary problem alluded to above is the expectation by theorists of one force and one theory. The problem arises upon attempts by theorists to combine the two successful theories into a single theory. It does not work! Such combined theories yield nonsense results. The combined theory requires what essentially is division by zero. The results are described as being rife with infinities. So I’m told; the calculations are way beyond my pay grade. A. Einstein spent the rest of his life in an unsuccessful effort to crack this nut and failed. Unification work continues using a new approach known as “String Theory”. The advantage of “String Theory” is the avoidance of division by zero. Good luck on that.

The “Big Bang Theory” followed the announcement by Edwin Hubble about 1922 that the Universe was not static; it was expanding! It was quickly realized if the Universe is expanding, it must have a beginning. The “Big Bang Theory” was the result. The pondering of the implications of “The Big Bang Theory” quickly lead to questions about the size of the Universe. Is it finite in extent? If so, what lies beyond? Is that infinite? If our Universe is not finite it must be infinite in extent. Does an infinite Universe mean somewhere in our Universe there is another “OLD MACHINIST” typing his Post? Was all of the energy in our Universe crammed into an infinitesimal volume? Was it zero volume? How do we deal with the concept of all the energy in the Universe in zero volume when we can not understand our ordinary everyday infinities? Small wonder Dr. Cantor went mad.

Mel,

According to String Theory the smallest measurement is the Planck Length (1.6 x 10-35 m). Anything smaller becomes meaningless as far as any discernible difference in space-time. Once a body moving from point A reaches any distance smaller than the Planck Length from point B, it would essentially be at point B.

I assume that is how String Theory avoids dividing by zero.

Jerome