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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

#91

Post by jordan_curve »

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…

He made novel contributions to computability theory. The first person to show that one-dimensional cellular automata could be turing-complete.

That being said, he’s very much a crackpot nowadays. Being sane in the past does not imply you will be sane in the future, no matter how significant your list of accomplishments.

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

#92

Post by jkhdigital »

Why does Stephen Wolfram have to be so long-winded? I’m sure there is something valuable buried in this post, but it is over 50000 words. I guess this is what happens when a researcher doesn’t frequently publish in peer-reviewed formats with page limits.

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

#93

Post by thwayunion »

jordan_curve wrote: post #91
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…

He made novel contributions to computability theory. The first person to show that one-dimensional cellular automata could be turing-complete. That being said, he’s very much a crackpot nowadays. Being sane in the past does not imply you will be sane in the future, no matter how significant your list of accomplishments.

His results on cellular automata are clearly solid contributions. Any academic would be proud to open a whole line of inquiry. I understand that he's also done good work in Physics.

I was just surprised to see this particular result about propositional logic in particular highlighted because, as a logician, I really don't care :)

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

#94

Post by thwayunion »

TheOtherHobbes wrote: post #89
thwayunion wrote: post #62

Earlier 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…

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 blobblyto…

Yes, the exact sense in which maps are bloogidy-blop to one another may vary considerably, but if they cover the same geographic region there'll be some bloogidy-blop, ie common reference points, which would relate to the maps. Maybe blobblytoids always go in valleys. Or whatever.

We seem in agreement on the object question in any case.

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

#95

Post by mxkopy »

TheOtherHobbes wrote: post #89
thwayunion wrote: post #62

Earlier 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…

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 blobblyto…

I think there is some sort of threshold analogous to Turing-completeness (it probably is just Turing-completeness) when it comes to intelligence, in that information is eventually accessible to any system that passes it.

If there is a species that thinks in terms of things that humans never could understand, then I would argue that species isn't part of our physical reality. If their expression of concepts is at all rooted in physical reality, then we, as physical beings, would have access to it. Maybe not the first person that sees it, nor the second, nor their great-great-great-grandchildren, but at some point their children could build things/begin to appreciate what was being communicated.

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

#96

Post by BreakfastB0b »

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.

Banach–Tarski relies upon the Axiom of Choice / Law of the Excluded Middle. Zermelo–Fraenkel set theory is independent of the Axiom of Choice and there's an entire field of Mathematics called Constructivist Mathematics which avoids including the Axiom of Choice / Law of the Excluded Middle.

I had a hard time grasping why the Axiom of Choice / Law of the Excluded Middle was so problematic until I heard it translated into a Computer Science context.

The Law of The Excluded Middle sounds very reasonable at first. For all propositions P, P ∨ ¬P. i.e. Every proposition is either true or false. Sounds fine right? But when viewed in the context of computer science via the Curry-Howard Isomorphism. A proposition is actually a program, and deciding the truth value of a proposition involves "running" that program. So The Law of the Excluded Middle is actually the Halting Problem! It's really saying that all possible programs terminate and yield true or false, but we know that some programs don't terminate, some propositions aren't true or false, but undecidable.

So circling back around to the Banach-Tarski paradox. I would be very skeptical of any paradoxes resulting from assuming the halting problem doesn't exist!

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

#97

Post by vba616 »

jonbronson wrote: post #3
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…

> 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…

>they do not invent the truths those mathematics represent

I got the idea somewhere that because Principia Mathematica was doomed to failure, that means any two "islands" of math are not necessarily related to each other.

So I would think that hypothetical aliens in different circumstances could in fact have math that didn't intersect with ours at all.

If they did, wouldn't it be possible to build one system up from foundations?

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

#98

Post by BreakfastB0b »

ganzuul wrote: post #59
mrob wrote: post #14

Earlier quoted context omitted.

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.

Underrated point. Conservation Laws emerge from symmetries via Noether's theorem. In particular, Conservation of Mass / Energy arises from Time Translation Symmetry. General Relativity doesn't have Time Translation Symmetry because the universe is expanding. Now the question is can we extract free energy from the expansion of spacetime and avoid the heat death of the universe?

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

#99

Post by JackFr »

mxkopy wrote: post #95
TheOtherHobbes wrote: post #89

Earlier quoted context omitted.

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 blobblyto…

I think there is some sort of threshold analogous to Turing-completeness (it probably is just Turing-completeness) when it comes to intelligence, in that information is eventually accessible to any system that passes it. If there is a species that thinks in terms of things that humans never could understand, then I would argue that species isn't part of our physical reality. If their expression of concepts is at all…

> If there is a species that thinks in terms of things that humans never could understand, then I would argue that species isn't part of our physical reality.

What if they think can think in our terms AND they can think thoughts we are physically unable to, thoughts that are literally inconceivable?

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

#100

Post by [deleted] »

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…

[deleted]
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