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The physicalization of metamathematics and the implications for its foundations

writings.stephenwolfram.com

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Re: The physicalization of metamathematics and the implications for its foundations

#71

Post by uoaei »

VirusNewbie wrote: post #58
pfortuny wrote: post #27

Earlier quoted context omitted.

Oh, dear. Truth being the operating word… There is no truth in a set of axioms we cannot even conceive properly (any infinite set has properties beyond what seems reasonable, even “just” the Natural numbers). From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent… We (I am a working mathematician) do not understand our objects, we can just make do. Only finite graph th…

> From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent… This is kind of wrong. Are you familiar with Godel's thoughts on this?

...are you?

Re: The physicalization of metamathematics and the implications for its foundations

#72

Post by uoaei »

danbruc wrote: post #28

But what our Physics Project suggests is that underneath everything we physically experience there is a single very general abstract structure—that we call the ruliad—and that our physical laws arise in an inexorable way from the particular samples we take of this structure. I call it the ruliad. Think of it as the entangled limit of everything that is computationally possible: the result of following all possible co…

> I can imagine a universe where what is computable inside the universe is not sufficient to describe the universe.

I can too. It's called "any mathematical system, period". Goedel showed as much.

Re: The physicalization of metamathematics and the implications for its foundations

#73

Post by uoaei »

meroes wrote: post #48

Here’s something I don’t get about Wolfram and insisting on a computation-like underbelly of the universe. Computation is built on the idea of Turing machine. But what is reading the tape in Turing’s analogy? A human! The tape and Turing machine are designed so that every human agrees upon its formal validity. It’s not a statement about mental states or computation based “reality”. More than all of those and first, i…

Computation is not built on the idea of the Turing machine... That is a very, very strong claim that I assume you did not mean to imbue with such heft.

The Turing machine, rather, is a Platonic ideal -- a model -- of a computing machine which is transparent to humans performing analysis on it. There are other models, and there are other means of computation.

Re: The physicalization of metamathematics and the implications for its foundations

#74

Post by SkyMarshal »

AdamH12113 wrote: post #61

Last I heard about Stephen Wolfram (10-15 years ago) he had gone down a rabbit hole of trying to use cellular automata to model all of reality. I guess this "ruliad" concept is where he ended up. I can never figure out whether there's any actual substance there -- every time something of his gets posted it's a long, abstract article that links to multiple other long, abstract articles, but I never get any sense of so…

Same. He really needs to learn how to omit the hype and self-aggrandizement and reduce his articles down to their scientific essence only. Stop burying the lede and put the main ideas in an abstract, then develop, support and critique them in the text.

That would reduce their length by roughly 50% and increase their readability and information density immeasurably. For all its faults, this is one thing the academic scientific publishing system does well.

He needs to just let his work speak for itself, instead of trying to be a salesman for it. His inability or unwillingness to do that is negative signal, no matter how positively he phrases it.

I continue waiting with an open mind to see if his techniques produce any kind of predictive theories. But without testable predictions, there's no way of knowing if ruliads and the like are a new and useful representation of reality, or just the rules for an alternate simulation wholely different from and unconnected to our reality.

It would be really cool if they turn out to be the former, and he's figured out a new symbolic+computational representation of Maxwell's equations and relativity. But we'll see.

Re: The physicalization of metamathematics and the implications for its foundations

#75

Post by VirusNewbie »

uoaei wrote: post #71
VirusNewbie wrote: post #58

Earlier quoted context omitted.

> From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent… This is kind of wrong. Are you familiar with Godel's thoughts on this?

...are you?

Very much so.

Re: The physicalization of metamathematics and the implications for its foundations

#76

Post by uoaei »

meroes wrote: post #68
VirusNewbie wrote: post #52

Earlier quoted context omitted.

I think you missed the forest for the trees. It may help to familiarize yourself with turing completeness and what wolfram calls computational equivalence.

Not the angle I’m going down. I’m purely talking about don’t confuse how to engineer a formal system with ontology. That is the hype about formal systems. They are formal, they do not model the world!

We could be a little more permissive here about what constitutes a "model".

    Models are not systems.  Maps are not territories.
Formal systems are models. They model the world. Depending on which formal system you use, one may model the world better than another. There is no "perfect" there is only "comprehensible", "insightful", and, when predicting the future, "accurate". All of these are measured in degrees, not kinds.

Models are not related to systems a priori. You are permitted to apply any model to any system. Whether the application is useful is only posterior to this first step.

Re: The physicalization of metamathematics and the implications for its foundations

#78

Post by [deleted] »

meroes wrote: post #67
zmgsabst wrote: post #51

Earlier quoted context omitted.

There are other formalisms, which better fit Wolfram’s claims — eg, lambda calculus or automata. A model of state transfers on encoded operators actually isn’t very far from “physics just happens to particles”.

Okay but no formalism (formal system) should take up the purpose of metaphysical or physical ontology. That is a very clearly a mistake.

[deleted]

Re: The physicalization of metamathematics and the implications for its foundations

#79

Post by dane-pgp »

stonemetal12 wrote: post #33
whatshisface wrote: post #7

Earlier quoted context omitted.

Any species would prove the exact same theorems given the same axioms, and since mathematicians only claim that their axioms imply their theorems, I think they are right to claim absolute truth.

That begs the question. Would any other species pick the exact same axioms? Why would they have the exact same theorems? Are you suggesting there is only one way to think logically?

Any species would face a selective pressure towards theorems which help them understand the world around them (if they have any motivation to prove theorems at all), and they will similarly face a pressure towards choosing the smallest/simplest set of axioms which allows all those theorems to be proven (and new ones to be discovered).

In fact, if we assume that neural networks are the only sorts of intelligence that can occur naturally in the universe and be sophisticated enough for arbitrary abstract calculation[0], then we might be able to infer things about the sorts of concepts they will develop and in what order. For example, having the concept of finite sums would likely occur before having the concept of infinite sums.

[0] I know that cellular automata can emulate a universal Turing machine, but I can't imagine a situation existing in nature where the cells evolve into an arrangement that produces a Turing machine, much less a machine running a program of instructions that lead to it generating mathematical theorems.

Re: The physicalization of metamathematics and the implications for its foundations

#80

Post by macrolocal »

whatshisface wrote: post #37
pfortuny wrote: post #27

Earlier quoted context omitted.

Oh, dear. Truth being the operating word… There is no truth in a set of axioms we cannot even conceive properly (any infinite set has properties beyond what seems reasonable, even “just” the Natural numbers). From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent… We (I am a working mathematician) do not understand our objects, we can just make do. Only finite graph th…

I would contend that A -> B can be true even if A is not true or more relevantly to this discussion if A is unknown. That's math's version of objective truth, where "A" is filled by our various axioms and rules of inference.

In ~300 BCE, Euclid decides to work with morphisms not objects.
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