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As much as I like PG's writing in general, this is him dipping his toes in waters that are already well studied and coming off as ignorant and/or presumptuous.

• danielvaughn 4y ago
  No shade to you personally, but I really dislike this frame of thought. There are mountains and mountains of literature about a great deal of things, no one human can be expected to be aware of them all. Since that is obviously true, the lack of awareness about a single topic isn’t “ignorance” - it’s not like PG studiously ignored all of that literature. More likely he just never came across it, so the more apt word is “unaware” rather than “ignorant”. To criticize someone merely for pondering about a subject they aren’t well-versed in is hostile gatekeeping. I think those who are educated in the relevant subject would do well to be more welcoming and informative to newcomers.
  • apeace 4y ago
    That's true, but this essay is written with an authoritative voice, not a pondering one. PG tends to write as if he is teaching his audience something profound. So I think it's fair to call out when he is wrong or misguided.
    • danielvaughn 4y ago
      This might just be a subjective thing then. My impression was that he was simply following a casual train of thought. I agree with the other comment that my words “hostile gatekeeping” were too strong. I just get irritated with philosophers specifically, because philosophy itself is a normal human experience. Everyone finds themselves thinking about things that veer into philosophical subjects. To even criticize people for doing so without reading the professional literature is just so annoying to me.
      • missingrib 4y ago
        I think philosophers get a bit annoyed about this because they've spent literal years and years thinking about something that might be quite esoteric/tricky, and then someone comes along and just assumes the answer as if there hasn't been decades of discourse about this exact thing. The subtext in your comment is that everyone does philosophy all the time, and there's no difference between professional and amateur philosophers.
        • jdthedisciple 4y ago
          > then someone comes along and just assumes the answer as if there hasn't been decades of discourse about this exact thing. I'd think that would present itself as a rather pleasant opportunity for any bypassing philosopher to stop and teach them about the "decades of discourse" on it rather than being annoyed?
          • danielvaughn 4y ago
            Yeah that’s what I’ve seen before from chemists, biologists, geologists, physicists, etc. They’re excited, not irritated, to share their knowledge with the public.
            • missingrib 4y ago
              Nobody does this stuff to those kinds of scientists.
  • deltasevennine 4y ago
    True. But PG's authoritative status is weakened due to his lack of awareness no matter how you change the wording. This entire essay is weaker because of it as my opinion about his knowledge. >To criticize someone merely for pondering about a subject they aren’t well-versed in is hostile gatekeeping. The words "hostile gatekeeping" makes me classify your statement as accusatory and an actual attack. The Op's statements are just criticism. Harsh but valid, it certainly isn't hostile gatekeeping. In fact, PG is the gatekeeper here. He owns the site.
    • danielvaughn 4y ago
      Yeah my words were too strong, I’ll own that. I didn’t mean to come across so aggressive. Another way to phrase it: if someone were to follow a thought experiment in a another subject like biology or geology, the response from the educated community would be different. I’ve seen it on here before. The response is something akin to “hey yep that’s an interesting thought. It’s been done before, here are some links if you’re interested in reading further”. I just tend to see a very different (much more critical) response from people in philosophy, often with overtones of condescension and smugness. Not saying OP was guilty of that, just that I mistakenly responded as if they did.
  • ASalazarMX 4y ago
    > the lack of awareness about a single topic isn’t “ignorance” Since everybody is nitpicking here, I'll add that this is indeed ignorance. We're know what we know, and are ignorant of what we don't. Being ignorant by itself shouldn't be viewed as a flaw, but an inevitable state of our limited capacity to learn. The real flaw is being proud of staying willfully ignorant.
    • danielvaughn 4y ago
      I agree with what you're saying, but practically speaking, using the word "ignorant" is universally interpreted as a negative term.
      • mechanical_bear 4y ago
        It is not universally interpreted as negative. I have often said in conversation; “Please explain [x] to me, as I’m ignorant as to how that fits in/works/etc.” I certainly don’t say it to demean myself.
  • mechanical_bear 4y ago
    Ignorant is unaware. I’m sure there are loads of things that you have ignorance of, such as the meaning of Ignorance: lacking knowledge, information, or awareness about a particular thing: "they were ignorant of astronomy"
• malux85 4y ago
  I agree with what you’re saying, at the thresholds of math (and even quite a bit before) it gets very murky very quickly, But I think in this context he may have been better to say “arithmetic” or maybe root logical operations, because he was trying to convey small axiomatic units of universal truth rather than pull an entire field “into purity” I could say the same about his comment about Carbon, the exact mass of carbon we use is influenced by isotopes and is not exactly standardised in all human chemistry - so it’s likely the aliens idea of “the mass of carbon” would also be slightly different. A better example would be “the atomic number of carbon” which is always the same because it’s definitional and integer, but the same point is there - he’s conveying the existence of the tiny axioms (alien truths), not trying to define them
  • TuringTest 4y ago
    > But I think in this context he may have been better to say “arithmetic” or maybe root logical operations, because he was trying to convey small axiomatic units of universal truth rather than pull an entire field “into purity” What makes you think that arithmetic and logic are universal? A culture with a completely different way of thinking may not arrive to a system similar to our mathematics based on logical reasoning and search of consistency. Heck, we can see it in a lot of human beings who are not capable of consistent logical reasoning, and they're not even alien.
    • spookthesunset 4y ago
      > Heck, we can see it in a lot of human beings who are not capable of consistent logical reasoning, and they're not even alien. All humans share this flaw (if it is a flaw). Not just some. We all do. It’s part of what makes us human—we have emotions that for better or worse transcend logic and reasoning. In fact I could maybe argue that emotion is an important part of logic and reasoning. Emotion leads to skepticism and thinking outside the box. Both traits are needed to advance our understanding of the world.
    • malux85 4y ago
      I was thinking of simple arithmetic like addition and simple logical operations like “and” and “or” I was illustrating the existence of these things - like PG was doing in the article - not attempting to define them
• nine_k 4y ago
  Math is a formal game. You can define vastly varied algebraic structures, and, as long as they are internally consistent, we could consider them "mathematical". But the physical reality makes some structures more important in practice. Hence natural numbers, or real numbers, or complex numbers stand out among the sea of possible group structures. Chances are very high that aliens live in the universe with the same laws of physics, which make the same, or isomorphic, structures important for them, and thus informing the development of their mathematics. But this is not a given, of course; there can potentially be different ways to describe physical reality which are comparably successful, but which grow from entirely different foundations, are internally consistent, and otherwise work like our mathematics, but are not connected to it (yet). Even more funny is that Gödel's incompleteness theorem guarantees that "our" (currently developed) mathematics contain true statements which cannot be proven either true or false within the set of axioms which produce our mathematics. One can imagine that these true statements can be used to successfully describe important aspects of physical reality and were discovered by aliens for this purpose, while remaining incomprehensible for us. (But this is softer sci-fi stuff, of course.)
  • pluijzer 4y ago
    > But this is not a given, of course; there can potentially be different ways to describe physical reality which are comparably successful, but which grow from entirely different foundations, are internally consistent, and otherwise work like our mathematics, but are not connected to it (yet). Are you or anybody else aware of any other way to describe physical reality? Maybe some culture took a different path. That would be very interesting. It seems almost impossible to imagine one for me though.
    • nine_k 4y ago
      Ted Chiang's short novel "Story of Your Life" [0], later turned into the movie "Arrival", describes an alien culture which uses variational calculus [1] as the normal way to describe physics, that is, their equations are mostly about finding functions with extrema of certain kind, not just derivatives (as are the normal PDUs we use in physics). The idea is that "our" equations mostly deal with the arrow of time and thus use terms like dx/dt extensively, while "their" equations are about the entire configuration of something in spacetime, not singling out the time dimension. [0]: https://en.wikipedia.org/wiki/Story_of_Your_Life https://en.wikipedia.org/wiki/Story_of_Your_Life [1]: https://en.wikipedia.org/wiki/Calculus_of_variations https://en.wikipedia.org/wiki/Calculus_of_variations
• patientplatypus 4y ago
  I agree. Consider the octopus - each of it's limbs contains it's own cortex and so it has in essence 8 separable brains. Or ants and bees which contain a hive intelligence. An ant has almost no neurons, but the collective number of neurons of an ant hive approaches that of a human brain. These are only creatures that are on our earth that have radically alternative ways of experiencing intelligence. Why would we expect aliens to have an intelligence that's at all comparable to humanity when even somewhat intelligent animals on our planet don't exhibit the same characteristics?
  • spookthesunset 4y ago
    There is also bats who basically navigate the world by pinging sound off things. Don’t forget microscopic creatures like Tardigrades. Their world is so small that the air around us acts like water.
• poulpy123 4y ago
  I don't know for the axiom of choice but I would be surprised if 1+1=2 is not universal, whatever way is used to express it
  • colordrops 4y ago
    Is that "math" though, or just a particular construct in the system of math? I could imagine an alien mind that does not break things down across integer boundaries, when you consider that any labelings and groupings are arbitrary. That 1 "hat" you are wearing plus that 1 "hat" your buddy is wearing are actually not 1 and 1 but a conglomeration of fields/particles in an arbitrary configuration that can be torn or shredded or turned into plasma.
    • lupire 4y ago
      You could construct such a mind in a simulation but I doubt one would arise naturally in the physical constraints of our universe. Approximate discretization is everywhere.
    • zdragnar 4y ago
      At the end of the day, you've still identified that particular configuration as a "hat"- a configuration forming a discrete unit. Without it, you have no concept of object permanence and no basis for comprehending any form of logic at all. After all, even within plasma, there are discrete units which are configurations of other discrete units.
      • colordrops 4y ago
        But these discrete units in the plasma are also model constructs of ours rather than fundamental properties of nature. The point is that any discrete lines we draw are ours, and not necessarily nature's. Of course we have no concept of object permanence without these, because concepts and objects are discrete models we create ourselves. It's tautological.
• dooglius 4y ago
  This seems like an uncharitable take, the set of axioms and definitions used doesn't really fall under "mathematical truths", it's about agreement on what is implied by a given set of axioms. Even on Earth many people work in alternatives to ZFC and generally don't really disagree about what is "true".
  • TuringTest 4y ago
    Yet "what is implied by a given set of axioms" is the most common definition of "what's true" in mathematics. Therefore, an alien culture using completely different axioms and rules could arrive to an extremely alien math. There might be some parts of it that are translatable to our own systems, but there's no guarantee that both systems will be ultimately compatible.
• jsbg 4y ago
  "The truths of mathematics would be the same, because they're true by definition" means that "(x => y) AND x" means y is true based on the definition of => and AND. There's no "philosophy" behind this.
  • lupire 4y ago
    Why would aliens have the same definitions? Your AND maybe wouldn't exist to them. They could have a more statistical/global non-binary/discrete mathematics.
    • vecter 4y ago
      I assume that PG was talking about "intelligent" aliens, which, for the sake of argument, let's say are those capable of interstellar communication (building radio telescopes and whatnot). I cannot fathom such a civilization not having discovered Boolean logic.
      • wizofaus 4y ago
        I don't see why it's so absurd to think another alien species could develop interstellar communication without even being sentient in the way we understand it at all, or at least have anything like the level of abstract self-awareness that humans do. They may just experience something more equivalent to "emotional" reactions when attempting to manipulate the natural world to create technology in such a way that it guides them towards a successful outcome. Indeed comprehending Boolean logic seems no more necessary for that than it is for a dog in order to understand that when something in front of them is person 1 OR person 2, AND that certain sounds are coming from that person, they should sit.
        • vecter 4y ago
          You're right about dogs, but dogs cannot build radio telescopes so I don't understand the point. To be capable of interstellar communication, you have to be able to build machines, and doing so requires a systematic engineering culture and understanding of at least basic principles. To put it differently, do you think such an advanced civilization would not be able to do something as simple as calculus, which would be far more advanced than manipulating a basic truth table? They almost certainly won't do it in any language that we understand on Earth, but they'll be manipulating and reasoning about the same concepts.
• simonh 4y ago
  I think, like many commentators here, you're taking a very narrow technical view of what PG is talking about in what is actually a very broad a high level discussion. Whether or not it's possible to prove a specific postulate in set theory is formally true (the axiom of choice, which was only formulated in 1904) isn't going to stop aliens counting objects and calculating the area of squares. They may not agree with us about formalism vs intuitionism, but it seems likely they would agree with us about a huge array of practical mathematical operations and results. PG is just asking what are the areas we would be extremely likely to agree on. Pointing out that there might be specific, advanced, highly obscure controversies we might disagree on isn't even in contention. Of course there are.
  • kmod 4y ago
    It's turtles all the way down -- we don't know what "truth" is, or how one can come to "know" truth (as opposed to just believe it). I'm less informed on these topics so I avoided them, but I've heard that our notion of truth is intimately tied to our experience of physical reality, and it seems easy to imagine aliens that have different experiences of reality. For example, what if aliens are so large that the non-euclidian nature of space becomes obvious to them? Perhaps they would never come up with euclidian geometry. Maybe they would never create the notion of pi. Yes it shows up in other areas but you can certainly imagine an alien civilization that has a different circumference-to-radius ratio or disavows the entire idea of having a constant circumference-to-radius ratio.
    • lupire 4y ago
      > imagine an alien civilization that has a different circumference-to-radius ratio Aliens still live in our Universe with our physics. Are these aliens bigger than the solar system? Maybe a different universe with different parameters could look m athematically different in major ways.
      • kmod 4y ago
        Again, talking a bit outside my comfort zone, but I believe our universe's physics say that circles only have a fixed circumference-to-radius ratio at "small" (astronomically) scales. At large enough scales space is curved and the ratio begins to change. In a similar way, a set of different physics would apply to small enough aliens that are principally governed by quantum mechanics. They might never come up with the real numbers because they do not believe that quantities are dense. We could think of more examples -- aliens who typically move at speeds close to the speed of light would have very different conceptions of time, etc.
• spookthesunset 4y ago
  > In particular, we humans still hotly debate which math is the "right" math. Is the axiom of choice "true"? Does it make sense to say that it has a truth value? I would love examples of this. Seems pretty cool to think about. I always assumed that “math” would be one of the constructs we could always reliably communicate with.
  • kmod 4y ago
    I really like the Stanford encyclopedia of philosophy. In this case: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/
• witherk 4y ago
  Math isn't choosing which axioms to use, is finding the consequences of a given set.
• golemotron 4y ago
  To be fair, someone writes this type of comment every time he posts. It's not a unique take. It falls into the "well studied" category you mention.
• AnimalMuppet 4y ago
  Take some number of things. Lay them out more or less in a straight line. Somewhere in that line, put a little gap. Count them from left to right. Then count them from right to left. You get the same number. That experiment is saying that addition commutes. If an alien ran that experiment, they would get the same answer. Lay out some things in regular rows in a rectangle. Count the things row by row. Then count the things column by column. You'll get the same answer. So will an alien. That is, multiplication commutes both for us and for the aliens. Now, as you go further into mathematics, things are true because they follow from the definitions, but the aliens may use different definitions. They may have a different definitions for, say, a commutative ring, and so abstract algebra may be different for them. But with the same definitions, we would arrive at the same conclusions.
  • Koshkin 4y ago
    > If an alien ran that experiment, they would get the same answer. This depends on the world they live in. If, for example, things were hard to control, tended to move, to disappear (and reappear in a different place), to look the same, etc., they would probably discover quantum field theory before arithmetic!
    • dagw 4y ago
      they would probably discover quantum field theory before arithmetic! Sure, but much like we eventually discovered quantum field theory, they'll eventually discover arithmetic. And even if for some weird reason they hadn't, they're probably smart enough to understand it once we demonstrate it to them.
    • vecter 4y ago
      That's logically impossible for many reasons, the first of which is that arithmetic is a prerequisite for quantum field theory, and that secondly quantum effects like those don't occur at scales at which intelligent organisms live.
  • wizofaus 4y ago
    Good analogy but is it so crazy to think the concept of counting in a particular direction wouldn't make sense to an alien intelligence? That the way their brain processes inputs means that the knowledge of the total number occurs to them either subconsciously or as the result of parallel processing that is beyond what human brains can even imagine being aware of? Even if the input signals being counted are separated by time (e.g. pulses of light or sound), such that their brains must logically store the previous count and increment it as new signals are detected, the concept of ordered addition (i.e. where we can conceive of "x plus y" and "y plus x" as separate concepts, even if they yield the same result) might simply not be necessary or possible in an alien brain. What even qualifies as "addition" at all is somewhat murky - e.g. "addition" of relativistic velocities is non-commutative (*), but is it meaningfully even addition in the same way adding integer counts is? * https://en.wikipedia.org/wiki/Velocity-addition_formula#Properties https://en.wikipedia.org/wiki/Velocity-addition_formula#Prop...
• mcqueenjordan 4y ago
  https://en.wikipedia.org/wiki/Principle_of_charity https://en.wikipedia.org/wiki/Principle_of_charity Or, infinitely caveating all of your writing is tedious and makes it less interesting to read. It's more respectful to your readers to assume they understand this and will apply the principle of charity.
• VyseofArcadia 4y ago
  > In particular, we humans still hotly debate which math is the "right" math. Is the axiom of choice "true"? Does it make sense to say that it has a truth value? Is such a truth value amenable to human discovery? Does PG know the answers to all of these questions? If we did encounter intelligent (and peaceful!) beings out in the universe, there would be a set theory gold rush as people race to put out papers that such-and-such alien axiom implies choice, diamond principle, etc., or the other way around. Every field of math would have such a gold rush really, on both sides, as we discover similarities and differences in techniques and formulations. And I bet famous unsolved problems on both sides would get solutions if not outright provided by the other species, then at least using novel tools from the other species. I can't imagine a more exciting time for math as a whole than first contact.
• benlivengood 4y ago
  I'd expect aliens to have discovered their equivalent of the Church-Turing thesis, Tarski's externality of truth, Godel's incompleteness theorems, etc. That would give them a strong ability to model our mathematics within their own and see its computational structure and begin finding correspondences to theirs. Similarly, we could begin modeling their conception of mathematics in a similar way and as soon as we both have arrived at a morphism between our representation and theirs of some particular theorem we could grow our knowledge and understanding of the other. The key is that mathematics flows from logic and logic flows from computation and computation is universal.
• solveit 4y ago
  > Is the axiom of choice "true"? Does it make sense to say that it has a truth value? Is such a truth value amenable to human discovery? I imagine PG would say that these are not purely mathematical questions, but rather metamathematical questions. Theorems remain theorems regardless of whether the axiom of choice is "true", although one may contend that theorems assuming that the AOC is false (or true) are vacuous. Aliens would surely agree with us on statements of the form "working in first-order logic (or whatever) formalized in XXX way, assuming axioms YYY and ZZZ, we may prove that...".
• WalterBright 4y ago
  Math has proven its predictive powers in the real world over and over. It's quite a stretch for it to be fundamentally wrong.
• pazimzadeh 4y ago
  Not that I agree with the argument, but this is relevant: https://www.quantamagazine.org/the-evolutionary-argument-against-reality-20160421 https://www.quantamagazine.org/the-evolutionary-argument-aga...
  • kubanczyk 4y ago
    A very interesting take. Your link is a worthy prize for enduring most of the other comments around this thread.