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.
The transcendental magic of pi aside, the eerie thing is how often it pops up in basic formulas. For instance, it shows up in Einstein’s General Relativity equation:
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.
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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.
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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.
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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.
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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.]
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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.
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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?
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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!
∇
Logos con carne 
March 5th, 2020 at 11:36 am
If it wasn’t clear, I’m saying both Platonism and nominalism are both correct and both incorrect. Math is a blend of abstractions based on reality patterns, and some of those abstractions directly correlate with aspects of reality whereas some of them don’t.
It is similar to literature in that some writing is about the real world and some writing is about imaginary things. Math can do both, too.
But it’s all based on the underlying patterns of reality. Which, at that, is something of an Aristotelian view. But those underlying patterns also comprise a kind of Platonic realm.
March 5th, 2020 at 1:14 pm
Put another way, 2D circles and 3D spheres are there to be discovered based on underlying principles of position and distance. Consideration of the 3D space we inhabit leads to an understanding of spaces with other dimensions (2D, 4D, etc).
Likewise cubes and the other Platonic solids. These are all based on very simple notions — equal distance notions — in 3D space. All math fundamentals are based on simple, often geometric, notions.
The real question isn’t about Platonism versus nominalism, but about why we live in a universe that isn’t just lawful and consistent, but in which simple rules (plus energy) result in complex structures emerging. Evolution leveraged that property of reality, and it led to us.
As I’ve mentioned before: Given the relentlessness of entropy, aren’t we lucky we live in a reality where complexity can emerge!
March 6th, 2020 at 10:28 am
I tend to think that both platonism and nominalism are wrong. Platonism because it just seems redundant, and nominalism because of the unreasonable effectiveness thing.
I think the foundations of mathematics are ultimately empirical. But those foundations enable us to engage in a priori reasoning and reach conclusions that may tell us about reality. In that sense, it’s just scientific theorizing by another name, except in the field of mathematics, we still value theories that have no apparent empirical correlates, at least if they’re logically coherent.
March 6th, 2020 at 2:02 pm
That’s pretty much what I’m saying here. I see both Platonism and nominalism as partly right, but there’s certainly no Platonic realm, and we do discover math’s foundations in our experiences.
Modern Platonism could be read as asking questions about the “unreasonable effectiveness” and elegance of math — about the underlying reason for it. (The answer I’m suggesting here is that underlying reality operates according to “mathematical” patterns that we discover.)
But we do make up a lot of stuff, too, using those fundamental building blocks. One of the eerie things is how often a made up “toy” — some mathematician’s idle fantasy — later turns out to be a useful tool. Which is one of those things that makes one wonder about nominalism.
There’s an extreme form, fictionalism, that says yes to Platonic abstractions, but denies that abstractions are real, thus all math statements are false. But that doesn’t mean they don’t tell a consistent logical story. Under fictionalism, new math is (of course) made up and false, but it must conform to the story told so far. (Like any good sequel.)
False in the sense that the statement, “Peter Pan’s Tinkerbell was a fairy,” is false — there are no such things as Peter Pan, Tinkerbell, or fairies. Yet in the context of the story, the statement is true. Math statements are true in the same sense.
I really like the comparison to fiction, but the view explicitly refuses to couple with the analogy. Still, I think the comparison highlights what I take from the view: the idea that some stories correspond to reality and some don’t. And, as in math, some stories that seem fantastic later turn out to be true. (And some never will.)
“In that sense, it’s just scientific theorizing by another name,”
What gives it the power to survive lacking empirical data is its logical rigor. Mathematical theories can stand on their own exactly because they can stand on their own logic.
And because they are seen as having value merely in virtue of existing — in a sense they explore some new part of the math landscape. One might explore a swamp of no perceived value, but at least it can be filled in on the map now.
March 6th, 2020 at 5:19 pm
The problem with nominalism vs Platonism is that it sees the world in terms of generalities vs particulars, as in which is more real. But do we have to see the world in terms of generalities? “Patterns” is an improvement, as you suggest. But “rules” make me queasy. As AI guru Judea Pearl puts it, “…rule-based systems never turn out well.”
March 6th, 2020 at 6:16 pm
Hello; welcome to my blog.
What I’m arguing, of course, is that both generalities and particulars have truth, although neither is exactly correct. I think we do need generalities, though. Science and math are founded on them, which means our technologies also depend on them. The trick, perhaps, is to keep an eye on the trees as well as the forest.
Science and math minded people have a different take on the word “rules” than people do for whom “rules” only ever meant “the laws of man.” Rules and law in science and math mean something very different than social rules and laws do.
For one thing, we can’t break science laws. That’s what makes one of my favorite physics quips funny:
Although a lot of folks sure wish we could break that law.
That said, it’s true rule-based AI never went anywhere (other than into winter). Learning-based AI is kicking some butt, though. It’s still rule-based, though; it’s just that the rules are micro-rules that apply to nodes rather than macro-rules that apply to the system. You can’t really escape rules; you can just push them down out of sight.
June 5th, 2021 at 3:11 pm
[…] idea (see the 2020 posts Rational vs Real and Number Islands) is that the real number system — an uncountable infinity — is an […]
January 25th, 2023 at 2:06 pm
[…] In the Rational vs Real post I mentioned that real numbers were each “an infinitely tiny island separated from direct contact with all other numbers.” The metaphor of each real number as an island comes from how, given any real number, it’s not possible to name the next (or previous) real number. […]