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> I think alien analogs to mathematics are unlikely to match ours. So aliens won't be able to count? They won't have a concept of zero? They won't have a conce

> I think alien analogs to mathematics are unlikely to match ours. So aliens won't be able to count? They won't have a concept of zero? They won't have a concept of 1=successor(0)? I find this very, very hard to believe, and a lot of mathematics follows from the structure of the natural numbers. If you accept evolution by natural selection is a universal law, then I think it naturally follows that ability to count must evolve. After all, it's pretty important to know whether there are 0, 1, or many predators/food/prey/enemies.

bmitc 4y ago
I think this is one argument that leads to the idea that the structures could be relatable, if a being could count. But who knows? Our mathematics relies strongly on the logical and axiomatic systems used. Things can get weird real quick with small tweaks to these systems, so it doesn’t seem like a stretch that whatever mathematical analogs aliens may possess may be wildly different. And there’s a lot of developments that our perception of reality is shaped by our biology in ways we barely understand. There are intelligent beings on Earth that don’t seem to even have analogs to human mathematics, at least that are apparent to us. We can barely communicate with a small subset of animals and plants on Earth. So I am just inherently skeptical of claims that alien thinking will bear any resemblance to human thinking.
- naasking 4y ago
  > Our mathematics relies strongly on the logical and axiomatic systems used. Yes and no. You don't need more structure than 0 and 1 to describe literally any form of information, and we're using machines right now that use such an encoding. The idea that any organism of sufficient complexity to have any kind of math won't have any notion of 0 and 1 is very implausible. That said, we certainly won't have the same syntactic descriptions of most structures, but they will certainly be relatable via isomorphisms. > We can barely communicate with a small subset of animals and plants on Earth. So I am just inherently skeptical that claims that alien thinking will bear any resemblance to human thinking. But what does that have to do with math? Math isn't about how thinking works, it's about how structures are related to each other. Structures and their relations don't depend on how one thinks. As above, how such structures are described/encoded probably depends on how one thinks (aliens maybe won't use pencil and paper), but the structure being described will be the same and so there will necessarily exist some kind of isomorphism between their "syntax" and ours, as syntax is a projection of the structure. Even plants have observable behaviour showing a distinction between 0 and 1: they observably move towards the sun when it's shining, and don't move when it's not. This isn't knowledge of "math", but simply to demonstrate that structure is everywhere and life simply must develop some intrinsic understanding of it.
  - bmitc 4y ago
    What do you mean by “no”? Computers and information theory most definitely rely on logical and axiomatic systems, and particular ones at that. Mathematics is also shaped by our thinking, which was my point. I think it’s a strong claim that aliens would even have a “mathematics”.
    - naasking 4y ago
      > What do you mean by “no”? Computers and information theory most definitely rely on logical and axiomatic systems, and particular ones at that. I mean "no" to your implicit assertion that such basic logical and axiomatic systems would not evolve in any alien species capable of mathematics. Any such alien will distinguish true and false, will have AND, OR and NOT connectives, and will understand a form of implication (it's inherent to causality). That's all you need to build an understanding of most of our formal systems. Yes the particular expression of our information theory and computer science depends on specific syntactic choices which implies a surface dissimilarity, but the underlying structure will be the same even when expressed in alien math. For instance, an alien species might evolve in an environment in which hyperbolic geometry is more natural (say a species large enough that they can sense gravity directly), and so they develop that geometry first. This will have an isomorphism to our formal model of hyperbolic geometry, and we can then explain Euclidean geometry to them from there. Edit: > Mathematics is also shaped by our thinking, which was my point. Yes, but ultimately irrelevant. This drives the pace of mathematical discovery, and what kinds of mathematical formulae we develop or find most interesting, but this is ultimately irrelevant to the fundamentals which underpin all math, which is what this really comes down to.
      - deleted 4y ago
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      - illogical_spock 4y ago
        > Any such alien will distinguish true and false, will have AND, OR and NOT connectives, and will understand a form of implication (it's inherent to causality). Implication doesn't have anything to do with causality and in fact the concept of implication in mathematical logic is broken. See: the paradoxes of material implication: https://en.wikipedia.org/wiki/Paradoxes_of_material_implication https://en.wikipedia.org/wiki/Paradoxes_of_material_implicat... To simplify, F -> T (true if false) is a true implication so, for example, I can say that "I am the pope therefore it rained yesterday" and, if it rained yesterday, then the implication is true even though I am not the pope. There has been endless grumbling among philosophers and mathematicians because of this kind of paradox but it is an inevitable result of the axiomatic definition of implication by means of a truth table, and there's no way to correct it without also changing the truth tables of disjunction and negation (because A OR NOT B is equivalent to NOT B THEREFORE A, i.e. because of the way disjunction and negation work, false implies true; you will have to work through this on your own and hit your head on your desk very hard, many times, just as I did when I first realised what a mess this is). In other words, either we accept human axioms of logic, and we have paradoxes of implication, or we don't have paradoxes of implication but then we don't accept human axioms of logic. An alien civilisation may well choose to not accept any axioms of logic that lead to paradoxes of material implication, so they won't have human axioms of logic and, if their formal system is sound, they won't have human logic, and therefore, no human mathematics. In other words, no, aliens will not necessarily have the same mathematics as humans.
Koshkin 4y ago
Counting implies the ability to perceive the discrete, but such discreteness may not be obvious to a shapeless creature living in a liquid or a gaseous substance.
- dagw 4y ago
  Most mathematical concepts are far from obvious to humans (lots of people seem to struggle with the continuum hypothesis for example), yet we can still work with them no problem. So even if this shapeless intelligent creature didn't start with discrete mathematics, they'd probably invent it eventually.
- naasking 4y ago
  > Counting implies the ability to perceive the discrete, but such discreteness may not be obvious to a shapeless creature living in a liquid or a gaseous substance. Is a shapeless creature even logically coherent? Intelligence needed for math requires making distinctions, and distinctions imply structure, and structure is logically incompatible with true "shapelessness".
  - caleb-allen 4y ago
    > Is a shapeless creature even logically coherent? Do you see how your argument is self-defeating? According to logic that humans have developed, there is such thing as a "shape". But Western philosophers have pondered the innateness of a "shape" or an "object" from very early on (Plato, through Leibniz, beyond). "Shape" and "logic" are both human constructs articulating "structure", another human construct. A shapeless creature doesn't need to be "logically coherent" to exhibit intelligence; logic, truth, and structure are features that have emerged from human intelligence. I wouldn't accept the argument that an entity must exhibit the same features to qualify as intelligent simply because humans have.
    - naasking 4y ago
      > According to logic that humans have developed, there is such thing as a "shape" There is such a thing as "structure", of which "shape" is an instance, yes. > "Shape" and "logic" are both human constructs articulating "structure", another human construct. Structure is not a human concept. We have particular conceptions of structure, but structure exists, period. 0 != 1, they have different structure. This is indisputable. > A shapeless creature doesn't need to be "logically coherent" to exhibit intelligence If you think that reality does not have to be logically coherent, or that that does not necessarily imply that any creatures within reality have to have a logically coherent description consistent with coherent natural laws, then you're talking about a fantasy world of your imagination and I don't think there's anything further to discuss.
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RxVd 4y ago
>They won't have a concept of zero? They won't have a concept of 1=successor(0)? I find this very, very hard to believe Most of the world did mathematics for a long time without zero (I hope you know that most number systems like Roman didn't have zero till that eventually came from India, and we evolved to have the current number system). Who knows what direction different number systems might have taken if they didn't come in contact with zero.
- naasking 4y ago
  That's mostly irrelevant. The naturals starting from 1 are isomorphic to the naturals starting at 0, which is why math didn't need zero for so long.
  - Koshkin 4y ago
    Not if you take into account the special properties of 0. (Generally speaking, a structure that admits a neutral element with respect to addition is not isomorphic to one that does not.)
    - naasking 4y ago
      You are correct, hence why I initially said it's mostly irrelevant. I should have qualified the claim about isomorphism as well. Still, quite a bit of math maps 1:1 without zero, so you can build a common understanding even if they don't have zero. I also don't think any alien species with which we will communicate will not understand zero. It just seems impossible. Before philosophers came up with zero in formal models, everyone intuitively understood the concept. Every animals knows when they have no food vs. when they have some food. Humans in ancient civilizations also couldn't just take something without paying.
      - illogical_spock 4y ago
  - RxVd 4y ago
deleted 4y ago
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bena 4y ago
Hacker News, the place where you will get told that we are definitely going to invent spacecraft that will be able to traverse the galaxy by solving the light speed issue, the gravity issue, and the radiation issue (among others) but that when we meet extraterrestrial lifeforms, they won't know how to fucking count. Sorry. That's the one. That's the one that broke me. Jeremy Bearimy
- bmitc 4y ago
  I don't believe we'll solve those issues anytime soon. But for aliens counting, which I think is itself arguable, it is not really under debate here. There's a vast gap between counting and what mathematics is and encapsulates.
  - naasking 4y ago
    > But for aliens counting, which I think is itself arguable I honestly can't imagine how you can reach this conclusion with any rigour. Do you agree that aliens will need to consume some energy source to stay alive, which we will call "food"? Do you agree that an understanding of "there's no food in my environment", "there's some food in my environment", and "there's lots of food in my environment" would be selected for? I certainly hope so, so at the very least they will understand the differences between zero, non-zero and "many". The only way this wouldn't happen is if the environment is so rich in abundance that there is never any absence of food. But this is impossible, because even single-celled life by necessity will reproduce to consume all available resources until it reaches an equilibrium matching the rate of food production. So any intelligent species will necessarily evolve in an environment of scarcity where zero and non-zero will be implicitly understood. Since intelligent life will necessarily evolve in scarcity, quantifying the amount of food is a useful trait that would be selected for. This is why we've now proven that numerous "non-intelligent" animals can count, including salamanders, chicks, mosquitofish, honeybees and more. Intelligent life needs to understand where they are, what they have and what they will need in the future. This involves quantifying, aka counting, no way to escape it. > There's a vast gap between counting and what mathematics is and encapsulates. Yes, but you posited intelligent aliens that have their own math. The conclusion that they would not understand zero and repeated application of a construction over zero to build non-zero quantities is impossible. It is the very root of building a theoretical structure of any kind, so if they have math of any kind, they have some kind of counting system that will have an isomorphism to ours.
zhynn 4y ago
What if aliens have no notion of discrete numbers, what if everything is probabilistic analog math? What about an organism that can see/focus/sense multiple things simultaneously, and a single "thing" is a set. What about a creature whose primary sensing organ is diffuse molecules (smell/taste) instead of sight and use light (instead of meat tentacles) to interact with matter. How might an organism that touches matter with laser fingers and smells the consequences count differently? I wouldn't have the first idea, honestly. There could be an entirely different paradigm to "counting" and consequently to the fundamentals of maths. The math that we invented is influenced by our biology and capacity to sense our environment. Our brains and how those brains work with our sense organs. This pattern is likely universal (all life will have methods of sensing their environment and interacting with it), but the methods might be very different.
- naasking 4y ago
  I'm sure you can imagine any kind of alien, but that doesn't make your imagined alien logically coherent or physically realizable and consistent with the theory of evolution by natural selection. Do you agree that these are real, physical constraints that any imagined alien species must satisfy? If not, then you have to explain how an alien species might develop that is not subject to physical constraints and evolution by natural selection. If so, then you must agree that any alien must be able to distinguish two scenarios, "I sense some food here" and "I sense no food here". The basic binary distinction is inescapable, and this is the foundation of true/false, 0 and 1, etc.
  - jarpadat 4y ago
    Consider an alien which subsists on photons, which is a form of life that exists today. We know from heseinberg that the sensing of this food "here" or "there" is nonphysical. Presumably our creature's civilization would require no heisenberg to discover what anyone can see from their own photosensor. Rather it is the concept of objects remaining in a single place that would require some real mathematical innovation to a creature with no experience of such an idea. And so this distinction of entirely separate logical states, far from being basic or inescapable, is our very human invention. It is useful for creatures like us, who perceive things in one place when they are not really so, who do their computing with sand in a region where it's bountiful, and who encode abstractions as software because doing so in dedicated hardware is more costly. While it is certainly possible that all intelligent life would have these constraints, there is no particular reason to expect it. What we can expect is that humans will expect others to be too much like ourselves; it's a well-known cognitive defect in our species.
geomark 4y ago
What if aliens did not perceive distinct objects, but rather that everything observable is part of a greater whole? Would they need counting numbers?
- naasking 4y ago
  You will have to explain how this property might be selected for by evolution by natural selection before I can even understand what you're suggesting.
- vecter 4y ago
  Counting numbers are such a basic foundational aspect of all life that it's hard to imagine any "intelligent" being not understanding the concepts of 1, 2, 3, etc.

bmitc 4y ago

I think this is one argument that leads to the idea that the structures could be relatable, if a being could count. But who knows? Our mathematics relies strongly on the logical and axiomatic systems used. Things can get weird real quick with small tweaks to these systems, so it doesn’t seem like a stretch that whatever mathematical analogs aliens may possess may be wildly different. And there’s a lot of developments that our perception of reality is shaped by our biology in ways we barely understand. There are intelligent beings on Earth that don’t seem to even have analogs to human mathematics, at least that are apparent to us. We can barely communicate with a small subset of animals and plants on Earth. So I am just inherently skeptical of claims that alien thinking will bear any resemblance to human thinking.

naasking 4y ago
> Our mathematics relies strongly on the logical and axiomatic systems used. Yes and no. You don't need more structure than 0 and 1 to describe literally any form of information, and we're using machines right now that use such an encoding. The idea that any organism of sufficient complexity to have any kind of math won't have any notion of 0 and 1 is very implausible. That said, we certainly won't have the same syntactic descriptions of most structures, but they will certainly be relatable via isomorphisms. > We can barely communicate with a small subset of animals and plants on Earth. So I am just inherently skeptical that claims that alien thinking will bear any resemblance to human thinking. But what does that have to do with math? Math isn't about how thinking works, it's about how structures are related to each other. Structures and their relations don't depend on how one thinks. As above, how such structures are described/encoded probably depends on how one thinks (aliens maybe won't use pencil and paper), but the structure being described will be the same and so there will necessarily exist some kind of isomorphism between their "syntax" and ours, as syntax is a projection of the structure. Even plants have observable behaviour showing a distinction between 0 and 1: they observably move towards the sun when it's shining, and don't move when it's not. This isn't knowledge of "math", but simply to demonstrate that structure is everywhere and life simply must develop some intrinsic understanding of it.
- bmitc 4y ago
  What do you mean by “no”? Computers and information theory most definitely rely on logical and axiomatic systems, and particular ones at that. Mathematics is also shaped by our thinking, which was my point. I think it’s a strong claim that aliens would even have a “mathematics”.
  - naasking 4y ago
    > What do you mean by “no”? Computers and information theory most definitely rely on logical and axiomatic systems, and particular ones at that. I mean "no" to your implicit assertion that such basic logical and axiomatic systems would not evolve in any alien species capable of mathematics. Any such alien will distinguish true and false, will have AND, OR and NOT connectives, and will understand a form of implication (it's inherent to causality). That's all you need to build an understanding of most of our formal systems. Yes the particular expression of our information theory and computer science depends on specific syntactic choices which implies a surface dissimilarity, but the underlying structure will be the same even when expressed in alien math. For instance, an alien species might evolve in an environment in which hyperbolic geometry is more natural (say a species large enough that they can sense gravity directly), and so they develop that geometry first. This will have an isomorphism to our formal model of hyperbolic geometry, and we can then explain Euclidean geometry to them from there. Edit: > Mathematics is also shaped by our thinking, which was my point. Yes, but ultimately irrelevant. This drives the pace of mathematical discovery, and what kinds of mathematical formulae we develop or find most interesting, but this is ultimately irrelevant to the fundamentals which underpin all math, which is what this really comes down to.
    - deleted 4y ago
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    - illogical_spock 4y ago
      > Any such alien will distinguish true and false, will have AND, OR and NOT connectives, and will understand a form of implication (it's inherent to causality). Implication doesn't have anything to do with causality and in fact the concept of implication in mathematical logic is broken. See: the paradoxes of material implication: https://en.wikipedia.org/wiki/Paradoxes_of_material_implication https://en.wikipedia.org/wiki/Paradoxes_of_material_implicat... To simplify, F -> T (true if false) is a true implication so, for example, I can say that "I am the pope therefore it rained yesterday" and, if it rained yesterday, then the implication is true even though I am not the pope. There has been endless grumbling among philosophers and mathematicians because of this kind of paradox but it is an inevitable result of the axiomatic definition of implication by means of a truth table, and there's no way to correct it without also changing the truth tables of disjunction and negation (because A OR NOT B is equivalent to NOT B THEREFORE A, i.e. because of the way disjunction and negation work, false implies true; you will have to work through this on your own and hit your head on your desk very hard, many times, just as I did when I first realised what a mess this is). In other words, either we accept human axioms of logic, and we have paradoxes of implication, or we don't have paradoxes of implication but then we don't accept human axioms of logic. An alien civilisation may well choose to not accept any axioms of logic that lead to paradoxes of material implication, so they won't have human axioms of logic and, if their formal system is sound, they won't have human logic, and therefore, no human mathematics. In other words, no, aliens will not necessarily have the same mathematics as humans.

Koshkin 4y ago

Counting implies the ability to perceive the discrete, but such discreteness may not be obvious to a shapeless creature living in a liquid or a gaseous substance.

dagw 4y ago
Most mathematical concepts are far from obvious to humans (lots of people seem to struggle with the continuum hypothesis for example), yet we can still work with them no problem. So even if this shapeless intelligent creature didn't start with discrete mathematics, they'd probably invent it eventually.
naasking 4y ago
> Counting implies the ability to perceive the discrete, but such discreteness may not be obvious to a shapeless creature living in a liquid or a gaseous substance. Is a shapeless creature even logically coherent? Intelligence needed for math requires making distinctions, and distinctions imply structure, and structure is logically incompatible with true "shapelessness".
- caleb-allen 4y ago
  > Is a shapeless creature even logically coherent? Do you see how your argument is self-defeating? According to logic that humans have developed, there is such thing as a "shape". But Western philosophers have pondered the innateness of a "shape" or an "object" from very early on (Plato, through Leibniz, beyond). "Shape" and "logic" are both human constructs articulating "structure", another human construct. A shapeless creature doesn't need to be "logically coherent" to exhibit intelligence; logic, truth, and structure are features that have emerged from human intelligence. I wouldn't accept the argument that an entity must exhibit the same features to qualify as intelligent simply because humans have.
  - naasking 4y ago
    > According to logic that humans have developed, there is such thing as a "shape" There is such a thing as "structure", of which "shape" is an instance, yes. > "Shape" and "logic" are both human constructs articulating "structure", another human construct. Structure is not a human concept. We have particular conceptions of structure, but structure exists, period. 0 != 1, they have different structure. This is indisputable. > A shapeless creature doesn't need to be "logically coherent" to exhibit intelligence If you think that reality does not have to be logically coherent, or that that does not necessarily imply that any creatures within reality have to have a logically coherent description consistent with coherent natural laws, then you're talking about a fantasy world of your imagination and I don't think there's anything further to discuss.

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RxVd 4y ago

>They won't have a concept of zero? They won't have a concept of 1=successor(0)? I find this very, very hard to believe Most of the world did mathematics for a long time without zero (I hope you know that most number systems like Roman didn't have zero till that eventually came from India, and we evolved to have the current number system). Who knows what direction different number systems might have taken if they didn't come in contact with zero.

naasking 4y ago
That's mostly irrelevant. The naturals starting from 1 are isomorphic to the naturals starting at 0, which is why math didn't need zero for so long.
- Koshkin 4y ago
  Not if you take into account the special properties of 0. (Generally speaking, a structure that admits a neutral element with respect to addition is not isomorphic to one that does not.)
  - naasking 4y ago
    You are correct, hence why I initially said it's mostly irrelevant. I should have qualified the claim about isomorphism as well. Still, quite a bit of math maps 1:1 without zero, so you can build a common understanding even if they don't have zero. I also don't think any alien species with which we will communicate will not understand zero. It just seems impossible. Before philosophers came up with zero in formal models, everyone intuitively understood the concept. Every animals knows when they have no food vs. when they have some food. Humans in ancient civilizations also couldn't just take something without paying.
    - illogical_spock 4y ago
- RxVd 4y ago

deleted 4y ago

[deleted]

bena 4y ago

Hacker News, the place where you will get told that we are definitely going to invent spacecraft that will be able to traverse the galaxy by solving the light speed issue, the gravity issue, and the radiation issue (among others) but that when we meet extraterrestrial lifeforms, they won't know how to fucking count. Sorry. That's the one. That's the one that broke me. Jeremy Bearimy

bmitc 4y ago
I don't believe we'll solve those issues anytime soon. But for aliens counting, which I think is itself arguable, it is not really under debate here. There's a vast gap between counting and what mathematics is and encapsulates.
- naasking 4y ago
  > But for aliens counting, which I think is itself arguable I honestly can't imagine how you can reach this conclusion with any rigour. Do you agree that aliens will need to consume some energy source to stay alive, which we will call "food"? Do you agree that an understanding of "there's no food in my environment", "there's some food in my environment", and "there's lots of food in my environment" would be selected for? I certainly hope so, so at the very least they will understand the differences between zero, non-zero and "many". The only way this wouldn't happen is if the environment is so rich in abundance that there is never any absence of food. But this is impossible, because even single-celled life by necessity will reproduce to consume all available resources until it reaches an equilibrium matching the rate of food production. So any intelligent species will necessarily evolve in an environment of scarcity where zero and non-zero will be implicitly understood. Since intelligent life will necessarily evolve in scarcity, quantifying the amount of food is a useful trait that would be selected for. This is why we've now proven that numerous "non-intelligent" animals can count, including salamanders, chicks, mosquitofish, honeybees and more. Intelligent life needs to understand where they are, what they have and what they will need in the future. This involves quantifying, aka counting, no way to escape it. > There's a vast gap between counting and what mathematics is and encapsulates. Yes, but you posited intelligent aliens that have their own math. The conclusion that they would not understand zero and repeated application of a construction over zero to build non-zero quantities is impossible. It is the very root of building a theoretical structure of any kind, so if they have math of any kind, they have some kind of counting system that will have an isomorphism to ours.

zhynn 4y ago

What if aliens have no notion of discrete numbers, what if everything is probabilistic analog math? What about an organism that can see/focus/sense multiple things simultaneously, and a single "thing" is a set. What about a creature whose primary sensing organ is diffuse molecules (smell/taste) instead of sight and use light (instead of meat tentacles) to interact with matter. How might an organism that touches matter with laser fingers and smells the consequences count differently? I wouldn't have the first idea, honestly. There could be an entirely different paradigm to "counting" and consequently to the fundamentals of maths. The math that we invented is influenced by our biology and capacity to sense our environment. Our brains and how those brains work with our sense organs. This pattern is likely universal (all life will have methods of sensing their environment and interacting with it), but the methods might be very different.

naasking 4y ago
I'm sure you can imagine any kind of alien, but that doesn't make your imagined alien logically coherent or physically realizable and consistent with the theory of evolution by natural selection. Do you agree that these are real, physical constraints that any imagined alien species must satisfy? If not, then you have to explain how an alien species might develop that is not subject to physical constraints and evolution by natural selection. If so, then you must agree that any alien must be able to distinguish two scenarios, "I sense some food here" and "I sense no food here". The basic binary distinction is inescapable, and this is the foundation of true/false, 0 and 1, etc.
- jarpadat 4y ago
  Consider an alien which subsists on photons, which is a form of life that exists today. We know from heseinberg that the sensing of this food "here" or "there" is nonphysical. Presumably our creature's civilization would require no heisenberg to discover what anyone can see from their own photosensor. Rather it is the concept of objects remaining in a single place that would require some real mathematical innovation to a creature with no experience of such an idea. And so this distinction of entirely separate logical states, far from being basic or inescapable, is our very human invention. It is useful for creatures like us, who perceive things in one place when they are not really so, who do their computing with sand in a region where it's bountiful, and who encode abstractions as software because doing so in dedicated hardware is more costly. While it is certainly possible that all intelligent life would have these constraints, there is no particular reason to expect it. What we can expect is that humans will expect others to be too much like ourselves; it's a well-known cognitive defect in our species.

geomark 4y ago

What if aliens did not perceive distinct objects, but rather that everything observable is part of a greater whole? Would they need counting numbers?

naasking 4y ago
You will have to explain how this property might be selected for by evolution by natural selection before I can even understand what you're suggesting.
vecter 4y ago
Counting numbers are such a basic foundational aspect of all life that it's hard to imagine any "intelligent" being not understanding the concepts of 1, 2, 3, etc.