The physicalization of metamathematics and the implications for its foundations
writings.stephenwolfram.com
The physicalization of metamathematics and the implications for its foundations
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Re: The physicalization of metamathematics and the implications for its foundations
#2I 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 somewhat common in formal mathematics and will invariably come out at relevant meetings/conferences after a few glasses of wine. Not this instantiation in particular, but a generally similar sort of metaphysics. I've always gotten church vibes from that sort of thing. (I never "got" church.)
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.
But the way that it happens does make sense. Intellectual curiosity is a valid work selection strategy and sometimes invention/discovery requires a leap of faith. Developing a predisposition to spiritual thought patterns while doing work that requires a leap of faith makes sense, I guess, but it's something I warn young mathematicians to guard against.
But then, I'm also not allergic to fava beans. Maybe math-as-spirituality and believing beans are evil are also just useful tools.
[1] https://reference.wolfram.com/language/ref/FindEquationalPro...
Re: The physicalization of metamathematics and the implications for its foundations
#3I 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 not change. Humans can create notation and formalisms, but they do not invent the truths those mathematics represent.
Re: The physicalization of metamathematics and the implications for its foundations
#4I 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.
Re: The physicalization of metamathematics and the implications for its foundations
#5I 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…
> 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.
Re: The physicalization of metamathematics and the implications for its foundations
#6I 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…
> 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 the meta stack does sort of confuse this initially if you don't stay grounded. I wonder if there is a field of meta-map-making and whether map makers sometimes confuse maps with territories when they start meta-map-making work.
> 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.
I didn't walk away (undergrad, PhD, PI, editorial committees, grant reviewer, ...). I'm about as far from walking away as is possible. But I suppose it is possible I'm an imposture :)
Re: The physicalization of metamathematics and the implications for its foundations
#7Earlier 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…
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.
Re: The physicalization of metamathematics and the implications for its foundations
#8Earlier 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.
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.
Re: The physicalization of metamathematics and the implications for its foundations
#9Is this ruliad the logos of philosophy?
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My own attempt at what Wolfram is doing resulted in:
1: f(a,a)
2: f(a,b)
I call the former introspection and the latter is regular old composition. This seems to be a taxonomy of everything my discerning mind is capable of and hence the limit of what aspects of human experience I can peer review. The rest (the bulk) is mysticism and always-valid personal experience.
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“Whatever we call reality, it is revealed to us only through the active construction in which we participate.” – Ilya Prigogine, Order Out of Chaos: Man’s New Dialogue with Nature (1984)
Re: The physicalization of metamathematics and the implications for its foundations
#10Wolfram is a famous crackpot. I wish a peer review before I start reading his essay. No doubt he is again advertising his Wolfram* products, New kind of science, etc. No?