Numbers are fake.
Especially infinity. But even four or seventeen.
I. Infinity is fake.
“The infinite is either that which is incapable of being traversed because it is not its nature to be traversed (this corresponds to the sense in which the voice is ‘invisible’), or that which admits only of incomplete traverse or scarcely admits of traverse, or that which, though it naturally admits of traverse, is not traversed or limited…”
— Aristotle, Metaphysics XI.10
Infinity isn’t real.
At least, one kind of infinity isn’t real, and that’s the regular kind — the kind we usually mean. There are other kinds. In the quotation above, Aristotle slices infinity into three kinds:
The intraversable. This is something that, by nature, you can’t cross, measure, or count at all. It’s not just that a journey across this infinite thing can’t end, it’s that the journey can’t even begin. At the risk of spoilers, I believe this kind of infinity is real. But it’s not what we usually mean when we say “infinity,” and it’s not the kind we mean in math class. So let’s set it aside for now.
Something that “scarcely admits of traverse” — I could be wrong, but I think this means finite but metaphorically or hyperbolically infinite. We may call something “infinite” and mean that, practically speaking, we can never traverse it, even though it is theoretically traversable. My children find “infinite” things to fight over, and I can’t count them, but in theory someone (or some ungodly digital thing) could have been keeping a running tally. For all practical purposes, the Horsehead Nebula is “infinitely” far from my house, but it’s actually a measurable distance (about 1,500 light years, i.e., about 8,820,000,000,000,000 miles).
The regular kind of infinity — ∞ / mathematical infinity. This kind of infinity means something that “naturally admits of traverse,” but “is not traversed or limited.” A journey across it can easily begin, but it can never be completed. You could spend a lot of time counting an infinite set of things in this sense of infinity, but you’d never finish counting them all. In numbers, for example, there is (according to conventional mathematical narratives) no “highest number,” because numbers just keep going up and never stop.
It’s that last kind of infinity — the traversable but inexhaustible — that I wish to slap across the face with a glove. J’accuse, ∞!
That kind of infinity just… isn’t real. It’s not just false but inherently illogical to define or apprehend. Infinity, per se, is nonsense.
Prove me wrong! Show me an infinity.
You can’t do it. You can’t do it because infinity is a scam, a con, a flim-flam. Infinity promises you that somewhere, somehow, there are things where you can Just Keep Counting them and the counting will Never End. But it's a lie. Count all you want; the counting will always end, I absolutely promise you. It will end because eventually the Earth will crash into the sun and you (even if you are an immortal robot vampire) will burn up in it. And if you have, by then, sailed far (but finitely) away, over every one of the 8,820,000,000,000,000 miles to the Horsehead Nebula, safe from this solar system’s demise, I promise that someday, some other kind of astronomical something will eventually come for you and end your counting.
But the thing is, far more likely than any of that, your counting will far sooner than aeons in the future at the deep freeze and/or heat death of the universe, because, in your actual, existing life, you will get bored of counting pretty quickly and decide to go have lunch. You will never get to this supposed “infinity” you’re counting toward — by which I mean that you will get somewhere else, and no further.
And let’s pause on that “somewhere else” to which you get, because it has a corollary: Not only is infinity fake, but there is, in fact, a Highest Number. Yes, Virginia, there is a Highest Number in the universe. Big Math wants you to think there is no Highest Number, but there is. No, I don’t know what it is. And no, it isn’t one fixed number that stays the same over time. But it’s there: The Highest Number is simply the highest number that anyone has ever said, written, or thought of. And, sure, as soon as you think of the Highest Number, you can immediately think, “That plus one,” or “That times itself, to the itselfth power,” and keep going on like that. But, eventually, as before with your (pathetic) attempt to count to “infinity”, you will again get bored and go have lunch — and wherever you left off before lunch, that is (for now) the Highest Number in the universe.
Look, the discipline of mathematics virtually admits that “infinity” leads us into contradictions, albeit without admitting the full implications of those contradictions (i.e., that infinity is an actual fraud). Mathematicians will tell you, with a straight face, that some infinities are bigger than others. Something that Georg Cantor decided to call ℵ0 (an infinity supposedly the size of the set of all natural numbers, even though there is no fixed quantity of natural numbers) means you keep counting and it will literally never end, and yet what he called ℵ1 (an infinity the size, purportedly, of the set of all real numbers, even though there is no fixed quantity of real numbers) is even larger than that.
The proofs behind this are very nice in terms of formal logical validity, but I still don’t buy them. Formal logical validity does not always equal truth. (I am fond of the old joke, which I first heard from my dad, proving by impeccable syllogism that a baloney sandwich is better than eternal happiness. How so? Well, nothing is better than eternal happiness; and a baloney sandwich is better than nothing.)
The idea that something can go on literally forever but still be smaller than something else that goes on literally forever is not an idea; it’s mouth sounds. It’s dogfood as language. It should be properly interpreted as the Console of Mathematical Logic displaying a 404 error message. These proofs about bigger and smaller infinities should be seen as hints that the thing we started with — regular infinity, a.k.a. ∞, a.k.a. ℵ0 — was always a mirage.
Let’s not confuse language for reality. “Whereof one cannot speak,” wrote Wittgenstein, “thereof one must be silent.” But Wittgenstein never met my kids. One should, perhaps, be silent about nonsense, but plenty of people (myself included) don’t let that stop them/us. Language can often refer to reality, but a language — any language! — can also output nonsense beefcake statutory blowtorch lethargically gorgonzola and. Beyond being merely hallucinatory or insane, a language can form the Epimenides paradox — put most simply, “This sentence is false” — which is perfectly cogent and absolutely grammatical, but has no truth value. It isn’t true, it isn’t false, and there is no in-between. With his system of Gödel numbering, Kurt Gödel showed that mathematics is a language that can generate the same kind of paradox.
But forget Gödel; my contention is that Cantor’s hierarchy of bigger and smaller infinities is also a bit like the Epimenides paradox — not a perfect parallel to it or isomorph of it, but like it simply in the sense that anything you try to say about “bigger and smaller infinities” has no truth value. Not because Cantor’s logical proofs are wrong, but because the object those proofs manipulate were already outside the bounds of reason before he touched them. Saying “some infinities are larger than others” is like saying “some square circles are rounder than others”. All the individual parts have meaning, but put together, it means nothing. A number is something you count; infinity means you can’t count it. Treat infinity (uncountable) like a number (that which is counted) and you already need your meds adjusted. You can’t rank things as “larger” or “smaller” if they have no measurable (traversable) quantity, because “larger” and “smaller” are properties that emerge directly and exclusively from quantity and mean nothing apart from quantity.
So the whole thing is a contradiction. This should not scandalize us; after all, this isn’t the only 404 error in math. If you try to divide 112 by zero, there is no “answer” to the question in mathematics. The answer to division by zero — which is not, properly speaking, an answer in mathematics, but rather an answer in metamathematics — is, “You just can’t do that.” But I say the same should apply to any statement about infinity! Once you go pushing the “8” over onto its side into “∞” and calling it a sign for something called “infinity,” you are already out of proper bounds; you’re not doing math, you’re doing mischief, and you deserve a good spanking.
Okay, fine… That was harsh of me.
Look, you can play with your supposed “infinities” if you really want to. Having imaginary friends can be fun. Just don’t take them too seriously.
In fact, another reason I was too harsh above is that infinity isn’t so special in being incoherent. After all, this whole business of mathematics is slightly suspect, seeing as how all numbers are fake.
II. Numbers are fake.
“One, two monkey!”
— My eldest daughter (at age 1)
When my eldest daughter was just over one year old, and talking up a storm but still very much working out what things meant, this seemed to be her first working hypothesis of what counting is:
Grownup asks you, “How many [X]?”
You say, “One, two [X]!”
That is, we noticed that if she was eating strawberries, and we said, “How many strawberries?” she would consistently answer, “One, two strawberry!” But this would bear no relationship to the actual number of strawberries at hand. She might have two strawberries on her little tray or thirty-four — the spoken answer was always “One, two strawberry!” And, with equal reliability, we could ask about things that were nowhere nearby, or even nowhere defined, let alone existent:
“How many Glorp?”
“One, two Glorp!”
And yes, I tried to get her to at least nominally add one more number to this procedure by repeatedly asking, “How many three?” (And it worked.)
Eventually, the girl got wise and revised the procedure:
Grownup asks you, “How many [X]?”
Identify item or group of items referred to by [X]
Point to each item in rhythm while reciting a chant that goes: “One, two, three, four, five, six, seven, eight, nine, ten!!”, putting a special emphasis on “ten” and cycling through the items as many times as are necessary to reach “ten”, because shouting “ten” seems to be the point of the exercise.
So the count grew to ten (regardless of how many items were there), which had a nice sound of completion every time. Yet this, too, seemed not always to please those inscrutable grownups. And so, at last, of course, she eventually brought the procedure to its final form:
Grownup asks you, “How many [X]?”
Identify the item, group of items, or lack of items referred to by [X]
If a lack of said items, say “None!”
If one or more items, begin the rhythmic chant, but (and this is important!) only chant enough times to point to each item exactly once. Then stop, wherever in the chant you are.
Using certain regular rules, the chant can generate as many sound patterns as you need to keep the chant going a long way beyond ten.
And look: That procedure there — that’s what a number is. That’s all a number is.
But that’s just me, talking out of parental experience and common sense. If we ask Big Math, they don’t like that definition.
Consider this video, which is very well made, but also made me very grumpy:
The first 23 minutes are just great. I particularly appreciate how, in minutes 8 and 9, he explains that, while the whole problem is trivial if you just start from a place where you understand zero and one, he doesn’t want to do that — he wants to get to a definition of zero and one as well. I appreciate this spirit of rigor, even though (as we shall see) it ends up leading him astray.
(Interestingly, in minutes 10-12, in a digression explaining why we can’t just trust intuition, the presenter explains how logic shows that the infinite set of all integers is the same size as the infinite set of all even integers, even though the former has all the evens and all the odds that the latter lacks. What we should derive from this digression is that infinity is nonsense, but we’ve already established that above, so never mind.)
Minutes 12-23 include (among other things) the delightful way Bertrand Russell first imported the Epimenides paradox into set theory, confounding much work that had come before.
The problem that interests me arises in minutes 24-29, when the presenter explains how Zermelo-Fraenkel Set Theory rebuilds what Russell’s paradox destroyed: a rigorous definition of numbers. And in the course of this — attempting to construct a basis for numbers that supposedly relies on the fewest assumptions possible — the presenter lays out five axioms that we must accept, unproven! From these, you can prove a sixth, and, from that, you can get to the concept of “one”.
But hold on a minute. Now we’re just assuming axioms and taking them on faith — five of them, no less! But remember, back in minutes 8 and 9, we declared it unacceptable to assume even two things, namely, zero and one! Once I’m giving you five freebie axioms, why don’t I just give you the concept of “zero” and “one” to begin with instead? It’s much cheaper! But no, the presenter, and the consensus of Big Math whom he represents, ask for more. They say we must assume five rules — each of which is no more proven beyond intuition than one or zero is!
This is balderdash. As I have written before, logic can never fully escape faith and intuition, but I do believe in trying to use logic to minimize the leaps of faith we must make, and this is the opposite of minimizing! We’re assuming five things, says this theory, so that we don’t have to assume two things!
But all of this is just groping towards what toddlers learn intuitively. The fancy Zermelo-Fraenkel Set Theory is very nice, but if it didn’t bring us to 1+1=2, we wouldn’t want it, and that’s because 1+1=2 isn’t anything we derived from the fancy theory. The fancy theory just confirms what we already knew. And, as that fact implies, the fancy theory is not what numbers are. It may be what mathematicians want numbers to be, post-facto. But it’s not what they are. It’s not where they come from, it’s not how we use them or think about them, and it’s not how or why the mathematicians first learned numbers or began to interact with them.
My daughter, by age two, knew what numbers are: Numbers are a chant we use while we point at things. We use the chant to measure the quantity of things. But numbers aren’t in the things; numbers are the chant. “Four” doesn’t exist outside the chant. “Four” is just a shortcut. If we’re counting cows, then there is no “number” of cows in baseline reality. There’s no Spirit of Four-ness hovering amongst the four cows, nor is there an Angel of Math hovering there holding a set of five axioms and nested abstract sets showing the succession from One to Four. “Four cows” is just a shortcut for saying what we really mean, which is: “Hey look, a cow, and another cow, and another cow, and another cow.”
Baseline reality doesn’t count them at all. But then, baseline reality doesn’t even lump them all together into the category called “cows,” which also doesn’t (at baseline) exist. “Cow” is a social construct. Categories are fake! People fight so much about race and gender being social constructs, and they are ( but let’s leave that can of worms for another time) — but so even is species. So even may be the supposedly rudimentary units of chemistry and physics! Cakes, cows, quarks — all of life is stuff we experience in artificially constructed groupings, a great lumping together of pluralities that need not, at baseline, be lumped.
That’s not to say that our mentally/socially constructed categories aren’t useful — they absolutely are, and so we rightly care about them and put a lot of energy into negotiating when and how it makes sense to use them. Categories can be supremely important on the practical level, and we can affirm that while still remembering that categories are always ultimately artificial. Baseline reality has no categories, no patterns, no laws, no species, no types or kinds of any fashion whatsoever. Baseline reality is only a Bunch of Stuff Being and Happening, each Thing and each Happening being distinctly itself. Two things we call oxygen atoms may be similar enough to our perceptions that they respond equally predictably to a fire, but at baseline, they are just… two things. Two distinct things, despite their similarity — for if they weren’t distinct, we couldn’t even begin to apply the artificial construct of “two” to them.
And yet… is even that idea of distinction, of plurality, real at the absolute baseline?
III. Everything except Infinity is fake.
“If you can quantify it, it’s not sacred.”
— Rick Rubin
We’ve had 25 centuries to do it, but no one yet has convincingly rebutted Zeno of Elea.
He’s the Greek fellow who used a series of paradoxes to argue (at least) that there are not many things in the universe, and (arguably) in favor of monism — the idea that there is only one thing in the universe. (Actually one can’t say “in” the universe; say rather, there is only one thing, which is the universe.)
And you know what? I think Zeno was right.
People have various alleged refutations of Zeno. Some such arguments are from calculus, but these seem to rely on various uses of infinite series or other forms of infinities — and you know, by now, how credible I deem mathematical infinities to be. Other arguments hold that the smallest parts of reality must not be divisible — that reality is, basically, pixelated — and this is called atomism. One problem with atomism is that no such smallest and indivisible atom/pixel of reality has ever been observed, and another is that modern physics depends on models that are quite different.
Then again, modern physics also relies on calculus and its infernal infinities… And, now that we come to it, Zeno and all his critics seem to rely on infinities.
All roads lead to infinity. Infinity is a contradiction. Infinity cannot reasonably be. But some kind of infinity or another seems inescapable.
For my money, the best refutation anyone could ever give Zeno was to gesture wildly at the world. If you throw me a ball, the ball actually arrives and is not, in fact, infinitely delayed by traversing infinite halfway points. But that’s not an explanation, just a perception. And perceptions can be flawed all the time (cf. dreams, optical illusions, etc.), so I, for one, can’t count Zeno out.
Cards on the table: For reasons unrelated to Zeno and unrelated to the contemplation of numbers, I believe plurality is fake. I have arrived at this conclusion not via math but via consciousness. A pantheistic/panpsychist monism — a single Consciousness, pervading or rather consisting of the Universe, which for cultural reasons I like to call “God” — neatly solves the “hard problem of consciousness” in a way that nothing else does.
But, having done so, this idea of a Unified and Conscious Cosmos can also cast Zeno’s paradoxes in a new light, while simultaneously suggesting a direction towards which (though not a means by which) Unified Field Theory might eventually solve the remaining mysteries of physics.
Hypothesis: Plurality is fake. Extension is fake. Motion is fake. Particles are fake. Spacetime is fake. Zeno of Elea was correct in implying that our perceptions of motion and plurality are illusory. There’s only One Thing in the universe. One Time. One Place. One Field. We are all just disturbances in that One Field, dreams that the One Indivisible Thing has about itself.
Yet, if all of existence is truly so unified and indivisible, then even to call that ultimate unity “one” is off-base. “One” is just the first of a list of numbers, after all, and numbers are fake; we’re talking about transcending quantity altogether.
And so, at long last, we return to Aristotle’s first and deepest definition of Infinity: not that which you can traverse but never completely get across, but that which you can’t even start traversing.
The One Thing that there is — God, the Unmoved Mover, the Cosmos, the Universe, the Unified Field — is infinite. Not infinite in the mathematical sense of counting without end, but infinite in the sense of being fundamentally uncountable.
Infinity is not fake. Infinity is the only thing that’s real.
“Hear, O Israel: YHWH is our God; YHWH is One.”
— Deuteronomy 6:4
“…I am First and I am Last…”
— Isaiah 44:6
“The idea which we need to form in our mind of unity is of oneness that is complete, a uniqueness that is absolutely devoid of composition or resemblance. Free, in every respect of plurality or number, that is neither associated with anything nor dissociated from anything.”
— Bahya ibn Paquda, Duties of The Heart, First Treatise on Unity 7:29