vol III Development:
Chapter 3: Cybernetics
page 5: Mathematics
Form and formalism
We might say that science is an answer to Parmenides' question: how can we have certain knowledge in a changing world? His answer has become standard: there is a central, eternal complete core of being which can be truly known by 'the path of conviction', then there is the ephemeral world of day to day events which we know through 'the way of mortals'. John Palmer - Parmenides
Through Plato and others, mathematics became associated with the way of conviction, since it appeared to deliver indisputable truths. In his dialogue Parmenides Plato expounds (through the character Socrates) his theory of forms, that 'there is a single, eternal, unchanging, indivisible, and non-sensible form corresponding to every predicate or property.' Samuel Rickless: Plato's Parmenides
We can identify two opposing concepts about the nature of mathematics: Platonism maintains that 'there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. . . . Mathematical truths are therefore discovered, not invented.' The alternative view is that mathematical statements are all our own work, invented rather than discovered. Øystein Linnebo
David Hilbert and Formalism
This second approach is often called formalism. Formalist mathematicians are not concerned with the reality or unreality of mathematical truths, but see mathematics as a field of formal games played according to certain axioms and rules of inference that sometimes lead to interesting results, and sometimes not. From this point of view, the only constraint on mathematics is that its axioms and rules be consistent and lead to consistent conclusions. Mathematical 'existence' is equivalent to formal consistency.
This approach was espoused by David Hilbert, among others, who was impressed by the discoveries of Georg Cantor, which he called 'Cantor's Paradise'. Although Cantor's transfinite numbers seem incredible at first sight, they involve no inconsistency. Hilbert saw these numbers as a hierarchy of function spaces, and von Neumann discovered that one form of function space, Hilbert space, is the natural habitat for quantum mechanics. Cantor's paradise - Wikipedia, von Neumann: Mathematical Foundations of Quantum Mechanics
Hilbert thought that the formal method would be so powerful that ultimately every mathematical problem would be solved. Gödel and Turing showed that this was not to be so: there are limits, known as incompleteness and incomputability, on formal mathematics. There are mathematical problems that have no consistent answer. Here we see these facts, when applied to the physical world, as the foundation of uncertainty and creativity. Gödel's incompleteness theorems - Wikipedia, Turing machine - Wikipedia
Eugene Wigner and the 'unreasonable effectiveness of mathematics'
Einstein 1954 to Besso: 'I consider it quite possible that physics cannot be based on the field principle, ie on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included. Kevin Brown: Reflections of relativity
Wigner highlights the miraculous correspondence between mathematical symbolism and observations of the physical Universe: . . . the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it. Eugene Wigner
Wigner's observation compels us to examine the relationship between mathematics and physics. Here I wish to suggest an explanation of Wigner's observation. The idea is essentially very simple. Science is devoted to detecting and connecting the fixed points of the Universe. Mathematics, on the other hand, represents the fixed points of a subset of the Universe, the mathematical community.Insofar as the Universe is one and consistent, it is not surprising that we find a considerable degree of isomorphism or symmetry between these two sets of fixed points. The challenge here is to build a bridge from physics through the mathematical community to the Universe as a whole that clearly illustrates that symmetry. Symmetry - Wikipedia
Quantum mechanics has taught us that all the observable features of the Universe (upon which science is based) are quantized. Nevertheless, quantum mechanics assumes that the mechanism underlying these quantized observations can be represented by continuous mathematics. Here I suggest that this assumption is false, and that we can better describe the Universe by assuming that it is digital 'to the core'.
This digitization suggests that we can see the Universe as a logical, rather than a geometric continuum. The mathematical representation of a logical continuum is the Turing machine, a stepwise digital process that leads deterministically from an initial condition to a final condition. We may see such logical connections as the fixed points in the universal dynamics which form the goal and substance of science. Linear continuum - Wikipedia
I guess that the theorems requiring fixed points in a dynamical system are indifferent to the complexity of the system, and I postulate an isomorphism between the dynamics of the mathematical community and the dynamics of the world. I propose that this isomorphism explains the 'unreasonable effectiveness' of mathematics in the sciences.
Is the transfinite network isomorphic with mathematics?
Here we accept the formalist view of mathematics introduced by Hilbert. Hilbert's approach distinguishes mathematics from the study of observable reality and sees it as a game played with certain symbols and certain rules. The symbols cannot move themselves to implement the rules: all the action comes from the mathematicians (and their computers) manipulating fixed and invariant symbols according to the rules of their current game. Richard Zach
The mathematics industry, like any other, thus comprises workers (mathematicians) who 'do' the mathematics and their mathematical communications which may be conversations, lectures, papers, books, models or any other means of mathematical communication.
Although mathematicians talk about continuous and infinite entities like lines and real numbers, all their communication is symbolic, that is finite and quantized. Everything that a mathematician needs can therefore be represented by finite strings of symbols manipulated according to finite sets of rules. This situation is reflected in the mathematical literature, which is finite even if when it talks about infinite entities like the set of natural numbers.
The transfinite network provides us with a model of the whole mathematical process. Mathematicians themselves are represented in network by nodes, which are themselves networks of cells. molecules and so on. These networks have a certain cardinality, the cardinal of humanity. Mathematicians communicate with one another using mathematical protocols which enable them to encode and decode ideas like 'complex numbers', 'Hilbert space', 'differential and integral operators' and so on.
The transfinite network is a formal model of the Universe, able to formally model its own dynamics, including the mathematics industry. We use this insight to found the claim that the transfinite network is in effect isomorphic to mathematics. Like mathematics, it has no boundaries except formal consistency and the rate at which its operators can work.
(revised 8 January 2019)
