Mathematics - innocence regained

Mathematics develops like anything else, as a resonating set of interrelated quanta, which are entangled with all things that are shaped by it.

Mathematics arises spontaneously

QO offers a way out of the mess of lost mathematical innocence. It postulates that mathematics arises like anything else, as a result of quantum mechanical processes. Individual components (concepts, axioms, rules of calculus) of a mathematical system may arise more or less independently, and then, by a process of stepwise resonance of pairs of components, build up a resonating whole. The mathematical system will persist if it comes to expression in things, because then there is no way back to nothingness. The system is true, from the time that it first resonates. Before that, the system has no meaning. Before 1 + 1 resonated to be 2, 1 + 1 had no meaning. But once it resonated to be 2 and this was expressed in the universe, it has unchangeably been 2. 

Okay, but how does this get mathematics out of the mess? Well, to start off with, it helps to avoid situations which have led to contradictions, as follows:

• All valid mathematics is finite. Any concepts of infinities or singularities are unsound. Continuity, which relies on infinite numbers of infinitely small steps, is an illusion similar to the illusion of continuous and smooth motion produced by a video film.
• Valid mathematical concepts cannot refer to themselves, either directly or indirectly. Therefore Russell's paradox fails (but also, I suspect, Gödel's proof).
• Only those mathematical operations which could be the result of the interaction of waves to produce new waves are permitted. For example, to produce the difference between two mathematical concepts is not necessarily a valid mathematical operation, just as the difference between two waves is not necessarily or even normally a wave.
• Only mathematics which expresses itself in the universe in some way may be expected to be sound. We have no guarantee that mathematics which we invent ourselves is free of contradictions.
• Only mathematical systems that resonate are valid. Why some systems resonate and others don't is perhaps beyond our abilities to explain, but is nevertheless a fact. Suffice it to say that resonance and mathematical elegance are closely related concepts.
• Only mathematics that can be constructed in a stepwise fashion can be presumed to be valid. Mathematical statements that can be proved only by reductio ad absurdum are therefore invalid. In any case, such arguments often depend on the assumption that if something is not true, it must be false, which assumption QO considers to be incorrect, because the statement might merely be undecided.

The QO approach prevents us from producing nonsense. It leads us to conclude that mathematical propositions which we cannot prove are neither true nor false, but undecided. And although that means that there will always be undecided statements, the ones that matter, namely the ones that express themselves in our universe, are in principle provable. For those that like applying labels, QO leads to a finitist, constructivist approach to mathematics.