Simply put, how can an object take an infinitely small step? The continuum of a line requires an infinite number of points. Assume a line that is one centimeter long; it has an infinite number of infinitely small points which are infinitely close together. Now consider another line that reaches from here to the Andromeda Galaxy. It too has an infinity of infinitely small points that are infinitely close together. This indicates the absurdity of thinking of infinity as something that actually exists.
Much of our math appears to be a method of making non-linear functions linear to be better able to work with them. This is not always a good idea and a way to examine this is by way of music theory and the problems that have occurred. It is not without reason that through much of its history music was considered to be a science.
A properly tuned chromatic scale is formed by three ratios; 16⁄15 25⁄24 and 135⁄128 . The sequential order of these three ratios in any given scale is determined by the controlling tonality. It is easy to see why it is impossible to build a keyboard instrument that can play every chromatic scale. The solution for this was debated for centuries with many big names getting into the act, some for a linear solution and some against. Once the market for keyboard instruments such as the harmonium became quite large the even-tempered scale became the only practical method of tuning these keyboard instruments. The even-tempered scale defines every chromatic semitone to be the twelfth root of two. That way we have a nice, easy to use, isometric scale, a thing that never appears in actual music. The problem is that everything is then out of tune.
This would have been fine but what happened is that the notion of an even tempered scale took over music theory and by the beginning of the twentieth century music theory now defined the minor second to be the twelfth root of two. This certainly simplified music theory. Arnold Schoenberg who, judging from his more conventional works was a minor composer with a bit of talent came up with a supposed system of musical composition. Chromatic writing was the going fashion then and so he devised an alleged system of composition based on the even-tempered scale. He espoused atonality even though atonality is physically impossible. He defined his system as using these twelve tones which, he said, that are related only to each other when in fact none of these tones have any relationship to each other at all.
This was a Godsend for both would be composers and the university scene as now one could become a composer with nothing more than a knowledge of musical rudiments and the ability to count to twelve. Then Joseph Schillinger wrote an enormous two volume set describing another twelve tone system of composition. He even went so far as to state that without the even-tempered scale harmony could not exist. Fortunately both twelve tone composition and the Schillinger system have drifted into blessed obscurity but music theory still defines the chromatic semi-tone as being the twelfth root of two.
Once this isometric scale system arose with it arose the tonometric system. This was designed to measure distances between tempered intervals. The tempered scale is linear of course but the tonometric system is also used to define 'distances' in the real music system even though you cannot measure non-linear functions with as linear ruler. The tempered semitone was now divided into 100 parts, the twelve-hundredth root of two. This they called a cent. It is endlessly quoted today in spite of the fact that it measures absolutely nothing. It surfaces endlessly but I have as yet to see one thing that it been used for apart from being quoted. All this from a well meaning but impossible attempt to linearize the non-linear music system.
It would seem that all of us, to one degree or another, have the need to be 'special', to belong to a real or imagined elite, not because they particularly enjoy opera. Many people go the opera simply because they feel that makes them part of an elite. The trends in music throughout the twentieth century were a haven for such desires. If you wrote music that people liked you were accused of pandering to the public. 'Avant-Garde' actually became defined as a style of composing. It is hard to imagine that science is free of such needs and it can easily become a great place for those seeking the elite, real or otherwise.
Scientists speak of seeking simplicity and yet we are burdened with a mathematic system whose complexities are mind-boggling. I think science has prospered so well is not because of its math but rather by the brilliance of those using it.
Being human it is very difficult for us to fashion an objective approach to reality but it may be safe to say that in the final analysis a continuous reality requires a denumerable infinity and a denumerable infinity is a fiction.
Tom
