Clearhat

No, one times one does not equal two. But here is an intriguing argument that one plus one is equal to three. It sounds absurd, so let's start small and carefully establish the logical elements which comprise this argument. Each element will be seen to be logically sound in themselves. Then we'll combine them and see if we have a sound argument.

In the end the reasonable reader should be able to agree that we have demonstrated one plus one is equal to three.

Along the way, we'll be learning to see through the lens of trinary logic instead of binary logic, but to keep things simple, that's all we need to know on that aspect til the end of this article.

Oneness is linked with the act of separation

We begin with the surprising fact that anything labeled "one" cannot exist by itself. Although this is a logically simple observation, you may have never noticed this, so let's look at a simple example: what happens when counting? Imagine you have a few marbles in the palm of your hand and you want to know how many. 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 else 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. Your map has a single thing on it: an abstract marble which you call "one." This is not a number built into nature. It is something your mind creates out of thin air and assigns to a part of the universe, while the rest of the universe remains "uncounted."

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 of a count beyond the fact that you continue iterating and the fact that you need a new name for each one^[1], so let's look more closely at that beginning step to understand what is happening in more detail. Once we know it better, we should be able to apply the 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. A curious side observation is that the separation between each counted item and the rest of the universe (including items previously counted) happens with each new number. In other words, what joins the series is not a feature of the counted items, but is another abstraction: One abstraction separates each item from the universe and assigns it a number. The other abstraction joins it with the other counted items. These happen simultaneously and hardly anybody ever thinks about them.

"One" turns out to be -- for something few people know, and which ought to be the simplest of simple -- not a simple division at all. So if it's not so simple, what is one?

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 considering 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" ironically relies upon division, as we've seen and now we'll look at from another angle.

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 know what to look for. While some argue about the obvious paradox of the phrase ("how could anything be beyond being?"), a more functional observation is that the phrase being beyond being is a way of extending the act of counting into a realm beyond counting. In this realm, beingness exists, but it has no other properties than pure simple being.

Ontologically speaking, being exists before counting exists. So the "one" exists in an "uncountable realm," yet the realm is obviously being counted in order to be called "one."

That's interesting, but a fairly simple semantic quibble: even if we acknowledge that "one" is just a label for something essentially uncountable, we still haven't addressed how calling The One "being beyond being" places a division between The One and ordinary countable being. In other words, we can move the essential division around, but we can't eliminate it. By saying "beyond" we're trying to transcend the countability of pure being, but really we're only shifting where we put the division. It would be simpler to just say One is being, and leave it at that, but there we are with the division again.

In short, counting is separate from being, and counting must begin ontologically after the essential beingness of the being beyond being. And counting requires a division. Apparently, we cannot count what pre-exists counting, because we must introduce division into the indivisible in order to do so.

Division, then, is a required part of counting, but not a required part of being. "One" requires division in order to exist. This can get confusing, so here's how the ontological stack is organized, if you want to get very linear about what's being said:

1. Being.
2. Division.
3. (0) Being beyond being = numbered "0" in this list, but also numbered "3" because it requires "2," division, to exist.
4. (1) Being.
5. (2) Another division.
6. (3) Counting.
7. (4) One.
8. (5) Two.
9. (6) Three.

Like Euclid's 5th postulate, that #3 looks mighty suspicious. It may take you a minute to see what is being said there, but it's correctly placed once you understand Being beyond being can only exist after division, which itself relies on something to divide. Eliminate it (and the then-redundant #4 and #5), and you get a cleaner, more honest list:

1. (0) Being.
2. (1/2) Division.
3. (0) Counting.
4. (1) One.
5. (2) Two.
6. (3) Three.

Occam's razor says this list is better since it is more simple. Some people might combine 2 and 3 into a single "1/2" step, but as I see it, there is a global division concept (Division) without implementation and a local division implementation (Counting) so I've got them separated. 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, a division 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, but no matter how clever you get at making it hard to find, there is still a division between the One and the Many, just as there are divisions between all the smaller ones which comprise 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 trinary logic way, which is how I frame things. But alas, 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 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. He wrote a book with 7 chapters. The 7th chapter is a single sentence long, saying "Whereof one cannot speak, thereof one must be silent," and then... he was silent. He left academia, gave away all his money, and went to live a simple life as a teacher of children in a small village, intending to never have anything to do with philosophy or math. The story ended differently than he intended^[2], but what we're talking about here is his intent: silence. Not the word silence, but the action.

In this way he went further "beyond being" than all the Neoplatonists who talk and write about being beyond being and relation beyond relata.

In other words, by doing what he was saying instead of merely saying it, and then going further in action without words, he showed how to go beyond language. Consider the difference between someone who repeatedly says: "we should all stop talking" and someone who says once: "let's stop talking" and then actually stops talking. Wittgenstein did the latter. Everybody else is still doing the former, unable to get the point of Wittgenstein even when he's right in front of them.

Think about that carefully, because most Wittgenstein scholars miss this point.

I'm guessing they miss it because they don't know how 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." If you want to know more, they change the subject. They know enough to mention it and then stop talking about it. This is a deep insight. Roughly speaking this is a Hebrew way of talking about the same underlying concept Neoplatonists call The One.

Let's look at it briefly.

Example two-and-a-half: the Hebrew understanding of Ein Sof

Rabbis are like Wittgenstein in the following way: 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 coincidentally the same word Wittgenstein used.

Wittgenstein was not intentionally invoking this interpretation by the rabbis, even though he did cryptically refer to the mystical in his short book. He was arguably doing what he had learned from Tolstoy, who had found the idea in a deep study of the Gospels. Although I've said little on this point, I've said enough.

Example three: The Tao Te Ching's understanding of the Tao that can be told

For the third and final example of the larger 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.

Wittgenstein wrote to the readers who would get the point of what he was saying without words. He knew this was a subtle point to those who don't get it, and observed that the people most likely to get it would be people who had come to a similar understanding on their own.

He was talking without talking. For those who have ears to hear, his silence was telling "the Tao that cannot be told," although that is not how he would have framed it.

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. His actions spoke.

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, hearing without ears, in a way.

Limiting language from within language

These three examples (the Neoplatonists, Wittgenstein and the rabbis, and Lao Tzu) are on the short list 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 the thousands of articles and books can all be grouped together. All these experts, exercising great cleverness, fail to do away with this primary division because they are using words:

Within the realm of words, the "division" is as much a part of "one" as the "oneness" is. And outside of words, there is no division.

If you're going to talk about "one" at the lowest, most fundamental levels of philosophy and language, you're essentially talking about a division between the speakable and the unspeakable while remaining within the speakable. The division is never within the one, rather, division is an aspect of that which speaks of the one.

Wittgenstein wrote about this very issue explicitly. He described limiting language from within language because you cannot limit it from without. By "without" he meant what we're saying here as "oneness," because oneness is "without," or "outside" what happens in the realm of the divided (it is also within, but that's the everywhereness of the oneness, another topic altogether). Oneness has no limit, so it cannot "limit from without." But language can be limited from within itself, which is what Wittgenstein was saying. Limit is an intrinsic part of language, not found in whatever came before language.

Division is an aspect of counting

My point by studying these three examples found in the philosophy and language around Oneness is that this same division most certainly happens when counting. Therefore: Division is an intrinsic part of counting, not found in whatever came before counting. In short, counting requires division.

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. (i.e. within counting only.) The marble itself is not "one," since it exists in the realm beyond words. Meanwhile, its "oneness" is a label which exists entirely within language, connected intimately with the act of separation. And that "oneness" label absolutely requires division in order to exist.

Back to "One plus One Equals Three"

If you're okay with what I've said so far, 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, the next part will break your brain even more. You may want to go back and read carefully to ensure we're on the same page before proceeding, or at least closer to the same page.

From the above, we can now see that division is inseparable from oneness, and that the pair always exist together within language. Even if you stop talking altogether, you only emphasize the place of the division, you do not eliminate it. Of course it should be noted that the division is inseparable from oneness but 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 every thing it separates.

Now look carefully and you'll see it: Whatever else division is, it is inseparable from all numbers. For a number to exist, division must be involved.

Now that's remarkable. Division is something that is united with everything it divides? This doesn't sound like division, this sounds like... something which is connecting all numbers... like...

Oneness?

Being?

Remember earlier when I separated "Division" from "Counting," the global concept from the local implementation in the numbered list above? What we're talking about here is the global concept, and how it is the same for all numbers. All numbers are connected to the same global division which separates them from each other. Thus the global concept of division is... unifying?

Could it be?

It could... it's another order of abstraction to do this, but it's technically possible, and quite similar to the abstraction which joins all counted "ones" into a series with a total, a sum which gathers together a bunch of individuals. So the question becomes: just how unifying is division?

A good way to begin a thought experiment is to consider the extreme. Is the act of separation which creates each-countable-one-within-language equally as unifying as the-uncountable-oneness-of-The-One...?

The answer yes seems quite possible.

Strange as it seems, separation may be equally unifying... just at a different level of abstraction than the actual oneness.

To illustrate the idea, we can consider the map and the territory mentioned earlier. The territory is an undivided oneness. We're proposing that the map is also, even if it has many labels and boundaries separating geographic features from each other.

The counting which separates things being counted all happens within a single layer of abstraction, which unites all the things being counted. Note that all this happens within a single mind while counting, but can be easily shared between minds.

Under close inspection, this thing we call division appears to unite equally as much as it divides. It holds everything together even as it cuts everything apart.

The paradoxical unity underlying division

I said earlier we'd come back to the difference between binary and trinary logic, and it's time for that now. Unlike binary logic, which demands that a word have exactly one definition in a given context (e.g. a proposition is true OR false, never both simultaneously), trinary logic allows a word to have two definitions simultaneously. Quantum physics calls this "superposition." Poetry calls this metaphor. Religion says you have a soul identical to your body inside your body, and some would say the spirit is the same to the soul as the soul is to the body. Superpositioned.

In this way trinary logic is more like poetry than it is like binary logic.

What we're investigating is called a paradox within binary logic but is perfectly reasonable within trinary logic.

Binary logic has a hard time with this unifying aspect of division, but trinary logic is okay with division both as "that which separates into multiple ones" and "that aspect of ones which connects with all the other ones."

The point here is easily counter-argued: "But each division is itself separate from other divisions. Divisions may be an inseparable aspect of the 'ones' they're dividing, but surely 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 changes, but the act of separation is the same every time. It's like using a single knife to cut an orange into four pieces. The knife divided the orange but it is itself a single thing. Its existence is something each orange slice has in common with the other orange slices.

A new name for the word "division"

We've now changed the meaning of the word "division" enough that in order to embrace this new superpositioned unifying aspect of division, we need to give this deeper understanding a different name.

For this reason, I'm now going to change the name of division 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... more this."

In this way, and simultaneously means division and unity. The hidden unity aspect of and is brought to the surface more easily than with the same hidden unity in the word "division."

The conclusion of the matter of one plus one

The last logical element of the argument is by far the simplest. The word "plus" has the same logical structure as "and." Instead of saying "one plus one" it is often said "one and one." Like and, plus divides, but it also unites. In other words, a plus is a division equally as much as it is an addition.

(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. We now can see how they are bound by being inseparably connected to the same division, the plus which unites them.

Together, they make a single greater "one," comprised of two parts. Like the Yin Yang symbol. Their total is commonly known as two, but we have just discovered that we must also include that which divides them into the total. At least partially, that is.

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:

1. The first "one" is one thing.
2. The other "one" is a second thing.
3. The plus, which unites these two things. It's a separate thing, yet it's also not. From a certain level of abstraction, it's a third thing.

Our total is three things. One plus one is three.

And now you have been introduced to trinary logic, where you're able to count things which are sometimes one thing and sometimes another. And sometimes both simultaneously, a concept for which there is no place in binary logic.

 

Footnotes:

1. ^{^} "To continue iterating" seems like a pretty simple act, but in fact, it's not always. There can be a variety of edge cases which arise 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 do so precisely, with ordinary language. We count things in rather loose categories in ordinary language.
2. ^{^} Thankfully, this wasn't the end of Wittgenstein's philosophical work. A group of students collected around his book (and grieviously missed the subtle point he was making, but that's another chapter in the story). They loved the insights he revealed and begged him to come back and teach more philosophy. Eventually, one of them travelled to the far away small village to convince him to return to teaching, which he eventually reluctantly did. Here's how we know that he had graduated from philosophy: Although he taught philosophy for the rest of his life, and is considered by many the greatest philosopher of his generation, he consistently encouraged his best students to find something better to do than philosophy. Others considered him a philosopher, but within his own paradigm, he had graduated from it when he published his book.