…x n ∈ x n-1 ∈ …x 3 ∈ x 2 ∈ x 1 .



Partly for this reason, it hasn't caught on amongst mathematicians.

Quine's new foundations might suit logical ETs to a T . Or they might devise some other system based on logic

that we haven't thought of that is even further from the ideas most mathematicians are used to, because of our fixation on the need to find a foundations for mathematics in our logic.

RUSSELL'S PARADOX

This is worth describing in detail because it uses such simple ideas, you'd think that just about all ETs would encounter it in their reasoning.

I like the way this is presented in wikipedia, so will just quote from the article on

"Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares in the plane. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal".



Now we consider the set of all normal sets, R. Determining whether R is normal or abnormal is impossible: if R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if R were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal: Russell's paradox."



As soon as you start thinking in terms of abstract concepts, and idea of a set, or collection of things - then Russell's paradox is not far away.

Human mathematicians didn't spot this paradox in our thinking until 1901. Though it's closely related to the ancient

The paradox got unearthed when Frege set out to do a methodical careful logical axiomatization of the whole of mathematics, using set theory in his great work, his life's work really, the

Just as the second volume of his great work was going to press, then he got a letter from Bertrand Russell about his paradox.

"Your discovery of the contradiction caused me the greatest surprise,and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic"

Why is it that if you just follow your nose and axiomatise everything carefully - then you fall straight into Russell's paradox, as Frege did?

Grundgesetze der Arithmetik

Why - philosophically - is the direct and most obvious route the wrong route to take?

There's no resolution to it, except to limit our reasoning to prevent it happening, with no really good mathematical or philosophical justification for doing that.

Is it possible that some ETs don't encounter Russell's paradox - if so - why, and how do they reason? Or do encounter it but for some reason don't find it paradoxical? Or is it a paradox for all ET mathematicians?

MATHS WITHOUT LINEAR TIME

More radically than that - ETs might not necessarily have a sense of linear time like us. We have a clear sense of past, present and future. And know exactly where we are in that time stream. But some ETs might live in a world where hardly anything changes from day to day. So is no need to remember when things happened but may be very important to know where they happened.

If so they might have a way of seeing the world that is spatially based - with linearly ordered time an abstract concept they find really hard to grasp. I can imagine e.g. if they live in the oceans beneath the surface of an icy moon like Europa, no idea that the rest of the universe exists, no seasons, nothing except gradients of temperature, and chemical gradients etc. They might have long term memory but no short term memory - as we understand their world.

After all in special relativity then time does play a rather strange role. It's not as easy to understand in a single ordered time stream.

Perhaps there are other ways of thinking about the universe that start from a more spatial basis - not that they have no idea of time at all - but - that they don't order it in a strictly linear way. What other ways of ordering it, they might have, I don't know.

MATHS THAT IS BASED ON A QUANTUM MECHANICS TYPE WAY OF EXPERIENCING THE WORLD

Or - do use linear time but are totally unable to experience it directly so is a strange very abstract concept - while at the same time maybe find some other ideas, e.g. quantum mechanics type ideas easier to understand.

Maybe think in terms of superpositions of many states at once - and collapsing of uncertainties. Maybe their maths then would somehow reflect that - they would know what a linear ordering is - but would not be like us where nearly all the most interesting mathematical spaces are based on notions of distance and linear orderings along lines - maybe they don't have geometry either as we have it but in some other form not based on Euclid's axioms.

MATHS WITH COUNTING AS AN EXTREMELY ABSTRACT CONCEPT RARELY USED AND HARD TO UNDERSTAND

And indeed (this is not necessarily the same ETS - these maybe - entities that live as gas clouds, or films like stromatolites, colonies of microbes that merge and separate and form greater or less intelligence depending how many individual microbes involved - sort of like sponges, can strain them through a sieve and they come together again as if nothing happened) - they could go as fundamental as different ideas about counting.

For creatures like that, topology could be fundamental to their maths, everything continuous, no discrete shapes. They might think naturally in terms of open and closed sets (regions with or without a boundary) - or some other topological primitives we haven't thought of yet.

Advance complex theorems in topology would be child's play to them like 1 2 3, while counting would be an incredibly abstract idea they could formulate mathematically but perhaps find hard to grasp.

MATHS WITH EXTREMELY SHORT DEDUCTION SPANS

Perhaps they can't make long deductions like we do. If they have hardly any time ordered short term memory - remember everything perfectly if they want to but not able to order it in time for more than a few seconds - then the very idea of chains of logical deduction may be alien to them, for anything more than a few deduction steps.

Instead they could rely extensively on seeing things at a glance. For instance with small numbers of things, we have the ability to see how many there are at a glance, without need to count them as 1, 2, 3.

See

When you are familiar with geometry, you can often see geometrical theorems at a glance.

If you are used to geometrical ideas, you may be able to see at a glance that both squares have the same total area, and that therefore the two white squares at the right add up to the same total area as the single white square to the left, and see also that this relationship between the area of the square on the diagonal and the square on the two shorter sides holds for any right angle triangle. This is the

Mathematicians often talk about suddenly seeing a proof of a theorem at a glance. Here is Professor Roger Penrose talking about one such moment:

A colleague (Ivor Robinson) had been visiting from the USA and he was engaging me in voluble conversation on a quite different topic as we walked down the street approaching my office in Birkbeck College in London. The conversation stopped momentarily as we crossed a side road, and resumed again at the other side. Evidently, during those few moments, an idea occurred to me, but then the ensuing conversation blotted it from my mind!



Later in the day, after my colleague had left, I returned to my office. I remember having an odd feeling of elation that I could not account for. I began going through in my mind all the various things that had happened to me during the day, in an attempt to find what it was that had caused this elation. After eliminating numerous inadequate possibilities, I finally brought to mind the thought that I had had while crossing the street- a thought which had momentarily elated me by providing the solution to the problem that had been milling around at the back of my head! Apparently, it was the needed criterion that I subsequently called a ‘trapped surface’ and then it did not take me long to form the outline of a proof of the theorem that I had been looking for. Even so, it was some while before the proof was formulated in a completely rigorous way, but the idea that I had had while crossing the street had been the key.



The Emperor's New Mind

What if the ETs can only do mathematics in that way - as sudden moments of insight?

If they also have topology as fundamental - things like intersection of sets and various distinctions of types of sets and how they can interact - their theorems might not use straight lines and circles.

Instead, maybe their advanced theorems consist of a huge Jackson Pollock type painting of blotches which interact in complex ways - which they can see at a glance but for us is almost impossible to understand.

Perhaps an ET might draw something like this, show it to us and say "This is the maths we use for constructing our spaceships" - and expect us to understand at a glance - and have no other way of presenting their maths.



Jackson Pollock - biography, paintings, quotes of Jackson Pollock



Interestingly, "Action painting" like this is based on the idea of trying to tap into an archetypal visual language.



Proving theorems for them might consist of spending hours, even days painting intricate patterns of blotches on a large canvas until they can step back and look at what they painted, and say "I see it now!".

MATHS AS SUDDEN INSIGHT AIDED BY PROOF

Less radical than that, we can imagine that ET mathematicians might have normal proof methods, as we do - but a far higher degree of sudden insight. What if they are all Ramanujans?

After all human mathematicians don't, in practice, make much use of formal proof. We work on mathematical intuition most of the time, informal deductions. Even the most detailed proofs of a working mathematician wouldn't count as a completely rigorous proof in first order formal logic. Yet, we have no doubt that these proofs are correct.

So, though their maths may be based on similar deduction methods to us, they might make so many intuitive leaps that it is really hard for a human mathematician to understand what's going on.

The Indian mathematician Srinivasa Ramanujan came up with pages of mathematical results which he recorded in his notebooks, with no mathematical proof. That's partly because paper was expensive, so he did his rough working on slate, and then just recorded the answers in his notebooks.

Still he also had a remarkable level of mathematical intuition, and intuited many results which he could not prove rigorously - most of which were proved later by other mathematicians. His notebooks, which were intended for his personal use, contain a few mistakes, but very few, nearly all his intricate and surprising formulae and results are correct. Many of them were startling new results in mathematics.



A devout Hindu, he attributed his results to inspiration from the goddess Namagiri Thayar, and also saw visions of some of the formulae in his dreams.

"While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing."

Perhaps this also might give us an idea of what ET maths might be like if they depend on sudden insight and a high level of mathematical intuition, with only a small amount of deductive proof.

Their communications could be filled with dense sheets of equations - and if they are all Ramanujans - just a single line on a single page, which they can see to be true instantly, requires hundreds, or thousands of lines of our more clumsy intuitive proof methods.

FRACTAL MATHS

They might also think in terms of

,- see fractals all around them, and classify fractals, and think of everything else in terms of these as their primitives.

This image done by Ondřej Karlík

I don't know how it would work, we don't have any maths like this as far as I know, but they might find fractals like this easier to understand than our triangles, squares and circles. And try to approximate a circle as a fractal.



The fractal, shown here is an example of a Mandelbulb - recently discovered type of 3D based on the Mandelbox - another 3D fractal discovered in 2010 by Tom Lowe.

ETS WITH DISCRETE GEOMETRY

When you think about geometry, you will probably have in mind continuous geometry with ideas of straight lines and points.

However - a less well known area of maths is taxicab geometry. For humans, this is mainly an area of interest to recreational mathematics. You can use squares, or hexagons or triangles as the building blocks.

But it's also the geometry used for cellular automata - and for discrete simulations of water flow, and many computer models.



Taxicab geometry - similar to routes traveled by taxis in modern grid network type cities. The three paths shown in red, blue and yellow are all the same length. Green path shows the distance in a continuous geometry.

So that's another possibility. ETs could make far more extensive use of discrete geometries, and might make hardly any use of continuous geometry.

It's not as if our space is continuous in any essential obvious way. We can't measure anything to infinite precision. So continuous space is as much of an approximation as a discrete space. But for some reason human mathematicians have settled on a continuous geometry as the "default" way of thinking about space.

Continuous geometry does have advantages of isotropy - hard to make an isotropic discrete geometry (e.g. one with no "preferred direction" for fast travel). But that again might not be impossible (I actually wrote a paper about isotropic discrete geometries, might have a go at publishing it, but haven't attempted to publish it yet - anyway - found that there are techniques you can use to create isotropic discrete geometries - that is - isotropic in the limit as the cells get smaller and smaller. It took a bit of lateral thinking, but once I got the idea - it wasn't that hard - I found two different ways to do it, maybe you can think of others? I think the main reason we don't study them is just because nobody is that much interested in them).

Another way is to use discrete gas cellular automata. There are exact solutions to equations of gas diffusion and incompressible liquid flow on hexagonal lattices. This lets you construct cellular automata evolving just according to rules about nearest neighbours, that have things like expanding circular waves. Here is an example of a gas automata

What if ETs think of it in terms of discrete geometry as their "default" way of thinking about the space they live in - and unlike us - do all their physics using discrete geometries like this.

They might have continuous geometry as a recreational area of maths similar to taxicab geometry. Again most ETs, possibly, might not even have heard of continuous geometry.

I don't know how likely or possible this is. Just putting it forward as a possible idea to think over - is it possible that ETs could have discrete rather than continuous geometry?

ETS WITH COMPLEX MATHS, BUT WITHOUT NUMBERS

All modern human societies have numbers in some form. Many different counting systems, and some of them are inefficient for counting large numbers - but they all have numbers.

Many birds and animals can also "count" to some extent.

So we tend to think that counting will be universal amongst ETs. But would it?

What about an intelligent slime mold? Or an intelligent creature that lives in a Europa type ocean, and has almost no short term memory? Would counting come naturally to them also?

They might think in terms of linear orderings instead for instance. And perhaps have fuzzy continuous geometrical primitives, or topologically equivalent sets as their primitives, understand everything in terms of topology instead of discrete sets.

You can go a long way in some areas of maths without ever mentioning numbers or counting things. Surely they'd have some equivalent but it might be as abstract for them as open and closed sets are for most of us. Could be that non mathematician ETs don't even know about numbers - and in maths, they use them only in particular specialized fields.

ETS WHO CAN'T ADD UP

Another possibility is - that they have numbers, but are no good at arithmetic, so again, numbers are things they use rarely, because they find them difficult to understand.

Like Emma King, cosmologist, "The Mathematician who can't add up".

Perhaps they are all like her, for them, dyscalulia might be the norm. We might seem prodigies to them, like "lightning calculators".

(Of course could equally be the other way that the ETs are all lightning calculators as the norm).

ETS WITHOUT MATHS

What if the ETs don't use maths at all, as a formal discipline at least?

After all many humans get by fine with very little use of maths. Suppose they do everything by biological engineering and analogue computing, they might have a poetic / artistic approach even to traveling between stars.

It's only recently that mathematicians have become common and important elements of society - not that long ago there would be only a few mathematicians in an entire country. Perhaps part of the reason we have so many mathematicians nowadays is because of the success of maths in technology.

So, suppose that the ETs don't need maths to build complex machines, even computers - but somehow - like slime molds perhaps - can do it instinctively.

They might not be as mathematical as humans are. Yet accomplish as much or more technologically. Or indeed also what about hive minds? Colonial ETs where no individual is intelligent, just the community as a whole. Would they be able to count?

Also, how limited is our vision of the range of possibilities for ETIs?

We have so many examples on Earth - slime moulds, ants, bees, dolphins, birds etc to use as analogies for ETS- but they all

use the same DNA

same biochemistry, same building blocks

all evolved under 1 Earth gravity, one atmosphere pressure, limited temperature range

on the surface of a planet of a G type yellow dwarf star with a large Moon etc etc

Of course, we can only reason by analogy from what we know.

But some ET life might be radically different in some way we haven't yet imagined in their fundamental biology or life processes, not closely resembling any of the creatures we know about on the Earth. So what might that do to their maths?

WHAT WOULD COMPUTERS BE LIKE FOR ETS WHO RARELY USE NUMBERS?

This is a somewhat forgotten episode of computing.

If you used the word "computer" in 1950, this is what they would think you are talking about. It's not a programmed Babbage type mechanical computer - rather - is an analogue machine, doesn't use numbers internally at all. Skip to 1.26 to see the computer in action. Just a minute or two of it.

At 1:45 "If you look inside a computer, you find an impressive assembly of basic mechanisms. Some of them are duplicated many times in one computer"



Wikipedia article about it, range keeper .



If they have no idea of numbers - or numbers are very abstract concepts for them - then they could still have analogue computers like this, as the computers are based on direct analogue connections between things and don't need to use numbers as such.

They could go on and develop analogue electronic computers also - instead of the numbers based digital computers we have. They'd have many challenges to meet - but then the early digital computers did also.

Hard to say if a technological society much like us that developed analogue computers instead of our digital computers would be further ahead than us or behind us by now.

Surely at any rate they'd be able to develop an analogue electronic computer based technology one way or another.

Here are a few things we are exploring as humans - which might also point the way to alternative histories for other ETs.

The last one points to a rather radical way ETs could be different. They might be slime moulds, able to just extrude parts of themselves to use as computing devices in machines.

This suggests the possibility of ETs that either have little by way of mathematics - or who have maths but not based on numbers, who might well have advanced technology including spaceships.

Or, they might not be technologically advanced. If not mathematically inclined, still they might be great philosophers, or artists, or poets or musicians, and might have long lived non technological civilizations. Could be by inclination, or could be for a simple matter that, for instance, they don't have hands - maybe like parrots, clumsy and not very strong - or like octopuses - live in the sea, not an easy place to develop technology (without fire) - or like dolphins - no hands or any easy way to build anything.

HUMANS WITHOUT MUCH MATHS BY COMPARISON



It may seem a little hard to imagine that ETs might, for instance, be unable to count, or have hardly any maths that we would recognize as such. Throughout the universe, two octopuses plus two octopuses = four octopuses. So wouldn't ET octopuses know that?



But - we humans ourselves managed for a long time, hundreds of thousands of years, with modern intelligence but hardly any maths.



Many things that to us are elementary concepts you teach to children - some of them even to very young children - were beyond the capabilities of even the most advanced thinkers in the past.



Including:

Positional notation with zeroes. The Babylonians had a primitive place notation - but without a trailing zero. They had base 60, but if we had the same system to base 10 you could distinguish 36 from 306 but couldn't distinguish 306 from 3060 (you understood what it was by context). See Positional notation

Idea of zero as a number in its own right - dates back to 9th century India 0 (number)

Negative numbers - most early civilizations treated them as absurd, and only came into common use after 7th century India for debts Negative number

Fractions - our modern idea that you can express the ratio of any pair of numbers e.g. 7/5 as a fraction is surprisingly recent - for a long time they worked with them in what seems to us a clumsy fashion using "Unit fractions" - only fractions with 1 on the top. For instance, 4/5 could be written as 1/2 + 1/4 + 1/20 (or as 1/3 + 1/5 + 1/6 + 1/10 etc.). See Unit fraction Fraction (mathematics)

Solution of the quadratic - a long history, originally many special cases and only some could be solved (made more complicated by the special cases you need to avoid negative numbers). Complete solution not until C16 Quadratic equation

Solution of the cubic - major problem in the C16, is now considered rather basic stuff. Cubic function

And - these are undoubtedly modern humans. You are talking here about entire cultures of intelligent, witty, clever, thoughtful people who never once thought of the idea of writing a fraction as 4/5, but instead always wrote it using the likes of 1/2 + 1/4 + 1/20 and similar expressions, and having to go through all sorts of hoops to multiply them together.



For centuries, entire cultures that did that. No-one in the entire world thought of the idea of writing it as 4/5 until modern times.



I think fair to say, that if you took any of us, and put us back into anywhere in the world prior to say around C7, even our most brilliant mathematicians and scientists - then if brought up in C7, anywhere in the world - they would not understand any of those concepts. Take Albert Einstein or Paul Erdos or Roger Penrose or Stephen Hawkings, or Évariste Galois or Srinivasa Ramanujan or Isaac Newton, or anyone you care to name who you happen to think is brilliant in the area of Maths or Physics.



If they were born and brought up in a culture with no historically developed maths - they would, almost certainly, not hit on the idea of a ratio, or a negative number, or positional notation.



If they became an extremely brilliant mathematician of their time - then they might perhaps come up with a novel way of solving the quadratic equation as a result of their life work in mathematics - or some new proof of a geometrical result or some such. Because that's what brilliant mathematicians did back then.



See for example al-Khwarizm who had six chapters devoted to the six then known types of quadratic equation (all written to make no use of negative numbers as mathematicians of his time didn't recognize them as valid numbers)



Here by squares, he means our x^2 and by roots, our x. His six chapters covered:

Squares equal to roots. Squares equal to numbers. Roots equal to numbers. Squares and roots equal to numbers, e.g. x^2 + 10x = 39. Squares and numbers equal to roots, e.g. x^2 + 21 = 10x. Roots and numbers equal to squares, e.g. 3x + 4 = x^2.

If you were a brilliant mathematician in 800 AD - this is the sort of thing you would study as your life's work.



See Quadratic etc equations



These various maths concepts are quite tricky to grasp even today for young children. If you haven't tried to teach them, and perhaps don't remember your own childhood too clearly - you might be surprised how very tough it is for young children to get these ideas. But it's not so surprising when you discover how long it took for humans to develop the ideas in the first place. Basically in our schools we attempt to fast forward our children through several thousand years of human mathematical development in a few years.



Put them into our culture, and they might learn our maths quickly enough.



They would then go on to prove results in advanced mathematics that we couldn't even begin to explain to our predecessors without giving them a modern education first.



I wonder sometimes - what simple basic concepts we might be missing that our descendants might learn as children.

SIMPLE COUNTING FOR HUMANS AND ETS



For humans - as it turns out, for us counting numbers has been something all cultures learn from an early age - to count at least as far as five or ten. Children do have to learn to count, is not "hard wired" but we learn at a very early age.



Hard to say what maths we had a couple of hundred thousand years ago. But quite likely we could count already way back then. At any rate all modern human cultures can count.



We have some cultures with systems that would be clumsy for counting large numbers e.g. in the thousands and higher. But all can count.



But for some ETs those might be concepts as advanced as negative numbers, ratios, positional zero and solving the quadratic was for us, or more so.



So then it would all depend - not on their intrinsic intelligence - but on whether they are likely to stumble on those concepts - and on whether they need them to build technology and spaceships.



If the answer to both those questions is No - then - I think you could have intelligent ETs with dyscalulia - or indeed without any concept of numbers at all.

HOW LIKELY IS ANY OF THIS?



It might be that out of a billion ETs, there is only one that can't count, or finds counting abstract or can't count far. Or we might be the whiz kids - of the galaxy and universe - one of very few with this ability - like those few children who are able to add, multiply, square roots etc for numbers with many digits in their heads.



Or that absolutely all can count. I'm not attempting to assess probabilities of these things here. Don't know indeed how anyone could do that. Just looking into conceptual possibilities.

IN SCIENCE FICTION



It's not much treated in science fiction. But there are a few stories. Have you read Ted Chiang's story, "Story of Your Life"?



Any suggestions of other good sci. fi. stories treating idea of ETs with maths different from Earth in intrinsic ways (more than just a different base or counting system which is common in sci. fi.) - do say in the comments, and I'll add some of them here.

EXPECT SOME GROUNDS FOR COMMUNICATION

If they do have maths, I think it is possible that ET maths could be so different from ours that it is hard to communicate to start with. But would be astonished if we don't eventually find close parallels here and there. Which might be counting. Or it might be topology. Or might be Gödel's theorem. Or might be quantum mechanics. Or might be Russell's paradox, or an alternative set theory that is used by only a dozen or so people in our society - or paraconsistent logic - or fractals - eventually expect we'd find some common area of maths.

Then once we've done that, especially since we do live in the same universe - would finally find a way to map almost everything into terms we can understand to some extent.

BUT MIGHT NOT FIND THEIR MATHS EASY TO UNDERSTAND

But I am not certain that we'd find the maths immediately easy to understand. Might or might not. Without any previous experience of ET maths I think hard to know for sure.

It is possible there are some ways of thinking that would involve many kinds of "aha's " of insight before humans can get what they are about. After all if you look at the history of human maths, many ideas that are commonplace to us now were not even thought of for centuries or millennia.

E.g. the concept of zero or of a negative number, or of a ratio, or of a uniform way to solve any quadratic equation - these are all things we teach nowadays - some at primary school and some at secondary school - but a few centuries ago these were advanced areas of maths that only a few humans in the whole world understood - and go back further and there were times before any of those concepts were understood.

A few millennia back - nobody in the world understood the mathematical idea of zero, their idea of ratios was very different from ours, they had no idea of solving the quadratic in its general case - they could solve a few special cases of the Pythagoras theorem by trial and error probably - and had bizarre ways of working with fractional amounts e.g. the unit fractions of the Sumerians - everything expressed as sums of reciprocals of whole numbers - seems very clumsy to us - did have some nice points about it - but main thing is - that was a whole society of humans - as intelligent as ourselves - who didn't think of any of the modern ideas of maths.

So - I think - there could well be similar concepts that ET mathematicians have that we haven't thought of yet.

And at the end of that -as in some ET stories, perhaps we'd no longer be thinking quite as humans do today. For good or for bad.

THEIR MATHS LIKELY TO BE MILLIONS OF YEARS FURTHER DEVELOPED THAN OURS

A mathematical ET might not be technological - can be mathematical without technology e.g. if don't have hands or for whatever reason can't manipulate their environment much.

If we meet an ET with maths - the chance they developed it in the last few thousand years of the billions of years since conditions suitable for evolution in our galaxy must be tiny - so small as to be almost impossible.

So, if we do encounter ET mathematicians, there is an excellent chance that they are using maths concepts that they have developed, not for our few millennia - but for millions of years, possibly even billions of years.

What will our maths be like a billion years from now? What concepts would every young school child understand then? Perhaps some of them things that our brightest minds haven't' thought of yet.

This began as an answer to a question on Quora:

HOW TO GET THIS ARTICLE AS A KINDLE EBOOK

You can get it here:



Maths From an Extra Terrestrial Civilization: What Could It Be Like (Amazon)