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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

#51

Post by zmgsabst »

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…

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”.

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

#52

Post by VirusNewbie »

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…

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

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

#53

Post by ganzuul »

vaillant wrote: post #43
ganzuul wrote: post #9

Is this ruliad the logos of philosophy? --- My own attempt at what Wolfram is doing resulted in: 1: f(a,a) 2: f(a,b) I call the former introspection and the latter is regular old composition. This seems to be a taxonomy of everything my discerning mind is capable of and hence the limit of what aspects of human experience I can peer review. The rest (the bulk) is mysticism and always-valid personal experience. --- “Wh…

I think that identifying the ruliad with the "logos" makes sense - the idea that there's some underlying abstract shape to the universe that makes rationality possible in the first place.

Perhaps we can back-propagate physics through mathematics to discover physical proof of logos? Then we could substantiate the claim that reality is not illusion composed of ever fluctuating qualia.

That discovery would certainly fill this one with wonder.

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

#54

Post by VirusNewbie »

thwayunion wrote: post #6
jonbronson wrote: post #3

Earlier quoted context omitted.

> But I ultimately think of mathematics as a just an invented tool whose only reason for existence is to solve concrete problems. This might be the source of disconnect. I frequently encounter this perspective and worry there's a fundamental problem with how mathematics is taught if so many people walk away believing this. Whether or not humans ever mastered mathematics, what is and isn't mathematically true would no…

> Humans can create notation and formalisms, but they do not invent the truths those mathematics represent. The land represented by a map exists independently of humanity. Another intelligent species would have to come up with a roughly isomorphic representation if they wanted a similar tool. Maps, to be clear, are just invented tools. They can be more or less right or wrong, but they are not the territory. Moving up…

'roughly isomorphic' would be saying 'not isomorphic', so I'm not sure what you're trying to say. Actually I'm frustrated with most physicists/math folks misusing this term 'isomorphism' to mean "a bijection".

Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them.

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

#56

Post by thwayunion »

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'm not sure that this objection has much practical significance even if it turns out to be true.

I think the more pressing concern with the ruliad program is that a description of all things possible is a also description of nothing in particular.

Other workers developing mega-logical type stuff, frameworks and so on, ran into similar problems when it came time to find actual utility for their work. Sure, you have this super expressive thing... but the things you're supposed to build in it are better built on their own terms and the super expressive framework doesn't buy you enough to be worth engaging with.

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

#57

Post by zmgsabst »

pfortuny wrote: post #27
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.

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…

The enduring appeal of both triangles and Frodo is that they relate a truth that is hard to see, because it’s diffuse and abstract.

But that “abstract truth” is why your pizza doesn’t flop if you fold it or Amazon can’t make a good LOTR.

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

#58

Post by VirusNewbie »

pfortuny wrote: post #27
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.

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?

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

#59

Post by ganzuul »

mrob wrote: post #14
jonbronson wrote: post #3

Earlier quoted context omitted.

> But I ultimately think of mathematics as a just an invented tool whose only reason for existence is to solve concrete problems. This might be the source of disconnect. I frequently encounter this perspective and worry there's a fundamental problem with how mathematics is taught if so many people walk away believing this. Whether or not humans ever mastered mathematics, what is and isn't mathematically true would no…

How is the Banach–Tarski paradox a truth that exists independent of humanity? It makes a physically implausible assumption (existence of infinitely small objects) and reaches a physically implausible conclusion (violation of conversation of mass). Mathematics is full of things like this. They all look like human inventions to me.

General Relativity does not conserve energy.

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

#60

Post by thwayunion »

boole1854 wrote: post #49
novaRom wrote: post #10

Wolfram is a famous crackpot. I wish a peer review before I start reading his essay. No doubt he is again advertising his Wolfram* products, New kind of science, etc. No?

It seems to me that 'crackpot' is a bit strong. Even if one thinks Wolfram's foundations of physics project will never bare useful fruit, it's undeniable that he has made progress in other fields that are of interest to many people. A simple case in point: the study of logic has been a interesting human endeavor for thousands of years, since at least the time of the Greek and Vedic schools. After thousands of years o…

> Even if one thinks Wolfram's foundations of physics project will never bare useful fruit, it's undeniable that he has made progress in other fields that are of interest to many people.

This is true, I'm sure, in Physics. But I don't think it's true in logic. Did anyone working in logic at the time care about this question? Does it have any practical significance? If experts at the time didn't care and it has no useful purpose, then why should this impress me? Doing novel work is really easy -- just work on stuff other people don't care about.

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