Perhaps it’s not that they are boring, but fearful. I can’t recall a single time I mentioned these words in a conversation that weren’t abruptly interrupted with “there is no absolute truth!”. If the rebranding helps us calm down and think about it without fear, it might be worth it.
Perhaps a little off topic, but I have thought about this a bit: the type of consciousness and ways of acting in physical reality would probably be very different for digital vs. biology-based life. As strangely different alien biological life might be, true AI will be stranger, unless it is designed and evolves to emulate us in some sense.
I am reading the sci-fi book The Sea of Rust right now, that takes place on earth after AI’s have finally killed the last humans. In this fictional work, AIs take on human traits and I don’t find that “believable” even for sci-fi.
Indeed. Intelligent alien life is likely to have evolved out of other forms of life, so concepts as competition, survival, cooperation, are likely to be innate somehow (not necessarily in a conscious way). AI does not need to share that basis at all.
I disagree. "AI" even if it were to exist independently of humans or other pre-existing life still needs to process and dissipate energy in order to perpetuate itself... ie it needs to be "life" first. Intelligence requires the processing of information, and the processing of information requires energy.
For example, even if silicon-based AI replaced humans on earth, it would eventually [0] need to find a way to power itself / continue itself. Ultimately, it would have to revert to solving the same "problem of life"... how do we transform energy/entropy available in the environment into something that "perpetuates the system." When that happens, this AI will itself become subject to the forces of natural-selection, and - over a long enough period of time - naturally-selected traits will be re-aquired (even if such traits were "lost" during a human-to-AI hand-over).
[0] Yes, of course, there could be a very large period of time during which currently constructed energy infrastructure continues on... and this period could be measured in hundreds/thousands of years... very long in terms of human lifespans, but not in geological terms.
That's a rather naive view of truth. For intuitionistic mathematics (which is very important for theoretical computer science), truth is a mental construct of a proof in a language, and only shared by communicating that construct to other minds.
The question is whether we can communicate our mathematics to non-humans, whose senses may be radically different from our own.
And likewise, would we be able to understand a non-human proof?
I'm a puzzled by the confidence here. I would assume pg is at least minimally familiar with some of the key philosophical themes and schools of thought around this topic.
> For intuitionistic mathematics ... truth is a mental construct of a proof in a language, and only shared by communicating that construct to other minds.
Even assuming you are fully and accurately representing the intuitionist view, you must be aware that there are competing schools of thought with strong pedigrees, like mathematical platonism, that are grounded in a more realist view of mathematical objects.
PG didn't go so far as to stake out that position here in the essay, but his thought experiment leverages a view of mathematical truth that hues closer to this (platonist) camp.
> The question is whether we can communicate our mathematics to non-humans ...
That is an interesting question, but pg did not ask that question in the essay, and its answer doesn't seem relevant to the point he was trying to make.
It's Graham's confidence that I (and I think OP) find puzzling. He doesn't say "According to Platonist ideas...", or "In my opinion...". Just "mathematical truths are true by definition". There's a brashness and lack of nuance there that makes me want to stop reading.
Some of that is a stylistic judgement call on the part of pg. He wants to move quickly to his central idea, not enter into a nuanced discussion of the philosophy of mathematics.
I think pg is educated, at least the broad outline, of the age-old controversies in this topic and I'm grateful as a reader that he spares me the details and assumes some preliminary context.
He’s clearly not, is the thing, unless he has acquired that education in the last year or two.
His style of pop philosophy is useful in startups because it does not require context. It is ungrounded in actual philosophy, and would struggle to find an audience outside of this group.
I appreciate what you're saying; it's only the first paragraph and he wants to nail some initial things down that he sees as straightforward context or examples.
But it is not straightforward to me. Or, in my impression, most philosophers. I'm reluctant to agree that aliens will assign any importance or even understand concepts like geometry or calculus. So to me, it is not preliminary context but it is a critical part of the discussion.
> I would assume pg is at least minimally familiar with some of the key philosophical themes and schools of thought around this topic.
But even a brief foray into discussions by actual philosophers will show that to not be true. What Paul Graham is doing is providing small subcultural insights tailored to an audience with a passing interest in philosophy.
> his thought experiment leverages a view of mathematical truth that hues closer to this (platonist) camp
I think the word in that idiom is 'hews', which means to adhere strictly to a standard, probably from the sense of the word meaning to strike or cut or beat - often used to talk about cutting a tree into shape.
> The question is whether we can communicate our mathematics to non-humans, whose senses may be radically different from our own.
I'm not sure why people find this so relevant. Regardless of the sense, there must be enough discernible structure to detect whether 0 of something is there, whether 1 of something is there, or whether there are many more than 1.
Human sense of smell might top out at differentiating maybe 4 or 5 different things, dogs can probably sense a lot more, but either way it lets us set up basic counting, and that's generally all you need for most of our math.
Well, first we could send a series of integers, each being a sequence of pulses, to convey the fact that we are talking about integers. Then express each axiom by a series of concrete examples. And repeat. Intelligent species should be good at recognizing patterns.
Why would we identify a sequence of pulses as integers if, for example, the passage of time is perceived differently by aliens? Or if mathematics within alien life is based on atomic units of group like set logic, and so the sequence of pulses itself is seen as "1".
The ancient Egyptians expressed axioms by a series of concrete examples. Stating them as actual axioms or theorems with proofs (by Greeks, Indians and Chinese, among others) was an improvement.
How can we communicate actual axioms to extra-terrestrials?
Presumably we can create a kind of Rosetta Stone with different representations of logical and algebraic expressions, and maybe they can decode that if they can figure out how to decode whatever "broadcast" representation of this we come up with. And that assumes that they recognize such a broadcast as something from intelligent life.
My dog is an intelligent being and he doesn't share our affinity for math. We have to acknowledge that our concept of intelligence is human-centric. What we interpret as knowledge is based on our specific limitations and perceptions. We don't know what intelligence is and won't necessarily know it when we see it.
Let's hope then that aliens coming here will be vegan.
We're not intelligent enough yet not to harm others, or to go to other star systems, or even other galaxies. Would we be considered intelligent, or a source of protein?
I appreciate what he is saying, but this also feels like the speech given at the beginning of a sci-fi/horror movie where you realize you’re completely wrong.
The beauty of thinking about other civilizations is it provides a due balance for materialist views, where instead of a universal truth, the question becomes what values and principles would be sufficient for us to co-exist with more (and less) advanced beings without respective reduction to foodstuffs, pets, or slavery.
To a more advanced civilization, we are chimpanzees who are both outwardly intelligent, but also tremendously dangerous, and so on what basis could they establish trust with us, or could we establish trust with a civilization of others? As Graham notes, math is one indicator that we are capable of apprehending the universe around us, but given the infinity of life and its necessary physical conditions of beginning and ending, and evolving in aggregate using tools and principles, it's not sufficient. Maybe one way to ensure trust is to share DNA, so that we become each other and we are all "us" - or, perhaps the Girardian mimetic concept generalizes such that it is better to preserve our differences so that we are not competitors for the same resources, and so that we can co-exist with an obvious other but without an existential threat or intrinsic power struggles.
Are there existing moral or philosophical systems that are suited to this problem? Probably, I'm not a religious scholar, but the golden thread that links them seems pretty consistent in attempting to derive alignment to an external truth. The proto-Christian tribe of Essenes, from whom John the Baptist originates and who was the one who baptized Jesus into what became Christianity (solving a weird bootstrapping problem, imo) espoused the values that became the first Church, so there is a historiographical way of looking at moral systems instead of as dogma. Outside religion, in the search for these values that would be suitable for a community of inhabitants, I've come to suspect this is what freemasonry is about, and while not about aliens, I was impressed by their allegorical emphasis on tools instead of doctrine as the landmarks for discovery.
The essential question to me is, once you have accepted there is an other that is greater, or a place that is elsewhere, does it matter whether it's a dude with a beard, multi-armed flying blue people, or an ineffable oneness? That there is a concievable elsewhere beyond your current limits, there must therefore be some point or idea to align and orient yourself to so as to be able to relate to the other beings who have discovered the same point outside our current perspective.
It's all very meta, but it implies a logical and even rational case for some guidance or alignment to this otherness to navigate our present, and that isn't material. The value of the idea of an "alien" truth is it is a means to reconcile secular rational thinking and moralism with universal, essential, or spiritual values, and that could be a very useful tool.
I wrote a scifi book in which I wrote 'heaven' as being an alien construct. Somewhere in the universe a tremendously technologically advanced civilization constructed a device which simulates 'heaven' for all people in the universe. Through 'enlightenment' the discoverer is able to bring knowledge of 'heaven' back to their people but in our history how could a figure like Buddha or Jesus post-enlightenment explain aliens and advanced technology. So you end up producing a story at a level of your current day scientific understanding.
Fundamentally your message to humanity post-enlightenment would be the rules on how to get to heaven. Which many world-religions classes go into depth. There are fundamental rules that benefit everyone to follow that wouldn't really be inherently human to follow.
>We'd probably share Occam's razor. There doesn't seem anything specifically human about any of these ideas.
Aliens will also have developed the piano and chess. They are inherent things to discover eventually.
Fundmantally a great way to analyze what the rules are would be impossible to list. Just look at the list of crimes in countries which are so large lawyers dont even know them all. So you need a system that's much more simply. Isn't that system 'karma'.
> Aliens will also have developed the piano and chess. They are inherent things to discover eventually.
How could that claim be true? We have highly intelligent beings (i.e., “aliens”) right here on Earth that have not developed these things.
These discussions on aliens are often off the rails from the start because they implicitly begin with the assumption that humans are the only intelligent beings on Earth.
>How could that claim be true? We have highly intelligent beings (i.e., “aliens”) right here on Earth that have not developed these things.
Are you using the 'illegal immigration' definition of alien?
>These discussions on aliens are often off the rails from the start because they implicitly begin with the assumption that humans are the only intelligent beings on Earth.
Do please elaborate because I don't share this opinion. Do you believe aliens live amongst us?
What is an alien other than a biological being from another planet? We have biological beings on Earth that share DNA with us but possess wildly different intelligences and cognitive systems. Is it a stretch to use these as examples that aliens may share little in common with us?
>What is an alien other than a biological being from another planet?
Alright, agreed. Which as far as I know we have no known aliens ever discovered.
>We have biological beings on Earth that share DNA with us but possess wildly different intelligences and cognitive systems. Is it a stretch to use these as examples that aliens may share little in common with us?
You're backpedaling pretty hard. You said there are 'highly intelligent beings on earth' besides us. I know of no known examples that fit your claim. Happy to listen.
It’s not backpedaling. Both things are true. There are wildly different “intelligences”. Plants, for example. As for highly intelligent, orcas are an example.
And all this relies on some definition of intelligence, which I don’t think we even have a good one for.
That is a work of stunning arrogance and foolishness.
Math is a game we play in our heads that represents a fictionalized ideal version of reality.
An alien intelligence might have realized that two plus two never equals four not because the underlying logic is wrong, but because two does not exist in reality.
The idea that the little game of math we play represents an immutable and universal truth is typical of the overwhelming anthropocentrism of our kind.
Just because some alien societies will not mimic our rules of addition, we do know for certain it is possible that other societies can build abstract concepts that are isomorphic to those we have. And many of these concepts, such as addition, are very useful.
Does this guarantee that aliens come up with the same stuff? No. Does it guarantee that if they did, they would these concepts to the same esteem? No. Is there an element of 'truth' here that can be replicated by others? Absolutely
I guess I disagree that math and physics are universal truths.
For maths, I would say 1+1=2 is a pretty universal truth (although it takes a while to get there in the principa mathematica), but didn't we just invent complex numbers because they are useful?
Same goes for physics, the speed of light is the same everywhere, but how quantum mechanics work is still subject to many discussions.
Love to hear some thoughts on this, as claiming a whole field as universal truth is something I'm a little uncomfortable with.
But the usefulness is objective, that is, it is not an arbitrary product of the mind but rather it is dictated by the logic of things once the goal is set, so invention (or discovery) of useful things is more or less unavoidable.
As to quantum mechanics, you are talking about the variety of interpretations which from the practical standpoint are simply different ways of looking at quantum behavior, which, in turn, sometimes leads to different methods of calculation.
> But the usefulness is objective, that is, it is not an arbitrary product of the mind but rather it is dictated by the logic of things once the goal is set, so invention (or discovery) of useful things is more or less unavoidable.
It certainly seems like it is objective, and often it probably is, but in a more general sense, any instance of "x 'is' y" very often turns out to be subjective very quickly. Even with "is useful", things get complicated if one explicitly injects the dimension of Time into the question (it is there in the first place implicitly, but is easily overlooked).
I think the argument is mostly that there are universal truths than math and physics describe not that our current level of math and physics are universal truths. So finding an example that we do not understand fully doesn't mean that there aren't truths in other aspects of math and physics. As for things like complex numbers there is an underlying debate that has been around in philosophy of science and math that distinguishes between discovery and invention. Our representation may have been invented but we discovered some thing that works historically and has predictive power.
> but didn't we just invent complex numbers because they are useful?
Not any more than we invented natural numbers because they are useful.
There are several ways to naturally derive complex numbers, either from mathematics or from physics.
For one, complex numbers are probably the simplest possible extension of the real numbers in which all real-valued polynomials have roots (for example, x^2 + 1 doesn't have a root if x has to be real). This is the same reason why the non-transcendental irrational numbers were invented (such as sqrt(2) ).
(Incidentally, the transcendental numbers (pi, e) are less justified than the complex numbers from this point of view - any polynomial of any rank whose coefficients are non-transcendental real numbers has roots that are either real non-transcendental numbers, or a complex number whose real and imaginary parts are real non-transcendental numbers )
For a physical explanation, complex numbers are the best way we know of describing wave mechanics (either classical or quantum), and in general periodic phenomena and how they compose.
I think you have the right idea. Most human truths are contingencies of our evolution and the evolution of life on earth. It's very hard to extrapolate from this to universal and alien truth.
>For example, I think we'd share the principle that a controlled experiment testing some hypothesis entitles us to have proportionally increased belief in it.
This isn't even a shared principle among humans. How many experiments does it take for you to have 50% belief in a hypothesis?. What is the number of experiments? It's literally impossible to answer. It's not even clear what "belief" is or what 50% means.
This ambiguity of the word isn't even the main problem. If I run the same experiment with perfect observational tools 10 billion times and it verifies my hypothesis. Does that raise my belief further? What if on the 10 billionth and first time the test shows a negative result? That literally invalidates the hypothesis. Keep in mind we are assuming my observational tools are perfect. Does this make my belief shoot down to zero?
If this possibility of a negative result remains true after any number of tests then what does it say about belief? Why should I believe anything if a single negative experiment can invalidate 10 billion positive experiments (assuming perfect observational tools of course)?
Let me bring a more concrete example. I hypothesize all zebras have stripes. I observe zebras 10 billion times. They all confirm my hypothesis. Then on the 10 billionth and first time I see a zebra with spots. My hypothesis is wrong. This can happen any time.
Anyway to bring it back to his point. Don't assume shared axiomatic truths. PG already assumed that it's shared among humans. He's wrong. The nature of science and the scientific method is not universally shared or even fully understood among humans. He's likely also wrong about aliens as he is about humans.
> This isn't even a shared principle among humans. How many experiments does it take for you to have 50% belief in a hypothesis?. What is the number of experiments? It's literally impossible to answer. It's not even clear what "belief" is or what 50% means.
A bit of a complicated read. Not sure what you're trying to say here and I don't even completely understand it. But anything that gets into bayesians and frequentists ends in a fundamental divide in humanity. Humans don't agree on which interpretation is correct. Which is the point of everything I wrote.
So why would aliens hold this "principle" the same if humans don't even agree on it? PG is wrong. His own principles upended not even by aliens, but by humanity, thus how accurate can his assumptions about universal principles even be? No that accurate imho.
> But anything that gets into bayesians and frequentists ends in a fundamental divide in humanity. Humans don't agree on which interpretation is correct.
There is no divide, there is the illusion of divide because we didn't have a rigourous formal model of how to build reliable knowledge and everyone focused on different but relevant aspects.
Bayesian reasoning is the correct way if you have justifiable priors, but we didn't have a way to calculate the correct prior.
Solomonoff showed us how with his theory: Kolmogorov complexity is a measure of parsimony, and this is how to select priors in a formal, rigourous way.
Solomonoff induction is to knowledge what Turing machines or the lambda calculus are to computation. Sure, aliens might not discover Turing machines exactly, or the lambda calculus exactly, but whatever they do build that's capable of universal computation, we already know it must be isomorphic to a Turing machine, because all constructions capable of computation must be by necessity.
The frequentist/Bayesian divide is a separate issue about how to interpret statistical data in useful ways, not specifically about how we know what we know and what confidence we should have in our knowledge, which is what you were asking about.
Interesting. Do you know of any popular science articles or books that can describe what you're talking about? Academic papers are fine too, just harder to parse.
Hard to find simple articles on such an esoteric topic as algorithmic probability, which cuts across subjects like probability, information theory and computation. This one seems to hit all the notes but who knows if it's as accessible as it's aiming to be:
> I hypothesize all zebras have stripes. I observe zebras 10 billion times. They all confirm my hypothesis. Then on the 10 billionth and first time I see a zebra with spots. My hypothesis is wrong. This can happen any time.
Even if the original belief turns out to be wrong, you only have to slightly weaken it and it will remain true: "the vast majority of zebras have stripes". Even if you discover a new continent full of hordes of uniformly-colored zebras, the true hypothesis becomes "the vast majority of zebras in my original continent are striped".
Essentially every observation brings proof for a whole family of hypotheses. We normally only talk about the strongest of these hypotheses, but that doesn't meant that a negative example rules out the entire family.
For example, even if we didn't find a deductive proof the Fermat's last theorem was wrong even after all of the empirical proof that it probably wasn't, a weaker version would have still remained true - the one validated by that empirical proof.
>Even if the original belief turns out to be wrong, you only have to slightly weaken it and it will remain true: "the vast majority of zebras have stripes". Even if you discover a new continent full of hordes of uniformly-colored zebras, the true hypothesis becomes "the vast majority of zebras in my original continent are striped".
The hypothesis does not remain true. It was never proven to be true and the new hypothesis is still not proven to be true. Science cannot prove anything to be true. I can find a cave full of of spotted zebras, and you have to further weaken your hypothesis of continents, I can then find that the stripes were actually microscopic spots and my perfect observation tool, though never wrong has limited resolution. Ad infinitum. Nobody ever considers your made up philosophy because it's changing the rules of the game. It's making a statement then adjusting your statement once it's proven wrong... people look down on that kind of thing.
What I'm writing here isn't something I pulled out of my ass. It's well known that in science, the scientific method, and reality itself, nothing can be proven. Proof is the domain of math and logic, not science. In science, things can only be falsified. To quote Einstein:
“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”
Einstein obviously isn't saying stuff like a single experiment causes me to adjust my hypothesis and divide split it into two different ones because it's kind of inconsistent.
There are people who truly understand science, but most of the population doesn't (including PG). I think what's going on with you is you're in the later camp, you've long held the incorrect belief that science can prove things and this long held ideology is coming into contact with the actual logic of the situation and your adjusting your belief to maintain a biased ideology.
Do you look up to PG? Bias can be corrected when an authority confirms the opposite. I quoted Einstein here. One of the ultimate authorities on science, a person who overturned the hypothesis about Newtonian physics being a model for motion. A single experiment proved it wrong and now Newtonian physics is simply an approximation that is ultimately wrong. Hopefully that will clear things up, if not... then you must be an Alien far more strange then what PG is describing.
Please don't act condescending. The problem of induction is well-known, and is closely related to what you are discussing here. I agree that science can't literally prove any hypothesis is true in the same sense that mathematics/logic can; but we also can't jump from here to considering inductive reasoning an entirely useless tool in the search for truth.
That is the point that I am trying to make: experimentation can bring proof to strengthen a hypothesis. Even if a later experiment invalidates a hypothesis, all of the previous experiments' results don't disappear, and any new hypothesis we formulate still needs to be coherent with them to have any value: we have actually learned something important from our thousand experiments, even if our 1001st showed that the hypothesis we had in mind was false.
Also, this is not unique to science. The same phenomenon can happen in mathematics or logic for theorems that have been neither proven nor disproven yet. We can perform numerical experiments to test a numerical theorem, and gain some amount of confidence in that theorem even if we haven't proven it to be true. We can often establish lower or upper bounds in the course of this experimentation, where we find that the theorem is True at least for some limited subset of all numbers - and this remains True and useful even if it later turns out that there exist counter-examples.
This observation is also very important for understanding why the history of natural philosophy is essentially one of continuous progress, with very little backtracking: even if induction is not good enough to know that we have a perfectly complete and consistent theory (and we will never have one), we always have something salvageable from all of the experimentation done so far. Even geocentric models with their epicycles were actually working models, which predicted the positions of planets in the next 1000 years to quite good accuracy, even if they were clearly wrong in the end.
Please don't accuse me of acting condescending. It's very offensive and hurts my feelings when I'm accused of something I'm not doing.
I am criticizing you, but I am not being condescending. There is a huge difference.
Perhaps the alien thing was bad. I apologize for that. The intent was a joke and was not condescension.
>That is the point that I am trying to make: experimentation can bring proof to strengthen a hypothesis. Even if a later experiment invalidates a hypothesis, all of the previous experiments' results don't disappear, and any new hypothesis we formulate still needs to be coherent with them to have any value: we have actually learned something important from our thousand experiments, even if our 1001st showed that the hypothesis we had in mind was false.
Yes but this was not part of the discussion. We're talking about science as a principle. Not what we have learned from the process of science.
>This observation is also very important for understanding why the history of natural philosophy is essentially one of continuous progress, with very little backtracking: even if induction is not good enough to know that we have a perfectly complete and consistent theory (and we will never have one), we always have something salvageable from all of the experimentation done so far. Even geocentric models with their epicycles were actually working models, which predicted the positions of planets in the next 1000 years to quite good accuracy, even if they were clearly wrong in the end.
Important or not, we diverged from the point. Whether Science is a valid principle shared by humans and aliens is the point. My point is, PG's view isn't even shared with humans, why should he assume it's going to be shared with aliens?
You're talking about the importance of science. The value of science. That's off topic.
no. Don't agree. Experimentation can't prove anything. It also doesn't strengthen anything. Proof is not a strengthening of something. If you prove something it means it's true.
>Well, the experimentation part, that can bring proof to strength a hypothesis, is something that you agreed to.
>So that part would be shared, that you agreed to.
Never agreed. You misinterpreted. I agreed to this: "we have actually learned something important from our thousand experiments". You learned that for 1000 experiments you observed something. That's it.
You're being pedantic. Obviously I agree with you something was learned. But I disagree with you that anything was proven or established by what was learned.
> Whether Science is a valid principle shared by humans and aliens is the point. My point is, PG's view isn't even shared with humans, why should he assume it's going to be shared with aliens?
What we're discussing is "the principle that a controlled experiment testing some hypothesis entitles us to have proportionally increased belief in it". Anyone that rejects this principle doesn't know if the sun will rise up tomorrow, is terrified that they may fall through the floor at any moment, or worse, drift off into the enormity of space.
What PG was essentially talking about was that all humans agree on the value of inductive reasoning ("experience") *, despite the philosophical problem of induction. This is far older than any notion of science, and is universal among not just humans, but also life forms on Earth in general (at least plants, fungi, and animals). I honestly very much doubt that it is even possible to function in the world, let alone to build interplanetary communication, if you reject inductive reasoning.
The fact that we can't reconcile inductive and deductive reasoning is a limitation of our philosophical/logical/mathematical systems, not some ultimate truth that invalidates the above principle.
Quite off-topic, but I will also note that I don't really like PG in general, and think much of his argumentation style is unpleasant and often makes undue assumptions. I just happen to strongly agree with (my interpretation of) this particular point.
* admittedly, he did go a step further by mentioning "controlled experiments", which induction doesn't rely on; but I don't think that really modifies the statement. As a toddler, when you place a cube on the ground, look away, and then look back, you're performing a controlled experiment to check if objects have permanence.
>The fact that we can't reconcile inductive and deductive reasoning is a limitation of our philosophical/logical/mathematical systems, not some ultimate truth that invalidates the above principle.
It's not reconciling those two systems. It's reconciling all of logic (induction and deduction) and reality itself. Logic, by logic itself is inapplicable to reality. We live in a universe of unknown domains and imprecise/inconsistent measurements. At any point in time we can make an observation that contradicts a previous observation. This makes proof impossible. While proof is impossible because of the possibility of a contradictory observation, falsification is very possible. The goal of science is falsification, not proof.
This is the logical conclusion of science and therefore reality. Logic, deduction and induction and proof are mostly the domain of mathematics or little axiomatic games we play where we artificially limit the domain. It's a Very very different domain from the one science operates in.
Most of humanity actually agrees deduction and induction are inapplicable to reality. Hence why science is, in the end, the most rigorous form of determining truth (despite the fact that it actually can't) instead of logic. This is in fact the conclusion reached ABOUT reality when we apply logic to it; that logic itself is inapplicable to reality as we know it.
When we check if a cube on the ground 100 times and see that it exists but we can't know what the next 10 billion observations will yield. Perhaps the 100 observations were biased, and the 10 billion subsequent observations yield that cube was a reflection, the toddler was mistaken and the situation did not exist long enough for the toddler to observe the cube past 100 observations.
> Anyone that rejects this principle doesn't know if the sun will rise up tomorrow, is terrified that they may fall through the floor at any moment, or worse, drift off into the enormity of space.
This is my issue with PG. If you look at science rigorously... we actually don't assume this is true. Science cannot verify whether the sun will rise tomorrow or whether or not we will or will not fall through the floor. That is science in a nut shell. PG is saying something WRONG about science and that Aliens will share a belief with us about it.
As for our day to day experiences, you're right. We all believe the sun will rise tomorrow, but this isn't science. This is simply bias, that all humans are born with. We ASSUME the sun will rise tomorrow, but there is no form of reasoning (scientific, deductive or inductive) that can lead us to that conclusion. PG was NOT talking about this. He was talking about Science and controlled experiments. Not shared assumptions about reality.
If PG said, "We assume that Aliens, like us, assume that when we're not looking the cube still exists even though we only observed it a couple of times." then I can probably get behind that, but it is an entirely different statement.
In addition to alien truth there are probably also alien games: ones so simple for their level of depth and enjoyment that you would expect them to be independently discovered. For example, Hex [1] is reasonably deep and has been invented at least twice. Go, with something like the Tromp-Taylor rules [2] might be as well? Probably not Chess, though!
> 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.
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.
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.
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.
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.
> 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.
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.
> 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.
> 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.
> 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:
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.
>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.
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.
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.
> 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".
> 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.
> 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.
>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.
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.)
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.
How is it irrelevant to this discussion? Parent proposed that aliens will have zero by posing that question. I gave an example from our own earth indicating intelligent life can manage without zero.
> I gave an example from our own earth indicating intelligent life can manage without zero.
Firstly, I disagree that humanity managed without zero. Literally everyone had an intuitive understanding of zero, they just didn't have it in their formal systems that were being studied by philosophers. For instance, try walking walking up to a vendor in Ancient Greece and just taking something without paying.
Secondly, it's largely irrelevant because a lot of math with zero can be mapped to math without zero with no loss of information, so even if aliens used math without zero there would be no trouble communicating as there would still be an understandable formal correspondence.
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
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.
> 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.
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.
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.
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.
> Consider an alien which subsists on photons, which is a form of life that exists today.
Plants don't just subsist on photons, there are many other ingredients.
> We know from heseinberg that the sensing of this food "here" or "there" is nonphysical.
I don't know what this means. How do you "non-physically" sense photons?
> 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.
Assuming you're talking about some alien made of bosons that aren't subject to the Pauli exclusion principle, you'll note that bosons still interact with fermions in which that principle does apply, so I don't think your argument follows. I admit I don't really understand your premises though so I have no idea what you really meant.
>What if aliens have no notion of discrete numbers, what if everything is probabilistic analog math?
0 and 1 are both valid probabilities.
>What about an organism that can see/focus/sense multiple things simultaneously, and a single "thing" is a set.
Its possible, using sets only containing other sets (or possibly the empty set), to construct the integers.
>There could be an entirely different paradigm to "counting" and consequently to the fundamentals of maths.
Systems of mathematical expressions are just like coding languages. The choice is arbitrary, one can always emulate the job of another. Just like how I did with your chosen examples, in principle one can always hack the integers out of whatever system you give me (or hack whatever system out if integers).
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.
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.
> 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.
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.
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.
> 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.
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.
> 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}}.
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.
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.
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?
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
I haven't read that book, but I did just read the wikipedia page for it, for what it's worth. Based on that I feel it's more that they're arguing that the path we've taken and the metaphors we've used while exploring our way to our current understanding of mathematics is based on our biology. Our biology has greatly affected the order we've discovered things and how we understand those thing and how we actually 'do' math day to day, and all that I agree with. It's also very likely that aliens will have taken a very different path and have very different metaphors and proofs for understanding and doing their version of what we call mathematics. Because of this it might very well be very difficult for us to initially understand each others mathematics.
Indeed looking through human history our philosophical understanding of mathematics fundamentally 'is' has changed many times. Yet mathematical truth's we've found along the way have always remained constant (barring errors in calculation or reasoning) even as our understanding of mathematics has changed.
I believe that once we gotten passed all that both us and the aliens will find that, at the core, we both agree on what is "mathematically" true.
I will however also concede that some of the arguments in this thread has made me slightly less sure than I was before, so that is something I guess.
> Mathematics exist independently of biology or perception of reality.
Even the concept of "number" itself is almost certainly an artifact of perception that enabled our ancestors to survive our niche, planetary environment, and not an inherent feature of objective reality.
Aliens, evolved to survive in another environment entirely with a different set of initial conditions, almost certainly would not have the same, nor even any, understanding of "number".
Would we consider such aliens a civilization or some kind of insensate "process"?
I'm so over this deconstructionist "reality is your perception" line.
Numbers are a thing. In fact they're one of the most basic observable things about the universe. And the concept if a number holds up all the way down to the quantum level. I.e. space and time are discrete and therefore space and time can both be COUNTED. Counting is literally one of the most basic and early achievements of human cognition and were gonna act like we just made it up?
> I'm so over this deconstructionist "reality is your perception" line.
What makes you completely reject that? I don't think it has as much to do with deconstructionism as it does embodied cognition. We keep learning more and more about how biology and physiology and evolutionary pressures affect and inform cognition and thus perception. I was recently reading about how there have been scientific studies that seem to suggest that certain animals seem to experience time differently that we do. So you could say "time is a thing", but yet, it appears that it is not the same thing across lifeforms. There are animals that sense gravitational and electromagnetic fields, something we cannot do. Would it make sense to them to say "all beings can read these fields because we do"?
I think the problem is that it is all too easy to fall into the trap in thinking that alien lifeforms would be like us. There's a multitude of evidence of that here on Earth in the variety of life, despite even coming from the same origin.
> There are animals that sense gravitational and electromagnetic fields, something we cannot do.
That actually proves the fact that reality is not simply someone's perception. (We humans do not perceive these fields, and so it took scientific advances for us to discover them as part of the objective reality.)
Yes, thank you. This line of reasoning is so painfully flawed and in a lot of cases outright dangerous or unhealthy.
There is an objective reality that exists beyond our perceptions. But our perceptions are based on that objective reality to some extent. We're not just making shit up. That doesn't make any sense.
> 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
> 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?
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.
> 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.
Where is this coming from? Is this relevant to what I wrote?
Well, sure, because the term "math" already presupposes a lot. There are two possibilities, here:
1) Math describes reality, so human math and alien math are the same thing, just different perspectives of reality and so, ultimately compatible.
2) Math is a language that humans use to describe their perceptions of reality to other humans. Perception is not reality, but a kind of isomorphism or pared-down heuristic, driven as it is by the evolutionary imperative to streamline for survival. Therefore a language that appears to be universal will only make sense to entities that employ the same perceptual framework for understanding, which is to say, human beings.
I agree that something that is "true" in human math will not be directly "false" in an alien math. 2+2 really does equal 4. The question is whether 2+2=4 is relevant or even understandable to an alien.
But see I don't think think what your saying contradicts the above post. If our subset intersects with the alien subset of perception, then we both will have a perception of an underlying truth, and we could conceivably understand the alien models and vice versa because they address the same subject.
UP said that mathematics is universal. I believe it models only our perception and understanding, which are informed by our biological, evolutionary imperatives.
Reality itself is beyond us. All that we have at our disposal are our perceptions, which is what we are modeling when we "do math". Mathematics is not universal, it's a language that we use to communicate what we perceive to other humans.
It's speculative, whether an alien species, with a perceptual and cognitive system evolved entirely elsewhere under other pressures, would have an understanding that intersects with our subset of reality. I personally think it's unlikely. What that would mean is that we wouldn't be able to communicate, never mind trade technology and mathematical ideas.
> It's speculative, whether an alien species, with a perceptual and cognitive system evolved entirely elsewhere under other pressures, would have an understanding that intersects with our subset of reality. I personally think it's unlikely. What that would mean is that we wouldn't be able to communicate, never mind trade technology and mathematical ideas.
I mean, maybe, but this is conveniently unprovable, much like the flying spaghetti monster, since you are saying we could not communicate because our slices of reality don't intersect. I disagree and think that most likely we would live in a reality that was largely the same, but I guess we'll never know!
The only thing that is "universal" is Nature itself. Mathematics, on the other hand, is a reflection of Nature in the human mind; or, put differently, it's Nature's language we humans are capable of understanding. It is therefore conceivable that other creatures, far removed from us, could "hear" a language that is just as far removed from ours.
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.
> 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.
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.
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.
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.
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.
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.
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.
> 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'...
> I wouldn't want to bet that all intelligent beings would understand the concept of justice, but I wouldn't want to bet against it either.
Given that even people (loosely) in the same culture often disagree about what constitutes "justice" and use the term in mutually exclusive ways, we should definitely bet against the proposition that "all intelligent beings" understand it.
For a person to have an opinion about what constitutes justice is for them to demonstrate an understanding of the concept of justice (assuming their opinion is cogent). So, if people are disagreeing about what precisely justice is, it actually means that they do understand the concept of justice.
Maybe for some highly abstract definition of justice. But for more everyday use, it's not hard to come up with examples that one society considers just while the other unjust.
Justice does not have to be cogent, which is defined as "(of an argument or case) clear, logical, and convincing.". There were and are justice systems that leave out one or more of these ingredients to some extent. Some leave out some logic by presuming the existence of a supernatural being. Some are more authoritarian and not very convincing.
* A society of one family isolated in nature, where each member is allowed to express their peculiarities and eccentricities, but never do each other any harm - not because of established rules, but because they truly love and care of each other.
I disagree. As long as I have a personal concept of 'good' and 'bad', and prefer it when 'good' things to happen to 'good' people (and vice versa) then I have a concept of justice.
Often, what is good for some is bad for others (and vice versa). Justice would be way too relative (subjective) outside law, so as to be devoid of any meaning, actually.
Laws are relative as well, they differ in every country. Why would justice only exist in another relative system? You know vigilante justice is a well accepted concept that exists outside of the law by definition.
Except 'vigilante justice' and 'justice' are concepts that have little to do with each other. You might as well be talking about the 'de facto law' (like for instance the "law" enforced by the local mafia) vs. the 'de jure law' here.
> Except 'vigilante justice' and 'justice' are concepts that have little to do with each other.
That depends on how well different people's definitions of justice line up. There are many things that vigilantes can enforce pretty well.
> You might as well be talking about the 'de facto law' (like for instance the "law" enforced by the local mafia) vs. the 'de jure law' here.
Sure, why not? Mafia law is often not justice, but I think it qualifies as law where sufficiently powerful. You seem to think this argument debunks itself?
Reputation and peer pressure are pretty effective at compelling people to act justly (according to the local consensus definition) even in the absence of a formal legal system.
If reputation and peer pressure scaled to group sizes bigger than Dunbar's number (i.e. about a hundred or so) then we probably wouldn't need laws at all.
I suppose you one might say that social expectations are just another kind of law, in which case, yeah it's hard to imagine any group of people without some kind of expectations of how each other will behave. That's kind of the basis of human relationships.
This is spot on. For concreteness, let me give a candidate definition for the virtue of justice:
Justice - rendering to each person what is owed to them.
It's obvious that we will very often disagree about 'what is owed', but doesn't our passionate disagreement in this case show that (1) we agree that practicing justice is good and (2) we are closely aligned on the existence of this thing called 'justice'?
Gotcha. I did preface this definition with the word 'candidate' and I acknowledge that there may be good alternate formulations. The spirit of this particular exchange is about whether or not 'justice' can be formulated as a universal.
I shared an argument above for why it can be viewed as a universal and judging by your comment above you are somewhat skeptical of this claim.
If we shift the discussion to allow conceptions of 'justice' that move away from the classical tradition and include modern ideas like 'climate' justice or 'social' justice, I will revert to agreeing with your skepticism.
I don't think anyone can plausibly claim that these more marxist-oriented modern definitions are universals.
I don't think the definition falls apart just because you might owe more than one person the same thing.
For instance, in climate justice we might say that we owe something to all of the people who are young now or haven't been born yet. In social justice you might owe something to a whole group of people. Just because they're groups doesn't mean the individual people that comprise the group dissolve into an abstract concept, even if that's how it might seem in our minds.
In some cases maybe the definition is too narrow because it depends on what is or isn't a "person" which we might define too narrowly. Do we owe things to animals? I'd say so, and I think most people would agree, depending on the animal. Do we owe things to plants? Maybe. Do we owe things to the planet Mars, assuming it's entirely devoid of life? I don't know, but I think it would offend people's sensibilities if we were to dump toxic waste all over it's surface, even if we were sure that humans are never going to settle there or use any of its resources. There's a view of climate justice that we owe things to the Earth directly rather than (or in addition to) owing something to the people that live there or will live there.
While I personally agree with the spirit of your definition (though I think it's incomplete) I don't think it's universal even amongst humans. You don't have to look far to see societies or people for whom "what is owed" isn't applicable. Eg. your conventional fascist society where justice means "what I have the power to do/take" (hence the Russia/Ukraine situation).
I agree with you so I was trying to think of some counter examples and the one that comes to mind would be species that have hive minds? Is justice as important if the many are considered part of the whole? If I drop a stone on my foot and lose my foot, justice is not involved
> we should definitely bet against the proposition that "all intelligent beings" understand it
Hey, we can always claim that those who disagree with us are not intelligent.
I'm only partially joking. Lots of today's "justices" have so many internal contradictions that I feel like we should separate them into their own category.
Tit-for-tat is a highly effective strategy when playing an iterated-prisoner's dilemma[0]... ie the "concept of justice" can emerge through natural selection if "intelligent beings" were forced to play such games on which their survival depends (a plausible model of "society").
I really liked this essay. It poses some interesting questions, but it's short and it doesn't try to do too much.
There is overlap between pg's ideas and what in the classical tradition is called Natural Law Theory. PG may or may not be interested in drawing out the connection, but since he references Aristotle I have to believe he is at least aware of a touch point.
To give a distilled definition, Natural Law Theory is the application of the laws of nature to rational creatures.
In the context in which NLT developed, the only free rational creature was the human being. But both AI development, and concepts from evolution through natural selection, potentially allow us to apply aspects of the theory to different rational agents.
> that it would be true for aliens that one can get better at something by practicing
I think that one depends a lot on how they learn or can transfer knowledge. We rely on language for it, and language is severely limited - We can't learn olympic gymnastics from watching TV, but a species that could directly transfer memories and behaviors would have a huge leg on us in that regard - I'd assume we'd quickly notice their fast-paced technological advance.
Or completely miss it, because they'd anyhilate themselves a couple hours after discovering the military use of nuclear fusion ;-)
If Occam's razor is presented as a truth (which is pretty subjective, as it depends on what we/someone considers "simple"). Then finding one case where what most people thought was the simple reason, was not actually the reason after evidence came to light, then Occam's razor can be rejected as a theory right?
I think this has long happened, and do not understand why this is still presented as "truth".
I prefer the standard of truth used by natural-sciences. The rest I find pretty bendable (virology included).
> If Occam's razor is presented as a truth (which is pretty subjective, as it depends on what we/someone considers "simple").
It doesn't though. Firstly, Occam's razor is not about simplicity but about "parsimony". Parsimony is calculated in information theory via Kolmogorov complexity:
> It doesn't though. Firstly, Occam's razor is not about simplicity but about "parsimony". Parsimony is calculated in information theory via Kolmogorov complexity
In theory perhaps, but in practice it is firstly calculated by the mind of the person who plays the Occam's Razor card, and then subsequently by people who ingest the claim, and typically all participants are performing their calculations using biased heuristics and flawed logic, and have negative interest in what is actually true, or if truth is even reachable.
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[ 5.4 ms ] story [ 316 ms ] threadI am reading the sci-fi book The Sea of Rust right now, that takes place on earth after AI’s have finally killed the last humans. In this fictional work, AIs take on human traits and I don’t find that “believable” even for sci-fi.
For example, even if silicon-based AI replaced humans on earth, it would eventually [0] need to find a way to power itself / continue itself. Ultimately, it would have to revert to solving the same "problem of life"... how do we transform energy/entropy available in the environment into something that "perpetuates the system." When that happens, this AI will itself become subject to the forces of natural-selection, and - over a long enough period of time - naturally-selected traits will be re-aquired (even if such traits were "lost" during a human-to-AI hand-over).
[0] Yes, of course, there could be a very large period of time during which currently constructed energy infrastructure continues on... and this period could be measured in hundreds/thousands of years... very long in terms of human lifespans, but not in geological terms.
The question is whether we can communicate our mathematics to non-humans, whose senses may be radically different from our own.
And likewise, would we be able to understand a non-human proof?
I'm a puzzled by the confidence here. I would assume pg is at least minimally familiar with some of the key philosophical themes and schools of thought around this topic.
> For intuitionistic mathematics ... truth is a mental construct of a proof in a language, and only shared by communicating that construct to other minds.
Even assuming you are fully and accurately representing the intuitionist view, you must be aware that there are competing schools of thought with strong pedigrees, like mathematical platonism, that are grounded in a more realist view of mathematical objects.
PG didn't go so far as to stake out that position here in the essay, but his thought experiment leverages a view of mathematical truth that hues closer to this (platonist) camp.
> The question is whether we can communicate our mathematics to non-humans ...
That is an interesting question, but pg did not ask that question in the essay, and its answer doesn't seem relevant to the point he was trying to make.
I think pg is educated, at least the broad outline, of the age-old controversies in this topic and I'm grateful as a reader that he spares me the details and assumes some preliminary context.
His style of pop philosophy is useful in startups because it does not require context. It is ungrounded in actual philosophy, and would struggle to find an audience outside of this group.
But it is not straightforward to me. Or, in my impression, most philosophers. I'm reluctant to agree that aliens will assign any importance or even understand concepts like geometry or calculus. So to me, it is not preliminary context but it is a critical part of the discussion.
But even a brief foray into discussions by actual philosophers will show that to not be true. What Paul Graham is doing is providing small subcultural insights tailored to an audience with a passing interest in philosophy.
I think the word in that idiom is 'hews', which means to adhere strictly to a standard, probably from the sense of the word meaning to strike or cut or beat - often used to talk about cutting a tree into shape.
I'm not sure why people find this so relevant. Regardless of the sense, there must be enough discernible structure to detect whether 0 of something is there, whether 1 of something is there, or whether there are many more than 1.
Human sense of smell might top out at differentiating maybe 4 or 5 different things, dogs can probably sense a lot more, but either way it lets us set up basic counting, and that's generally all you need for most of our math.
How can we communicate actual axioms to extra-terrestrials?
Presumably we can create a kind of Rosetta Stone with different representations of logical and algebraic expressions, and maybe they can decode that if they can figure out how to decode whatever "broadcast" representation of this we come up with. And that assumes that they recognize such a broadcast as something from intelligent life.
peano arithmetic (an axiomatiazation of counting) isn't how we count. It is how we make other things count for us.
We're not intelligent enough yet not to harm others, or to go to other star systems, or even other galaxies. Would we be considered intelligent, or a source of protein?
If so, then he understands basic calculus.
He can't write the equations, but he can solve them.
To a more advanced civilization, we are chimpanzees who are both outwardly intelligent, but also tremendously dangerous, and so on what basis could they establish trust with us, or could we establish trust with a civilization of others? As Graham notes, math is one indicator that we are capable of apprehending the universe around us, but given the infinity of life and its necessary physical conditions of beginning and ending, and evolving in aggregate using tools and principles, it's not sufficient. Maybe one way to ensure trust is to share DNA, so that we become each other and we are all "us" - or, perhaps the Girardian mimetic concept generalizes such that it is better to preserve our differences so that we are not competitors for the same resources, and so that we can co-exist with an obvious other but without an existential threat or intrinsic power struggles.
Are there existing moral or philosophical systems that are suited to this problem? Probably, I'm not a religious scholar, but the golden thread that links them seems pretty consistent in attempting to derive alignment to an external truth. The proto-Christian tribe of Essenes, from whom John the Baptist originates and who was the one who baptized Jesus into what became Christianity (solving a weird bootstrapping problem, imo) espoused the values that became the first Church, so there is a historiographical way of looking at moral systems instead of as dogma. Outside religion, in the search for these values that would be suitable for a community of inhabitants, I've come to suspect this is what freemasonry is about, and while not about aliens, I was impressed by their allegorical emphasis on tools instead of doctrine as the landmarks for discovery.
The essential question to me is, once you have accepted there is an other that is greater, or a place that is elsewhere, does it matter whether it's a dude with a beard, multi-armed flying blue people, or an ineffable oneness? That there is a concievable elsewhere beyond your current limits, there must therefore be some point or idea to align and orient yourself to so as to be able to relate to the other beings who have discovered the same point outside our current perspective.
It's all very meta, but it implies a logical and even rational case for some guidance or alignment to this otherness to navigate our present, and that isn't material. The value of the idea of an "alien" truth is it is a means to reconcile secular rational thinking and moralism with universal, essential, or spiritual values, and that could be a very useful tool.
Fundamentally your message to humanity post-enlightenment would be the rules on how to get to heaven. Which many world-religions classes go into depth. There are fundamental rules that benefit everyone to follow that wouldn't really be inherently human to follow.
>We'd probably share Occam's razor. There doesn't seem anything specifically human about any of these ideas.
Aliens will also have developed the piano and chess. They are inherent things to discover eventually.
Fundmantally a great way to analyze what the rules are would be impossible to list. Just look at the list of crimes in countries which are so large lawyers dont even know them all. So you need a system that's much more simply. Isn't that system 'karma'.
How could that claim be true? We have highly intelligent beings (i.e., “aliens”) right here on Earth that have not developed these things.
These discussions on aliens are often off the rails from the start because they implicitly begin with the assumption that humans are the only intelligent beings on Earth.
Are you using the 'illegal immigration' definition of alien?
>These discussions on aliens are often off the rails from the start because they implicitly begin with the assumption that humans are the only intelligent beings on Earth.
Do please elaborate because I don't share this opinion. Do you believe aliens live amongst us?
Alright, agreed. Which as far as I know we have no known aliens ever discovered.
>We have biological beings on Earth that share DNA with us but possess wildly different intelligences and cognitive systems. Is it a stretch to use these as examples that aliens may share little in common with us?
You're backpedaling pretty hard. You said there are 'highly intelligent beings on earth' besides us. I know of no known examples that fit your claim. Happy to listen.
And all this relies on some definition of intelligence, which I don’t think we even have a good one for.
Math is a game we play in our heads that represents a fictionalized ideal version of reality.
An alien intelligence might have realized that two plus two never equals four not because the underlying logic is wrong, but because two does not exist in reality.
The idea that the little game of math we play represents an immutable and universal truth is typical of the overwhelming anthropocentrism of our kind.
Just because some alien societies will not mimic our rules of addition, we do know for certain it is possible that other societies can build abstract concepts that are isomorphic to those we have. And many of these concepts, such as addition, are very useful.
Does this guarantee that aliens come up with the same stuff? No. Does it guarantee that if they did, they would these concepts to the same esteem? No. Is there an element of 'truth' here that can be replicated by others? Absolutely
For maths, I would say 1+1=2 is a pretty universal truth (although it takes a while to get there in the principa mathematica), but didn't we just invent complex numbers because they are useful?
Same goes for physics, the speed of light is the same everywhere, but how quantum mechanics work is still subject to many discussions.
Love to hear some thoughts on this, as claiming a whole field as universal truth is something I'm a little uncomfortable with.
But the usefulness is objective, that is, it is not an arbitrary product of the mind but rather it is dictated by the logic of things once the goal is set, so invention (or discovery) of useful things is more or less unavoidable.
As to quantum mechanics, you are talking about the variety of interpretations which from the practical standpoint are simply different ways of looking at quantum behavior, which, in turn, sometimes leads to different methods of calculation.
It certainly seems like it is objective, and often it probably is, but in a more general sense, any instance of "x 'is' y" very often turns out to be subjective very quickly. Even with "is useful", things get complicated if one explicitly injects the dimension of Time into the question (it is there in the first place implicitly, but is easily overlooked).
Not any more than we invented natural numbers because they are useful.
There are several ways to naturally derive complex numbers, either from mathematics or from physics.
For one, complex numbers are probably the simplest possible extension of the real numbers in which all real-valued polynomials have roots (for example, x^2 + 1 doesn't have a root if x has to be real). This is the same reason why the non-transcendental irrational numbers were invented (such as sqrt(2) ).
(Incidentally, the transcendental numbers (pi, e) are less justified than the complex numbers from this point of view - any polynomial of any rank whose coefficients are non-transcendental real numbers has roots that are either real non-transcendental numbers, or a complex number whose real and imaginary parts are real non-transcendental numbers )
For a physical explanation, complex numbers are the best way we know of describing wave mechanics (either classical or quantum), and in general periodic phenomena and how they compose.
This isn't even a shared principle among humans. How many experiments does it take for you to have 50% belief in a hypothesis?. What is the number of experiments? It's literally impossible to answer. It's not even clear what "belief" is or what 50% means.
This ambiguity of the word isn't even the main problem. If I run the same experiment with perfect observational tools 10 billion times and it verifies my hypothesis. Does that raise my belief further? What if on the 10 billionth and first time the test shows a negative result? That literally invalidates the hypothesis. Keep in mind we are assuming my observational tools are perfect. Does this make my belief shoot down to zero?
If this possibility of a negative result remains true after any number of tests then what does it say about belief? Why should I believe anything if a single negative experiment can invalidate 10 billion positive experiments (assuming perfect observational tools of course)?
Let me bring a more concrete example. I hypothesize all zebras have stripes. I observe zebras 10 billion times. They all confirm my hypothesis. Then on the 10 billionth and first time I see a zebra with spots. My hypothesis is wrong. This can happen any time.
Anyway to bring it back to his point. Don't assume shared axiomatic truths. PG already assumed that it's shared among humans. He's wrong. The nature of science and the scientific method is not universally shared or even fully understood among humans. He's likely also wrong about aliens as he is about humans.
https://en.wikipedia.org/wiki/Solomonoff%27s_theory_of_induc...
So why would aliens hold this "principle" the same if humans don't even agree on it? PG is wrong. His own principles upended not even by aliens, but by humanity, thus how accurate can his assumptions about universal principles even be? No that accurate imho.
There is no divide, there is the illusion of divide because we didn't have a rigourous formal model of how to build reliable knowledge and everyone focused on different but relevant aspects.
Bayesian reasoning is the correct way if you have justifiable priors, but we didn't have a way to calculate the correct prior.
Solomonoff showed us how with his theory: Kolmogorov complexity is a measure of parsimony, and this is how to select priors in a formal, rigourous way.
Solomonoff induction is to knowledge what Turing machines or the lambda calculus are to computation. Sure, aliens might not discover Turing machines exactly, or the lambda calculus exactly, but whatever they do build that's capable of universal computation, we already know it must be isomorphic to a Turing machine, because all constructions capable of computation must be by necessity.
The frequentist/Bayesian divide is a separate issue about how to interpret statistical data in useful ways, not specifically about how we know what we know and what confidence we should have in our knowledge, which is what you were asking about.
https://www.lesswrong.com/posts/Kyc5dFDzBg4WccrbK/an-intuiti...
Even if the original belief turns out to be wrong, you only have to slightly weaken it and it will remain true: "the vast majority of zebras have stripes". Even if you discover a new continent full of hordes of uniformly-colored zebras, the true hypothesis becomes "the vast majority of zebras in my original continent are striped".
Essentially every observation brings proof for a whole family of hypotheses. We normally only talk about the strongest of these hypotheses, but that doesn't meant that a negative example rules out the entire family.
For example, even if we didn't find a deductive proof the Fermat's last theorem was wrong even after all of the empirical proof that it probably wasn't, a weaker version would have still remained true - the one validated by that empirical proof.
The hypothesis does not remain true. It was never proven to be true and the new hypothesis is still not proven to be true. Science cannot prove anything to be true. I can find a cave full of of spotted zebras, and you have to further weaken your hypothesis of continents, I can then find that the stripes were actually microscopic spots and my perfect observation tool, though never wrong has limited resolution. Ad infinitum. Nobody ever considers your made up philosophy because it's changing the rules of the game. It's making a statement then adjusting your statement once it's proven wrong... people look down on that kind of thing.
What I'm writing here isn't something I pulled out of my ass. It's well known that in science, the scientific method, and reality itself, nothing can be proven. Proof is the domain of math and logic, not science. In science, things can only be falsified. To quote Einstein:
“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”
Einstein obviously isn't saying stuff like a single experiment causes me to adjust my hypothesis and divide split it into two different ones because it's kind of inconsistent.
There are people who truly understand science, but most of the population doesn't (including PG). I think what's going on with you is you're in the later camp, you've long held the incorrect belief that science can prove things and this long held ideology is coming into contact with the actual logic of the situation and your adjusting your belief to maintain a biased ideology.
Do you look up to PG? Bias can be corrected when an authority confirms the opposite. I quoted Einstein here. One of the ultimate authorities on science, a person who overturned the hypothesis about Newtonian physics being a model for motion. A single experiment proved it wrong and now Newtonian physics is simply an approximation that is ultimately wrong. Hopefully that will clear things up, if not... then you must be an Alien far more strange then what PG is describing.
That is the point that I am trying to make: experimentation can bring proof to strengthen a hypothesis. Even if a later experiment invalidates a hypothesis, all of the previous experiments' results don't disappear, and any new hypothesis we formulate still needs to be coherent with them to have any value: we have actually learned something important from our thousand experiments, even if our 1001st showed that the hypothesis we had in mind was false.
Also, this is not unique to science. The same phenomenon can happen in mathematics or logic for theorems that have been neither proven nor disproven yet. We can perform numerical experiments to test a numerical theorem, and gain some amount of confidence in that theorem even if we haven't proven it to be true. We can often establish lower or upper bounds in the course of this experimentation, where we find that the theorem is True at least for some limited subset of all numbers - and this remains True and useful even if it later turns out that there exist counter-examples.
This observation is also very important for understanding why the history of natural philosophy is essentially one of continuous progress, with very little backtracking: even if induction is not good enough to know that we have a perfectly complete and consistent theory (and we will never have one), we always have something salvageable from all of the experimentation done so far. Even geocentric models with their epicycles were actually working models, which predicted the positions of planets in the next 1000 years to quite good accuracy, even if they were clearly wrong in the end.
Please don't accuse me of acting condescending. It's very offensive and hurts my feelings when I'm accused of something I'm not doing.
I am criticizing you, but I am not being condescending. There is a huge difference.
Perhaps the alien thing was bad. I apologize for that. The intent was a joke and was not condescension.
>That is the point that I am trying to make: experimentation can bring proof to strengthen a hypothesis. Even if a later experiment invalidates a hypothesis, all of the previous experiments' results don't disappear, and any new hypothesis we formulate still needs to be coherent with them to have any value: we have actually learned something important from our thousand experiments, even if our 1001st showed that the hypothesis we had in mind was false.
Yes but this was not part of the discussion. We're talking about science as a principle. Not what we have learned from the process of science.
>This observation is also very important for understanding why the history of natural philosophy is essentially one of continuous progress, with very little backtracking: even if induction is not good enough to know that we have a perfectly complete and consistent theory (and we will never have one), we always have something salvageable from all of the experimentation done so far. Even geocentric models with their epicycles were actually working models, which predicted the positions of planets in the next 1000 years to quite good accuracy, even if they were clearly wrong in the end.
Important or not, we diverged from the point. Whether Science is a valid principle shared by humans and aliens is the point. My point is, PG's view isn't even shared with humans, why should he assume it's going to be shared with aliens?
You're talking about the importance of science. The value of science. That's off topic.
So then you agree that "experimentation can bring proof to strengthen a hypothesis"
> Whether Science is a valid principle shared by humans and aliens
Well, the experimentation part, that can bring proof to strength a hypothesis, is something that you agreed to.
So that part would be shared, that you agreed to.
>Well, the experimentation part, that can bring proof to strength a hypothesis, is something that you agreed to. >So that part would be shared, that you agreed to.
Never agreed. You misinterpreted. I agreed to this: "we have actually learned something important from our thousand experiments". You learned that for 1000 experiments you observed something. That's it.
Ok great, so then you agree that this is a principle that would be shared, which is the point.
So then yes, that is an agreement that at least to that statement.
> But established nothing.
Well it established that at least you agree with that statement.
> Because I disagree with you.
actually you said this "I agreed to this:"
This is you saying that you agreed with the statement that followed that. So yes, you used the word agree.
Ok, you agree with this then.
Yes, that is my point. This is what you agreed with.
You are agreeing that at least something has been learned.
That is a valuable point to make, in and of itself, that at least something has been learned.
Valuable,.depends on your opinion, useless to proving a point yes.
What we're discussing is "the principle that a controlled experiment testing some hypothesis entitles us to have proportionally increased belief in it". Anyone that rejects this principle doesn't know if the sun will rise up tomorrow, is terrified that they may fall through the floor at any moment, or worse, drift off into the enormity of space.
What PG was essentially talking about was that all humans agree on the value of inductive reasoning ("experience") *, despite the philosophical problem of induction. This is far older than any notion of science, and is universal among not just humans, but also life forms on Earth in general (at least plants, fungi, and animals). I honestly very much doubt that it is even possible to function in the world, let alone to build interplanetary communication, if you reject inductive reasoning.
The fact that we can't reconcile inductive and deductive reasoning is a limitation of our philosophical/logical/mathematical systems, not some ultimate truth that invalidates the above principle.
Quite off-topic, but I will also note that I don't really like PG in general, and think much of his argumentation style is unpleasant and often makes undue assumptions. I just happen to strongly agree with (my interpretation of) this particular point.
* admittedly, he did go a step further by mentioning "controlled experiments", which induction doesn't rely on; but I don't think that really modifies the statement. As a toddler, when you place a cube on the ground, look away, and then look back, you're performing a controlled experiment to check if objects have permanence.
It's not reconciling those two systems. It's reconciling all of logic (induction and deduction) and reality itself. Logic, by logic itself is inapplicable to reality. We live in a universe of unknown domains and imprecise/inconsistent measurements. At any point in time we can make an observation that contradicts a previous observation. This makes proof impossible. While proof is impossible because of the possibility of a contradictory observation, falsification is very possible. The goal of science is falsification, not proof.
This is the logical conclusion of science and therefore reality. Logic, deduction and induction and proof are mostly the domain of mathematics or little axiomatic games we play where we artificially limit the domain. It's a Very very different domain from the one science operates in.
Most of humanity actually agrees deduction and induction are inapplicable to reality. Hence why science is, in the end, the most rigorous form of determining truth (despite the fact that it actually can't) instead of logic. This is in fact the conclusion reached ABOUT reality when we apply logic to it; that logic itself is inapplicable to reality as we know it.
When we check if a cube on the ground 100 times and see that it exists but we can't know what the next 10 billion observations will yield. Perhaps the 100 observations were biased, and the 10 billion subsequent observations yield that cube was a reflection, the toddler was mistaken and the situation did not exist long enough for the toddler to observe the cube past 100 observations.
> Anyone that rejects this principle doesn't know if the sun will rise up tomorrow, is terrified that they may fall through the floor at any moment, or worse, drift off into the enormity of space.
This is my issue with PG. If you look at science rigorously... we actually don't assume this is true. Science cannot verify whether the sun will rise tomorrow or whether or not we will or will not fall through the floor. That is science in a nut shell. PG is saying something WRONG about science and that Aliens will share a belief with us about it.
As for our day to day experiences, you're right. We all believe the sun will rise tomorrow, but this isn't science. This is simply bias, that all humans are born with. We ASSUME the sun will rise tomorrow, but there is no form of reasoning (scientific, deductive or inductive) that can lead us to that conclusion. PG was NOT talking about this. He was talking about Science and controlled experiments. Not shared assumptions about reality.
If PG said, "We assume that Aliens, like us, assume that when we're not looking the cube still exists even though we only observed it a couple of times." then I can probably get behind that, but it is an entirely different statement.
[1] https://en.wikipedia.org/wiki/Hex_(board_game)
[2] https://senseis.xmp.net/?TrompTaylorRules
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.
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.
https://blogs.scientificamerican.com/cross-check/was-philoso...
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-scie...
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.
https://cosmosmagazine.com/technology/artificial-intelligenc...
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.
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.
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.
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”.
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.
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_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.
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.
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.
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".
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.
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.
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.
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.
Firstly, I disagree that humanity managed without zero. Literally everyone had an intuitive understanding of zero, they just didn't have it in their formal systems that were being studied by philosophers. For instance, try walking walking up to a vendor in Ancient Greece and just taking something without paying.
Secondly, it's largely irrelevant because a lot of math with zero can be mapped to math without zero with no loss of information, so even if aliens used math without zero there would be no trouble communicating as there would still be an understandable formal correspondence.
Sorry. That's the one. That's the one that broke me. Jeremy Bearimy
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.
And you believe that aliens that can count is something that is "arguable".
You're the dot.
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.
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.
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.
Plants don't just subsist on photons, there are many other ingredients.
> We know from heseinberg that the sensing of this food "here" or "there" is nonphysical.
I don't know what this means. How do you "non-physically" sense photons?
> 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.
Assuming you're talking about some alien made of bosons that aren't subject to the Pauli exclusion principle, you'll note that bosons still interact with fermions in which that principle does apply, so I don't think your argument follows. I admit I don't really understand your premises though so I have no idea what you really meant.
0 and 1 are both valid probabilities.
>What about an organism that can see/focus/sense multiple things simultaneously, and a single "thing" is a set.
Its possible, using sets only containing other sets (or possibly the empty set), to construct the integers.
>There could be an entirely different paradigm to "counting" and consequently to the fundamentals of maths.
Systems of mathematical expressions are just like coding languages. The choice is arbitrary, one can always emulate the job of another. Just like how I did with your chosen examples, in principle one can always hack the integers out of whatever system you give me (or hack whatever system out if integers).
you said every thing which implies discreteness. Also, probabilities are probabilities of an event (read: something discrete) happening
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.
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.
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.
That's a strong statement. I'd probably talk about homomorphism.
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.
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.
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-Euclid...
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.
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.
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?
I haven't read that book, but I did just read the wikipedia page for it, for what it's worth. Based on that I feel it's more that they're arguing that the path we've taken and the metaphors we've used while exploring our way to our current understanding of mathematics is based on our biology. Our biology has greatly affected the order we've discovered things and how we understand those thing and how we actually 'do' math day to day, and all that I agree with. It's also very likely that aliens will have taken a very different path and have very different metaphors and proofs for understanding and doing their version of what we call mathematics. Because of this it might very well be very difficult for us to initially understand each others mathematics.
Indeed looking through human history our philosophical understanding of mathematics fundamentally 'is' has changed many times. Yet mathematical truth's we've found along the way have always remained constant (barring errors in calculation or reasoning) even as our understanding of mathematics has changed.
I believe that once we gotten passed all that both us and the aliens will find that, at the core, we both agree on what is "mathematically" true.
I will however also concede that some of the arguments in this thread has made me slightly less sure than I was before, so that is something I guess.
Even the concept of "number" itself is almost certainly an artifact of perception that enabled our ancestors to survive our niche, planetary environment, and not an inherent feature of objective reality.
Aliens, evolved to survive in another environment entirely with a different set of initial conditions, almost certainly would not have the same, nor even any, understanding of "number".
Would we consider such aliens a civilization or some kind of insensate "process"?
I'm so over this deconstructionist "reality is your perception" line.
Numbers are a thing. In fact they're one of the most basic observable things about the universe. And the concept if a number holds up all the way down to the quantum level. I.e. space and time are discrete and therefore space and time can both be COUNTED. Counting is literally one of the most basic and early achievements of human cognition and were gonna act like we just made it up?
Absolute silliness.
What makes you completely reject that? I don't think it has as much to do with deconstructionism as it does embodied cognition. We keep learning more and more about how biology and physiology and evolutionary pressures affect and inform cognition and thus perception. I was recently reading about how there have been scientific studies that seem to suggest that certain animals seem to experience time differently that we do. So you could say "time is a thing", but yet, it appears that it is not the same thing across lifeforms. There are animals that sense gravitational and electromagnetic fields, something we cannot do. Would it make sense to them to say "all beings can read these fields because we do"?
I think the problem is that it is all too easy to fall into the trap in thinking that alien lifeforms would be like us. There's a multitude of evidence of that here on Earth in the variety of life, despite even coming from the same origin.
That actually proves the fact that reality is not simply someone's perception. (We humans do not perceive these fields, and so it took scientific advances for us to discover them as part of the objective reality.)
There is an objective reality that exists beyond our perceptions. But our perceptions are based on that objective reality to some extent. We're not just making shit up. That doesn't make any sense.
You imagined what I meant so vividly that you literally made up a quote!
Try responding to what I actually wrote, and if it's unclear, asking with some humility.
I'm not even sure what your objection is, exactly. Nothing you wrote contradicts what I wrote.
Kant would have a field day with this statement
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
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?
Where is this coming from? Is this relevant to what I wrote?
1) Math describes reality, so human math and alien math are the same thing, just different perspectives of reality and so, ultimately compatible.
2) Math is a language that humans use to describe their perceptions of reality to other humans. Perception is not reality, but a kind of isomorphism or pared-down heuristic, driven as it is by the evolutionary imperative to streamline for survival. Therefore a language that appears to be universal will only make sense to entities that employ the same perceptual framework for understanding, which is to say, human beings.
I agree that something that is "true" in human math will not be directly "false" in an alien math. 2+2 really does equal 4. The question is whether 2+2=4 is relevant or even understandable to an alien.
Reality itself is beyond us. All that we have at our disposal are our perceptions, which is what we are modeling when we "do math". Mathematics is not universal, it's a language that we use to communicate what we perceive to other humans.
It's speculative, whether an alien species, with a perceptual and cognitive system evolved entirely elsewhere under other pressures, would have an understanding that intersects with our subset of reality. I personally think it's unlikely. What that would mean is that we wouldn't be able to communicate, never mind trade technology and mathematical ideas.
I mean, maybe, but this is conveniently unprovable, much like the flying spaghetti monster, since you are saying we could not communicate because our slices of reality don't intersect. I disagree and think that most likely we would live in a reality that was largely the same, but I guess we'll never know!
Perhaps. I was responding to the assertion that Mathematics exist independently of biology or perception of reality which is at least as unprovable
The only thing that is "universal" is Nature itself. Mathematics, on the other hand, is a reflection of Nature in the human mind; or, put differently, it's Nature's language we humans are capable of understanding. It is therefore conceivable that other creatures, far removed from us, could "hear" a language that is just as far removed from ours.
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.
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.
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.
https://cosmosmagazine.com/technology/artificial-intelligenc...
Different truths that describe the same system.
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.
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.
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'...
More at: https://corinth.kardianos.com/
Given that even people (loosely) in the same culture often disagree about what constitutes "justice" and use the term in mutually exclusive ways, we should definitely bet against the proposition that "all intelligent beings" understand it.
Justice does not have to be cogent, which is defined as "(of an argument or case) clear, logical, and convincing.". There were and are justice systems that leave out one or more of these ingredients to some extent. Some leave out some logic by presuming the existence of a supernatural being. Some are more authoritarian and not very convincing.
What do you mean by that?
* A society of one.
* A society of one family isolated in nature, where each member is allowed to express their peculiarities and eccentricities, but never do each other any harm - not because of established rules, but because they truly love and care of each other.
The second example is virtually unreal (and even expressions of 'love' and 'care' can be harmful).
So does the definition of justice.
> vigilante justice
Except 'vigilante justice' and 'justice' are concepts that have little to do with each other. You might as well be talking about the 'de facto law' (like for instance the "law" enforced by the local mafia) vs. the 'de jure law' here.
That depends on how well different people's definitions of justice line up. There are many things that vigilantes can enforce pretty well.
> You might as well be talking about the 'de facto law' (like for instance the "law" enforced by the local mafia) vs. the 'de jure law' here.
Sure, why not? Mafia law is often not justice, but I think it qualifies as law where sufficiently powerful. You seem to think this argument debunks itself?
If reputation and peer pressure scaled to group sizes bigger than Dunbar's number (i.e. about a hundred or so) then we probably wouldn't need laws at all.
I suppose you one might say that social expectations are just another kind of law, in which case, yeah it's hard to imagine any group of people without some kind of expectations of how each other will behave. That's kind of the basis of human relationships.
Justice - rendering to each person what is owed to them.
It's obvious that we will very often disagree about 'what is owed', but doesn't our passionate disagreement in this case show that (1) we agree that practicing justice is good and (2) we are closely aligned on the existence of this thing called 'justice'?
I shared an argument above for why it can be viewed as a universal and judging by your comment above you are somewhat skeptical of this claim.
If we shift the discussion to allow conceptions of 'justice' that move away from the classical tradition and include modern ideas like 'climate' justice or 'social' justice, I will revert to agreeing with your skepticism.
I don't think anyone can plausibly claim that these more marxist-oriented modern definitions are universals.
For instance, in climate justice we might say that we owe something to all of the people who are young now or haven't been born yet. In social justice you might owe something to a whole group of people. Just because they're groups doesn't mean the individual people that comprise the group dissolve into an abstract concept, even if that's how it might seem in our minds.
In some cases maybe the definition is too narrow because it depends on what is or isn't a "person" which we might define too narrowly. Do we owe things to animals? I'd say so, and I think most people would agree, depending on the animal. Do we owe things to plants? Maybe. Do we owe things to the planet Mars, assuming it's entirely devoid of life? I don't know, but I think it would offend people's sensibilities if we were to dump toxic waste all over it's surface, even if we were sure that humans are never going to settle there or use any of its resources. There's a view of climate justice that we owe things to the Earth directly rather than (or in addition to) owing something to the people that live there or will live there.
This is extremely subjective.
Hey, we can always claim that those who disagree with us are not intelligent.
I'm only partially joking. Lots of today's "justices" have so many internal contradictions that I feel like we should separate them into their own category.
[0] https://en.wikipedia.org/wiki/Prisoner%27s_dilemma#The_itera...
There is overlap between pg's ideas and what in the classical tradition is called Natural Law Theory. PG may or may not be interested in drawing out the connection, but since he references Aristotle I have to believe he is at least aware of a touch point.
To give a distilled definition, Natural Law Theory is the application of the laws of nature to rational creatures.
In the context in which NLT developed, the only free rational creature was the human being. But both AI development, and concepts from evolution through natural selection, potentially allow us to apply aspects of the theory to different rational agents.
I think that one depends a lot on how they learn or can transfer knowledge. We rely on language for it, and language is severely limited - We can't learn olympic gymnastics from watching TV, but a species that could directly transfer memories and behaviors would have a huge leg on us in that regard - I'd assume we'd quickly notice their fast-paced technological advance.
Or completely miss it, because they'd anyhilate themselves a couple hours after discovering the military use of nuclear fusion ;-)
I think this has long happened, and do not understand why this is still presented as "truth".
I prefer the standard of truth used by natural-sciences. The rest I find pretty bendable (virology included).
It doesn't though. Firstly, Occam's razor is not about simplicity but about "parsimony". Parsimony is calculated in information theory via Kolmogorov complexity:
https://en.wikipedia.org/wiki/Kolmogorov_complexity
This then lets us describe how to do induction so it provably converges in the fastest way possible, Solomonoff Induction:
https://en.wikipedia.org/wiki/Solomonoff%27s_theory_of_induc...
Arguably, this can be seen as a formalization of the scientific process.
In theory perhaps, but in practice it is firstly calculated by the mind of the person who plays the Occam's Razor card, and then subsequently by people who ingest the claim, and typically all participants are performing their calculations using biased heuristics and flawed logic, and have negative interest in what is actually true, or if truth is even reachable.