In the recent post Inevitable Math I explored the idea that mathematics was both universal and inevitable. The argument is that the foundations of mathematics are so woven into the fabric of reality (if not actually being the fabric of reality) that any intelligence must discover them.
Which is not to say they would think about or express their mathematics in ways immediately recognizable to us. There could be fundamental differences, not just in their notation, but in how they conceive of numbers.
To explore that a little, here are a couple of twists on numbers:
Rather than the ten-symbol positional notation system we use, which expresses numbers as strings using those ten symbols, Heechee numbers are strings of symbols representing prime numbers.
In the Heechee system, the order of symbols isn’t significant. In ours, which is called “positional” for good reason, it’s crucial! On the other hand, the Heechee system requires a lot more than ten symbols.
One problem is that the list of primes is (countably) infinite. That implies a need for an infinite numbers of symbols. Presumably, for the bigger primes, the Heechee used some form of compound notation, but this points out the first (of many) problems with this number system.
Each prime (up to a point) requires its own distinct symbol or notation. If we consider just the first ten primes, we need symbols for these values: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
We need a symbol for 1 to stand for the value one. (Although one is the loneliest number, it’s not considered prime and, of course, isn’t useful in creating products.)
Let’s assign the following eleven symbols, respectively, to the list of the ten primes (plus one): 1:(─), 2:(┌), 3:(╓), 5:(╒), 7:(╔), 11:(┐), 13:(╖), 17:(╕), 19:(╗), 23:(└), 29:(╘)
So Heechee counting goes:
─ = 1
┌ = 2
╓ = 3
┌┌ = 2×2 = 4
╒ = 5
┌╓ = 2×3 = 6 (or: 3×2 = ╓┌)
╔ = 7
┌┌┌ = 2×2×2 = 8
╓╓ = 3×3 = 9
That should give you the idea. The next number, 10, would be ┌╒ (2×5) (or ╒┌ (5×2) — but maybe the Heechee had a rule about lowest to highest).
The infamous 42 is┌╓╔ (2×3×7), 100 is ┌┌╒╒ (2×2×5×5), which makes 1000 ┌┌┌╒╒╒ (2×2×2×5×5×5)!
On the other hand, 99 is just ╒╒┐ (3×3×11). Some large numbers are fairly short: 841 is just ╘╘ (29×29) (and 881 is prime, so it’s just a single symbol, maybe something like: ╫).
Negative numbers are just a matter of a minus sign.
Rational numbers might need to be always expressed in a/b form because it’s hard to see how to implement decimal expansion. A number like 1.55 means 55/100 — the number of decimal digits determining the denominator. What would be the implicit denominator in the Heechee system?
Real numbers seem even a bit more challenging, but perhaps an expert mathematician (which I am not) would see a way. Maybe the Heechee had a different number system for the domain of the reals.
Perhaps a species might think in vectors and angles.
Remember how a 3D XYZ point is a location vector but also has an associated magnitude vector that represents the point’s distance and direction from the origin? (See Scalars & Vectors if you don’t.)
That magnitude vector (in 3D space) encodes three numbers which we can think of as two angles and a distance. Even a 2D version, one angle plus a distance, encodes two numbers. Suppose a species thought in terms of such vectors?
Their mental image of “numbers” would be of arrows swinging around in some multi-dimensional space. It need not be limited to 2D or 3D space (such as we are). Maybe they can think of arrows swinging around in 7D space. Such arrows would encode six angles plus the distance.
However this has an interesting limitation in that angles have a limit — they only express a distinct value from zero to some maximum, and then they “wrap around” and repeat. For example, 1° degree and 361° degrees (and 721° and more) are all the same angle.
On the other hand, our positional notation also has limits in that individual digits only go from 0 to 9. Imagine a number form in which each angle is one “digit” of the value, and these digits potentially have any value along the circle.
That would allow analog numbers of high complexity and precision. On the other hand, talking about and notating such numbers might be a challenge.
But the fact that I can imagine and describe these shows that they’re built of more fundamental concepts — concepts we know and understand.
We know a lot about prime numbers and angles. We know about surreal numbers! We even know about the mathematics of 7D space.
We know about hyperbolic geometry (which is pretty weird), for example, even though it doesn’t describe any actual reality we know.
The point is that all these more complex ideas are built on the basic foundations of mathematics.
In terms of calculations with numbers, Information Theory says it all boils down to ones and zeros. The ultimate number base is binary — the others are just prettier.
That foundation starts with the natural numbers and the idea of operations on those numbers. That leads to the rational numbers. The real numbers come along through analysis of the rationals, or through a posteriori experience. (Geometry, likewise, can be based on a priori or a posteriori analysis.)
The bottom line to me is that there are many ways to think about math, just as there are many spoken languages. Both are (in our case) human inventions designed to let us think and talk about the ideas the languages represent.
We live in a universe in where there is a fundamental divide between the realm of discrete objects and the realm of continuous objects (although it’s possible that if you get small enough, nothing is actually continuous).
As I wrote in Smooth or Bumpy, we find those worlds in many places around us (even in sports). Transfinite mathematics aside, I’ve long believed there are just two domains: the discrete and the smooth. My guess is that countable and uncountable infinity is the fullness of it.
Therefore, no matter how a species thinks about or notates its mathematical languages, under the hood we’ll find the same old relationships that our ancient civilizations found.