Among those who try to imagine alien first contact, many believe that mathematics will be the basis of initial communication. This is based on the perceived universality and inevitability of mathematics. They see math as so fundamental any intelligence must not only discover it, but must discover the same things we’ve discovered.
There is even a belief that math is more real than the physical universe, that it may be the actual basis of reality. The other end of that spectrum is a belief that mathematics is an invented game of symbol manipulation with no deep meaning.
So today: the idea that math is universal and inevitable.
A while back I wrote about how we may have discovered the basics of mathematics in our need to count things (from sheep to ships). That progression assumed humans on Earth (with sheep and ships).
This time I want to try starting from first principles with as few assumptions as possible. I’m going to start by assuming physical existence, intelligent self-awareness, and minimal sensation.
Let’s see where that takes us.
The One True Fact that we can know regarding intelligent self-awareness comes from René Descartes’ famous statement: Cogito ergo sum — “I think, therefore I am.” From this fact, and our self-aware thought process, what might we derive?
We can notice that we seem to experience the passage of time. There are thoughts we had earlier. There are the thoughts we are having now. We may even realize there are thoughts we haven’t had yet.
A sense of time passing leads to the need to measure its passage. We might do that by noting heart beats or breaths; for us time’s basic units are on the order of seconds or minutes.
A being capable of sensing chemical processes might measure time on a much smaller scale.
A being capable of sensing only the passing of days (or seasons) might measure it on a very large scale.
But sensing the passing of time leads to the need of a metric — a means of measuring, and also recording, the passage of time.
“Yesterday” and “an hour ago” are relative measures that reference the current moment. Dates (and times) are more absolute measures, although even those are relative to some “mile marker” moment.
Time implies the concepts of before and after — relationships between moments in time.
The sense of units of time, whether vibrations of electrons, heartbeats, mere thoughts, or the passing of seasons, seems to lead directly to the need to count those units, to the idea of the natural numbers.
Merely thinking thoughts, let alone noticing the passing of time, introduces the key ideas of objects and — crucially — classes of objects.
Not only do there exist things, but some things are similar enough to be grouped together.
The passing heartbeats, or the sensation of passing days, each pulse is like another.
To recognize multiples of a type of thing is to recognize the natural numbers.
But what if there are beings who sense time only as a smooth continuum? It’s hard to see how a mind would never discover periodicity or multiple objects, but that could be a failure of my imagination.
Perhaps such a being might see past events as magnitudes of time from its current moment. Even so, there are still relationships between events and the idea of intervals of time between them.
We have the idea of a continuum of time, and that gives us before and after as well as shorter and longer.
If we consider two past events, even if we sense the time spans as, say, an electrical potentials in our minds — if all we can say is that event A happened longer ago than event B — we still have the two events!
If we can remember specific moments, or specific thoughts, then some form of counting seems inevitable. Remember that the natural numbers are really set size cardinals. Counting is nothing more than considering how many objects are in the set.
So far we haven’t opened our eyes. We’re just inside our own heads noticing that our thoughts occur in time. And sensing how our bodies feel passing time.
Yet we have almost certainly recognized objects of thought — ideas and memories.
It would seem hard for intelligence to progress from that without the natural numbers. All it takes is the recognition of objects (which seems impossible not to have) and the recognition of classes (which might be an indicator of intelligence — it is something of a distinction between humans and animals).
Now we have sets, and sets have cardinality, and — voila — out pops the natural numbers. (Given the obvious question of what’s between two natural numbers, it’s likely we’re also onto the rational numbers or even the real numbers.)
We have also discovered relationships, which, with a little thought, lead to the idea of operators.
All without opening our eyes or experiencing the physical world around us.
Let’s open our eyes now and experience the physical world.
Just as we realized we exist in time, now we see that we exist in space. If we have any experience of space at all, what might be basic to that experience is a sense of distance (of spatial extent).
Exactly as time separates points in time, spatial extent separates points in space.
Recognizing spatial extent leads us to the need for a spatial metric, a way of measuring space.
As with time, we can ponder the idea of a mind that doesn’t see distances as having units, but as magnitudes.
Perhaps they can only be compared as greater or lesser (but again comparing magnitudes implies distinct, thus countable, objects: the magnitudes themselves).
These metrics of time and space allow more than measuring. They allow us to label, and thus remember, think about (calculate with!), and discuss, these concepts.
Even if an alien race consisted of a single vast individual or mind-sharing collective (think hive), remembering and thinking about time and space requires labels.
So it almost seems impossible to think about these things without discovering (some might say “inventing”) mathematics — the language necessary for those thoughts!
The German mathematician, Leopold Kronecker, famously said: “God made the integers, all else is the work of man.”
This recognizes how fundamental the natural numbers are (they are called “natural” for good reason).
In a universe of discrete objects, shared properties create classes (sets) of all objects having that property (e.g. “redness” or “spaceshipness”).
Sets of objects not only can be counted, they demand to be counted. The one thing all sets have is a size, a cardinality!
The natural numbers, either through the division operator, or through pondering natural steps between integers, lead fairly directly to the rational numbers. The world of countable objects seems intimately part of a universe of recognizable objects.
Arriving at the real numbers — the world of uncountable objects — is another milestone along the journey, and there are multiple ways to get there. A species who thinks naturally in magnitudes might even get there instinctively.
The same “what’s in between” logic that leads from the naturals to the rationals can lead to the reals. A growing ability to measure magnitudes with precision can also lead to the reals — or if not, at least to the rationals (which just adds to their inevitability).
A self-imposed word limit prohibits getting too deep into the inevitability (and universality) of geometry now, but careful consideration of a circle tells us something weird is going on — something beyond the discrete world of countable objects.
The universe’s duality turns up once again. It’s filled with both countable and uncountable things. As such, experiencing the universe in any meaningful and intelligent way seems to lead naturally — inevitably — to mathematics.
The other question involves the universality of math. I’ve written before about the difference between how we spell numbers versus what those numbers actually mean. As words are just local handles to ideas, physical numbers are just handles on mathematical ideas.
These ideas, at the very least, have the reality of ideas and in many cases represent real-world objects. Taking the metaphor a step further, mathematics is the language that uses those words.
But just as different languages express the same idea different ways, different mathematics can only express the same mathematical ideas in unfamiliar ways.
Put simply, 2+2=4 regardless of how it’s expressed!
What might happen — what does happen in language — is that one may express ideas unknown to a culture that hasn’t encountered them (so far).
Certainly an advanced culture might have advanced mathematics, but they wouldn’t expect to use that in first contact (any more than we would).
Further, they still have to balance their alien expense reports, so ordinary mathematics won’t have gone away (any more than it has for us).
For me, the question isn’t whether mathematics is inevitable and universal. It’s how can it not be?
Stay numerate, my friends!
 Note that we are adding another assumption: That all intelligent minds experience time in the same serial fashion that we do. Could there exist beings who experience time all at once or that have no (or just a different) sense of past or future?
Yet time seems to be a universal aspect of reality. Chemistry, and thus biology, is a process that occurs in time. Motion also occurs in time (in fact, you can’t define it without time).
The idea of thought without the experience of time seems incoherent to me, so I’m going to claim that time is inherent to thought. (If anyone disagrees, we can consider it Debate Point #1!)
 Yet another implicit assumption: That intelligence comes with memory, but I think this is also a reasonable assumption. Memory is necessary to learning (and judging comparisons) and therefore, surely, intelligence. (Alternately: Debate Point #2!)
 Maybe intelligent trees (who communicate via chemicals wafted on the wind) sense only the passing days and the longer rhythm of seasons. But these are distinct, very similar, objects of perception, so even intelligent trees with no language (written or verbal) might still discover counting and, thus, mathematics.
 If you are disinclined towards “objects of thought” consider that, really, everything is an object of thought (even your sensations).
 It starts by realizing we can add two numbers together. That leads directly to subtraction. Multiplication is just serial addition. Division is a bit more of a leap, but it follows from multiplication (and division gets you to the rational numbers.). Exponents are just fancy multiplication.
We’ve already recognized some relational operators: greater-than, less-than, and all the others. These lead to the idea of logical operators: not, and, or, xor. At this point you’ve discovered arithmetic.
 We might ponder whether a species necessarily discovers the idea of three-dimensional space. Could a physical being exist without a notion of volume? Might a being only be aware of two-dimensional space (surfaces) or even one-dimensional space (lines)?
That seems unlikely to me, but I won’t rule it out entirely. It does seem that any sense of extent, or of movement in space, must lead to the recognition of dimensions. After all, we ponder 4+ dimensional space. (I’ve written about dimensionality before. See Dimensional Coordinates.)