It's turtles all the way down -- we don't know what "truth" is, or how one can come to "know" truth (as opposed to just believe it). I'm less informed on these topics so I avoided them, but I've heard that our notion of truth is intimately tied to our experience of physical reality, and it seems easy to imagine aliens that have different experiences of reality.

For example, what if aliens are so large that the non-euclidian nature of space becomes obvious to them? Perhaps they would never come up with euclidian geometry. Maybe they would never create the notion of pi. Yes it shows up in other areas but you can certainly imagine an alien civilization that has a different circumference-to-radius ratio or disavows the entire idea of having a constant circumference-to-radius ratio.

We only need to evaluate what is practically useful. At the end of the day that's all science does - it's a methodology for generating guidelines for what actually works in the real world and for which accuracy, or truth, can be verified. We call those guidelines scientific 'laws', but truth in the absolute sense is less critical than is often made out by both proponents and critics of science.

Newton's laws of motion strictly speaking are not true, Einstein proved this, but they are incredibly useful as is the mathematics we use to formalise them. We know that the quantum mechanics and relativity theories we have now and not complete, but that's beside the point.

PG is simply pointing out that since aliens, if they exist, live in the same real world we do they will discover a lot of the same practical results we do, and then goes further to suggest they might agree with us about a lot of the less strictly provable ones as well. Going "Oh well but this obscure question here would still be in contention" is, well, somewhat sailing past the point by a little smidgenette. It's criticising a claim I don't think it's reasonable to think PG is making.

Agree to disagree, I suppose. Another comment in this thread suggested using the value of pi to demonstrate intelligence, but you can imagine alien civilizations without the same reverence of pi as we do (if they are large enough). I mentioned in another comment that you can also imagine small enough aliens not inventing a real number line because they do not have the experience that everything is infinitely divisible.

I believe our difference is that you are presupposing that aliens will have essentially the same experience with reality that we do. I agree that if somehow a separate human civilization evolved in another galaxy, their math would likely look similar to ours. But you can imagine aliens that have very different experiences of "what actually works in the real world" (due to experiencing a different set of physics, such as at a different physical scale) and thus come up with different math.

We already have very robust descriptions of how the world works at every scale, from the planck length all the way up to the limits of the observable universe, and in extremes of environment from the surface of neutron stars, to the roiling virtual particles of empty space. We can calculate behaviours in these environments very precisely right now. Sure, there are things about these environments we still need to learn, but they're not utterly intractable or inconceivable to us.

There's a character Rimmer in a comedy series Red Dwarf that's always going on about how aliens would be unimaginable to use because they're alien and everything they do would be... well.. 'alien'. It's poking fun at SF that presents aliens as inherently utterly unintelligible to us simply by virtue of being alien, but frankly that's absurd. Physics doesn't care who or where you are, the same rules apply.

Your brain is lying to you.

It's not your brain's fault, really. It doesn't know any better. Your brain has a way of interpreting the sensory perceptions communicated to it. It conflates this with 'reality' because it doesn't know anything thing else. And because it doesn't have anything else, it also declares itself the universal understander. Just like there are sounds you cannot hear, and chemicals you cannot taste, wavelengths you cannot see -- consider that there may be thoughts that you physically cannot think.

No magic and no spooky mysterious stuff. Just that it seems quite possible that our brains are limited in their ability to construct a representation of the universe, and insofar as the representation is true, we have kno way of knowing if it's the only true one.

If you know what Turing complete means, and what a universal Turing machine is and can do, and that we are examples of such, then you must realise that’s not true. Provably. Unless you’re a dualist of some sort and believe thinking is some sort of spiritual woo, in which case I can’t help you. But I can imagine what that’s like (see what I did there?).

Seriously, there are sensory experiences I can’t have, but in principle no thoughts that are intractable to analysis and understanding given enough information.

Our brains are technically only "turing complete" if you ignore their finite memory. (Edit: which is what is commonly meant by turing complete.)

It seems pretty obvious that there are thoughts that are too big to fit inside our heads.

Again this is descending into nitpicking at the extremes. PG is not saying everything we understand will be in common with aliens, or that our cognitive match with them will be 1 to 1, just that there is likely to be overlap. That given the universality of human cognition, there is likely to be thoughts we can mutually appreciate. I am not claiming that human brains are capable of all thought possible to aliens, only that there will be common ground. That’s all. It’s a very minimal claim.

Pointing out possible edge cases is a non sequitur. Of course there will be things we don’t have in common. Yes, we know that. The fact we’re explicitly looking at common ground from first principles implicitly acknowledges there will be ground not in common. That’s given.

You said that JackFr was provably incorrect. I'm not pointing out an edge case but pointing out the massive hole in your argument.

As a limited computing system, our brain has been optimized to process the world in specific ways using various heuristics and compression techniques. Alien brains would presumably be optimized in different ways. Given our limited capabilities, it seems quite possible that translating any concepts between the two systems would be unfeasible for either our brain or the alien one.

I would like to think that there would common ground but that isn't something that can be known a priori, and is thus not presently "proveable".

Edit: We do have some fairly "alien" brains here on earth but we haven't had much luck teaching any mollusks any of our mathematical truths.

OK, then walk out the window of my old apartment.

I see it as being on the third floor, but that's only my brain lying to me, right?

Yes, but the point I think is that the rules of symbolic reasoning are not like the rules of physics; formal systems have an arbitrary shape, so exploring the universe from a different starting point could yield a completely different method of representing it and processing information about it. And such methods needs not be compatible with our understanding of math.

You've already presupposed that experimenting across all those scales is somehow an integral (necessary?) part of being the kind of life that we might encounter.

That could be possible, but I don't think that we can assume that.

Pi is the human joke of the galaxy--they all use Tau.

> We only need to evaluate what is practically useful.

As simple as this may seem, a few problematic points:

- who decides who gets included in "we"?

- who decides what "evaluate" is composed of?

- who decides what is practical, and useful?

- there is an implicit dimension of Time: for example, many things were initially categorized as useful, but then it can turn out to be a lot more complicated (Thalidomide, fossil fuels, arguably democracy & journalism (at least as they are practised, which seems to be fairly immutable))

Of course, these issues can be easily dismissed ("pedantic!"), but that doesn't make them go away.

> At the end of the day that's all science does - it's a methodology for generating guidelines for what actually works in the real world and for which accuracy, or truth, can be verified. We call those guidelines scientific 'laws', but truth in the absolute sense is less critical than is often made out by both proponents and critics of science.

Another problem we have is language (and various other things, semantics, semiotics, etc) - in this example, a reader could easily take away a few not necessarily correct beliefs from the way you've worded it:

- it could be (very poorly) interpreted to mean science does only this (and nothing else)

- it could be interpreted to mean science alone does this

- it could be interpreted to mean that the output of science (what actually works in the real world) is comprehensive, as in "all that works has been discovered by science - and if it isn't, science would (or will) find it"

- it could be interpreted to mean that science alone is able to produce accuracy, truth, and verification

- it could be interpreted to mean (or, a person may just not consider) that "what works" is not necessarily a constant over time (see: Thalidomide, fossil fuels)

To be clear, I'm not saying you're asserting these things, I'm just pointing out that language is extremely ambiguous and can be misleading.

> PG is simply pointing out that since aliens, if they exist, live in the same real world we do they will discover a lot of the same practical results we do

This isn't even accurate for the different peoples on our one planet.

It really is, that’s why the basic mathematical principled discovered in India, Mesopotamia and Greece have spread and been further developed throughout the world. Because they work just as well everywhere. The Maya discovered the natural numbers and zero, did arithmetic and performed calendrical and astronomical calculations. China developed their numerical system independently, calculated Pi, performed division, root extraction and linear algebra and many original techniques. The forms of expression and some techniques were unique but maths is maths.

I'm saying that beings developing "basic math" isn't even a given on this planet.

All of your examples come from societies where not being able to count and record with perfect precision meant that you were going to be screwed by people who could at the market. They're also no strangers to conflict, and math wins battles. They share characteristics because they had similar pressures and desires.

Imagine a thriving group of people who only use the numbers "one", "two", and "many". What sort of environment and point of view would lead to such a situation, given equivalent intelligence? How do they navigate their universe so successfully?

Unfortunately most of the research I've read in this area has focused on what people like this can't do relative to our framework, and hasn't been especially curious about what we can't do relative to theirs.

All those cultures were running on the same hardware, so to speak. Let's see what the squids have to say.

We’ll, there’s some evidence octopi and squid can count. If so then at least there’s some elementary maths in common. There’s no reason to suppose super intelligent cephalopod creatures would have particular difficulty with things like natural numbers, for all their different experience of the world. Two shrimp plus two shrimp is still the same amount of food as one shrimp plus three shrimp, no matter how many arms you have.

To a whale there would be either many shrimp or zero shrimp.

> It's turtles all the way down -- we don't know what "truth" is, or how one can come to "know" truth

> I've heard that our notion of truth is intimately tied to our experience of physical reality

It's been thousands of years since the first retorsion arguments were recorded, and yet I still see obviously self-refuting or self-undermining claims being made and quite often. Like these.

> imagine an alien civilization that has a different circumference-to-radius ratio

Aliens still live in our Universe with our physics. Are these aliens bigger than the solar system?

Maybe a different universe with different parameters could look m athematically different in major ways.

Again, talking a bit outside my comfort zone, but I believe our universe's physics say that circles only have a fixed circumference-to-radius ratio at "small" (astronomically) scales. At large enough scales space is curved and the ratio begins to change.

In a similar way, a set of different physics would apply to small enough aliens that are principally governed by quantum mechanics. They might never come up with the real numbers because they do not believe that quantities are dense.

We could think of more examples -- aliens who typically move at speeds close to the speed of light would have very different conceptions of time, etc.

Our best estimate for the topology of the universe at the largest scale is that it is 'flat'. If it has curvature, it's below the margin of error we are able to measure it at. The only thing we know that can curve space significantly is gravity, but it would take truly stupendously strong gravity fields to significantly change basic results from classical geometry, such as those close to the event horizon of a black hole. Even then we can calculate it accurately nowadays so our results would still correspond to theirs even in those circumstances.

If we aren't talking about the same things, we wouldn't be talking about the same things.

Part of establishing communication is ensuring we're talking about the same things.

Therefore, once we've established communication, we'd be talking about the same things, such as what a flat plane is.

If you think humans are the only species capable of comprehending Euclidean geometry, say so, and defend the assertion.

Squares do not exist outside of abstract axioms. Solids projected from them share the same status. A mason making a tile forces reality to approximate the concept. Even crystalline structures which suggest them are too approximate. Why is a square implicit to an alien?