One of the great philosophical conundrums involves the origin of numbers and mathematics. I first learned of it as Platonic vs Aristotelian views, but these days it’s generally called Platonism vs Nominalism. I usually think of it as the question of whether numbers are invented or discovered.
Whatever it’s called, there is something transcendental about numbers and math. It’s hard not to discover (or invent) the natural numbers. Even from a theory standpoint, the natural numbers are very simply defined. Yet they directly invoke infinity — which doesn’t exist in the physical world.
There is also the “unreasonable effectiveness” of numbers in describing our world.
For example, any ballistic object, from a rifle bullet to a thrown baseball (or an ICBM), follows a parabolic path. The math describing a parabola couldn’t be simpler; it’s basically x-squared.
Squaring something is a natural geometric idea. If we have some length, say ten feet, squaring it gives us — literally — a square with ten-foot sides.
If we use one-foot square tiles to pave the area, it would take 100 of them. Ten squared is one-hundred. A simple natural geometric idea. (Which generalizes to rectangular areas.)
This simple squaring idea lurks behind every thrown (ballistic) object big or small. (There are other factors that apply — wind resistance, for instance — and gravity is assumed, but basically ballistic objects follow a parabola.)
(The x-squared parabola is also useful as a focusing reflector.)
Or consider pi, which is another simple geometric idea: the ratio between a circle’s diameter and circumference.
This simple geometric idea (of pi) shows up in most sinuous or cyclical processes. Which makes sense. What’s a little eerie is how it shows up in formulae that don’t seem to involve cyclical processes.
There is also the idea of infinity. (Which gets worse deeper into math.)
Nothing is infinite in the real world, and this seems to tag math as at least somewhat imaginary. (Math even has things called “imaginary numbers”!)
Infinity allows weird things like the Hilbert Hotel. In general, infinite subsets of infinite sets are equally infinite — are the same size. Divide infinity by a million, and it’s still infinity.
It really makes one wonder just how “real” math can be.
And the infinity stuff gets worse the deeper into math we explore. The first step into the imponderable is from plain old infinity to the continuum of real numbers — each of which is an infinitely tiny island separated from direct contact with all other numbers.
From there the path leads to math using orders of infinities, and I get off the bus before then. I’m not sure there is more than countable infinity and uncountable infinity. Anything beyond that is a fever dream.
The thing is, even from the beginning, math is an abstraction of the real world. The natural numbers come from recognizing the idea of sets of things — four dinner plates, 50 sheep, 300 fruit trees.
Everything else grows from that seed, which has a simple, obvious, downright physical theoretical definition:
- An empty bag.
- Adding one thing to the bag.
Axiom #1 gives us zero. Axiom #2 gives us the rest of the natural numbers.
A few more axioms give us basic mathematics (addition, subtraction, and so on).
There seems to be a strong case that math is invented, that it’s an abstraction we create of the physical world.
But why is this invention so effective in describing the physical world?
We designed it for that purpose, but even so the key seems to fit the lock weirdly well. And why does the key seem so elegant and simple? Again and again, math equations for real world processes pass Ockham’s razor with flying colors.
There is also that the idea of a sphere or cube seems too general to be invented. Such fundamental geometric ideas do seem somehow to live in Plato’s realm of perfect forms.
It does seem we must discover such basic ideas.
Discovering circles leads to discovering transcendental magical pi (once you think really hard about circles), and that seems to argue all of math must be something we can discover.
Math is thought to be so universal, so inevitable, that it can be a first language for contact with technological alien civilizations.
Indeed, while the natural numbers are an abstraction based on the ideas of sets, this also seems like such an obvious observation that any intelligence must discover it.
Or it’s so obvious any intelligence must invent it. It could work either way.
But what makes it so obvious, so inevitable?
Having it actually exist in some Platonic sense would explain that. It would explain why it’s there to be discovered.
The problem is that it’s hard to make sense of a Platonic realm.
[Neal Stephenson took on the idea in his novel, Anathem, but it was actually one of the weaker aspects of the story. Fortunately it wasn’t central to the plot. Most of the book works just fine if the “aliens” are from our universe.]
Ultimately, I think the question of a Platonic realm isn’t meaningful. (It’s certainly philosophically problematic.)
I think what’s happening is that reality works according to patterns, and these patterns are based on simple underlying rules or principles.
As such, it’s not surprising that the language we invent to capture and describe these patterns turns out to itself have interesting and useful patterns.
And it’s not surprising we see fundamental patterns (like the ratio of a circle’s diameter and circumference) popping up in myriad places. Those patterns are fundamental to reality itself.
What we’re seeing in physics laws and mathematics is the lawful behavior that governs reality. Math, in some sense, gives us access to the machinery of the universe.
When it comes to math, we’re playing with the gears of reality.
One of my favorite science quotes is due to mathematician Leopold Kronecker, who said: “God made the integers, all else is the work of man.”
I think there is truth to this.
The integers come from basic observation. The need to count things is fundamental. Another fundamental need, dealing with parts of things (like half a loaf of bread), takes us to the rational numbers, the fractions.
There is no requirement forcing us to accept the real numbers and the continuum. Energy and matter are quantized, so values associated with them can, in principle, be expressed by rational numbers.
Many scientists believe that gravity must be quantized, and it’s possible that time and space are also quantized. At the least, the Planck Length prohibits us from seeing anything below that length.
The thing is, the continuum, while it seems a reasonable abstraction of numbers, leads to all sorts of numerical conundrums.
Is it possible the patterns of reality are not expressed in real numbers, but only in rational numbers? Is it possible we really did make up the real numbers?
If so, it would suggest the odd turn of phrase:
Reality isn’t real,… but it is rational.
And apparently complex.
But that’s a whole other post.
Stay rational, my friends!