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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

#21

Post by whatshisface »

thwayunion wrote: post #15
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.

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" :)

Nothing can be objectively interesting, only objectively true. Just because math is objectively true does not mean you've been robbed of your license to decide if you think it's interesting. :)

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

#22

Post by eli_gottlieb »

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…

>I never got the "emergent properties have special aesthetic and nearly spiritual significance" or "everything is just an " cognitive confusion that so many mathematicians (especially formalists) seem to have.

Say what you like about formalists, but at least they're not goddamn Platonists.

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

#23

Post by thwayunion »

whatshisface wrote: post #21
thwayunion wrote: post #15

Earlier quoted context omitted.

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" :)

Nothing can be objectively interesting, only objectively true. Just because math is objectively true does not mean you've been robbed of your license to decide if you think it's interesting. :)

Agreed. Similarly, just because I don't "get" church doesn't mean I can prove God doesn't exist. And it certainly doesn't mean I should stand in the way of others enjoying the experience of going to church regardless of their beliefs. It just means I don't "get" it.

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

#24

Post by whatshisface »

thwayunion wrote: post #23
whatshisface wrote: post #21

Earlier quoted context omitted.

Nothing can be objectively interesting, only objectively true. Just because math is objectively true does not mean you've been robbed of your license to decide if you think it's interesting. :)

Agreed. Similarly, just because I don't "get" church doesn't mean I can prove God doesn't exist. And it certainly doesn't mean I should stand in the way of others enjoying the experience of going to church regardless of their beliefs. It just means I don't "get" it.

Religion is slightly different though, they claim actual direct truth (not mere truth of implication given certain assumptions) which makes their claims more interesting but prevents them from claiming automatic objective truth. The Formal Gospel would go, "If God so loved the world that he gave his only begotten son, ..." ;)

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

#25

Post by marcosdumay »

thwayunion wrote: post #6
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…

> 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…

> I wonder if there is a field of meta-map-making

There certainly is, although I'm not sure it has a name. Kids gets introductions to it on schools, when they have classes about how to read a map in Geography.

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

#26

Post by thwayunion »

whatshisface wrote: post #24
thwayunion wrote: post #23

Earlier quoted context omitted.

Agreed. Similarly, just because I don't "get" church doesn't mean I can prove God doesn't exist. And it certainly doesn't mean I should stand in the way of others enjoying the experience of going to church regardless of their beliefs. It just means I don't "get" it.

Religion is slightly different though, they claim actual direct truth (not mere truth of implication given certain assumptions) which makes their claims more interesting but prevents them from claiming automatic objective truth. The Formal Gospel would go, "If God so loved the world that he gave his only begotten son, ..." ;)

That's why I used words like "spirituality" and "faith" instead of "religion" and "dogma".

I'm describing a shared social/psychological phenomenon, not a shared epistemological status.

Read my posts less as works of the philosophy of mathematics and more as anthropological musings.

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

#27

Post by pfortuny »

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.

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 theory has a chance of being “real”. And it stops being finite very soon.

And we certainly should be honest enough to admit that our “science” says very little about the “real” world, where truth lies.

Maths is just a tool. Funny, exciting and even in some sense beautiful. But “truth” does it not contain. Except, I insist, in very specific finite constructions.

Statements hold but they are not “true” because they do not relate to the real world (otherwise, Frodo reaching Mount Doom would also be “true”).

There are no continuous functions out there. Bolzano’s theorem is not “true”.

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

#28

Post by danbruc »

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 computational rules in all possible ways.

My initial objection is the following. I can imagine a universe where what is computable inside the universe is not sufficient to describe the universe. The universe might, for example, run on real numbers but due to something vaguely resembling the uncertainty principle those can not be fully used for computations within that universe and so the most powerful computational device within the universe ends up being something discrete like a Turing machine.

Admittedly those two quotes are essentially everything I have read about this topic and this might be addressed somewhere, maybe my objection itself is not consistent, but I think one needs a good justification why computability within a universe is essential for understanding or explaining that universe.

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

#29

Post by arka2147483647 »

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…

Most people are intrested in what they can do with mathematics, and therefore, for them, it is a tool.

Because of this, they are more likely to ’get’ mathemathics if it is presented to them as a tool, instead of as an abstract truth-of-everything.

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

#30

Post by JackFr »

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 would prove the exact same theorems given the same axioms

Assume there exists at least one species....

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