Improvisation versus Order

Abstract: This article discusses what seems to be a paradoxal situation, i.e. that DonParagon has objections against maths, yet uses what appear to be mathemetical calculations in his composing of songs.



A. Improvisation versus Order

Improvisation and order are often regarded as opposites. Improvisation is something that defies rules and structured planning. For people who like order and rules, improvisation is seen as disruptive, provocative behavior that has no place in a civilized society, except perhaps as a default response in emergency situations.

For people like DonParagon, improvisation is not something negative at all, but is instead the ultimate way to act in the future. For DonParagon, improvisation is part of his Vision of the Future, in which the law, rules, order, science and religion and similar ideologies and belief systems will be replaced by optionality in future times that Don refers to as Improvisation Time.

As one can imagine, scientifically oriented people reject Don's Vision. They often point at maths as the justification of their scientific orientation, as the evidence that there is "order in the Universe".

This calls for a more detailed analysis. Don's Vision focuses on cultural differences and seems to "hinge on moral and ethical arguments", as one of Don's critics said recently. Therefore, let's first go all the way into the holy grail of maths and look into mathematical manipulations such as counting, addition and multiplication.

B. The Basics of Maths

In the lower school grades, children are taught to count. Initially, this just seems like giving names to fingers. But if maths is nothing but a method to shorten sentences, then it is nothing more than rhetoric. After all, if counting is just giving things different names, then maths is simply a subset of language. The question is whether there is more logic behind maths than there is substance behind rhetoric.

Having learned to give their fingers different names (i.e. the names of numbers), children will first try to work out questions such as 1+2=? and 2+3=? by using their fingers. The problem is that, in such equations, the name three (that first belonged only to one specific finger, i.e. the middle finger) is suddenly applied to a group of (three) units. Few people seem willing to imagine that children have problems with this. They see this as the basics (thus easiest part) of maths. Yet, some children regard counting and grouping as two entirely different things. The equation 1+1+1=3 appears confusing to them. The problem that such children have with the equation 1+1+1=3, is that three different fingers are apparently each given the same name (i.e. number one, as they each count for one unit). Subsequently, the same finger (the third one, the middle finger, which was to be regarded as only one finger, just like the other ones) is suddenly given the name three like it used to have. For such children, counting and grouping are two entirely different things that are confusingly mixed at school.

Teachers, often without realizing this, present children with a number of different processes during the first few school years. There is counting. There is grouping of units. Group-counting, i.e. counting in groups of two, three or more units, could again be regarded as a separate process. Skip-counting, i.e. counting while each time skipping one unit, could also be regarded as a separate process that introduces the concept of odd and even numbers. Similarly, the concept of prime numbers could be treated as the result of a separate process.

Most teachers present grouping of units as a logical step that is implied in counting itself. Most teachers regard group-counting and skip-counting as an abstraction from counting, i.e. from the use of fingers to imaginary sequences or groups of units. For such teachers, differences between children's ability in dealing with maths are just a question of mathemetical talent. They believe some children are simply stupid, unwilling to accept the obviousness of maths. They will accept that some children are better in mental arithmatic, while other children prefer to see the sums in front of them. They will accept that different cultures produce different maths results in the classroom. But they will not accept that anyone can have any philosophical problems with maths.

In school, children are first taught to count. Adding up is then presented as a manipulation that is implied in counting. In an equation such as 3+4=7, the 3 and the 4 are both regarded as groups of units that are each counted individually within their group. The summation and equation symbols (+ respectively =) then instruct all those units to be regarded as part of one single line. Philosophically, this is not just a huge step, it is a questionable act. At first the groups of 3 and 4 units each had their own counting lines, each counting back to one. At the other side of the equation there suddenly is only one counting line.

Teachers like to present multiplication as just another case of counting. The question what six times three is, is tackled by creating six groups of three units each. Such an equation is presented as a stretched-out exercise in counting, resulting in eighteen units to be counted in total. In this approach, the symbols 6x3 stand for a queue of units that are counted and add up to eighteen. Again, the separate counting lines within groups are magically transformed into a single counting line.

Maths combines processes such as counting, adding and multiplying into a single system, as if all such processes are manipulations of numbers that are all lined up and named in a uniform way. Maths takes things that are clearly separate things to start with, and then unifies them in some magical way. It is a trick, the great unifying trick. The worst part is that maths has no philosophical scruples about this. Maths simply presents this as the truth, the one and only truth. Maths does not even accept any alternative model of approaching things. Order is what children are taught at school, not as a choice, but as the truth.

C. The Logic of Maths

One may wonder why school encourages the use of fingers in counting, given the potential to reveal philosophical inconsistencies with school's dogma. There is an insidious purpose in this. The use of fingers or other physical objects is a seductive method; it suggests that there is some physical reality supporting the exercise of giving each finger a different name; it suggests that there is logic behind the rhetoric; it suggests that using numbers is a reflection of nature, of physical reality. Maths as a system embraces all manipulations of numbers, as if all such manipulations are part of one uniform, logical framework that makes up the building blocks of reality.

At an early age, children are told to use their fingers to count, to create the impression that there is some physical reality behind counting. Subsequently, when children are expected to have accepted this dogma, they are confronted with addition and multiplication. If they, by that time, are still using their fingers to count, they are told not to use their fingers any more, as if this is just for babies.

The deeper logic behind maths is linear and singular. The essence of maths is a line with an infinite amount of points, each point on this line having a different name and each point positioned on this line at an equal distance to their neighbours. Addition is moving up on this line, while the opposite - subtraction - is going down. The line may be imaginary, but for scientifically oriented people, the logic regarded as real. Central in the logic of counting and in logic of adding and subtracting numbers is the concept of zero, the absence of anything as the balance of it all. The zero is the perfect center of this line. The philosophy of the zero is accepted like the gospel, it is the holy truth of science.

D. Don Paragon's Music

DonParagon is someone who rejects this philosophy. Don regards conformity with supposedly universal rules as a state of paralysis imposed by the philosophy of the zero. Instead, Don regards improvisation as the best way to act. Don plays his music not as if there is one, single perfect composition carved into stone. Instead, Don improvises, always keen to try out new combinations of sounds and rhythms, changes in volumes and tempos, etc.

Given Don's objections against maths, some people may wonder why Don appears to resort to mathematical calculations to work out his compositions. For some compositions, Don has worked out frequency ratios with great precision, as discussed in the article Improvisation in Music.

The reason for this is that the keyboard of instruments such as the piano and electronic organs is normally tuned in one specific way, called the equally tempered tuning system. This system fixes the pitch of keys at specific frequencies, e.g. the A (a specific key in the middle of the keyboard) at 880Hz. This system is also reflected in MIDI on electronic keyboards and on computers. Don is not happy with the uniformity that this system expresses. Don argues that it sounds better to work out more harmonious frequency ratios. Don rejects the suggestion that this is the same as using a different tuning system and using maths to work it out. Don does not accept any fixed frequency points as if they all line up on a single number line. Don does emphasize specific relationships between sounds, but such relationships are relative, rather than absolute.

As an example, at one stage in a song sound A may relate to sound B. They (A and B) may sound nice together, as they relate to each other in terms of simple frequency ratios. Subsequently, sound B is sustained and combined with sound C. Sound B may also relate to sound C in the same simple frequency ratios and thus they also sound nice together. But that does not imply that sound A and sound C would sound nice together. In fact, they may sound horrible when they are played together. In some compositions, Don deliberately chooses sounds such as the A, B and C in the above example. This way, Don expresses his rejection of the idea that all sounds in a song should conform to a single line of fixed frequencies. Don does not count the points on such a single line to work out specific frequencies. The frequencies of sounds in Don's songs are not absolute. Instead, sounds are related to other sounds, perhaps directly or perhaps indirectly. A change in one sound may therefore imply changes in many other sounds. This is what makes improvisation so interesting - a seemingly minor change may turn out to have a major impact.


Appendix A. Choice versus Uniformity

The sounds and symbols used in counting, as a sequential naming system, may differ from one language to the other. But there is a limit to the amount of names one can remember and apply in daily use. Instead of giving each unit an individual name, it makes sense to name groups of units, on order to reduce the amount of different names and symbols one has to use. Such grouping of units can be done in various ways. In some cultures, it was common to use the alphabeth; other cultures preferred less digits, e.g. the decimal system, the "Imperial System" and the hexadecimal system, in which numbers are grouped in subsequently ten, twelve and sixteen units. Computers are even more radical and prefer a binary system of only two distinctive digits (hence the word 'bits').

A zealous use of fingers to count, together with dictators' lust for standards, have benefited the metric system (i.e. decimal grouping) to the extent that the metric system is often regarded as the only possible system. But in reality, there is choice between such naming systems and such choice appears rather arbitrary. Mathematicians are quick to admit that there are various ways to group numbers, but they will still maintain that there is only one way in which such numbers line up.

This example shows the differences in perspective. From the dictatorial perspective of maths, all naming systems should conform to the logic of a single line, just like a dictator likes to use a single line to impose decrees. But from a practical perspective, time and again it turns out that there are choices.


Appendix B. Does Maths make things easier?

Few people seem to bother about the intricacies of such early maths, yet the problems people have with maths are widespread. A simple household task such as programming a videorecorder for time recording seems to complicated for many people. On older VCRs one had to key in the amount of minutes that it took to broadcast the program. To work this out required subtraction of the start time from the end time. Watching the clock on the wall usually means one has to count in steps of five. Furthermore, many people have problems with the fact that there are sixty minutes in one hour, i.e. time seems to progress in conflict with the metric system. Such issues appear too complex for many people. Newer VCRs tackle this problem by asking for both the start time and the end time of the program. Some models allow for bar-code entry of program details.

The fact that it has taken so long for manufacturers of VCRs to come up with more friendly ways of time programming, is indicative for the established attitude towards maths. Maths is essentially a way to abbreviate words. Teachers did not like to write out long sentences on the blackboard. If sentences became too long, they did not fit on the blackboard and they became too hard too read anyway. So teachers started to introduce abbreviations and symbols. This is all that maths is. Teachers can capture a lot of rhetoric in a single formula. Teachers therefore present maths as something that makes things easier.

The problem with maths is that it expresses the philosophy of uniformity, of universal truth, of singularity, etc. Some people may agree to use a symbol or abbreviation to refer to a specific process. This may be useful in whatever they are doing. But what maths does is to elevate such an abbreviation or symbol to a uniform system that everyone has to use, as if it is the one and only true reflection of universal reality. Of course, this is nothing but dictatorship, which may seem an easy way for the dictator to deal with society, but which clearly is detrimental for all involved, including that very dictator.




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