- #71
apeiron
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Ken G said:Ah, interesting question. No doubt this is indeed the central basis of the Platonist idea that mathematical truths lie at the heart of reality, such that when we discover those truths, we are discovering reality.
Plato was really more concerned with the form of things than mathematical truth as such. But maths is the science of patterns, so there is a large overlap.
The modern way of viewing this would be that the realm of form (as the counterpart to the realm of substance) is about self-organisation. Or global constraints (yes, sigh). It is about all the self-consistent patterns that can exist. And so it is all about symmetry principles. Changes which are not a change.
Maths looks like reality as maths creates a library of possible pattern descriptions and reality is a self-organising pattern.
Wolfram's cellular automata project was so funny because he took a very simple pattern generator and then just exhaustively generated every possible pattern to see which ones resembled reality.
But in general, this is what maths does. It defines some broad axiomatic truths (it creates some global constraints) and then generates all the possible patterns made possible by those constraints.
The problem then is that the axioms can't be known to be true (even if the consequences that flow from the axioms are deterministic, or at least taken to be proven step by step).
So the forms are real. But the maths is the modelling of forms.
However, maths can also hope to model self-organisation itself. Which is where chaos theory for example comes in as a hugely successful way of modelling "random nature".
Key constraints (on linearity for instance) are relaxed. The system is then allowed to organise its own constraints as part of what it does.
This is why maths can make historical progress. The early work was overloaded with presumed constraints (such as a presumption space was flat, it had just three dimensions, time was a separate clock, etc). Too much was taken as globally axiomatic when it was only specific to a local system.
But maths has just kept relaxing the constraints so as to arrive at the fewest and simplest axioms. And then more recently started to do the other part of the job - invent models of constraint development. Models of global self-organisation.
So first strip the constraints out, then find the way they can build themselves back in. Once maths reaches this level of modelling, it really will be powerfully close to the truth of a self-organising reality.