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

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

#31

Post by housecarpenter » Thu, Apr 07, 2022, 5:13 PM UTC

whatshisface wrote: post #7

thwayunion wrote: post #6

Earlier 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…

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.

Any species could prove the same theorems given the same axioms, but (besides the fact that they might not choose the same axioms) I'm not sure if they would prove the same subset of theorems that we have proven/will prove. Perhaps they'd have different ideas about what is interesting.

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

#32

Post by pas » Thu, Apr 07, 2022, 5:15 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…

Something simila: soap bubbles and minimal Steiner trees: https://scottaaronson.blog/?p=266

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

#33

Post by stonemetal12 » Thu, Apr 07, 2022, 5:15 PM UTC

whatshisface wrote: post #7

thwayunion wrote: post #6

Earlier 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…

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?

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

#34

Post by ganzuul » Thu, Apr 07, 2022, 5:15 PM UTC

thebooktocome wrote: post #5

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…

> Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change. Humans can create notation and formalisms, but they do not invent the truths those mathematics represent. We quite literally have no way of ever knowing this. This proposition and its negation are both beyond the scope of human knowledge.

It's almost as if there is a phase diagram with impedance mismatches between systems of belief.

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

#35

Post by vaillant » Thu, Apr 07, 2022, 5:18 PM UTC

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 theory's bogged down in issues of axioms in the wake of Cohen's proof of the indecidability of the continuum hypothesis. It's probably better to see these foundational projects as providing windows into the mathematical universe instead of being the actual substance of mathematics.

After all, we do mathematics without pure logic or sets all the time. Axioms are chosen for their elegance and ability to describe conceived mathematical concepts, not the other way around.

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

#36

Post by vaillant » Thu, Apr 07, 2022, 5:19 PM UTC

alangibson wrote: post #4

I have no special knowledge, but I've read that Wolfram has been shopping his grand unified theory around for a while with no takers in academia. He reportedly started his own business to be free to pursue his research program on his own terms, but you have to wonder if the bumpers of peer review and academic respectability aren't there for a reason.

It's not complete bullshit, it just fails to engage with the already existing work on the subject. He's creating his own entire system instead of looking at similar work by others and fitting his theory in with those. If you don't play philosophy on their turf, philosophers won't engage with you.

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

#37

Post by whatshisface » Thu, Apr 07, 2022, 5:20 PM UTC

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…

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.

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

#38

Post by meroes » Thu, Apr 07, 2022, 5:21 PM UTC

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…

How can you explain appealing to these “unreal objects” (real numbers, set theory, arithmetic) * does* help science? (Effectiveness maybe)

I see you are also a non realist about science.

But even the methodological naturalist (one who takes natural empirical science to be the best method but not an ontology) must wonder how we are uncovering and putting more precision to more and more of the world.

I don’t think we can currently explain why this made up tool “works”.

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

#39

Post by whatshisface » Thu, Apr 07, 2022, 5:21 PM UTC

housecarpenter wrote: post #31

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.

Any species could prove the same theorems given the same axioms, but (besides the fact that they might not choose the same axioms) I'm not sure if they would prove the same subset of theorems that we have proven/will prove. Perhaps they'd have different ideas about what is interesting.

Human mathematicians are already fanning out into other systems of deduction (constructive mathematics being a great example), and given enough time the mathematicians of each galaxy will eventually discover the other galaxy's mathematics, even if it perhaps happens in a different order.

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

#40

Post by zozbot234 » Thu, Apr 07, 2022, 5:23 PM UTC

thwayunion wrote: post #2

I 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…

Note that "well-behaved" rewriting systems are usually confluent; the nLab wiki has a useful description of confluent categories[ https://ncatlab.org/nlab/show/confluent%20category ]. In general, category theory has plenty to say about any mathematical structures where simple operations may be arbitrarily "composed" in sequence to build more complex ones, and rewrite systems seem to be one example of this (if perhaps one where physical substrates that directly reflect that structure may be easier to come across).