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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

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/FindEquationalProof.html https://reference.wolfram.com/language/ref/FindEquationalPro...

jonbronson 5y 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.
- thebooktocome 5y 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.
  - whatshisface 5y ago
    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.
    - thebooktocome 5y ago
      OP is explicitly not talking about something open to human interaction (“saying”, “symbols”, “moving (symbols) around” are all human operations).
      - whatshisface 5y ago
        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.)
  - ganzuul 5y ago
    It's almost as if there is a phase diagram with impedance mismatches between systems of belief.
- thwayunion 5y 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 :)
  - whatshisface 5y ago
    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.
    - thwayunion 5y ago
      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" :)
      - whatshisface 5y ago
        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. :)
        thwayunion 5y ago
        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.
- mrob 5y 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.
  - ganzuul 5y ago
    General Relativity does not conserve energy.
    - BreakfastB0b 5y ago
      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?
      - deleted 5y ago
        [deleted]
  - BreakfastB0b 5y ago
    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!
- arka2147483647 5y 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 5y 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?journalCode=ajp https://aapt.scitation.org/doi/abs/10.1119/1.2402156?journal...
  - thwayunion 5y ago
    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.
- vba616 5y 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 5y 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 5y 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 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).
- thwayunion 5y 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 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"
vaillant 5y 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.
deleted 5y ago
[deleted]

jonbronson 5y 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.

thebooktocome 5y 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.
- whatshisface 5y ago
  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.
  - thebooktocome 5y ago
    OP is explicitly not talking about something open to human interaction (“saying”, “symbols”, “moving (symbols) around” are all human operations).
    - whatshisface 5y ago
      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.)
- ganzuul 5y ago
  It's almost as if there is a phase diagram with impedance mismatches between systems of belief.
thwayunion 5y 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 :)
- whatshisface 5y ago
  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.
  - thwayunion 5y ago
    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" :)
    - whatshisface 5y ago
      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. :)
      - thwayunion 5y ago
        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.
mrob 5y 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.
- ganzuul 5y ago
  General Relativity does not conserve energy.
  - BreakfastB0b 5y ago
    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?
    - deleted 5y ago
      [deleted]
- BreakfastB0b 5y ago
  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!
arka2147483647 5y 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 5y 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?journalCode=ajp https://aapt.scitation.org/doi/abs/10.1119/1.2402156?journal...
- thwayunion 5y ago
  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.
vba616 5y 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 5y 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 5y 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 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).

thwayunion 5y 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 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"

vaillant 5y 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.

deleted 5y ago

[deleted]