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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

#11

Post by thebooktocome »

whatshisface wrote: post #8
thebooktocome wrote: post #5

Earlier quoted context omitted.

> 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.

We do have a way of knowing this, it is as simple as saying that the finish line of a symbol game will stay the same given the initial symbols and the rules that can be used to move them around.

OP is explicitly not talking about something open to human interaction (“saying”, “symbols”, “moving (symbols) around” are all human operations).

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

#12

Post by whatshisface »

thebooktocome wrote: post #11
whatshisface wrote: post #8

Earlier quoted context omitted.

We do have a way of knowing this, it is as simple as saying that the finish line of a symbol game will stay the same given the initial symbols and the rules that can be used to move them around.

OP is explicitly not talking about something open to human interaction (“saying”, “symbols”, “moving (symbols) around” are all human operations).

None of those are human-specific, machines and animals can do them. (Except saying, but I'm the one doing that verb, talking about moving symbols.)

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

#13

Post by ganzuul »

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.

Most responses are clouded by emotion still. This is a kind of war on reality since the proposal is to pass the torch of Truth-saying from one institution to another. That no heads are being lopped off in more than verbiage is truly great progress in our quest for self-knowledge.

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

#14

Post by mrob »

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…

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.

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

#15

Post by thwayunion »

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.

This is either an extremely obvious and boring observation or the basis for a metaphysical trip, depending on how pre-disposed you are to "mathematical spiritualism" :)

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

#16

Post by porcoda »

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.

I think he started his company to sell Mathematica - the research agenda stuff has been more recently emphasized. When I first started using Mathematica almost 30 years ago the company was pretty much just selling the tool for technical computing. It was after his book came out around 20 years ago that things drifted more towards his weird research, which really amped up in the last decade.

The biggest complaint I’ve seen about his work, especially the new Wolfram Physics stuff is that it’s a theory without predictions. He pretty much exclusively talks about representing things we already know but in terms of his world of cellular automata and rewriting systems. Until it is applied to make a testable prediction, it’s not as much science as it is gratuitous programming, visualization, and grandiose claims.

It is impressive the sheer volume of self-congratulating text that he can produce. It’s quite hard to read since it spends nearly as much time talking about himself and how insightful he is as it does the technical topics. From the things I’ve read, there isn’t much of value buried in there.

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

#17

Post by adamnemecek »

I have thought about this thing for some time and I think that the unifying idea of math, physics (quantum shit in particular), programming, type theory, probability etc is the idea of fixed points which is just ubiquitous. Fixed points are based on the idea of an adjoint (1-to-many relationship) and a norm (many-to-1 relationship).

I have written up something about this https://github.com/adamnemecek/adjoint

Jon Claerbout has website that explains the concept of adjointness https://reproducibility.org/RSF/book/bei/conj/paper_html/pap...

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

#18

Post by m0llusk »

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 other in interesting ways. Will anything important ever come of String Theory? Maybe, but from a mathematical point of view no theory needs to correctly model what is real or have demonstrated application to be worthy of exploration and study.

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. There is no particular reason to choose one or the other, but set theory is the most common foundational substrate. Depending on how this is done it may be more simple, easy, or direct to express or make proofs regarding some particular idea or domain. But Wolfram seems to be seeing things in terms of there being a big space with defined paths through it. Instead of leaning on set theory or logic theory to get to a particular expression or result there should be a global basis for expressions and results. That different models can be used for the same phenomena doesn't seem to be considered valuable or even a possibility.

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

#20

Post by ganzuul »

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…

You insert "an observer like us" and see if the universe it generates looks like ours. If it does, we learn about the observer. This was the original project of natural philosophy. - Know thyself.
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