“How can you map or define the space of solutions?” [closed]

Question

Particular solutions, or solutions in general, can be thought of as a combination of waves within a finite period of time. The creation of the waves are generally constrained by limits of physics of oscillators. We also typically associate certain types of solutions with number of particles and certain types of oscillators.

Due to the definable finiteness, it would seem that one could create a mathematical space that was sufficiently large such that each conceivable solution produced by oscillators in the universe would be defined as a point in that space. So in some wild way, it is in fact the determination and mapping of the space of solutions that that creates the solutions, and it is merely up to a group of scientists to later discover the points of interest in that space.

Not being a scientist myself, what sort of physical parameters and variables would I consider to define the space of solutions in physics?

Answer 1

I premise that your question is a bit different to normal question about physics here, For a composer (I'm a composer for example... ;-) ), characters of sound for our hearing are:

-intensity: that is connected to intensity of the wave (amplitude, frequency, density of air...and you must include the fact that the response of our ears is logarithmic (Weber-Fechner law)...);

-lenght;

-height: that is connected to frequency, but here the human ears have logarithmic response...

-tone-colour: a violin has a different sound to a piano...this depend from fourier spectrum of the waves produced from instruments.

These 4 charachters are specified by composer with musical notation. So, because they are associated with physical and measurable magnitudes, you can associate them to numbers. If we consider only intensity, lenght and height, we associate every wound at a point with values of these parametres as coordinates in a space (I think it will have strange topologycal proprieties, that sure won't vectorial space...). A melody is rapresented so with a set of points in this space. When you have many voices (many instruments, or a piano or a organ that are polyphonic instrument) you must consider cartesian product of many of that spaces.

The tone colour complicate the problem, because the number of its fourier armonycs is infinite;-) But I think that with a laborious work you can define it in this situation too.

Your question is so connected with another question: "maths is invented or discovered?" I think the first option... ;-)

Answer 2

This is a philosophical question on the lines of: "how many monkeys would it take and for how long to reproduce the works of Shakespeare".

It also borders on the question of whether consciousness can be described with mathematical tools or it is a meta level to the mathematical descriptions. Example of meta-level: the alphabet and "Romeo and Juliet". The latter is a meta level to the level of alphabet.

In that sense the songs are meta levels to sounds and words, and the rules of meta levels are different from the rules of the base. Example from physics: statistical mechanics was found to be the base level of Thermodynamics, each framework having independent axiomatic and mathematical descriptions. So even though the particles participating in an ensemble define a mathematical space on which thermodynamics is based, the thermodynamic description has a mathematical order with independent solutions appropriate to the thermodynamic view.

The same holds for songs from your hypothetical space.

"Discovery" in the everyday sense has no meaning because one would have to explore zillions of ensembles to hit on one that has a good solution for a song. Probability 0 .