The physicalization of metamathematics and the implications for its foundations

https://writings.stephenwolfram.com/2022/03/the-physicalization-of-metamathematics-and-its-implications-for-the-foundations-of-mathematics/

by SvenSchnieders • 4 years ago

97 98 4 years ago

writings.stephenwolfram.com

thwayunion14• 4 years ago

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 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 <insert structure here>" 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...

jonbronson• 4 years ago

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.

thebooktocome2• 4 years ago

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.

thwayunion14• 4 years ago

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 :)

mrob• 4 years ago

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.

arka2147483647• 4 years ago

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.

JumpCrisscross• 4 years ago

Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change...[we] can create notation and formalisms, but they do not invent the truths those mathematics represent

This is a big philosophical question. (Kant would agree with you. Some of his detractors would not.)

Setting that aside, I agree with your criticism of the claim that mathematics only exists to solve concrete problems. Mathematical fiction, e.g. exploring how a system following nonsense rules might behave, is perfectly good math. It's interesting, potentially beautiful, first and foremost; it might also be useful, though that's of secondary concern. (It has an uncanny knack for being so [1].)

To say math must serve physical reality is to discard its artistic side, perhaps essence; that's disappointing, debilitating and reductive.

[1] https://aapt.scitation.org/doi/abs/10.1119/1.2402156?journal...

vba616• 4 years ago

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?

eli_gottlieb• 4 years ago

I never got the "emergent properties have special aesthetic and nearly spiritual significance" or "everything is just an <insert structure here>" 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.

zozbot234• 4 years ago

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).

thwayunion14• 4 years ago

That's certainly a description, but I'm not sure it's terribly useful for anyone who doesn't already know what confluence means in the context of a rewriting system.

For the casual programmer who's seen a lambda calculus at some point and is interested in understanding confluence in some detail, I recommend https://www.cs.cornell.edu/~kozen/Papers/ChurchRosser.pdf

If that work is too dense, all you need to know is the first paragraph. If that's all greek to you, it can be expanded as follows: Suppose you have a program e and an interpreter that executes e by applying a big set of rules that rewrite e until e becomes a value (such as a number, string, function, etc.).

So, for example, (function add(x) { x+x }; add(5)) -> 5+5 -> 6+4 -> 7+3 -> ... -> 10.

Here, "->" is thedefined by a big set of rules that basically pattern match on the syntax of the left-hand term and produce a corresponding right-hand term. So, for example, we have rules like:

(function F(x) { B[x] }; F(y) -> B[y]

and

n+m -> (increment of n) + (decrement of m)

and

n+0 -> n

where n,m are defined to be non-negative natural numbers.

Your interpreter is just a big set of these sorts of rewriting rules, and there's no "ordering" on the rules. Any rule that is applicable to the left-hand side could be used at any point where it's applicable.

Imagine, now, that you define your rewrite rules and then notice that at some points more than one rule might be applicable to the left hand side!

That could be bad if a single program could compute different values depending on which rule was chosen!

In general: Suppose there are some rewrite rules such that that e -> ... -> e1. (Which we sometimes write as e ->^* e1 for readability.) Suppose there are also some rules such that e -> ... -> e2, where e1 and e2 are different and might not be values (ie there's still more rules applicable to e2 and e3).

Your system is confluent -- i.e., not bad in the above sense -- if whenever the above happens there are also some rules such that e1 -> ... -> e3 and e2 -> ... -> e3. Then you know "there are many possible executions of the interpreter but at the end of the day the interpreter always spits out the same value"

vaillant4• 4 years ago

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

People just like spiritual beliefs. There's a reason why a majority of human beings hold non-rational beliefs like religion or money having intrinsic value.

alangibson• 4 years ago

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.

ganzuul8• 4 years ago

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.

porcoda• 4 years ago

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.

vaillant4• 4 years ago

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.

ganzuul8• 4 years ago

Is this ruliad the logos of philosophy?

---

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.

---

“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)

vaillant4• 4 years ago

I think that identifying the ruliad with the "logos" makes sense - the idea that there's some underlying abstract shape to the universe that makes rationality possible in the first place.

ganzuul8• 4 years ago

Perhaps we can back-propagate physics through mathematics to discover physical proof of logos? Then we could substantiate the claim that reality is not illusion composed of ever fluctuating qualia.

That discovery would certainly fill this one with wonder.

novaRom• 4 years ago

Wolfram 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?

boole18543• 4 years ago

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 of study, a major breakthrough was made with the first formalization of propositional logic (as Boolean algebra) by George Boole in 1854. Since then, there has been a search for the simplest foundational formal axioms from which all of propositional logic could be derived. This project ended in 2000 with Wolfram's discovery of, and proof that, [1] is the shortest possible single axiom that can be used as a foundation for all of propositional logic.

[1] ((a⁍b)⁍c)⁍(a⁍((a⁍c)⁍a)=c, where ⁍ is NAND

thwayunion14• 4 years ago

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 useful purpose, then why should this impress me? Doing novel work is really easy -- just work on stuff other people don't care about.

pierrebai• 4 years ago

The tricky bit is that "simplest" has no formal definition. Here Wolfram claims to simplicity is having the least number of axiom, even if that axiom is very complicated.

Unfortunately, that is a perversion of the idea of axiom. An axiom should be as simple as it can be and ideally self-evident. Being self-evident is a strong requirement because axioms are not proven but accepted as true.

Clearly, no-one would say that Wolfram's axiom is self-evident. As a matter of fact, even Wolfram does not find it self-evident: that it is equivalent and sufficient as a basis needed to be proven, via proving it can generate all other set of axioms.

Basically, once a field has matured enough and we found a simple set of axioms, one can come up with a new set. All set of axioms must encode the same information, so you can either have multiple simple one or a single complex one. The quantity of information that is encoded must be constant.

So having a single axiom is not simpler, except by the very naive measurement of count.

adamnemecek• 4 years ago

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...

ganzuul8• 4 years ago

Is this the magic behind the kernel trick?

m0llusk• 4 years ago

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.

ganzuul8• 4 years ago

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.

vaillant4• 4 years ago

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.

ogogmad• 4 years ago

the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure

sigh I find these discussions tedious, but oh well...

I've heard this before and it seems wrong and rooted in a misunderstanding. Can someone (not necessarily the person I'm replying to) explain why you believe this?

solveit• 4 years ago

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.

No, most of mathematics can be expressed in set theory. It's like saying every program can be written in C. It's more or less true, but the philosophical implications are overblown. That is, it's important that set theory and C are so powerful, but there's nothing[1] special about them in particular, we could just as well choose different foundations/Turing complete languages.

[1] Disclaimer: I am not a set theorist and I presume there's a reason set theorists study ZFC and its more powerful cousins so intensively.

boole18543• 4 years ago

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.

You say this as if it contradicts some point that the author is trying to make or some key assumption that he holds.

Yet on the contrary, the author himself makes a similar point in the article:

But the way we’ve modeled mathematics here has been much more about what statements can be derived (or entailed) than about any kind of abstract notion of what statements can be “tagged as true”. In other words, we’ve been more concerned with “structurally deriving” that “” than in saying that “1 + 1 = 2 is true”.

That is, math is not necessarily about truth or usefulness in some (meta)physical sense, it is about entailment from certain axioms via formal rules.

danbruc• 4 years ago

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.

pas• 4 years ago

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

thwayunion14• 4 years ago

I'm not sure that this objection has much practical significance even if it turns out to be true.

I think the more pressing concern with the ruliad program is that a description of all things possible is a also description of nothing in particular.

Other workers developing mega-logical type stuff, frameworks and so on, ran into similar problems when it came time to find actual utility for their work. Sure, you have this super expressive thing... but the things you're supposed to build in it are better built on their own terms and the super expressive framework doesn't buy you enough to be worth engaging with.

uoaei6• 4 years ago

I can imagine a universe where what is computable inside the universe is not sufficient to describe the universe.

I can too. It's called "any mathematical system, period". Goedel showed as much.

lol_what• 4 years ago

no need to imagine this, this is already the case in this universe. at least if we're talking about actually computing something, not just writing down the equations on paper.

for example from everything we know at least so far there are some truly continuous non-quantized quantities yet all numerical solutions can ever produce is an ever increasingly good approximation of something.

some constants are irrational so we can never get true values of certain physical constants, etc...

nyc1112• 4 years ago

When we do physics, the traditional approach has been to start from our basic sensory experience of the physical world, and of concepts like space, time and motion—and then to try to formalize our descriptions of these things, and build on these formalizations. And in its early development—for example by Euclid—mathematics took the same basic approach.

I don’t think Euclid started from “basic sensory experience”. Euclid based his mathematics on non-existent things, neither point nor line exist in this world as defined by Euclid: A point is that which has no part. A line is breadthless length.

Also, in physics “space” does not exist as a quantity, only distance exists. Space is an abstraction.

nyc1112• 4 years ago

that we call the ruliad—and that our physical laws arise in an inexorable way from the particular samples we take of this structure.

What is a physical law? To me a physical law is a proportionality, that is, an equality of ratios. When we find something that stays constant while something else is changing, we call this a law. But it is really a proportonality. So, mathematics and physics is tied by proportionality. They have proportionality in common.

meroes5• 4 years ago

Here’s something I don’t get about Wolfram and insisting on a computation-like underbelly of the universe.

Computation is built on the idea of Turing machine. But what is reading the tape in Turing’s analogy? A human! The tape and Turing machine are designed so that every human agrees upon its formal validity.

It’s not a statement about mental states or computation based “reality”. More than all of those and first, it is a about how society can use social rule-following and basic step-by-step processes (using language) to create formal systems.

A human reading the tape or even a human with pencil and paper. That is the main analogy.

So why do so many like Wolfram think computation is reality? It seems from the get go he is headed down the wrong track.

You and I are computers and computers can think according to Turing. But TM’s came about to develop formal systems.

To me Wolfram and the Churchlands seem to have completely unjustified claims.

zmgsabst2• 4 years ago

There are other formalisms, which better fit Wolfram’s claims — eg, lambda calculus or automata.

A model of state transfers on encoded operators actually isn’t very far from “physics just happens to particles”.

meroes5• 4 years ago

Okay but no formalism (formal system) should take up the purpose of metaphysical or physical ontology.

That is a very clearly a mistake.

VirusNewbie6• 4 years ago

I think you missed the forest for the trees. It may help to familiarize yourself with turing completeness and what wolfram calls computational equivalence.

meroes5• 4 years ago

Not the angle I’m going down. I’m purely talking about don’t confuse how to engineer a formal system with ontology.

That is the hype about formal systems. They are formal, they do not model the world!

uoaei6• 4 years ago

Computation is not built on the idea of the Turing machine... That is a very, very strong claim that I assume you did not mean to imbue with such heft.

The Turing machine, rather, is a Platonic ideal -- a model -- of a computing machine which is transparent to humans performing analysis on it. There are other models, and there are other means of computation.

Invictus0• 4 years ago

/r/iamverysmart

AdamH12113• 4 years ago

Last I heard about Stephen Wolfram (10-15 years ago) he had gone down a rabbit hole of trying to use cellular automata to model all of reality. I guess this "ruliad" concept is where he ended up. I can never figure out whether there's any actual substance there -- every time something of his gets posted it's a long, abstract article that links to multiple other long, abstract articles, but I never get any sense of something like "...and this is how you get special relativity" or "this is how you get Maxwell's Equations", or even something abstract but relatively concise like Noether's Theorem. Maybe I just don't know enough higher math and physics -- is there a more concrete explanation of any of this?

SkyMarshal• 4 years ago

Same. He really needs to learn how to omit the hype and self-aggrandizement and reduce his articles down to their scientific essence only. Stop burying the lede and put the main ideas in an abstract, then develop, support and critique them in the text.

That would reduce their length by roughly 50% and increase their readability and information density immeasurably. For all its faults, this is one thing the academic scientific publishing system does well.

He needs to just let his work speak for itself, instead of trying to be a salesman for it. His inability or unwillingness to do that is negative signal, no matter how positively he phrases it.

I continue waiting with an open mind to see if his techniques produce any kind of predictive theories. But without testable predictions, there's no way of knowing if ruliads and the like are a new and useful representation of reality, or just the rules for an alternate simulation wholely different from and unconnected to our reality.

It would be really cool if they turn out to be the former, and he's figured out a new symbolic+computational representation of Maxwell's equations and relativity. But we'll see.

motoboi• 4 years ago

I remember reading something concrete last year maybe was this: https://www.wolframphysics.org/bulletins/2020/05/event-horiz...

jkhdigital• 4 years ago

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.

wly_cdgr• 4 years ago

Stephen Wolfram often comes off as the smartest insane homeless man you've ever met

The physicalization of metamathematics and the implications for its foundations

Related Stories