Prof. Eric Steinhart (C) 1999
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I cite sections of Leibniz's Monadology in quotes, then follow them with comments. The translation is by Robert Latta (1898) with my emendations; Latta's translation is, to the best of my knowledge, in the public domain. This is a work in progress.
1. "The Monad . . . is nothing but a simple substance, which enters into compounds. By 'simple' is meant 'without parts.'"
1. Monads are physically simple: it is not possible to divide a monad into spatial or temporal parts. Monads have no parts, but that does not stop them from having many interrelated qualities. Monads do not have any shape, size, mass, charge, or energy.
Physical simplicity does not rule out formal complexity. Each monad is formally (that is, mathematically) complex. The formal structure of each monad is complex. Monads have qualities and their qualities are mathematically interrelated. Monads are not physical machines; they are logical machines. Monads are like little animated equations; they are like equations that do things, and what they do is to cause physical things to exist and change. I treat monads as immaterial computers. Monads are functionally complex.
2. "there must be simple substances, since there are compounds; for a compound is nothing but a collection or aggregatum of simple things."
3. "where there are no parts, there can be neither extension nor shape nor divisibility. These Monads are the real atoms of nature and, in a word, the elements of things."
3. Monads are extensionless and figureless: they are not spatial and they are not in space. Since they have no size and no shape, monads do not form physical things (with size and shape) by accumulating like material particles. The sum of infinitely many zeros is zero; objects without shape or size cannot be spatially arranged to make shapes with sizes. So monadic aggregation is not spatial composition. The monads are not in space; yet they must have something to with space, for if they did not, there would not be anything spatial at all. There is only one alternative: if the monads are not in space, space must be in the monads. Monads are real; space is virtual. Monads are not material atoms; they are logical atoms. Monads are the logical (real) basis for the physical (virtual) world.
4. "No dissolution of these elements need be feared, and there is no conceivable way in which a simple substance can be destroyed by natural means."
5. "For the same reason there is no conceivable way in which a simple substance can come into being by natural means, since it cannot be formed by the combination of parts [composition]."
6. "Thus it may be said that a Monad can only come into being or come to an end all at once; that is to say, it can come into being only by creation and come to an end only by annihilation, while that which is compound comes into being or comes to an end by parts."
4,5,6. Monads are created logically (not physically) at the start of the world and destroyed logically (not physically) at the end of the world. The start and end of the world are logical limits of the world and are not in the world. Monads are not subject to natural (i.e. physical) creation and destruction, so they are not in time at all. Monads are not in time, time is in the monads. Monads are real; time is virtual.
7. "there is no way of explaining how a Monad can be altered in quality or internally changed by any other created thing; since it is impossible to change the place of anything in it or to conceive in it any internal motion which could be produced, directed, increased or diminished therein, although all this is possible in the case of compounds, in which there are changes among the parts. The Monads have no windows, through which anything could come in or go out. Features cannot separate themselves from substances nor go about outside of them . . . Thus neither substance nor feature can come into a Monad from outside."
7. There are no causal interactions among monads. The elimination of moving parts in the monad means there is no internal motion; so there are no mechanical interactions within monads, so there cannot be mechanical interactions between monads. The fact that feature are not communicated from one monad (substance) to another means that even non-mechanical exchanges of information are not possible. Monads are entirely causally isolated.
There are relations between monads. Relations between monads are based on their individual properties. A physical analogy helps: if my height is 68 inches and yours is 72 inches, then you are taller than me and I am shorter than you; but these relations only hold between us because of our individual properties. All relations among monads are based on their internal properties; the relations are virtual.
8. "the Monads must have some qualities, otherwise they would not even be existing things. And if simple substances did not differ in quality, there would be absolutely no means of perceiving any change in things. For what is in the compound can come only from the simple elements it contains, and the Monads, if they had no qualities, would be indistinguishable from one another, since they do not differ in quantity. Consequently, space being a plenum [entirely filled], each part of space would always receive, in any motion, exactly the equivalent of what it already had, and no one state of things would be discernible from another."
8. Monads have no parts; but if they have no qualities, then there is no way for them to differ, so there is only one monad; but there is more than one monad, so they do have qualities, and their qualities are not all the same. Every monad has infinitely many qualities. The qualities of the monads are not physical; rather, physical qualities are constructed from the qualities of monads.
Every monad has qualities. Each quality determines the monad in some specific way. To understand qualities, think of the qualities of physical objects. For instance, red is a specific determination of the quality color; triangular is a specific determination of the quality shape. Suppose this piece of paper is white and square. The quality "color-of" associates the object "this piece of paper" with the specific determination "white"; the quality "shape-of" associates "this piece of paper" with the specific determination "square". These qualities can be written using a functional notation: color-of( paper ) is white; shape-of( paper ) is square. The qualities color-of and shape-of are variables whose specific determinations are their values.
I'll give the qualities of monads numbers: Q1, Q2, and so on. Each quality is the quality OF some monad: Q1-of( this monad). Since I'm treating monads computationally, I treat the qualities Q1, Q2 as bits, that is, as binary variables whose values (specific determinations) are either 0 or 1. So: Q1-of( this monad) is 0, while Q2-of( this monad) is 1, and so on for all the qualities.
The qualities of monads are somehow the logical basis for motion in physical space. Every monad has enough qualities to simulate all the spatial content and change of the whole universe from its own point of view (it's unique identity). Think of a chess board. Physically, it has 64 square cells; but logically these cells are only qualities. So if a monad has 64 qualities Q1, Q2, . . . Q64, and if all these qualities have geometric relations, then the logical structure of the qualities of the monad is the basis for the physical structure of the chess board. You can think of chess pieces as different specific determinations of the qualities: Q1-of( this monad) is white-queen.
The qualities of monads are interrelated. Physical things illustrate how qualities are interrelated. Consider some volume of gas, such as ordinary air in a tire. The relation pv = nrt says that the pressure of the gas times the volume of the gas equals the constant nr times the temperature of the gas. Increase the temperature and either the volume or the pressure or both have to increase. As it gets hotter, the size and pressure of your tire increase: too hot, it explodes. So qualities of monads are interrelated too. The interrelations are a system of equations whose variables are qualities.
Suppose Q1, Q2, . . . Q16 are qualities of some monad. Examples of mathematical relations are: Q1 = Q2 + Q3 + Q4, and Q2 = Q3 + Q4 + Q5, Q3 = Q4 + Q5 + Q6, and so on. If the formula F that relates qualities of this monad takes 3 qualities and adds them together, then we write it like this: F(x, y, z) = x + y + z where x, y, and z are some qualities. So: Q1 = F(Q2, Q3, Q4); Q2 = F(Q3, Q4, Q5), Q3 = F(Q4, Q5, Q6). In this case, each quality Q1 . . . Q16 is related by F to 3 others.
The qualities of the monads are logically interrelated bits. They are the qualities of immaterial computers. These logically interrelated bits are a program. Monads execute programs. The programs of all the monads define the spatio-temporal physical world. The program that a monad runs somehow simulates the whole physical world.
9. "Indeed, each Monad must be different from every other. For in nature there are never two beings which are perfectly alike and in which it is not possible to find an internal difference, or at least a difference founded upon an intrinsic quality [unique identity]."
9. Every monad differs from every other monad. Every monad has some unique identity (its "denomination"). Its unique identity is the logical distinctiveness of the program that it runs. No two monads run the same program or compute the same function. All monads are functionally unique; the functional or computational uniqueness of each monad is its identity. Any relations among monads are based on the differences and similarities of their computations. The unique identity of each monad is its perspective on the whole universe.
10. "every created being, and so the created Monad, is subject to change, and further that this change is continuous in each."
10. Monads change, and they change continuously. It's not possible to make a computational model in which there is continuous change, so I leave the change discrete. My model explicitly fails to satisfy Leibniz's Monadology in this respect. Time for my computational monads occurs in discrete clock ticks.
For there to be any internal change in the monads, the qualities of the monad must be variables whose values differ from moment to moment in time. So, each quality is a variable that at each moment actually takes on some one of many possible values. So, at time t quality Q1 of this monad is 0, while at time t+1 quality of Q1 of this monad is 1, so there is change from time t to time t+1.
In the chess board analogy, each quality of a monad is a square of a virtual chess board and the specific determination of each quality is the piece on that square. If quality Q1 of this monad at time t is white-queen and quality Q2 of this monad at time t is empty, and quality Q1 of this monad at time t+1 is empty and quality Q2 of this monad at time t+1 is white-queen, then the white queen has moved from Q1 to Q2 from time t to t+1. Of course, the white-queen and the motion of that piece are all virtual: just qualitative change of the monad whose logical structure simulates a chess board.
Every monad runs at least something like a virtual reality program that generates the appearance of the whole physical universe from its unique point of view. George Ross puts it like this:
Computer graphics can be used to create animated film sequences representing the changing shapes and positions of imaginary objects from particular perspectives. We can imagine an infinity of such films, each from infinitesimally different viewpoints, all being run simultaneously. Even though the objects and their interactions are entirely fictional, it will be as if there had been infinitely many cameras filming one and the same scene from different points of view. The simplest way of describing what they portrayed would be by adopting that fiction, even though its only reality would be as a formula in a computer program. (G. MacDonald Ross, Leibniz (New York: Oxford University Press, 1984), p. 98.)
11. "the natural changes of the Monads come from an internal principle, since an external cause can have no influence upon their inner being. (Theod. 396, 400.)"
11. There is some natural power in each monad that causes it to change. This power is logically internal to the monad: it arises from the logical structure of the monad. For instance, in the paradox "This sentence is false" there is some sort of internal logical motion as the sentence changes its own truth-value; but that is not physical motion, and the power that causes the changes is due to the internal logical structure of the sentence. The change in monads is likewise due to their inner logical-mathematical transformations.
The internal principle of change is the rule that determines how the qualities of the monad change their values. In technical terms, it is an algorithm. The algorithm is an aspect of the monad's nature, its unique identity. Each monad computes its own algorithm, performs its own mathematical transformations of its qualities, independently of all other monads. All monads are synchronized like watches.
Monads are self-powered. They have their own internal eternal batteries. More accurately: they require no energy, since energy is physical. The paradox "This sentence is false" does not require any energy to change its truth-value. The change is formal, not physical. I think such mathematical (formal) change must be real, since physical change is mathematically ordered and so presupposes the existence of mathematical structure and dynamics. The algorithm in a monad is like its own internal logos, concept, or form. Monads are like self-moving numbers or self-moving computers.
12. "besides the principle of the change, there must be some detail of that which changes [un detail de ce qui change], which constitutes, so to speak, the specific nature and variety of the simple substances."
12. The internal complexity is the detail of the monad. The detail of a monad is the system of mathematically interrelated qualities. The system of interrelated qualities is a system of equations. The system of equations is the program that the monad runs or executes.
The qualities of the monad are variables able to take on different values. For example, the color-of a physical thing is a variable that takes on the values red, green, or blue; the shape-of a physical thing is a variable that takes on the values square, round, triangular; so the system of these qualities shape-of and color-of varies over complex states like red-square and green-round. Just so, the whole system of interrelated qualities of a monad varies as different qualities take on different values. The variations are mathematically ordered.
Monads differ because their detail differs; either the differences are in the detail itself (different qualities), or in the content of the detail (differeng values for those qualities); if the differences were in the detail itself, monads would have nothing in common when they were compared and so would not be able to aggregate into composites. So, monads have the same kind of detail, while differing in the contents of their details. Moreover, the differentiations of their details must be sufficient to constitute the entire physical world, particularly the spatio-temporal changes (motions) of that world.
Some qualities of monads serve as the logical basis for the spatial structure of the physical world. These qualities are interrelated by geometric relations. Suppose you interpret the qualities as spatial points and their relations as spatial relations such as the directions north of, east of, south of, and west of. If you add northeast, southeast, southwest, and northwest, you have the 8 directional relations that arrange the spatial qualities into a 2 dimensional grid like a chess board. If each quality has all these 8 relations with other qualities, then each quality has 8 neighboring qualities. For instance, if Q1 is to the north of Q2, then Q1 is the northern neighbor of Q2 and Q2 is the southern neighbor of Q1.
If each quality has 8 neighbors and all the qualities are arranged by these relations into a structure like a chess board, then each quality has square shape just like a cell in a chess board. But the shapes of the qualities are not physically real. Shapes are implied by neighbor relations: the qualities have virtual shapes. The shape of any quality of a monad (but not of the monad itself!) is an n-sided polygon.
If we give monads just 3 neighbor relations, then we get cells with triangular shape. If we give monads just 6 neighbor relations (call them up, down, left, right, before, behind), then we get cells with a hexagonal shape in a honeycomb lattice.
The qualities of monads also have virtual sizes, in the sense that the sides of their virtual polygons determine the basic units of length of those qualities. If qualities have 8 neighbor relations, so that they are virtual squares, then the length of the side of each virtual square is the fundamental constant L. All physical units of measurement are defined by the length L of the side of the virtual polygon. If space is only finitely divided, L is the atomic unit of length so L = 1; if space is infinitely divided (as Leibniz says), L is infinitesimal.
So, physical qualities (shape and size) have been completely defined by purely abstract relations. Abstract immaterial patterns produce concrete material things! Physical reality is virtual reality.
13. "The internal detail [of each monad] integrates a multiplicity into the unity or simplicity [of the monad]. For, as every natural change takes place gradually, something changes and something remains unchanged; and so a simple substance must be affected and related in many ways, although it has no parts."
13. The detail of each monad is complex: it contains many relations and properties. The internal detail of each monad is sufficiently complex to serve as the logical basis for the space-time world. The whole spatio-temporal world is constituted in each monad, in the detail of its programming. It is enfolded in that detail without filling the monad with space-time like a balloon is filled with air. Monads contain space-time without themselves being extended. Monads contain space-time as a system of interrelated qualities.
Suppose a monad has qualities Q01 to Q16, and that each quality is related to 8 other qualities, as shown below. The relations between qualities allow them to be arranged in a grid (the grid is closed -- each edge has neighbors on its opposite side). So each quality interacts with its 8 neighbors as determined by formula F.
Qualities in monads whose interrelations arrange them into a regular extended structure serve as the logical basis for physical space. The qualities are points or cells of space. Each point (quality) interacts with its neighbors as determined by the formula F.
We can easily add time to the monad by letting each quality on the right-hand side of the equals sign be the old value of that quality and the quality on the left-hand side be the new value of that quality. So, if Q1 = F(Q16, . . . Q6), then Q1's new value is defined in terms of the old values of Q16, . . . Q6. When you consider some variables (qualities of monads) as points or cells in space, and let their old values determine their new values according to some rule F, you get what is called a cellular automaton.
In my model monads, the monadic qualities are binary variables (bits) whose values are either 0 or 1. So at any moment, the spatial structure of a monad has some pattern of 0s and 1s on it. If you think of 0 as being a "dead" quality and 1 as being a "live" quality, then at any moment the spatial structure appears as a grid occupied by a pattern of living qualities (living cells or points) on a dead background. The pattern of live cells at one moment determines the pattern at the next. As time goes by the patterns change. Since a grid like this can be interpreted as consisting of a living pattern on a dead background, it's called a life grid.
Patterns on the life grid change according to the rule F that says how individual cells (monadic qualities) change their values. Whether or not a cell is alive (value = 1) or dead (value = 0) at the next moment on two factors: (1) its current state (alive or dead); and (2) the total number of neighbors that are alive. A cell has neighbors on all sides and at all corners, for a total of 8 neighbors. If a cell is alive now, then it will be alive in the next moment only if it is surrounded by either 2 or 3 live neighbors; otherwise it dies. If a cell is dead now, then it will be alive in the next moment only if it is surrounded by exactly 3 live neighbors; otherwise it stays dead. Figure 1 shows the rule (with live as 1 and dead as 0). The operation of the rule on a simple pattern is shown in Figure 2. Black squares are live cells; white squares dead. Numbers are live neighbor counts. The rules imply that a horizontal bar of 3 cells changes to a vertical bar and then back to a horizontal bar, over and over again.
This rule is known as the game of life. Since the life grid running the game of life produces a series of changing patterns, it's a good way to model the changing internal detail of a monad. The system of qualities arranged by the game of life rule F into a grid is the virtual reality (VR) program in each monad that generates the appearance of the space-time world.
Figure 1. The game of life rule in tabular format.
Figure 2. Transformations of patterns on the life grid.