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> The truths of mathematics would be the same, because they're true by definition. Ditto for the truths of physics These are pretty strong statements for which

by bmitc 4y ago

> The truths of mathematics would be the same, because they're true by definition. Ditto for the truths of physics These are pretty strong statements for which there’s no arguments provided for but serve as assumptions for the rest of the article. I don’t think there’s consensus among mathematicians, philosophers, cognitive scientist, or biologists on this. Mathematics is most definitely a human endeavor, and so we can’t really make claims about its existence in the universe independent of humans. I think alien analogs to mathematics are unlikely to match ours. If we are lucky, I think it could be the case that the various structures could be similar, but the likelihood the implementations resemble each other are slim. It’s even a stretch to assume the structures would relate. Even humans do not fully agree on mathematics. There is no “one” mathematics because mathematics is the human exploration of idealized objects using a variety of human logical systems. And then there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand.

cloogshicer 4y ago
Fully agreed. These beliefs you quoted from the article, which unfortunately most people don't even recognize as beliefs, form the basis of the dominant religion of the western world (scientism). The worrying thing is that the majority of people who believe in this religion don't even realize they are believers.
- bmitc 4y ago
  If I am not mistaken, this is similar to what Feyerabend seemed to be on about. https://blogs.scientificamerican.com/cross-check/was-philosopher-paul-feyerabend-really-science-s-worst-enemy/ https://blogs.scientificamerican.com/cross-check/was-philoso...
- gregmac 4y ago
  Science doesn't have "facts" or "truths". It is based on falsifiability: for a theory or hypothesis to be considered scientific it must be able to be tested and conceivably proven false. This is the key difference from religion, which has no such "falsifiability" equivalent. The closest thing to "beliefs" is probably an individual following which of several competing theories is most likely correct -- but there's always the underlying basis that any of them might have evidence showing they're incorrect at any time, and one's view should adjust as a result. Often this comes in the form of deferring to other people or a consensus view, which could be construed as "faith" but is different: If you asked me how the universe exists, I'd say the big bang theory is the best answer we have, but I don't understand enough about the underlying science to explain why nor can my brain comprehend the reality of it. I have no loyalty or allegiance to this view, though; I could be swayed to another theory if the big bang is ever proven false or if a better theory arises. Further reading: https://www.theatlantic.com/science/archive/2015/11/why-scientific-faith-isnt-the-same-as-religious-faith/417357/ https://www.theatlantic.com/science/archive/2015/11/why-scie...
- simonh 4y ago
  The thing that differentiates science from religion is repeatability. With religion everyone has their own opinion, people out of contact with each other come up with radically different religious beliefs and there is no way to bridge between those beliefs. If we forgot everything we know about religion, in a thousand years we might rediscover religion again but they'd be entirely different religions from what we have now. With science it doesn't matter who does a given experiment, anyone else doing the same experiment will get the same results. There's no scope for disagreement about verifiable scientific facts. Just do the experiment and find out. If we forgot everything we know about science, in a thousand years if we rediscovered science, very quickly we'd rediscover all the exact same facts about the world again.
tambourine_man 4y ago
Yeah, lots of strong baseless assumptions in the first paragraph made me stop reading.
adrianN 4y ago
I'm pretty sure that any sufficiently advanced species that does any kind of mechanical engineering will have maths that is homomorphic to analysis.
exitb 4y ago
Isn't the basic math largely driven by attempts to understand and quantify the world around us? In such case, it depends on more universal concepts like distance, time, speed, acceleration etc. Concepts which I would imagine to be familiar to any intelligent being that takes a physical form. I can't imagine an alien wondering about a period of a pendulum and arriving at an answer that's really different from ours.
- p0pcult 4y ago
  Funny that you chose pendulums: https://cosmosmagazine.com/technology/artificial-intelligenc https://cosmosmagazine.com/technology/artificial-intelligenc...
naasking 4y ago
> 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.
LocalH 4y ago
> Mathematics is most definitely a human endeavor, and so we can’t really make claims about its existence in the universe independent of humans. I think the idea is that math is not created by humans, but documented by humans. Sure, the specific terminology may be our invention, but there are basic mathematical properties that seem (from our perspective) like they should be universal. For example, whatever names a being has for the numbers 1 and 2, if you take that 1 and add 1 more, you must get 2 (or the "local equivalent") as the result. My guess is that, if what we call math isn't truly universal, it's probably at least universally true within the realm of physical life, and there's likely some massive causal chain from the root properties of physics itself to the mathematical properties that we call "math". When it comes to raw, untethered "consciousness" (or whatever one would prefer to call it), this may not hold true even in the slightest. Yes, this comment steps slightly outside what could ever be determined purely by the scientific method at the end. I feel it is useful to do so in discussion, even when that cannot directly enter into research. There are some truths to the larger universe that I don't think the scientific method will ever truly be able to uncover, just due to it's rigor. Some aspects of the universe are just simply not falsifiable, but they're still worthy of discussion with an open mind.
- dTal 4y ago
  There is most certainly a massive causal chain between physics and what humans call "math", because everything humans do is determined by the laws of physics. The causal chain leads through millions of years of evolution and tens of thousands of years of culture. "The numbers 1 and 2" are a complex web of analogies that have not actually been demonstrated to "exist" outside of our minds, so evidence that alien mathematicians would have words for them is much weaker than our intuition would suggest. The question that must be answered is "Given the constraint of precisely modelling the world in a useful way, to what extent are all rule-based systems isomorphic?" Responding to your second point, I'm afraid I can't agree with you that unfalsifiable propositions are "useful" discussion contributions - especially not with an "open mind". The only criterion on which such propositions can be judged is whether they are fun to believe, and that is a very dangerous muscle to flex.
dagw 4y ago
I think alien analogs to mathematics are unlikely to match ours. Almost certainly not, but they're probably isomorphic. And either way if we show them our axioms they will be able to validate our mathematics and vice versa. The truths of mathematics are of the form 'if A then B'. Even if they don't start at A or even accept A as true, but will should still get B if they assume A.
- Koshkin 4y ago
  > they're probably isomorphic That's a strong statement. I'd probably talk about homomorphism.
mr_mitm 4y ago
> Even humans do not fully agree on mathematics. There is no “one” mathematics because mathematics is the human exploration of idealized objects using a variety of human logical systems. Maybe not in its entirety, but I find it hard to imagine that any civilization as advanced as ours (let's say a civilization that manages to harness nuclear fission, just to set a baseline) will not come up with concepts such as prime numbers, real number, complex numbers, calculus, etc. If they do that, they will inevitably find the same structures we found using function theory. They will know about differential equations and prove similar theorems about them as we did. And the Pythagorean theorem is a universal truth that holds everywhere.
- lostmsu 4y ago
  In that list prime numbers might not be a thing.
- lmkg 4y ago
  > real number, complex numbers, calculus, etc Some humans believe that the prominence of real numbers is a historical accident. It seems quite plausible to me that a human society, much less an alien one, would go down a mathematical evolutionary path based on the constructable numbers and the computable numbers. Regardless of whether we eventually find the same structures, there are things that we might consider basic which they find esoteric and vice-versa. Heaven forbid we encounter an alien civilization that discovered an O(log n) algorithm for integer factorization before they invented steam power.
  - bmitc 4y ago
    > Some humans believe that the prominence of real numbers is a historical accident. That is a great point. The continuum is contentious and pretty highly debated philosophically. I also encourage people interested in that to also learn about smooth infinitesimal analysis. Just a small tweak in the underlying logic and model yields unique mathematics and questions established assumptions.
- illogical_spock 4y ago
  > And the Pythagorean theorem is a universal truth that holds everywhere. Or at least in every world where there exist straight lines, yes? For instance: > The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate.[59][60] Thus, right triangles in a non-Euclidean geometry[61] do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to π/2, and all its angles are right angles, which violates the Pythagorean theorem because {\displaystyle a^{2}+b^{2}=2c^{2}>c^{2}}. https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclid...
agent008t 4y ago
The statement that "the truths of mathematics would be the same, because they are true by definition" is correct. Mathematics exist independently of biology or perception of reality. It is just a collection of arbitrary abstract definitions and what follows from them. The alien species may come up with different base definitions that they find more useful or interesting. But they would derive the same conclusions as we would if they were starting from the same definitions and applying the same abstract rules.
- bmitc 4y ago
  > Mathematics exist independently of biology or perception of reality. Is there a “proof” of that? How could a proof exist? You’d probably win several awards if you had one. > It is just a collection of arbitrary abstract definitions and what follows from them. … created by humans.
  - dagw 4y ago
    Is there a “proof” of that? It's how we define the concept "mathematics". If a result was dependant on "biology or perception of reality" or anything else outside its defining axioms, it wouldn't be mathematics.
    - bmitc 4y ago
      I'm sorry, but I just disagree that that's how mathematics is defined and that it doesn't depend on our biology and perception, because we are making those definitions. A book I might recommend and that I'm going through at the moment is Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by George Lakoff and Rafael Nunez. The origin and meaning of mathematics is strongly influenced by cognitive sciene, and thus biology. I've been downvoted, but this is not a totally novel or off the rails idea. It is basically accepted in robotics that embodied cognition is how you get a robot to understand and perceive its environment. Where do you think that idea came from?
      - dagw 4y ago
rendall 4y ago
> there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand. Almost certainly true. We as evolved creatures do not perceive objective reality, but only enough reality as to enable our ancestors to survive our very niche environment (niche relative to the entirety of the universe). Our science and mathematics model only our perceptions of reality, and not reality itself. It is folly to assume that an alien, evolved along an entirely different set of initial conditions, would share our perceptions of reality. Our mathematics, modeling as it does our perceptions, serves human needs and perceptions only.
slibhb 4y ago
While it's true that there's no consensus on this topic, that doesn't imply that people can't make claims one way or the other. In fact, the claims that the essay makes (Platonism) are very commonly made.
dbsmith83 4y ago
> Mathematics is most definitely a human endeavor, and so we can’t really make claims about its existence in the universe independent of humans. Can you give an example of a mathematical concept which could be different? I believe that math is universal. We may use models to understand it (infinity, perfect circles, etc.), but the underlying mathematical truth is independent of humans. The same is true for science. There are physical laws which we look to discover. We use models in science to understand them, but the models are not the same as the underlying truth
- rendall 4y ago
  > I believe that math is universal. We may use models to understand it (infinity, perfect circles, etc.), but the underlying mathematical truth is independent of humans. Humans (probably) perceive and understand a slim subset of reality. We have an illusion of universalism of our perceptions because we perceive nothing outside of them. Also, we are the dominant species of the planet, which gives us a reason to believe that our perceptions are "more accurate" than, say, a bat's. Personally, I don't think an alien's perceptions and understanding would contradict our own, but if it's given that our perceptions are a subset of reality, then an alien's understanding might include elements of reality that we literally cannot perceive or even understand. > Can you give an example of a mathematical concept which could be different? By definition, no. But, hmm. What would number and a mathematical system look like from creatures who thought in logarithms? Or in primes? Or that had no concept of "number"? What if even "greater than" and "less than" had no relevance to an alien civilization?
  - dagw 4y ago
    All of this is arguments for why our understanding of science and the nature of the universe might differ. It can also explain how our mathematics might evolve very differently and we will have made different mathematical discoveries and do math in very different ways (all of which I agree are very likely). However it doesn't explain how an alien race will look at one of our mathematical proofs that we have proven True and be able to prove that it is False.
pattt 4y ago
Fully agreed, the way I'd approach this would be that the said mathematical proofs about certain truths "by definition" rely on human logic as the main building block and substrate. Logic is a human basis of agreeing which seems necessary evolutionary. Counting and separating observable objects turned out to be quite necessary for survival as well. Hence this statement seems to imply that aliens would need to have a corresponding logic reasoning system and observational abilities. If that was the case perhaps there would be a strong inclination to believe that the isomorphic reasoning would be deduced.
pfortuny 4y ago
I guess he was thinking of arithmetic. In any case, any model which includes infinity (and Peano arithmetic already does) is pure convention and unconsciously assumes a lot of things. Finitary induction may make sense as something "universal". Further than that, we are making things up as we go (and I am a professional mathematician). The fact that they work to solve real problems does not make them more real.
- illogical_spock 4y ago
  More to the point, our mathematics work to solve real problems up to a certain point. For instance our mathematics have not yet been able to identify polynomial-time solutions to problems in the class NP and it's possible that this is exactly because our mathematics are inadequate to express such solutions, if such solutions do exist (and Donald Knuth, for example, believes they do). In which case we'll never know whether P = NP (or we will, but it won't be any use, similar to what Knuth, again, suggests). It is a tautology that the famous incompleteness results in mathematics and computer science are the result of the axioms of arithmetic used to derive them. Would Gödel be able to derive his incompleteness result without Peano's axiomatisation of arithmetic? Not really. Arithmetic is axiomatic and our axioms of it are arbitrary and ad hoc. Because they're axioms. Who says aliens would come up with the same ones? There are huge assumptions made in this thread that only indicate the brief time that the contributors have given to thinking about all this stuff. If you think about it for a couple of minutes, sure, it all feels very natural. Zero, infinity, division, mathematical logic, set theory, etc. But if you think about it a bit more, and then do a bit more than think, and go read about it, it's obvious that those are just the ideas that we chose to go with, not the only ones that exist, and certainly not the only ones proposed by mathematicians, logicians, computer scientists and philosophers over the years. For instance, Hilbert was a finitarian, division doesn't work with zero, zero doesn't work with division, dividing an infinity multiplies it, material implication leads to counter-intuitiveness paradoxes, set theory with only the membership relation leads to paradoxes, etc etc etc. Mathematics is full of unnatural holes that need constant patching up, and there is nothing to say that it is in any way, shape or form "real", let alone universal as so many people in this thread seem to be saying.
p0pcult 4y ago
I also take issue with the author's assumptions. Consider the case of AIs trained to identify new variables in a system of pendulums: https://cosmosmagazine.com/technology/artificial-intelligence-physics-variables/ https://cosmosmagazine.com/technology/artificial-intelligenc... Different truths that describe the same system.
deleted 4y ago
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sovande 4y ago
Somethings in mathematics are constant, both here and on alpha centauri, like the circumference of a circle divide by its diameter is Π or the hypotenuse squared is the sum of each leg squared in a right triangle etc.
- lmkg 4y ago
  But neither of those facts are true in non-Euclidean geometries like Spherical Geometry or Hyperbolic Geometry. The jury is still out on whether the universe is flat or has some sort of curvature. Meanwhile spherical geometry is fundamentally useful because we live on a sphere, not on a plane, and it is more accurate at modeling the 2-D space that we navigate in. My point being, assumptions get baked-in to systems in surprising ways. Even something seemingly-objective like math. Especially when you're using it as the basis for communication, then what counts as "basic" or "fundamental" or "standard" reflects a perspective, not a fundamental truth. It's likely their mathematical systems would eventually reach the same conclusions as ours, but the prominence or significance of fields or results (like circles and triangles) might be radically different. Even though we view those components as absolutely fundamental, it's possible an equivalent system could be built from different primitives.
carapace 4y ago
I think mathematics could be defined as that part of philosophy which is self-evident and universal. The value of Pi isn't contingent? Aliens may have different biochemistry, but it would be made from the same chemical elements as ours. Likewise their formal systems may be wildly different from ours, but they will still be based on form (even implication is ultimately a very simple formal structure. Math doesn't even require causality as a prerequisite!) Last but not least. many people (Kurt Gödel among them) believe that mathematical thought is actually perception of real phenomenon in a "higher" plane of reality, which, if true, seems to me to imply that alien mathematicians would be perceiving the same phenomenon as humans, literally. In this view, the "truths of mathematics" are literally the same "objects" for them and for us.
rewgs 4y ago
> There is no “one” mathematics because mathematics is the human exploration of idealized objects using a variety of human logical systems. But isn't the whole point to to do our best to bypass human-centric systems of understanding, and arrive at the "core truth" of the matter? Whether that's possible is another matter, but even if it's not possible, surely it's something that can be theoretically approached, and I would wager is precisely what PG means by "one mathematics." > And then there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand. Yes, but also no. Consider some first principles: -We have every reason to believe that any and all life would not live forever, or if the life in question is "intelligent" (a nebulous/human-centric term, for sure) would at the very least conceive of other things not lasting forever (such as stars, or even the universe itself [or, if you want to be really generous, "this current iteration of the universe"]). -Therefore we can reasonably assume that all "intelligent" life in the universe would understand the concept of scarcity (either via finite lifespans/time, food/energy sources, both, or something else), non-infinity. I'd go so far as to say that any life form that doesn't understand its own mortality or other such limits should be not be considered "intelligent," at least for the reasons of this discussion. -Therefore we can reasonably assume that said intelligent life would somehow conceptualize a binary state (you're either alive or you're not, you either have access to an energy source or you don't, etc), and consequently would somehow or another understand the concept of "zero," "nothing," etc, as well as its opposite, "something." And from there, would necessarily discern the differences between two states of "somethings" (the state of "something" that is "one" is different than the state of "something" that is "two"). I know I'm using a lot of loaded terms here -- "reasonably," "assume," "discerning" -- but just like we look for life by looking for the markers of life that we know were necessary for Earth (carbon, water, etc), we can look for intelligence that exhibits the properties that we understand it to have. We need some sort of frame of reference, after all, if we are to do anything other than simply flail. If that frame of reference is to be proven wrong, that's wonderful, but until that's the case, I don't think it's unreasonable to assume that the universe's "primitives" would be perceived in any truly, truly different way such that the species' interpretation would cause humans to rethink our own understanding of the universe's "primitives" from the ground up. After all, conceiving a difference between hydrogen and helium requires being able to tell the difference between one and two (electrons, as well as separate elements themselves). And considering we have every reason to believe that those make up the majority of the mass in the universe, any "intelligent" life (there's that human-centric term again) can be expected to somehow conceptualize that difference, and thus, do something like counting, and thus, approach the same primitives of mathematics that we do. The approach might be different, but what they're approaching -- the very fabric of reality, hopefully as objectively as possible -- must be assumed to be the same (again, that is, until we're given compelling evidence to believe otherwise). That said, I've never studied the philosophy of mathematics, so I could be talking out of my ass here, this is just the reasoning of a layman after all. If anyone reads this and goes "no you're way off base," I'd love to hear it!

by bmitc 4y ago

cloogshicer 4y ago

Fully agreed. These beliefs you quoted from the article, which unfortunately most people don't even recognize as beliefs, form the basis of the dominant religion of the western world (scientism). The worrying thing is that the majority of people who believe in this religion don't even realize they are believers.

bmitc 4y ago
If I am not mistaken, this is similar to what Feyerabend seemed to be on about. https://blogs.scientificamerican.com/cross-check/was-philosopher-paul-feyerabend-really-science-s-worst-enemy/ https://blogs.scientificamerican.com/cross-check/was-philoso...
gregmac 4y ago
Science doesn't have "facts" or "truths". It is based on falsifiability: for a theory or hypothesis to be considered scientific it must be able to be tested and conceivably proven false. This is the key difference from religion, which has no such "falsifiability" equivalent. The closest thing to "beliefs" is probably an individual following which of several competing theories is most likely correct -- but there's always the underlying basis that any of them might have evidence showing they're incorrect at any time, and one's view should adjust as a result. Often this comes in the form of deferring to other people or a consensus view, which could be construed as "faith" but is different: If you asked me how the universe exists, I'd say the big bang theory is the best answer we have, but I don't understand enough about the underlying science to explain why nor can my brain comprehend the reality of it. I have no loyalty or allegiance to this view, though; I could be swayed to another theory if the big bang is ever proven false or if a better theory arises. Further reading: https://www.theatlantic.com/science/archive/2015/11/why-scientific-faith-isnt-the-same-as-religious-faith/417357/ https://www.theatlantic.com/science/archive/2015/11/why-scie...
simonh 4y ago
The thing that differentiates science from religion is repeatability. With religion everyone has their own opinion, people out of contact with each other come up with radically different religious beliefs and there is no way to bridge between those beliefs. If we forgot everything we know about religion, in a thousand years we might rediscover religion again but they'd be entirely different religions from what we have now. With science it doesn't matter who does a given experiment, anyone else doing the same experiment will get the same results. There's no scope for disagreement about verifiable scientific facts. Just do the experiment and find out. If we forgot everything we know about science, in a thousand years if we rediscovered science, very quickly we'd rediscover all the exact same facts about the world again.

tambourine_man 4y ago

Yeah, lots of strong baseless assumptions in the first paragraph made me stop reading.

adrianN 4y ago

I'm pretty sure that any sufficiently advanced species that does any kind of mechanical engineering will have maths that is homomorphic to analysis.

exitb 4y ago

Isn't the basic math largely driven by attempts to understand and quantify the world around us? In such case, it depends on more universal concepts like distance, time, speed, acceleration etc. Concepts which I would imagine to be familiar to any intelligent being that takes a physical form. I can't imagine an alien wondering about a period of a pendulum and arriving at an answer that's really different from ours.

p0pcult 4y ago
Funny that you chose pendulums: https://cosmosmagazine.com/technology/artificial-intelligenc https://cosmosmagazine.com/technology/artificial-intelligenc...

naasking 4y ago

> 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.

LocalH 4y ago

> Mathematics is most definitely a human endeavor, and so we can’t really make claims about its existence in the universe independent of humans. I think the idea is that math is not created by humans, but documented by humans. Sure, the specific terminology may be our invention, but there are basic mathematical properties that seem (from our perspective) like they should be universal. For example, whatever names a being has for the numbers 1 and 2, if you take that 1 and add 1 more, you must get 2 (or the "local equivalent") as the result. My guess is that, if what we call math isn't truly universal, it's probably at least universally true within the realm of physical life, and there's likely some massive causal chain from the root properties of physics itself to the mathematical properties that we call "math". When it comes to raw, untethered "consciousness" (or whatever one would prefer to call it), this may not hold true even in the slightest. Yes, this comment steps slightly outside what could ever be determined purely by the scientific method at the end. I feel it is useful to do so in discussion, even when that cannot directly enter into research. There are some truths to the larger universe that I don't think the scientific method will ever truly be able to uncover, just due to it's rigor. Some aspects of the universe are just simply not falsifiable, but they're still worthy of discussion with an open mind.

dTal 4y ago
There is most certainly a massive causal chain between physics and what humans call "math", because everything humans do is determined by the laws of physics. The causal chain leads through millions of years of evolution and tens of thousands of years of culture. "The numbers 1 and 2" are a complex web of analogies that have not actually been demonstrated to "exist" outside of our minds, so evidence that alien mathematicians would have words for them is much weaker than our intuition would suggest. The question that must be answered is "Given the constraint of precisely modelling the world in a useful way, to what extent are all rule-based systems isomorphic?" Responding to your second point, I'm afraid I can't agree with you that unfalsifiable propositions are "useful" discussion contributions - especially not with an "open mind". The only criterion on which such propositions can be judged is whether they are fun to believe, and that is a very dangerous muscle to flex.

dagw 4y ago

I think alien analogs to mathematics are unlikely to match ours. Almost certainly not, but they're probably isomorphic. And either way if we show them our axioms they will be able to validate our mathematics and vice versa. The truths of mathematics are of the form 'if A then B'. Even if they don't start at A or even accept A as true, but will should still get B if they assume A.

Koshkin 4y ago
> they're probably isomorphic That's a strong statement. I'd probably talk about homomorphism.

mr_mitm 4y ago

> Even humans do not fully agree on mathematics. There is no “one” mathematics because mathematics is the human exploration of idealized objects using a variety of human logical systems. Maybe not in its entirety, but I find it hard to imagine that any civilization as advanced as ours (let's say a civilization that manages to harness nuclear fission, just to set a baseline) will not come up with concepts such as prime numbers, real number, complex numbers, calculus, etc. If they do that, they will inevitably find the same structures we found using function theory. They will know about differential equations and prove similar theorems about them as we did. And the Pythagorean theorem is a universal truth that holds everywhere.

lostmsu 4y ago
In that list prime numbers might not be a thing.
lmkg 4y ago
> real number, complex numbers, calculus, etc Some humans believe that the prominence of real numbers is a historical accident. It seems quite plausible to me that a human society, much less an alien one, would go down a mathematical evolutionary path based on the constructable numbers and the computable numbers. Regardless of whether we eventually find the same structures, there are things that we might consider basic which they find esoteric and vice-versa. Heaven forbid we encounter an alien civilization that discovered an O(log n) algorithm for integer factorization before they invented steam power.
- bmitc 4y ago
  > Some humans believe that the prominence of real numbers is a historical accident. That is a great point. The continuum is contentious and pretty highly debated philosophically. I also encourage people interested in that to also learn about smooth infinitesimal analysis. Just a small tweak in the underlying logic and model yields unique mathematics and questions established assumptions.
illogical_spock 4y ago
> And the Pythagorean theorem is a universal truth that holds everywhere. Or at least in every world where there exist straight lines, yes? For instance: > The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate.[59][60] Thus, right triangles in a non-Euclidean geometry[61] do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to π/2, and all its angles are right angles, which violates the Pythagorean theorem because {\displaystyle a^{2}+b^{2}=2c^{2}>c^{2}}. https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclid...

agent008t 4y ago

The statement that "the truths of mathematics would be the same, because they are true by definition" is correct. Mathematics exist independently of biology or perception of reality. It is just a collection of arbitrary abstract definitions and what follows from them. The alien species may come up with different base definitions that they find more useful or interesting. But they would derive the same conclusions as we would if they were starting from the same definitions and applying the same abstract rules.

bmitc 4y ago
> Mathematics exist independently of biology or perception of reality. Is there a “proof” of that? How could a proof exist? You’d probably win several awards if you had one. > It is just a collection of arbitrary abstract definitions and what follows from them. … created by humans.
- dagw 4y ago
  Is there a “proof” of that? It's how we define the concept "mathematics". If a result was dependant on "biology or perception of reality" or anything else outside its defining axioms, it wouldn't be mathematics.
  - bmitc 4y ago
    I'm sorry, but I just disagree that that's how mathematics is defined and that it doesn't depend on our biology and perception, because we are making those definitions. A book I might recommend and that I'm going through at the moment is Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by George Lakoff and Rafael Nunez. The origin and meaning of mathematics is strongly influenced by cognitive sciene, and thus biology. I've been downvoted, but this is not a totally novel or off the rails idea. It is basically accepted in robotics that embodied cognition is how you get a robot to understand and perceive its environment. Where do you think that idea came from?
    - dagw 4y ago

rendall 4y ago

> there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand. Almost certainly true. We as evolved creatures do not perceive objective reality, but only enough reality as to enable our ancestors to survive our very niche environment (niche relative to the entirety of the universe). Our science and mathematics model only our perceptions of reality, and not reality itself. It is folly to assume that an alien, evolved along an entirely different set of initial conditions, would share our perceptions of reality. Our mathematics, modeling as it does our perceptions, serves human needs and perceptions only.

slibhb 4y ago

While it's true that there's no consensus on this topic, that doesn't imply that people can't make claims one way or the other. In fact, the claims that the essay makes (Platonism) are very commonly made.

dbsmith83 4y ago

> Mathematics is most definitely a human endeavor, and so we can’t really make claims about its existence in the universe independent of humans. Can you give an example of a mathematical concept which could be different? I believe that math is universal. We may use models to understand it (infinity, perfect circles, etc.), but the underlying mathematical truth is independent of humans. The same is true for science. There are physical laws which we look to discover. We use models in science to understand them, but the models are not the same as the underlying truth

rendall 4y ago
> I believe that math is universal. We may use models to understand it (infinity, perfect circles, etc.), but the underlying mathematical truth is independent of humans. Humans (probably) perceive and understand a slim subset of reality. We have an illusion of universalism of our perceptions because we perceive nothing outside of them. Also, we are the dominant species of the planet, which gives us a reason to believe that our perceptions are "more accurate" than, say, a bat's. Personally, I don't think an alien's perceptions and understanding would contradict our own, but if it's given that our perceptions are a subset of reality, then an alien's understanding might include elements of reality that we literally cannot perceive or even understand. > Can you give an example of a mathematical concept which could be different? By definition, no. But, hmm. What would number and a mathematical system look like from creatures who thought in logarithms? Or in primes? Or that had no concept of "number"? What if even "greater than" and "less than" had no relevance to an alien civilization?
- dagw 4y ago
  All of this is arguments for why our understanding of science and the nature of the universe might differ. It can also explain how our mathematics might evolve very differently and we will have made different mathematical discoveries and do math in very different ways (all of which I agree are very likely). However it doesn't explain how an alien race will look at one of our mathematical proofs that we have proven True and be able to prove that it is False.

pattt 4y ago

Fully agreed, the way I'd approach this would be that the said mathematical proofs about certain truths "by definition" rely on human logic as the main building block and substrate. Logic is a human basis of agreeing which seems necessary evolutionary. Counting and separating observable objects turned out to be quite necessary for survival as well. Hence this statement seems to imply that aliens would need to have a corresponding logic reasoning system and observational abilities. If that was the case perhaps there would be a strong inclination to believe that the isomorphic reasoning would be deduced.

pfortuny 4y ago

I guess he was thinking of arithmetic. In any case, any model which includes infinity (and Peano arithmetic already does) is pure convention and unconsciously assumes a lot of things. Finitary induction may make sense as something "universal". Further than that, we are making things up as we go (and I am a professional mathematician). The fact that they work to solve real problems does not make them more real.

illogical_spock 4y ago
More to the point, our mathematics work to solve real problems up to a certain point. For instance our mathematics have not yet been able to identify polynomial-time solutions to problems in the class NP and it's possible that this is exactly because our mathematics are inadequate to express such solutions, if such solutions do exist (and Donald Knuth, for example, believes they do). In which case we'll never know whether P = NP (or we will, but it won't be any use, similar to what Knuth, again, suggests). It is a tautology that the famous incompleteness results in mathematics and computer science are the result of the axioms of arithmetic used to derive them. Would Gödel be able to derive his incompleteness result without Peano's axiomatisation of arithmetic? Not really. Arithmetic is axiomatic and our axioms of it are arbitrary and ad hoc. Because they're axioms. Who says aliens would come up with the same ones? There are huge assumptions made in this thread that only indicate the brief time that the contributors have given to thinking about all this stuff. If you think about it for a couple of minutes, sure, it all feels very natural. Zero, infinity, division, mathematical logic, set theory, etc. But if you think about it a bit more, and then do a bit more than think, and go read about it, it's obvious that those are just the ideas that we chose to go with, not the only ones that exist, and certainly not the only ones proposed by mathematicians, logicians, computer scientists and philosophers over the years. For instance, Hilbert was a finitarian, division doesn't work with zero, zero doesn't work with division, dividing an infinity multiplies it, material implication leads to counter-intuitiveness paradoxes, set theory with only the membership relation leads to paradoxes, etc etc etc. Mathematics is full of unnatural holes that need constant patching up, and there is nothing to say that it is in any way, shape or form "real", let alone universal as so many people in this thread seem to be saying.

p0pcult 4y ago

I also take issue with the author's assumptions. Consider the case of AIs trained to identify new variables in a system of pendulums: https://cosmosmagazine.com/technology/artificial-intelligence-physics-variables/ https://cosmosmagazine.com/technology/artificial-intelligenc... Different truths that describe the same system.

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sovande 4y ago

Somethings in mathematics are constant, both here and on alpha centauri, like the circumference of a circle divide by its diameter is Π or the hypotenuse squared is the sum of each leg squared in a right triangle etc.

lmkg 4y ago
But neither of those facts are true in non-Euclidean geometries like Spherical Geometry or Hyperbolic Geometry. The jury is still out on whether the universe is flat or has some sort of curvature. Meanwhile spherical geometry is fundamentally useful because we live on a sphere, not on a plane, and it is more accurate at modeling the 2-D space that we navigate in. My point being, assumptions get baked-in to systems in surprising ways. Even something seemingly-objective like math. Especially when you're using it as the basis for communication, then what counts as "basic" or "fundamental" or "standard" reflects a perspective, not a fundamental truth. It's likely their mathematical systems would eventually reach the same conclusions as ours, but the prominence or significance of fields or results (like circles and triangles) might be radically different. Even though we view those components as absolutely fundamental, it's possible an equivalent system could be built from different primitives.

carapace 4y ago

I think mathematics could be defined as that part of philosophy which is self-evident and universal. The value of Pi isn't contingent? Aliens may have different biochemistry, but it would be made from the same chemical elements as ours. Likewise their formal systems may be wildly different from ours, but they will still be based on form (even implication is ultimately a very simple formal structure. Math doesn't even require causality as a prerequisite!) Last but not least. many people (Kurt Gödel among them) believe that mathematical thought is actually perception of real phenomenon in a "higher" plane of reality, which, if true, seems to me to imply that alien mathematicians would be perceiving the same phenomenon as humans, literally. In this view, the "truths of mathematics" are literally the same "objects" for them and for us.

rewgs 4y ago

> There is no “one” mathematics because mathematics is the human exploration of idealized objects using a variety of human logical systems. But isn't the whole point to to do our best to bypass human-centric systems of understanding, and arrive at the "core truth" of the matter? Whether that's possible is another matter, but even if it's not possible, surely it's something that can be theoretically approached, and I would wager is precisely what PG means by "one mathematics." > And then there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand. Yes, but also no. Consider some first principles: -We have every reason to believe that any and all life would not live forever, or if the life in question is "intelligent" (a nebulous/human-centric term, for sure) would at the very least conceive of other things not lasting forever (such as stars, or even the universe itself [or, if you want to be really generous, "this current iteration of the universe"]). -Therefore we can reasonably assume that all "intelligent" life in the universe would understand the concept of scarcity (either via finite lifespans/time, food/energy sources, both, or something else), non-infinity. I'd go so far as to say that any life form that doesn't understand its own mortality or other such limits should be not be considered "intelligent," at least for the reasons of this discussion. -Therefore we can reasonably assume that said intelligent life would somehow conceptualize a binary state (you're either alive or you're not, you either have access to an energy source or you don't, etc), and consequently would somehow or another understand the concept of "zero," "nothing," etc, as well as its opposite, "something." And from there, would necessarily discern the differences between two states of "somethings" (the state of "something" that is "one" is different than the state of "something" that is "two"). I know I'm using a lot of loaded terms here -- "reasonably," "assume," "discerning" -- but just like we look for life by looking for the markers of life that we know were necessary for Earth (carbon, water, etc), we can look for intelligence that exhibits the properties that we understand it to have. We need some sort of frame of reference, after all, if we are to do anything other than simply flail. If that frame of reference is to be proven wrong, that's wonderful, but until that's the case, I don't think it's unreasonable to assume that the universe's "primitives" would be perceived in any truly, truly different way such that the species' interpretation would cause humans to rethink our own understanding of the universe's "primitives" from the ground up. After all, conceiving a difference between hydrogen and helium requires being able to tell the difference between one and two (electrons, as well as separate elements themselves). And considering we have every reason to believe that those make up the majority of the mass in the universe, any "intelligent" life (there's that human-centric term again) can be expected to somehow conceptualize that difference, and thus, do something like counting, and thus, approach the same primitives of mathematics that we do. The approach might be different, but what they're approaching -- the very fabric of reality, hopefully as objectively as possible -- must be assumed to be the same (again, that is, until we're given compelling evidence to believe otherwise). That said, I've never studied the philosophy of mathematics, so I could be talking out of my ass here, this is just the reasoning of a layman after all. If anyone reads this and goes "no you're way off base," I'd love to hear it!