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 something like "...and this is how you get special relativity" or "this is how you get Maxwell's Equations", or even something abstract but relatively concise like Noether's Theorem. Maybe I just don't know enough higher math and physics -- is there a more concrete explanation of any of this?
The physicalization of metamathematics and the implications for its foundations
61–70 of 106 posts
Re: The physicalization of metamathematics and the implications for its foundations
#62Earlier quoted context omitted.
> 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 'm…
It's not a bijection either. If we take your level of pedantry seriously, it's nothing. There are no mathematical structures in play. I'm using natural language words to describe informal ideas outside of any mathematical system.
Let's use bloogidy-blop to avoid silly arguments over sequences of characters that have clear contextual meaning which you refuse to acknowledge for some reason :)
I'm not sure what your second point is supposed to be. Here's the relevant quotes:
>>> 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.
> 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.
Right... but I didn't say that there was a bloogidy-blop between maps and land. I said that a useful alien map would have to be roughly bloogidy-blop to our maps. So I'm not really sure what you're trying to say here that is relevant to my original post.
Re: The physicalization of metamathematics and the implications for its foundations
#63Earlier quoted context omitted.
'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 'm…
It's not a bijection either. If we take your level of pedantry seriously, it's nothing. There are no mathematical structures in play. I'm using natural language words to describe informal ideas outside of any mathematical system. Let's use bloogidy-blop to avoid silly arguments over sequences of characters that have clear contextual meaning which you refuse to acknowledge for some reason :) I'm not sure what your sec…
Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no?
Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality.
Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.
Re: The physicalization of metamathematics and the implications for its foundations
#64Earlier quoted context omitted.
It's not a bijection either. If we take your level of pedantry seriously, it's nothing. There are no mathematical structures in play. I'm using natural language words to describe informal ideas outside of any mathematical system. Let's use bloogidy-blop to avoid silly arguments over sequences of characters that have clear contextual meaning which you refuse to acknowledge for some reason :) I'm not sure what your sec…
Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no? Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality. Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we…
> Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no?
No, their maps could be VERY different but be used in roughly the same way and therefore useful in the same way. There is an operator involved -- reading/navigating/interpreting -- which is why I chose "isomorphism" instead of "equivalent".
Also, "isomorphism" IS natural language that is used in many fields outside of mathematics and also has a vernacular sense to it. The word happens to also be used to describe certain formal constructions by mathematicians from time to time, but it is natural language.
Again, this is all very pedantic and silly. I insist on bloogity-blop. If we're going to have silly arguments, let's use silly words :)
> Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality.
Reality isn't constrained by maps.
Useful maps are constrained by reality.
> Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.
Huh? I can't use reality the way I use maps because I'm not always able to fly into the air and look around before navigating to the supermarket or a trail head.
Reality is not constrained by my inReach. I promise you it's the other way around. And I promise you I can't fly, which means I need maps, even if flying high into the air would make it way easier to find a trail head than following a not-great trail map.
Maps are useful. Very useful. But they DO NOT constrain reality.
A belief to the contrary in the case of mathematics and physics is quite spiritual. Which was kind of my original point :)
Re: The physicalization of metamathematics and the implications for its foundations
#65Earlier 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?
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Re: The physicalization of metamathematics and the implications for its foundations
#66I have a lot of background in programming language theory and mathematical foundations, which is sort of one half of the topic that's explored in this post. Two thoughts: 1. Rewriting systems are very useful tools. One of the things I learned from this post was about the existence of FullEquationalProof [1], which I think is pretty darn neat and super useful. 2. This post is imbued with a latent metaphysics that is s…
> 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…
> Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change...[we] can create notation and formalisms, but they do not invent the truths those mathematics represent
This is a big philosophical question. (Kant would agree with you. Some of his detractors would not.)
Setting that aside, I agree with your criticism of the claim that mathematics only exists to solve concrete problems. Mathematical fiction, e.g. exploring how a system following nonsense rules might behave, is perfectly good math. It's interesting, potentially beautiful, first and foremost; it might also be useful, though that's of secondary concern. (It has an uncanny knack for being so [1].)
To say math must serve physical reality is to discard its artistic side, perhaps essence; that's disappointing, debilitating and reductive.
[1] https://aapt.scitation.org/doi/abs/10.1119/1.2402156?journal...
Re: The physicalization of metamathematics and the implications for its foundations
#67Here’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”.
Okay but no formalism (formal system) should take up the purpose of metaphysical or physical ontology.
That is a very clearly a mistake.
Re: The physicalization of metamathematics and the implications for its foundations
#68Here’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.
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!
Re: The physicalization of metamathematics and the implications for its foundations
#69Earlier 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…
> Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change...[we] can create notation and formalisms, but they do not invent the truths those mathematics represent This is a big philosophical question. (Kant would agree with you. Some of his detractors would not.) Setting that aside, I agree with your criticism of the claim that mathematics only exists to solve concrete…
With respect to aesthetics, to each their own. I do find some mathematics beautiful. In fact, "I am bored" is a real problem and mathematics can be used to solve that problem by being a tool that tickles our brains in pleasant ways.
The "tool" and "problem" here are meant as comments on the metaphysical content of mathematics, not some sort of statement that mathematics is for engineering and that's all.
In particular: I'm commenting on the imbued/latent metaphysics of Wolfram's post, which goes beyond mere artistic appreciation. If his framing were "and look how pretty cellular automata are!" then I guess my reaction would be "yeah they are quite cool aren't they?"
I find Church-Rosser quite beautiful and also think Wolfram puts way too much metaphysical weight into the behavior of confluent rewrite systems. Similarly, some Psalms are beautiful and the story of Jesus is very nice but god does not actually exist. There's no contradiction there -- you can take the beauty and spit out the metaphysics.
Re: The physicalization of metamathematics and the implications for its foundations
#70Earlier quoted context omitted.
> 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 'm…
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