Counting to Abstraction - Evolution of Number !!

Yesterday, we were watching the film The Martian. Not the first time. Each time one gets caught in the magnificence of Mars, its harsh conditions and how the protagonist leverages his engineering and botany skills to survive. But what caught my attention this time was his attempts to communicate and, in the process, his use of hexadecimals.

And it got me thinking. You ever stop to think about how wild it is that we can just... write down a negative number? Or use hexadecimal to communicate? We take it for granted now. But there is something deeper going on here.

From not being able to count, to creating number systems with base 10, base 2, base 16, base 8, to negative numbers, to irrational numbers, to complex numbers... we have come a long way... more than just the mathematical evolution... rather about how human consciousness itself has expanded over the centuries.

Think about it. Each breakthrough probably felt radical at the time. Moving from concrete counting... five fingers, five objects... to positional notation with zero? That was extraordinary. It opened up the ability to represent infinity, to do operations that were simply impossible before. Then different number bases emerged because, well, sometimes you need them.

But then things got weird.

Negative numbers bothered people for ages. Seriously. How do you have "less than nothing"? It sounds absurd. And yet, once we wrapped our heads around it, we could represent debts, temperature below zero, direction and motion. The mind had to expand. It had to accept that a number could be abstract... not tied to something you could actually hold in your hand.

Irrational numbers? That's where things get properly mind-bending. They showed us that not everything reduces to simple ratios. That geometry itself had mysteries built into it. It didn't just add a new category of numbers... it changed what we thought we knew about reality.

But then we went further. We asked: what's the square root of negative one? Mathematicians looked at that and said, "That doesn't exist. It's impossible." And yet... what if we just defined it anyway? Let's call it "i" And from there, we get complex numbers... numbers that exist in two dimensions, that blend the real with the imaginary. What a cocktail! This is where it gets almost philosophical. Complex numbers don't correspond to anything in the physical world the way five apples does. They are not even abstract-but-intuitive like negative numbers eventually became. They seemed purely invented, a mathematical game with no connection to reality. Mathematicians in the 18th century thought they were clever tricks, nothing more. Except then something extraordinary happened. It turned out complex numbers were essential for physics. Wave equations, quantum mechanics, electrical engineering—all of it relies on complex numbers. Reality uses them. The universe apparently thinks in terms of imaginary numbers.

Here's what gets me: each time we expanded the definition of "number," we weren't just adding more symbols. We were fundamentally changing what "number" is. We went from counting as a mirror of the physical world to working with objects that only exist in abstract space... and then even further, into spaces that seem to have no physical analogue at all. Yet there they are, woven into the fabric of how things actually work.

A Roman could not operationalize zero the way you do. Not because (s)he was less intelligent... it is because the conceptual scaffolding did not exist yet. You can't think about something until you have the mental tools to hold it. A civilization without negative numbers couldn't even pose certain physical problems. A culture without calculus couldn't reason about motion the way Newton could. A physics without complex numbers couldn't describe light or atoms. The abstractions we build become the boundaries of what we can understand.

And here's the strange part: we're not just discovering these things out there in the universe, waiting to be found. We are creating them. We invented zero. We constructed negative numbers. We defined imaginary numbers out of sheer mathematical audacity. We defined hexadecimal so machines could talk to each other. We made these things up.

And then they turned out to perfectly describe how reality actually works.

Why? Why should our invented abstractions align so eerily with physical reality? It's the kind of question that keeps you up at night.

There's something cognitive happening here too. Watch a kid struggle with negative numbers or try to wrap their head around π. That struggle isn't just them being bad at math... it's a real conceptual barrier. They're retracing steps that humanity took centuries to figure out. Their mind is literally expanding in the same way the collective human mind did. Next time be patient with them and take time to explain :-)

So here's what I think: we're not passively receiving eternal mathematical truths. We're actively creating conceptual frameworks, and in doing so, we're expanding what we can even think about. We build the ladder, we climb it, and from higher up, we see things that were always there but invisible from where we stood before. Sometimes we build ladders that seem to lead nowhere—and then we discover they lead exactly where the universe was already going.

Next time you use a number... whether it's the decimal you use every day, the hexadecimal encoding a secret message, or even an imaginary number in some equation... you are holding evidence of human consciousness learning to think bigger... a proof that reality itself is more expansive than our intuition initially allows.