Probably due to the influence of Plato, mathematics is widely conceived of as universal. In fact, it is widely accepted that the constant π would be equally well known to a reasonable advanced alien species as it is to us, albeit likely in a different base — an idea that has a recurring theme both in fiction and serious discussion about communication with an alien species.

I’m sceptical. Not because I think that the value of π is subjective, but because it is not at all clear to me that aliens would share our abstraction of numbers.

Two things have brought me to this view: reading about how humans count, and reading about the development of numbers.

Experiment with how long it takes to count the number of dots on a page and you will notice an odd pattern: when the number of dots increases past three or four, the amount of time it takes to count them suddenly jumps, while accuracy falls and brain activity changes. (You can try the test in this BBC article) While the small numbers can be almost instantly determined, with the larger ones people generally have to shift their focus on to small groups and increment a number in their head. This ability to rapidly recognize the number of something there is is called subitizing, and it is observed not just in adults, but in babies before they can speak and in non-Human animals (though the threshold at which subitization stops occurring varies between species). This evidence for an intrinsic biological root (almost for a numerical equivalent of Chomesky’s linguistic innateness hypothesis) casts doubt on the view that any reasoning species would develop numeracy.

However, evidence against some sort of universal literacy extends well beyond mere theoretical arguments: despite the advantage of subitizing, there are some very real counterexamples to numeracy. One need only look at the historical development of numbers: it was no trivial task for zero and negative numbers to be accepted as numbers, and people literally died over whether irrational numbers like e and π were numbers. And even in the modern world there are societies lacking our modern conception of numbers — the Piraha tribe lack words for precise numbers greater than two and its members have difficulty working with even modest numbers.

Still, it is reasonable to be sceptical, as many I describe this view to are, regarding whether a society could become advanced without numbers. Don’t we need them for trade? For Chemistry, Physics, Engineering and much of Biology? Where would computers be without numbers?

What alternative could there possibly be?

It is hard, of course, for me to come up with an alternative to numbers. Even if they weren’t, to some extent, hard-wired into my brain at birth, one of the most parts of elementary education is the development of the numerical abstraction — it is ingrained deeply into our thought process so that we use it without consciously deciding to (and while I may not like how modern math education works, that ingraining is a good thing! Most of the utility and beauty of math is only available when it becomes part of the way you think rather than just something you know). One can easily imagine aliens with their own abstraction, as foreign to our thought-process as numbers are to theirs.

However, while it is impossible for me to fathom how deeply alien an alien’s numerical abstraction alternative might be, it is possible to construct something. And so we might imagine a species of photosynthetic aliens with square bodies in a square environment. We shall dub them the Squariens. The Squariens spend their lives sitting in tightly packed grids, jumping up in the air and turning or flipping to face other neighbours. They have little concern for the amount of food they must eat, but are deeply concerned with the symmetries of the square, to us the dihedral group of order 8. And just like we may occasionally muse about other bases, they occasionally spend their time imagining the symmetries of other shapes, of triangles and pentagon, and so on: other dihedral groups. Eventually the Squariens discover group theory — they find the integers mod n in the rotational subgroups. At some point, a Squarien mathematician thinks about the symmetries of something with infinitely many sides… and when they consider the rotational subgroup, the discover a group isomorphic to the integers! Except they don’t think of them like that, don’t think of them like we do: just like we write elements of dihedral groups in terms of numbers (what else is s¹r³?) they describe their integer isomorphic group in terms of how they think of dihedral groups… But while these aliens might find our obsession with numbers odd (and our obsession with prime numbers even odder…) and we might find their obsession with symmetries strange, we’d still be able to understand each other quite easily.

However, that is a rather silly example. And I’m afraid I can’t provide a better example of an alternative to numbers. However, I believe that I can provide a serious and satisfying alternative to polynomials. It is my hope that the reader might take this a slight piece of evidence for the existence of alternatives to numbers.

When I was younger, I remember wondering why we were so concerned with polynomials. Why did we care about things of the form … rather than … ?

At that age, polynomials won me over when I realized that I could `push’ them into arbitrary shapes by using higher degree terms to control the shape farther from zero. What I had discovered was that one can find a polynomials arbitrarily close to any continuous functions of a closed interval (for a large variety of meanings of close, in fact). But what I didn’t realize at the time was that this was very much non-unique to polynomials — for example, it is also true for things of the form … (this follows easily from the Stone-Weistrass Theorem).

A deeper and more satisfying answer lies in Taylor’s Theorem. If we know the derivatives of a function at a point k, the natural way to approximate it is:

A polynomial! So polynomials are a natural result of the idea of a derivative. And while I have nothing more than anecdotal evidence for this, I’d suggest that, while many people may not know the formalities of calculus or the word derivative, everyone innately understands the idea. Certainly, young children understand ideas like speed. And the `rules of differentiation’ can be told to you by a lay person, if you ask in the right manner (sum rule: speed of a person walking on a ship, chain rule: speed of person in a movie when you fast-forward, and so on…).

But what is a derivative? One could just say that it is the rate at which something is changing and be done, but let’s go a little deeper. The definition of a derivative is usually:

It’s the ratio between change in y and change in x. And so if the derivative is A, we write dy/dx = A, meaning that A is the ratio between dy (the change in y) and dx (the change in x). One may also write it in the `differential form,’ dy = Adx, meaning that the change in y is A times the change in x. This latter form leads to a natural way to describe the derivative: it, times the change in x, is the amount one needs to add to move forward.

But there’s another interesting question: how much does one need to multiply by to move forward? This question gives birth to what we now call multiplicative or geometric calculus (Wikipedia deleted the main article, sadly, but this one still has useful content).

In many circumstances, multiplicative calculus is highly natural; for example, the decay of radioactive materials or the unconstrained growth of bacterial colony have constant multiplicative derivatives.

Just as normal derivatives naturally give rise to polynomials, multiplicative derivatives naturally give rise to things of the form … So you see, if we naturally thought of rates in terms of “how much do I need multiply by to move forward” instead of “how much do I need to add to move forward” things would be rather different. We live in a society with a very particular way of thinking, one might call us `polynomial centric’ or `linear approximation focused’.

Now an argument can be made that multiplicative derivatives would make standard physics problems more challenging. While I think it is important to keep in mind that this is simply a tradeoff and that there are other problems that are easier with the multiplicative approach, it is a valid point to consider. The easy solution would be for the Aliens to measure distance and related values in exponential form, in which case the problems would become computationally isomorphic to our normal ones. Still, there’s a lot of anthropomorphism in the assumption that they’d have the same “standard problems” as us. We like to consider point sources and things moving at constant speeds because they are simple and intuitive cases, but what’s to say that those would be the `simple’ cases they’d consider?

Regardless, there is clearly a serious contender as an alternative to polynomials that we humans can see. One of the things that really excites me about the possibility of Humanity contacting an Alien species (even if they are at a similar level of technology to us and are too far away for us to ever physically interact) is that it seems very likely that they will have a profoundly and inconceivably different perspective, and by extension abstractions, than us. I want to learn alien mathematics! And, if nothing else, I think that seeing another way of thinking will give us a lot of insight into our own. And on a similar note, I think a lot of the value of thought experiments like this one is the insight they can give us into our selves…

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