The physicalization of metamathematics and the implications for its foundations • Hacker News Board

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

#81

Post by boole1854 » Thu, Apr 07, 2022, 8:19 PM UTC

thwayunion wrote: post #60

boole1854 wrote: post #49

Earlier quoted context omitted.

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 use…

Workers in the field of logic made steady progress in shortening the axioms throughout the 20th century, after Alfred N. Whitehead's axiomatization in 1898. Edward Huntington reduced it to three axioms using fourteen instances of two operators (OR and NOT) in 1933. Herbert Robbins conjectured it could be reduced to thirteen instances of those operator but could not prove it. Alfred Tarski also investigated and could not prove it. In 1967, Carew Meredith proved the axiomatization could be reduced twelve instances of OR and NOT (with only two axioms). Later, Meredith shows it could be reduced to two axioms with ten instances of a single operator (NAND). Several single-axioms systems were later found, but they were very long axioms. William McCune proved the Robbins conjection in 1996. And finally Wolfram and McCune apparently independently discovered and proved that the shortest possible single-axiom has only six instances of a single operator (NAND).

That this list of workers includes names like Whitehead and Tarski suggests that this was in fact something that (some) significant people did care about. I'm not sure how much value to give to this heuristic, but every name on the list of workers above is an 'important enough person' to have their own Wikipedia article. Certainly when I look discrete math classes in university, this was discussed as if it was an interesting topic (and I personally was interested).

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

#82

Post by VirusNewbie » Thu, Apr 07, 2022, 8:44 PM UTC

uoaei wrote: post #77

VirusNewbie wrote: post #75

Earlier quoted context omitted.

Very much so.

So lording esoteric knowledge over the anonymous rabble is more productive than just explaining what is the issue?

ok fair enough. There is no issue with proving arithmetic consistency on a finite number of symbols. Incompleteness entirely relies on the unbounded induction step.

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

#83

Post by ogogmad » Thu, Apr 07, 2022, 8:46 PM UTC

vaillant wrote: post #35

m0llusk wrote: post #18

There is some interesting material here, yet at a basic level this sounds like it is all steeped in the kind of misunderstanding of mathematics that is common among physicists. In physics there are real tests and relations that have meaning so it makes sense to ask if String Theory is correct or useful. In mathematics there are complex structures built from axioms and sometimes these structures can be related to each…

> The particular thing that comes to mind repeatedly when reading this is the fact that more or less all of mathematics can be derived starting either from set theory or from logic theory. I don't know if this is the actual foundation of mathematics though - we're seeing more advances in category theory, the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure, and set the…

> the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure

sigh I find these discussions tedious, but oh well...

I've heard this before and it seems wrong and rooted in a misunderstanding. Can someone (not necessarily the person I'm replying to) explain why you believe this?

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

#84

Post by lol_what » Thu, Apr 07, 2022, 9:03 PM UTC

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…

no need to imagine this, this is already the case in this universe. at least if we're talking about actually computing something, not just writing down the equations on paper.

for example from everything we know at least so far there are some truly continuous non-quantized quantities yet all numerical solutions can ever produce is an ever increasingly good approximation of something.

some constants are irrational so we can never get true values of certain physical constants, etc...

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

#85

Post by tromp » Thu, Apr 07, 2022, 9:18 PM UTC

boole1854 wrote: post #81

thwayunion wrote: post #60

Earlier quoted context omitted.

> 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 use…

Workers in the field of logic made steady progress in shortening the axioms throughout the 20th century, after Alfred N. Whitehead's axiomatization in 1898. Edward Huntington reduced it to three axioms using fourteen instances of two operators (OR and NOT) in 1933. Herbert Robbins conjectured it could be reduced to thirteen instances of those operator but could not prove it. Alfred Tarski also investigated and could…

I wonder if this is related to the shortest one combinator basis being λx λy λz. x z (y (λ_. z)), whose type is (a -> b -> c) -> ((d -> a) -> b) -> a -> c.

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

#86

Post by motoboi » Thu, Apr 07, 2022, 9:24 PM UTC

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…

I remember reading something concrete last year maybe was this: https://www.wolframphysics.org/bulletins/2020/05/event-horiz...

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

#87

Post by thwayunion » Thu, Apr 07, 2022, 9:33 PM UTC

ogogmad wrote: post #83

vaillant wrote: post #35

Earlier quoted context omitted.

> The particular thing that comes to mind repeatedly when reading this is the fact that more or less all of mathematics can be derived starting either from set theory or from logic theory. I don't know if this is the actual foundation of mathematics though - we're seeing more advances in category theory, the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure, and set the…

> the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure sigh I find these discussions tedious, but oh well... I've heard this before and it seems wrong and rooted in a misunderstanding. Can someone (not necessarily the person I'm replying to) explain why you believe this?

AFAICT among non-academics it's mostly rooted in pop sci story telling. Same genre as “Godel went insane because of his impossibility result” and nonsense like that.

Among academics (esp. mathematicians) this impression comes from the fact that if you look at almost any mathematics department, there aren't many people working in/on formal logic. But that's mostly because all of the mathematicians working on/in formal logic suffer the humiliation of sitting in the fancy new CS building with higher salaries and lower teaching loads ;)

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

#88

Post by solveit » Thu, Apr 07, 2022, 9:42 PM UTC

vaillant wrote: post #35

m0llusk wrote: post #18

There is some interesting material here, yet at a basic level this sounds like it is all steeped in the kind of misunderstanding of mathematics that is common among physicists. In physics there are real tests and relations that have meaning so it makes sense to ask if String Theory is correct or useful. In mathematics there are complex structures built from axioms and sometimes these structures can be related to each…

> The particular thing that comes to mind repeatedly when reading this is the fact that more or less all of mathematics can be derived starting either from set theory or from logic theory. I don't know if this is the actual foundation of mathematics though - we're seeing more advances in category theory, the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure, and set the…

> The particular thing that comes to mind repeatedly when reading this is the fact that more or less all of mathematics can be derived starting either from set theory or from logic theory.

No, most of mathematics can be expressed in set theory. It's like saying every program can be written in C. It's more or less true, but the philosophical implications are overblown. That is, it's important that set theory and C are so powerful, but there's nothing[1] special about them in particular, we could just as well choose different foundations/Turing complete languages.

[1] Disclaimer: I am not a set theorist and I presume there's a reason set theorists study ZFC and its more powerful cousins so intensively.

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

#89

Post by TheOtherHobbes » Thu, Apr 07, 2022, 10:25 PM UTC

thwayunion wrote: post #62

VirusNewbie wrote: post #54

Earlier 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…

This is clearly not true. If the aliens happened to be 10% or 1000% of our size, their concept of a relevant physical feature would be very different.

Maps are basically feature extraction -> data compression. The feature extraction part is subjective and depends on the experience of a species.

A map of cellphone towers is useless to a cat. A map of blobblytoids is useless to a human who doesn't know what a blobblytoid is or how to recognise one.

A human may have some vague awareness that something is there, but it's also possible that blobblytoids look like random noise, or like weird probabilistic anomalies that travel around inside a multidimensional space, or like something completely unimaginable.

So in the limit features can't be extracted because they are invisible to a different consciousness. They can still be physically present, but their meaning as a feature of interest depends on having a subjective referent for them.

This seems to be something many humans struggle with. We assume everyone else - including other humans - has the same set of referents, and therefore our personal feature maps are somehow universal.

Of course they aren't. They aren't even universal among humans, never mind a completely unknown alien species.

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

#90

Post by meroes » Thu, Apr 07, 2022, 10:59 PM UTC

uoaei wrote: post #76

meroes wrote: post #68

Earlier quoted context omitted.

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 kin…

To me this a problematic viewpoint. All these formalisms are not meant to be models of the world. I’d even argue for anything to be accepted as a formal system it must be far removed from taking stances on reality.