(written cir. Spring 2001 or 2002)
See here for a much shorter version of this.
A multinity is a many-in-one. A trinity is a three-in-one even as a binity is a two-in-one and a unity is a one-in-one; the former two are multinities, the third not, for it is only one. Our question is this: is a multinity real or even possible, or is it an absurdity? It is certainly absurd to think 1+1=1.
We are not speaking of a triangle as a many-in-one, for example, for the three sides of a triangle constitute parts of the triangle, and so here the summation of the parts renders a total of 1+1+1 = 3, and the term triangle then is nothing more than a reminder that we are speaking of three straight line segments, each end point of each being an end point of two.* But the notion of a multinity excludes any consideration of parts.
* This definition will not render a single straight line divided into two parts. Consider a single straight line AC divided into two segments at B. It might be argued that since line AB and line BC and line AC are different, that they would sum to three lines. Logically they are three. For even if all the points of AB are points of AC, the fact that some points of AC are not points of AB proves that AB and AC, while each is a thing, they are not one and the same thing, and so logically they are two. Etc. But if we take a look at these lines we see that we are dealing with only one line and so our logic does not correspond here to the reality. And hence we can say that a triangle is a 1+1+1=3, and thus not a multinity, for the expression 1+1+1 does not express a single line divided into two parts, i.e., three lines are not parts of each other and can only be summed into a triangle.**
** For this reason, by the way, arithmetic is not composed analytically but only synthetically (thanks to Kant for this insight). For sometimes two things can be unified in a whole which is less then the constituent parts, e.g., in the above case the sum of AB and AC is AC. And so in order to know that 1+1=2 we have to make the assembly ourselves in space and take a look and notice that we are passing the 1 and going on to the 2.
In contrast to a many-in-one it is easy to speak of a one-in-many, even if the math of 1=1+1 does not follow for an analogy. This is a very common way of speaking. This is a table and that is a table, and so we have two tables. This is a triangle and that is a triangle, and so we have two triangles. In this way we mean to say that the concept of triangle just cited in the previous paragraph, is represented in each of these two figures or things. Hence we can conclude: many triangles are contained under this concept of triangle, even as we could imagine an infinite count of tables that could huddle under the concept of table.
There are a few concepts where we cannot speak of a many contained under, but only of a many contained in, a concept. Space is an example. We dont speak of this and that space as representing space in the way that we mean when we say this or that triangle or table might represent its respective concept, but rather as already contained in a singular, all-encompassing space as parts or, better said, as limitations or delineations of space. And so when we say that many are contained in one, we mean many limitations of the one and same space give us this infinitude of different things, e.g., the space delineated by a triangle or occupied by a table. Time is another example of a many in one. Here different times are not representations or examples of time, but rather are merely limitations of a single, all-encompassing time, e.g., yesterday or today or two hours from now. But even though this is closer to a multinity and we can see the meaning of the terms, this is still not what is meant by that term, for the multinity excludes any suggestion of a limitation as much as it excludes any suggestion of a part. And so space and time, though containing an infinite of things within each, are no more multinities than are triangles and tables and other things composed of component parts.
What we are looking for here is a true many-in-one, i.e., a something which is both plural and singular at the same time and without contradiction, involving neither parts nor limitations.
For the sake of simplicity of understanding I shall assume that for every concept, e.g., that of a table, an omnipotent God can created in a single act a perfect manifestation of that concept, what we might call the Archetype of that concept (as Plato might have put it). For example God could have conceived of a perfect man and then create him as the Archetype of Man. Thus this one thing is contained in this one concept, which constitutes a unity. If it were possible for two identical things to be contained in a single concept, i.e., two different archetypes of that identical concept, then we would have a multinity, i.e., a many-in-one. And in this case God would be able to engage in an extremely curious act of creating two different things in a single act of creation, each one a perfect representation of the concept, and yet each one different; two different archetypes of one and the same concept. And that this is the case we shall now see.
Let us consider the concept of a hand, and let it be described perfectly in the mind of God as a squarish and thick composite of bone and flesh which we shall call the base of the hand, and with one side a palm and the other side a back of the base. Now let there be four fingers in a row on one edge of this base and a thumb on another edge which is adjacent to the first edge. And let the fingers and thumb curl toward the palm side of the base. Now let the concept then also include the various proportions, e.g., that the middle finger is 10% longer than the index finger, and let the index finger lie closest to the thumb, and a middle finger next to and to the other side of the index finger from the thumb, etc. Now assuming the description of the material and all else that is needed to determine a hand, let God create that hand as an archetype in a sovereign, omnipotent act. And what do we have? Oddly enough when we have this single creation, this archetypical hand, we also necessarily must conceive of a second hand to oppose it, for any hand is necessarily either left or it is right, and there is nothing which is hand which is not also either left or right; and yet there is absolutely nothing in the concept of hand which can express left or right and so there is no concept of a left hand or of a right hand (and these can only be discerned in a sighting), but only of a hand. And so by the identical act of creation God could have just as immediately created a right hand, which would be identical in EVERY conceptual respect with the left hand, but still with a difference which is internal to the hand, but which cannot be understood intellectually, but only spied externally with respect to both in space, namely that despite the fact that the points making up the two hands can be substituted for each other in every respect, and despite the fact that the relationships of the parts are identical in every respect, still the hands are different hands.* ** Thus we have a true multinity, a two-in-one, two perceptible different hands which cannot occupy the same space (or glove) and yet each of which is a perfect archetype of the thing conceived.***
* Perhaps an easier way to visualize this situation is by means of a certain two triangles. Follow this construction. Lay out a segment of a great circle on the surface of a sphere. Label the ends A and B. Center a circle at A with a radius of AB and a circle at B with a shorter radius. The two circles will intersect at two places C and C. Joint A and C and B and C with segments of great circles of that sphere. Likewise join A and C and B and C. We now have two triangles ABC and ABC and they are so much alike that the parts of the one could be substituted for the parts of the other without making the least difference whatsoever, i.e., AC might be removed and put in the place of AC, and AC in the place of AC, etc. And so their parts are identical in every respect except location. And yet ABC cannot occupy the space of ABC for one is the mirror image of the other and so like the left and right hands cannot wear the same glove.
** This is not the case if instead of a single creation God decided to repeat himself (an absurd idea) and to create the same thing twice over. Then we would have two hands which could occupy the same space and so are identical in every respect, such that they might wear the same glove. And this would be merely a one-in-many, namely many examples of one and the same thing as indicated above in the discussion where many triangles and many tables can represent their respective concepts.
*** As Kant puts it in his Prolegomena (How Is Mathematics Possible, Par. 10): "What can be more similar in every respect and in every part more alike to my hand and to my ear, than their images in a mirror? And yet I cannot put such a hand as is seen in the glass in the place of its archetype; for if this is a right hand, that in the glass is a left one, and the image or reflection of the right ear is a left one which never can serve as a substitute for the other. In this case there are no internal differences which our understanding could determine by thinking alone. Yet the differences are internal as the senses teach, for, notwithstanding their complete equality and similarity, the left hand cannot be enclosed in the same bounds as the right one (they are not congruent); the glove of one hand cannot be used for the other." What we are calling here a binity, Kant called "incongruent counterparts".
The hand in 3-D space, therefore, gives us the fact of a true multinity, namely a single concept with two archetypes, each one a perfect representation of the same concept, and yet each different from the other. And in the same way we can imagine a trinity, namely three persons like three archetypes, as it were, under a single concept of God. The concept would indicate only a single, undivided being (as does also the concept of hand) and the count of its archetypes, to borrow the expression as a figure, would be three different beings, even as the concept of hand renders two unique archetypes.
This does not prove that a trinity is real, but only that it is possible and plausible, for the left and right hands defy all logic, and still they exist as a binity. Therefore mutlinities do in fact exist despite all logic to the contrary. And hence also then a trinity is no more absurd than a binity; and a binity is actual as evidenced by the left and right hands.