In the Alien Civilizations thread I suggested that we would be able to talk meaningfully with aliens only about mathematics and science. (I should also have said engineering.)

This is because physics, and engineering (which is applied physics) are both rooted in mathematics. So the question on the floor is whether the mathematics of an alien civilization would be different from ours.

My answer is, Yes but ...

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Since I expect most alien civilizations to be more advanced than we are, I would expect their mathematics also to be more advanced. However, advances in mathematics don't invalidate previous advances, they add to them.

So aliens' ideas about number theory are likely to be more advanced than our own, and they may very well be able to explain why prime numbers tend to cluster in integer number space, for example. But all numbers that are prime for them will be prime for us, and vice-versa.

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Some people, including some mathematicians, believe that mathematics is a creation of the human mind and is not a set of objective truths waiting to be discovered. The development of Riemannian (non-Euclidean) geometry is often cited as an example by non-mathematicians.

But Reimannian geometry falls naturally out of the axioms if you remove the Euclidean geometry assumption that parallel lines can never meet. Now some people will say "They can't both be true. The world is either one way, or it is the other."

Well, the world "is" in fact Riemannian as best we know today, but Euclidean geometry is equally true even though it no longer applies to the real world. You see, the truth content of mathematics is simply logical consistency. There can never be a world in which the integer one, when added to itself, produces some integer other than two.

So what is invented is the correspondence between mathematics and the real world, and not the mathematics itself, which is objectively true and lies before us, a vast and in fact infiinite ocean of mathematical truth that we do not and never will fully explore.

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What do I mean by the correspondence between mathematics and the real world?

Take Newtons laws. Two hundred years ago they appeared to be self-evidently "true", with no possibility of their being "wrong". Yet along came Riemannian geometry, which served as the basis for General Relativity.

But Newton's laws aren't "wrong" today, they are simply a special case of General Relativity -- the case where spacetime is flat.

Every major advance in physics contains the earlier physics as a special case. But the earlier mathematics were never "wrong", they are simply seen as inapplicable in the general case. The mathematics are always "right" because they are always internally consistent.

Goedel showed that there is no way to prove that in the general case a system of mathematics (a set of axioms and the consequences that flow from them) results in mathematical consistency, but to me this is like saying that there is no way to prove that pi will never terminate with a repeating sequence -- so what?

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An issue that is more interesting to me is whether so-called proof by computer is valid. For example, there are theorems in mathematics for which no proof exists, but for which there has been exhaustive exploration of the solution space without counterexamples ever being found.

This is a revolution in epistemology -- the science of how we know what we know -- because it makes statements similar to those of thermodynamics. For example ...

It can easily shown mathematically that it's possible that one hour from now all of the air in the world will suddenly escape to outer space, asphyxiating us all. But so what? Our behavior does not change just because this event is theoretically possible.

Similarly, if a theorem has been shown not to be violated in one billion samples of its solution space, then for all practical purposes we ought to behave as if the theorem is true even though we do not know this.

This is definitely an area in which aliens might have a great deal to teach us. Their sciences may be based in part on theorems, and therefore on physics, which are unknown to us today and which may never be shown to be "true" or "untrue".

My own guess is that the disconnect between quantum mechanics and general relativity is going to be resolved in exactly this way -- new mathematics which will not be proved in the traditional way but which will be so overwhelmingly likely to be true that we should behave as if it is true.