Numbers - PentadecaTET
Composition
The 1st part of the Arecibo message is a sequence of numbers in binary format from 1-10, which is to be read from top down. The bottom row blocks are markers to indicate the beginning of each of the numbers.
It's a very basic representation of our number system. It doesn't include any additional detail about combining these numbers nor any rules to follow. Without rules/axioms to govern the use of these numbers, they are meaningless - just like having the letters of the alphabet without any knowledge of what they sound like and how they phonetically fit together.
My first composition converts the rules of TNT and Zermelo-Fraenkel Set Theory into music. By amplifying the sound of pencil markings and mapping those sounds to harmony, I have been able to transcribe the 15 axioms into music. These axioms hold the fundamental basis of all mathematical principles both in arithmetic and set theory. Having this knowledge alongside the numbers described in the message, extra-terrestrial life would be able to formulate further theories and thus explore the infinite world of mathematics.
The full score provides details on the electronic setup and how to execute a live performance or recording of the work. Below are some details on the mathematical theories behind the work.
It's important to note that our number system (0,1,2,3,4,...) is represented by successors in TNT, denoted by 'S'. So 0 is represented by 0 and 1 is represented by S0 then SS0 for 2 (the successor of the successor of 0) etc...
For example, Axiom 1 states that for all values of a it is not true that the successor to a is 0 (we are restricting the domain to natural numbers). The other axioms introduce the notion of addition and multiplication. From these axioms further mathematical theories can be formed and proved.
The Metalogic about this theory is that it is a sound theory but it is not complete nor decidable. Gödel showed that TNT is incomplete and that it cannot prove its own consistency.
Using notation that is fairly similar are the 9+1 axioms of Zermelo-Fraenkel Set Theory. This set of axioms assert the existence of sets with various properties. From the collection of all sets, we carve out the usual inhabitants of the mathematical universe, not just the various number systems but also pairs, finite sequences, relations and functions. [2]
More details on these axioms and their respective symbols can be found here: ZF
The final axiom of the work is The axiom of choice. The axiom of choice (AC) is independent of
the other axioms of set theory. ZFC set theory consists of the ZF axioms together with AC. No
logical inconsistencies arise if we accept or reject AC. It is generally accepted, and many results
about infinite sets depend on it for their proofs. This has similar metalogic to TNT. [3]
On reflection, the work sits well in the bracket of experimental pieces that have the purpose of explaining scientific theories. What makes this work different to other compositions that use mathematics is that the beauty of the theory is not hidden or difficult to find but is instead shown throughout the work for the listener to follow and appreciate.
[1] Hofstadter, D. (1999), Gödel, Escher, Bach P204-213
[2]Avigad, J., 2017. 23. Axiomatic Foundations - Logic And Proof 3.18.4 Documentation. [online] Leanprover.github.io. Available at <https://leanprover.github.io/logic_and_proof/axiomatic_foundations.html>
[3] Fisher. D., (2016) MAT2051 Numbers and Sets Lecture Notes P9
(577 words)