Here is a small proof that one plus one is equal to three. That is a statement which on the surface seems completely absurd, so to alleviate that first impression, let's carefully establish the logical elements which comprise this proof. Each element will be seen to be logically sound. Then we'll combine them.
In the end any reasonable reader will agree that we have not only proven one plus one is equal to three, but that we have done so sensibly. At heart, we'll be looking at the three words "one plus one" in a way which is not exactly common, but is well within the bounds of reason.
Along the way, we'll be looking at things through the lens of ternary logic instead of binary logic, but to keep things simple, that's all we need to know til the end of this article.
Oneness is linked with the act of separation
We begin with the surprising fact that anything called "one" cannot exist by itself. You likely never noticed this, so I'll give a simple example: what happens when counting? When you begin counting a series, you say "1,2,3,..." while pointing at the first item, then the next, then the next.
At the moment you say "one" you are assigning the number one as an act of separation.
A division is occurring, where you are separating whatever you call "one" from everything in the entire universe. You're creating a layer of abstraction over what already exists in the universe, like a map is to a territory. In short, "one" is not a number built into nature. It is something your mind is assigning to a part of the universe.
This act of separation is repeated for whatever you call "two," then "three," and so on, until you stop counting. There's not much new in each iteration beyond the fact that you continue iterating[1], so let's look more closely at that beginning step to understand what is happening. Then we can apply the same understanding to the other iterations.
The real novelty happens with the first thing, that which you call "one." This is where the key separation, the division, of "one" from "everything" happens. Few people know of the division involved in thinking of "one." It turns out to be -- for something few people know, and which ought to be the simplest of simple -- not a simple division at all.
Let's go deeper, and you'll see how true this is.
Example one: The Neoplatonist understanding of The One
Neoplatonists know as much as any school of thought in Western thinking about something they call The One, so we'll start by analyzing what is happening with their concept. I'm not a Neoplatonist, so I'm going to do an amateur's job of summarizing them, but I do know enough to make the point I want to make. While Platonism traces back to Plato, it's the followers of the centuries-later Plotinus who most extensively talk about a boundaryless One, a concept which precedes any possibility of division.
In short, the Neoplatonist One is a philosophical idea of an underlying Oneness behind everything, out of which all the diversity of the Many have arisen. They think of this Oneness as being before division, and yet, at heart, their use of the word "One" relies upon division, as we'll see.
One popular way of talking about the Neoplatonist One is to describe it as "being beyond being." Look at that tiny phrase there, and you can already see the division if you are looking for it. While some argue about the obvious paradox of the phrase ("how could anything be beyond being?"), a deeper observation is that the phrase beyond being is a way of extending the act of counting into a realm beyond counting.
In other words, counting itself ceases before, or as, the essential beingness of the being beyond being begins.
If that sentence seems... strange... it is, but it's also an everyday example of the kind of language Neoplatonists use when talking about The One. There are many clever ways Neoplatonists have of talking about the difference between The One and All Other Things, but they all boil down to a separation which defies what they're attempting to say about nonseparation. Let's look at some examples so you can see how this happens.
Eliminating the division is a noble effort but doesn't go all the way
Sometimes they want to eliminate all boundaries, as in the example of the author of this paper: "Everything is Flat: The Transcendence of the One in Neoplatonic Ontology." Flattening everything is a clever way of doing away with the boundary, effectively pushing it out to the edges, but no matter how clever you get at making it hard to find, there is still a division between the One and the Many.
A better approach is found in the paper "Relationality as the Ground of Being: The One as Pure Relation in Plotinus." Here the author tries to do away with the separation by talking about the fundamental nature of pure relation, even before there are any relata. He overturns Aristotle's thoughts on the order of relation and relata while making his point.
I personally like this approach. It comes nearest to framing things in a ternary logic way, which is how I frame things, but the fundamental nature of the separation is so entangled in the concept of relata in relationship we really haven't broken as free as we'd like. We can debate this, and you might win, but before we do, I think there's a slightly better approach.
Example two: Wittgenstein's understanding of the end of language
In my estimation, Wittgenstein came the closest of anyone in the West to identifying the essential difference between the Neoplatonist One and the Many. Note that he was most certainly not a Neoplatonist. He was simply the one who had the insight to literally stop talking as a way of... well, saying by showing that which can't be put into words. In this way he went further "beyond being" than all the Neoplatonists who talk and write about being beyond being and relation beyond relata.
Wittgenstein's approach is similar to rabbis who will talk -- only briefly -- about what they call Ohr Ein Sof, which is Hebrew for "Light Without End." Roughly speaking this is a Hebrew way of talking about the same underlying concept Neoplatonists call The One.
Example two-and-a-half: the Hebrew understanding of Ein Sof
Rabbis are like Wittgenstein in this respect: The first thing one learns about the unspeakable Ein Sof is that there are no words to describe the true endlessness of the Holy One, and thus it is better not to talk about it. In their eyes, anyone going on about Ein Sof is speaking nonsense, which is the same point Wittgenstein also made explicitly. He talked briefly about what constitutes nonsense (basically saying that even his own words were nonsense) and then ended his Tractatus Logico-Philosophicus with a final chapter consisting of a single sentence:
"Whereof one cannot speak thereof one must be silent."
That's the entire chapter 7 of a densely written book. Think about a chapter that is one sentence long, and you'll see that Wittgenstein was showing that the substance of what he had to say... is best said without speaking. Note that he was not copying the rabbis when he made their same point more effectively. Although some have noticed mystical implications in what Wittgenstein was saying, most do not. He had arrived at this insight independently (it can be shown that he was inspired here by Tolstoy, but that is a long story better told elsewhere).
Example three: The Tao Te Ching's understanding of the Tao that can be told
For the third and final example of the point being made here, let's look at how this paradox is handled in Eastern thinking. It is easily seen in the ancient book of ineffable riddles known as the Tao Te Ching, by Lao Tzu. The first sentences of the book align well with Wittgenstein's final sentence. Here is Gia-Fu Feng's translation:
The Tao that can be told is not the eternal Tao.
The name that can be named is not the eternal name.
The nameless is the beginning of heaven and Earth.
The named is the mother of the ten thousand things.
The Tao that can be told is not the eternal Tao because the eternal Tao exists beyond language.
It can be said that Wittgenstein began talking in the language beyond language after he very intentionally stopped talking. He intended the reader to get the point of what he was saying without words. Those who get the point, see that he was talking without talking. For those who have ears to hear, his silence is telling "the Tao that cannot be told."
Do you see how Wittgenstein came closer than anyone to saying what cannot be said? Even the Tao Te Ching uses words to describe what words cannot describe. He went further, and literally stopped using words... if such a thing can be said to be literal...
In effect, he's "counting into a realm beyond counting" to use the phrase from earlier. He comes right up to the border of the "being beyond being" from Neoplatonism, and if you listen after he stops talking, you can hear what is being said without words. He did say that only people who have independently arrived at his insight would truly understand what he was saying, so if you're finding this kind of talk to be incoherent, at least be comforted by the obvious fact that it's... not easy to grok.
Limiting language from within language
These three examples (the Neoplatonists, Wittgenstein, and Lao Tzu) are pretty canonical among the many thousands of articles and books published discussing this same "paradox," or "riddle" or "unity of oneness with division," whatever you want to call it. But they can all be grouped together with the following observation. All these experts, exercising great cleverness, fail to do away with this primary division because they are using words:
The "division" is as much a part of "one" as the oneness is.
If you're going to talk about "one" at the deepest levels of philosophy and language, you're essentially talking about a division between the speakable and the unspeakable while remaining within the speakable. Wittgenstein talked about this explicitly in the Tractatus when he described limiting language from within language because you cannot limit it from without.
My point by studying these three examples found in the deep philosophy and language around Oneness is that this same division most certainly happens at the superficial level of counting.
Once you see it, you'll see it everywhere. Going back to our original example, we can count how many marbles are in the palm of our hand by separating the first marble from the whole universe, calling it "one." But we have separated it within language only. The marble itself is not "one," since it exists in the realm beyond words. Meanwhile, its "oneness" exists entirely within language, connected intimately with the act of separation, which also exists only within language.
Back to "One plus One Equals Three"
The next logical element of the proof is going to be easier to grasp, but you should know that if you're already having a hard time with the preceding, it will break your brain even more.
We're going to give the concept of "division" a different name. There is a little-known aspect to the act of language-based separation which we can bring into focus by giving it a different name. Here's how it works: We can see that division is inseparable from oneness, and that the pair always exist together in language. The division is inseparable from twoness as well, and all the other numbers.
If you think about it, this is a curious thing for something which separates to be inseparably connected to everything it separates.
Now look carefully and you'll see it: Whatever else division is, it is inseparable from all numbers.
Now that's remarkable. Division is a thing that is inseparable from everything it divides? This doesn't sound like division, this sounds like... something which is connecting all numbers... like...
Oneness?
Could it be? Is the act of separation which creates oneness more uniting than oneness?
In fact, it is.
The paradoxical unity underlying division
Unlike binary logic, which demands that a word have exactly one definition in a given context (e.g. a statement is true OR false), ternary logic allows a word to have two definitions simultaneously. Quantum physics calls this "superposition." Poetry calls this metaphor. In this way ternary logic is more like poetry than binary logic.
Binary logic has a hard time with it, but ternary logic is okay with division as: "the inseparable part of oneness" which ... connects with all the other parts. The point here is easily counter-argued: "But each division is separate from other divisions. They may be inseparable from the 'ones' they're dividing, but they're separable from each other."
But think about that for a moment. Are they separate? Each division used in counting is the same act, implemented over, and over, and over. The number is different, but the act of separation is the same every time. They aren't so separate as we might assume if we hadn't gone through the deep dive into the nature of oneness and division earlier.
For this reason, I'm going to change the name of division (don't worry, it's just a name change) to an even simpler word that more accurately conveys this hidden idea of connecting things while simultaneously dividing them.
The new name for division is going to surprise you, so I hope you're reading carefully, and didn't skim ahead. Are you ready?
The new name for division is: "and".
"And" is a simple word that conveys the concept of being connected. It also conveys a division, just like "this and that" are separated into two distinct things by the "and." This is why we say "this and that," not "this and more this."
In this way, and simultaneously means division and unity. The hidden unity aspect is brought to the surface more than with the word "division."
The conclusion of the matter of one plus one
The last logical element is by far the simplest. The word "plus" has the same structure as "and." It divides, but it also unites. Your brain may be experiencing something like a "slippery slope" at this point, but fear not... the continuum will catch you if you fall...
At this point we can see that each "one" in "one plus one" is not entirely separate from the other. They are bound by being inseparably connected to the same division.
Together, they make a single greater "one," comprised of two parts.
And there you have it. All the pieces are in place. Knowing all this, it's time to reconsider "one plus one equals three."
Let's count:
- The first "one" is one thing.
- The other "one" is a second thing. That makes two.
- And the two ones together, united within a single greater one. That makes three.
Our total is three things[2].
Footnotes:
- ^ "To continue iterating" seems like a pretty simple act, but in fact, it's not always. There can be a whole lot happening when you count. In the simple case of "ten rocking chairs made by the same craftsman," it is in fact simple to count. But as soon as you begin counting things which are not identical, or you start moving toward infinity, you run into all kinds of subtleties regarding the similarity of the "next" thing with the ones which preceded it. A question arises: where do you stop counting? You run into famous paradoxes, for example, where you're counting "people who are bald" and the question is whether a person who has a single hair on the side of their head is bald or not. You might think "a single hair counts a person as not bald," but then again, we commonly call people who have many hairs on the sides of their head "bald," so where do we draw the line? It turns out, you can't, with ordinary language. We count things in rather loose categories in ordinary language, so "to continue iterating" is not as simple as it seems.
- ^ Some will say this is a word game, not a logical proof. At the very least, it's both. The logic of this proof can be made rigorous, but that is another day's work.
