Alien mathematics.

[weemadando]

Seeing as it's not my area of expertise, I was wondering, just how different can mathematics be across cultures - both on earth and even with theoretical "alien" mathematics?

Naturally, certain universal factors must be represented, but how universal do numbers have to be?

I know this may be a stupid question, but hey, I figure someone here must have an idea.

[Ender]

Also not a mathematician, however as math follows the rules of logic and can break down into the basic fundamental of "1+1+1+1+1+..." if you go down far enough, then I would imagine the differences would be confined to the base system (e.g. base 10, binary, etc) and whether or not they are further along then us. But I could be way off.

[Gullible Jones]

Number systems may differ. Mathematics cannot.

(And for those of you ready to mention Greg Egan - please explain how large arithmetical operations can produce paradoxical results without violating conservation of energy.)

[Surlethe]

The fact the universe is rational will force alien logic to work in pretty much the same way as our logic. From there, if you have basic axioms of set theory, you'll get the same math. (Even if they don't, they'll still probably have at least calculus.)

[Jaepheth]

Number systems may differ. Mathematics cannot.

(And for those of you ready to mention Greg Egan - please explain how large arithmetical operations can produce paradoxical results without violating conservation of energy.)

I'm not familiar with Greg Egan. Are you talking about something like the Banach-Tarski Paradox?

[Kuroneko]

It depends. The fact that virtually all of our mathematics past a certain level is based on set theory, which is in turn framed in terms on Frege's logic probably doesn't mean anything more than that thinking in those terms makes sense to humans, but certain results of mathematics are practically inevitable for reasons already stated. So even if they have a different starting point for their mathematics, there still should be a very large overlap (for reasons already mentioned) that makes figuring out the rest much easier, even if they're "different" in that sense.

In the end, mathematics is just the study of abstract structure. We consider set theory fundamental and frame almost everything in else in those terms. But even if it is not true for aliens, it is virtually guaranteed that the different systems are mutually comprehensible because they will all be based on self-consistent principles, and that's all that's necessary for doing mathematics. In this sense, mathematics really can't be "different".

P.S. Who is Greg Egan?

[weemadando]

Just to frame it a bit more - I obviously thought through the basic stuff (like of course pi is pi by any other name due to the inherent nature of the ratios involved) and of course, they'd *have* to have zero in their numerical system (or whatever version of numerical theory that they use) if they want to advance their development significantly.

It's quite interesting some of the stuff in my links. Some is beyond me utterly, some I start to comprehend, but it's all very useful - thank you.

[Gullible Jones]

Number systems may differ. Mathematics cannot.

(And for those of you ready to mention Greg Egan - please explain how large arithmetical operations can produce paradoxical results without violating conservation of energy.)

I'm not familiar with Greg Egan. Are you talking about something like the Banach-Tarski Paradox?

Does this paradox produce two balls from one which are each some fraction of the mass, or do they only need to be identical in terms of volume?

[Gullible Jones]

Ghetto edit: I was talking about Egan's short story Luminous, in which the protagonists have discovered that very large arithmetical operations don't produce the results that they should. It's an entertaining story, although it leaves out a) how they figured this out, as to the human brain the "wrong" result would seem correct, and b) how this would fail to violate conservation of mass/energy, charge, and other things.

[Kuroneko]

Does this paradox produce two balls from one which are each some fraction of the mass, or do they only need to be identical in terms of volume?

Mass has absolutely nothing to do with this. The partitioning and re-assembly is rigid, with no distortions whatsoever, so even if you consider the balls to have some mass, the result is not due to simply re-shuffling and halving the mass of the ball to make two. That would be completely uninteresting.

But it's only a paradox if one insists that the pieces each have a volume, which is not the case (in mathematical jargon, they're not measurable, which is distinct from having zero measure).

[Kodiak]

my brother and I have been working on a sci-fi book in which humans have to reverse-engineer some alien technology that gives them a headache until they figure out that everything is in base 8 math. Aside from that, a couple of things which might give difficulties in math/science are:

1. Different conventions in the "order of operations". We do first, exponents, roots, and parentheses; second, multiplication and division; third, addition and subtraction; and lastly, we always work from left to right. This is, I believe, an arbitrary and agreed upon convention.

2. Decimal System. Part of the base-ten math is the decimal system. If the math were a different base, there would be a very different decimal and percentage scale.

3. Time. IIRC, the aztecs (or mayans), had some sort of time which was radically different than our current system. Also, there's the possibility that aliens would be on metric time. Either of these could cause great confusion.

4. Right-hand rule. While not specifically math, this is an example of one of many conventions used today for math and science that is completely arbitrary and could cause significant difficulties relating to the discussion of 3-d vectors as well as the behavior of EM fields.

These are just a few, but as has been pointed out they can be resolved with relative ease by simply realizing that our way isn't the "only way" or even the "best" way of doing things.

[Surlethe]

Does this paradox produce two balls from one which are each some fraction of the mass, or do they only need to be identical in terms of volume?

Mass has absolutely nothing to do with this. The partitioning and re-assembly is rigid, with no distortions whatsoever, so even if you consider the balls to have some mass, the result is not due to simply re-shuffling and halving the mass of the ball to make two. That would be completely uninteresting.

But it's only a paradox if one insists that the pieces each have a volume, which is not the case (in mathematical jargon, they're not measurable, which is distinct from having zero measure).

It might be interesting to note that in proving the 'paradox', you use the Axiom of Choice on infinite sets, which means a priori that the result is probably physically meaningless. Another interesting tidbit: the existence of unmeasurable sets (and hence the Banach-Tarski Paradox) is almost equivalent to the Axiom of Choice.

[Junghalli]

Hmm, many human cultures lacked a concept of zero, the Greeks and Romans for instance. Would it be possible to develop a technological civilization without first discovering zero?