Sunday, 3 September 2017

What is notation? (in Maths and Music)

When we are taught notations, whether in music or mathematics, we are always taught what certain symbols denote. In music, students have to work out which note is meant to be played when, and all of these "instructions" are contained in the way the note is written. In maths, we are taught which symbol means what and how they can be combined, and how strings of symbols can be manipulated and related to other symbols. So we learn that 2 + 3 = 5 is a legitimate use of the symbols 2, 3, 5, +, =, but 2 + 3 = 6 is not. When we learn maths, we are conditioned to mistake the symbols from the meaning. The meaning we only learn by playing with the symbols and working out what is legitimate and what isn't. What does "legitimate" mean? It must be some kind of social expectation: mathematicians coordinated their "dances with symbols" with the dances of other mathematicians. Without this coordination, there is really no maths at all.

It's the same with music. In any notated music, we are told which notes to play and in what order, and in what time. There is much that we are not told. The symbols are really an attempt to convey the constraints within which one might express oneself freely in music to be coherent with others expectations. The notation tells us what not to do.

Is the flow of logic a flow of constraint? What not to do at time t1 is not the same as what not to do at time t2. When solving a mathematical problem, or doing some kind of formal logical proof, there is a fluctuation in what not to do. To indicate these is to coordinate a common set of constraints between mathematicians.

Notation indicates constraints. But it produces its own constraints. Whatever is the reality of number lies in the common lifeworld which is experienced between people manipulating representations of number. But if notation is assumed to be real of itself, it will produce unexpected results which lead to confusion.

An example (from Lou Kauffman): if Euler's identity is:

which then means
How can an imaginary number equal a real number?

Musicians don't get caught in this. They coordinate their expectations at a deeper level. The mathematical example is produced because there is a double-layer constraint (much like a double-bind). There is constraint between the coordination of expectations about number at one level, and coordination of expectations about the notation at another.

1 comment:

Oleg said...

Why would you think the recurring exponential is real? Is it because we are initially taught that imaginary numbers need to have an 'i' - but later life becomes more interesting?

BTW it's similar to an irrational number being expressed as a recurring fraction:

(sqrt(5) - 1)/2 = 1/(1+1/(1+1/(1+....)))...