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> 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 f

> 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.
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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.
    - naasking 4y ago
      > Implication doesn't have anything to do with causality and in fact the concept of implication in mathematical logic is broken I disagree, it's seems very obvious that if-then connectives are a crude causal description. Yes, the crude form is problematic because it's crude.
mjan22640 4y ago
>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. All the things you see around you are an outcome your brain processing. That applies to any structures that you abstract from that as well. Math is an exploration of how the brain does that.

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.
  - naasking 4y ago
    > Implication doesn't have anything to do with causality and in fact the concept of implication in mathematical logic is broken I disagree, it's seems very obvious that if-then connectives are a crude causal description. Yes, the crude form is problematic because it's crude.

mjan22640 4y ago

>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. All the things you see around you are an outcome your brain processing. That applies to any structures that you abstract from that as well. Math is an exploration of how the brain does that.