Comment by thwayunion | Hacker News Reader

Comment by thwayunion | Hacker News Reader

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...

> 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.

> 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).

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.)

It's almost as if there is a phase diagram with impedance mismatches between systems of belief.

> 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 :)

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. :)

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.

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”.

I would contend that A -> B can be true even if A is not true or more relevantly to this discussion if A is unknown. That's math's version of objective truth, where "A" is filled by our various axioms and rules of inference.

In ~300 BCE, Euclid decides to work with morphisms not objects.

How can you explain appealing to these “unreal objects” (real numbers, set theory, arithmetic) * does* help science? (Effectiveness maybe)

I see you are also a non realist about science.

But even the methodological naturalist (one who takes natural empirical science to be the best method but not an ontology) must wonder how we are uncovering and putting more precision to more and more of the world.

I don’t think we can currently explain why this made up tool “works”.

The enduring appeal of both triangles and Frodo is that they relate a truth that is hard to see, because it’s diffuse and abstract.

But that “abstract truth” is why your pizza doesn’t flop if you fold it or Amazon can’t make a good LOTR.

> From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent…

This is kind of wrong. Are you familiar with Godel's thoughts on this?

...are you?

Very much so.

So lording esoteric knowledge over the anonymous rabble is more productive than just explaining what is the issue?

ok fair enough. There is no issue with proving arithmetic consistency on a finite number of symbols. Incompleteness entirely relies on the unbounded induction step.

> Any species would prove the exact same theorems given the same axioms

Assume there exists at least one species....

Any species could prove the same theorems given the same axioms, but (besides the fact that they might not choose the same axioms) I'm not sure if they would prove the same subset of theorems that we have proven/will prove. Perhaps they'd have different ideas about what is interesting.

Human mathematicians are already fanning out into other systems of deduction (constructive mathematics being a great example), and given enough time the mathematicians of each galaxy will eventually discover the other galaxy's mathematics, even if it perhaps happens in a different order.

Surely intergalactic mathematicians already know that the only time is now? =)

> As Prigogine explains, determinism is fundamentally a denial of the arrow of time. With no arrow of time, there is no longer a privileged moment known as the "present," which follows a determined "past" and precedes an undetermined "future." All of time is simply given, with the future as determined or as undetermined as the past. With irreversibility, the arrow of time is reintroduced to physics.

That begs the question. Would any other species pick the exact same axioms? Why would they have the exact same theorems? Are you suggesting there is only one way to think logically?

Any species would face a selective pressure towards theorems which help them understand the world around them (if they have any motivation to prove theorems at all), and they will similarly face a pressure towards choosing the smallest/simplest set of axioms which allows all those theorems to be proven (and new ones to be discovered).

In fact, if we assume that neural networks are the only sorts of intelligence that can occur naturally in the universe and be sophisticated enough for arbitrary abstract calculation[0], then we might be able to infer things about the sorts of concepts they will develop and in what order. For example, having the concept of finite sums would likely occur before having the concept of infinite sums.

[0] I know that cellular automata can emulate a universal Turing machine, but I can't imagine a situation existing in nature where the cells evolve into an arrangement that produces a Turing machine, much less a machine running a program of instructions that lead to it generating mathematical theorems.

> 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.

'roughly isomorphic' would be saying 'not isomorphic', so I'm not sure what you're trying to say. Actually I'm frustrated with most physicists/math folks misusing this term 'isomorphism' to mean "a bijection".

Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them.

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 second point is supposed to be. Here's the relevant quotes:

>>> 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.

> Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them.

Right... but I didn't say that there was a bloogidy-blop between maps and land. I said that a useful alien map would have to be roughly bloogidy-blop to our maps. So I'm not really sure what you're trying to say here that is relevant to my original post.

Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no?

Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality.

Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.

> Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no?

No, their maps could be VERY different but be used in roughly the same way and therefore useful in the same way. There is an operator involved -- reading/navigating/interpreting -- which is why I chose "isomorphism" instead of "equivalent".

Also, "isomorphism" IS natural language that is used in many fields outside of mathematics and also has a vernacular sense to it. The word happens to also be used to describe certain formal constructions by mathematicians from time to time, but it is natural language.

Again, this is all very pedantic and silly. I insist on bloogity-blop. If we're going to have silly arguments, let's use silly words :)

> Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality.

Reality isn't constrained by maps.

Useful maps are constrained by reality.

> Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.

Huh? I can't use reality the way I use maps because I'm not always able to fly into the air and look around before navigating to the supermarket or a trail head.

Reality is not constrained by my inReach. I promise you it's the other way around. And I promise you I can't fly, which means I need maps, even if flying high into the air would make it way easier to find a trail head than following a not-great trail map.

Maps are useful. Very useful. But they DO NOT constrain reality.

A belief to the contrary in the case of mathematics and physics is quite spiritual. Which was kind of my original point :)

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 blobblytoid is or how to recognise one.

A human may have some vague awareness that something is there, but it's also possible that blobblytoids look like random noise, or like weird probabilistic anomalies that travel around inside a multidimensional space, or like something completely unimaginable.

So in the limit features can't be extracted because they are invisible to a different consciousness. They can still be physically present, but their meaning as a feature of interest depends on having a subjective referent for them.

This seems to be something many humans struggle with. We assume everyone else - including other humans - has the same set of referents, and therefore our personal feature maps are somehow universal.

Of course they aren't. They aren't even universal among humans, never mind a completely unknown alien species.

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.

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.

> 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?

If the thoughts have any interaction at all between each other, we would be able to leverage that to understand those higher-level thoughts to the degree that they affect the ones on our level. As an analogy, we can project an N dimensional object onto N! planes and get a complete, but not intuitive, description of what it is.

Maybe they know the exact value of Chaitin's constant, but at least we know its properties.

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?

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!

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.

> 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...

With respect to aesthetics, to each their own. I do find some mathematics beautiful. In fact, "I am bored" is a real problem and mathematics can be used to solve that problem by being a tool that tickles our brains in pleasant ways.

The "tool" and "problem" here are meant as comments on the metaphysical content of mathematics, not some sort of statement that mathematics is for engineering and that's all.

In particular: I'm commenting on the imbued/latent metaphysics of Wolfram's post, which goes beyond mere artistic appreciation. If his framing were "and look how pretty cellular automata are!" then I guess my reaction would be "yeah they are quite cool aren't they?"

I find Church-Rosser quite beautiful and also think Wolfram puts way too much metaphysical weight into the behavior of confluent rewrite systems. Similarly, some Psalms are beautiful and the story of Jesus is very nice but god does not actually exist. There's no contradiction there -- you can take the beauty and spit out the metaphysics.

>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?

> 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.

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).

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"

>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.

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