They could have differing ideas about infinity, Godel’s theorem, linearity and time, and counting, to mention a few.
With no experience of ET mathematicians, we haven’t got much to go on. But, let’s take a look at a few of the ways ET maths could take different approaches from ours, or be hard for us to understand.
This is meant as no more than a light hearted exploration of these ideas, and if it stimulates some interesting thoughts, I’ve more than done my job.
Note, this answer runs to an estimated 46 printed pages. You can now get it as a kindle book, if you prefer,.
INFINITY, SETS AND LOGICAL PARADOXES
This is an area of maths (use of sets or infinity or both) – that for us is full of paradoxes – such as Russell’s paradox, various Cantor’s paradoxes, the Banach Tarski paradox etc.
Some say the paradoxes have been solved.
Yes our maths is elegant in a way, and if you follow the rules carefully you don’t get any contradictions (at least as far as we know). A mathematician might say that the paradoxes have all been “solved”.
However, if you look at those rules from a philosophically unattached standpoint you may get a different impression, which may perhaps give us some insights into some ways ET maths could differ from ours.
Modern set theory with
- The puzzling impossibility of counting many fundamental things in mathematics – as in – ordering them into an unending list.Yet everything “interesting” can be counted. Ratios, finite decimals, square roots, more generally, solutions to polynomial and trig equations – everything like that can be counted easily. If you haven’t come across this before, see
(just a short summary I did, linking to the material on the subject).
Our maths is so “Heath Robinson” at least from a philosophical point of view, why this need to include so many things you never need in everyday mathematical life? It’s a bit like this potato peeling machine:
We have all this apparatus of higher orders of infinity, just to include a whole bunch of obscure numbers that nobody ever needs as working mathematicians. That is to say – they never need any of them as individual numbers, just need to know, for logical reasons only, that all those uncountably many things exist.
Why? It seems so clumsy
(that is from a historical and philosophical viewpoint)
Perhaps all these uncountably infinite sets do exist in some sense – but if so – why are we so disconnected from them? Why is it that so much of maths is, in a sense, forever way beyond our grasp?
Some mathematicians such as removed them altogether of course, leaving only potential infinity. A few, such as .
It is even stranger when you find out about – that if somehow “behind the scenes”, you replace all those uncountable infinities by other (rather intricate) finite and countable things, all the same results still hold true about them.
That is – so long as the maths is expressible in a straightforward way using a finite number of symbols and proofs are easy to verify – “first order” maths
Techy detail for logicians: – you can avoid the paradox, technically, with a “second order” formal language with uncountably many distinct symbols. Which doesn’t really solve the philosophical issue of course.
Any human or ET mathematician will only be able to distinguish a (small) finite number of symbols from each other. It’s a general issue for any – it needs a proof theory before mathematicians can use it in practice – and when you do that, the paradox surfaces again.
An ET could reinterpret our maths in this way and their theorems would match ours in every detail.
Perhaps the way we do maths here on Earth is universal and all ETs do it this way. But these possibilities do suggest other possibilities.
- Would ETs follow the usual approach of human mathematicians – that most numbers and mathematical entities can’t be counted?
- Or take other views on infinity like some human mathematicians – perhaps very practical “constructive” in their approach to maths for instance, so the question doesn’t arise (more on that later)?
- Or – reinterpret all our maths in some complex abstract way, as in the – but for them it’s not a paradox, just how they think about maths?
- Or does the question just not arise for them for some other reason we haven’t thought of yet, or have some other meaning for them?
- Or, like us, have lots of points of view on the subject? An unending philosophical debate that’s gone on for millions of years?
- Could they have some other take on the whole question which we haven’t thought of?
- Continuum hypothesis – why does our maths say that we can never know whether or not there are other orders of infinity between the number of ratios or whole numbers, and the number of infinite decimals like pi?
- Axiom of choice – given infinitely many pairs of shoes, it is easy to choose one of each – for instance choose the left hand shoe each time. But for indistinguishable socks – is it possible to choose one from each pair?
(photo by Hoi Ito)
When you have a mathematical equivalent of infinitely many pairs of shoes, there is no problem picking out one of each. It’s easy, for instance, just choose the left one out of each pair.
But it gets far harder to cope with the mathematical equivalents of infinitely many pairs of socks.
That’s because they are identical to each other (you can swap your left and right socks and not notice that anything has changed). Our maths doesn’t let us pick out one of each – unless we add in an extra axiom, the axiom of choice.
It seems an obvious axiom, innocuous even – that if you have infinitely many pairs, you can choose a singleton from each one. However, it turns out that if you add it in, this leads – not to inconsistencies quite – but to results so strange that they seem paradoxical to human minds.
For instance, one of many famous puzzling consequences – it lets you split a sphere into a small number of geometrical “pieces” – and combine them together to make two spheres of same volume as the original – without any gaps.
If you accept it, you end up with maths that is more powerful – but let’s you prove these unintuitive results such as, that it’s possible to dissect a sphere geometrically into a small number of “pieces” (discontinuous but “rigid”) and re-assemble it to make two spheres of the same volume, without gaps.
As another example – it lets you fill 3D space entirely with radius 1 circles – with none of them intersecting, yet no gaps, a sort of 3D space filling chain mail. Again most would find that paradoxical…
Why does this axiom keep cropping up in Maths (from a philosophical point of view that is) – and should we use it – or is it too powerful since it lets us prove paradoxical seeming results?
Why does it matter, since in practice nobody ever is able to choose an infinite number of anythings in the real world? Nobody ever has an infinite number of pairs of socks, or of anything. So why do mathematicians need to think so much about their mathematical equivalents?
Would ETs use the axiom of choice? If so, what do they make of its paradoxical results? Or is it not even an issue for them for some reason?
- The arbitrary rules we use to keep maths consistent.For instance in one of the most popular ways of creating a logical foundations for maths, ZF, large sets are called “classes” and a class can’t be a member of a set. There is no good mathematical reason for this. It is just a “kludge” – we have to do it or we end up with an inconsistent theory.You do it just because, if you don’t keep to the rules that have been worked out and just “follow your intuitions” about sets you end up with contradictory results and pardoxes. Genuine unresolvable paradoxes.The most famous one,
(more about this later in this page).
(Techy note – actually you can have to work with classes indirectly in ZF, as its axioms refer to sets only – its theory of classes can be axiomatized using )
The whole thing is really a bit of a kludge (the axioms I mean) viewed somewhat dispassionately with your philosopher’s hat on rather than with your mathematician’s hat on.
When you invent a radically new axiom system – it’s not enough to create axioms that look good and work well together, because that could take you straight to a paradox as happened to Frege. The system could seem perfect to your mathematical intuitions, but that is not enough. You have to go one step further – usually – by proving relative consistency with ZF or some other established theory.
By Gödel’s theorem you know that you can’t prove that your new axiom system is consistent. But what you can do is to prove it is “as good as ZF”. You can prove that if it did fail, that failure would bring down ZF as well – which is generally thought to be good enough to establish it as an okay theory as regards consistency.
It seems to work okay and is beautiful in its way. Maths within the system may be elegant, lovely even. But is this really the best that we can do?
And whether or not – is it such an obvious way of proceeding that ETs would have to end up with the same system, with all the same mathematical and philosophical ideas as ourselves?
Would they come up with the same “kludges”.
Or, might they come up with something different?