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.
56. "Now this connexion or adaptation of all created things to each and of each to all, means that each simple substance has relations which express all the others, and, consequently, that it is a perpetual living mirror of the universe. (Theod. 130, 360.)"
56. Every monad is a "perpetual living mirror" of the whole universe; this phrase is a poetic symbol of a beautiful mathematical structure. The poetry needs to be clarified mathematically because the whole universe is nothing but monads. So it's a system of mirrors reflecting one another, a system of mirror-images. But mirror-images are just structure-preserving functions. Each such function maps the detail inthe VR subprogram in each monad onto the VR detail in another. We've already seen that each monad mirrors itself, in the sense that its AI program contains (an infinitely complex) internal map of itself; but the VR detail is able to be mapped onto the VR detail in every other monad, although it is only God's mind that actually contains the totality of these maps (which are the adjustments of the monads to one another). Thus we can even define the AI program as that aspect of a monad's internal complexity that is self-mirroring, ad the VR program as that aspect of a monad's internal complexity that is other-mirroring. (Indeed, the mind-body harmony will be an example of the AI program mirroring the VR program, another kind of internal self-mirroring).
The fact that monads (in their VR detail) mirror what is going on in one another does not mean that they see into one another (they have no windows). It just means that there are formal relations between them that enable them to deduce what is going on in their neighbors. The formal relations are the structure-preserving maps.
Another name for a structure-preserving map is an analogy. An analogy between monads is a map from the points in the one onto the points in other that preserves the positions of points relative to one another while transforming the positions of cells relative to the grid. The analogy preserves the structure of the neighbor relations.
Every mirroring (i.e. analogy) between two monads is equivalent to a transformation that rearranges the VR detail in the one into the VR detail in the other. The transformation must not destroy neighbor relations. Figure 1 shows how one monad mirrors another by means of an analogy. The grid wraps around, that is, points on the bottom have neighbors on the top and points on the left edge have neighbors on the right edge. The shift clearly preserves all neighbor relations, in that if N(x,y) is a neighbor relation N between points x and y in monad 1, then N(x,y) is also a neighbor relation between points x and y in monad 2. For instance, B is to the left of C in both monads 1 and 2; A is above E in both monads 1 and 2. Thus each point in each monad has the same set of neighbors, as an examination (for instance) of point K reveals clearly. Figure 2 shows how the analogy shifts the pattern on a monad. Figure 3 states the analogy explicitly as a relation by showing how points have their contents shifted.
Figure 1. Shifting the points in VR detail.
Figure 2. Shifting the pattern in VR detail.
Figure 3. Analogy between two monads.
The structure-preserving transformations of the VR detail of monads are the natural transformations of the monads. If the points in the VR detail have neighbor relations that make them square cells in a grid (like the life grid), then there are 8 natural transformations: shift right, shift left, shift up, shift down, rotate clockwise, rotate counterclockwise, reflect vertically, reflect horizontally. These are the internal symmetries of the VR detail's geometry. If the geometry of the VR detail is closed by wrap-around, we don't need all 8 natural transformations, since a series of transforms in one direction is equivalent to a series in the other direction (for instance, in an N by N grid, N left shifts are equivalent to 1 right shift). So in the sequel I'll deal only with shifting down (D) and right (R).
Each monad runs the same life grid from the same initial pattern; all monads run the same VR algorithm (the life transition function). But each monad has its own special focal cell which is in the center of its perceived life grid. But by shifting its focal cell, one monad is able to perceive what some other is perceiving. Each pictures what is going on in every other monad without looking into any other monad.
57. "And as the same town, looked at from various sides, appears quite different and becomes as it were numerous in aspects [perspectivement]; even so, as a result of the infinite number of simple substances, it is as if there were so many different universes, which, nevertheless are nothing but aspects [perspectives] of a single universe, according to the special point of view of each Monad. (Theod. 147.)"
57. Every monad is a perspective on the whole world. The ordinary way to think of perspectives is to think of looking at an object from an external point of view. Every shift in your position (to the left, right, up, or down) yields a different image of the object from a different point of view. If you laid on your side or stood on your head (if you rotated your body), you'd get other images from these unusual points of view. Each monad is a point of view on the whole universe, and there are as many monads as there are perspectives.
The difference with monads is that their points of views are not external to anything; rather, they are internal points of view (only God is external to the universe). To understand this difference, it's necessary to focus on the natural transformations (shifts, rotations, reflections) instead of the appearances. Since there is no one thing external to all monads that they are all mirroring (I don't interpret the monads as all mirroring God), the monads are all mirroring one another. It's like mirrors reflecting other mirrors, and the things in the mirrors (material things) are the appearances of the mirrors to one another. The patterns in the VR detail are mirror-images of other patterns in VR detail.
Perspectives are natural transformations. Thus the perspective of one monad on another might be to-the-right-of, or down. If the VR detail of monad X is mapped onto the VR detail of monad Y by a natural transformation R, then R is the perspective of monad X on monad Y. A point of view (a perspective) is not an appearance of a thing, but a transformation of one appearance into another.
Algebraic closure is perspectival closure. A set of monads is perspectivally closed if for each monad x and for each perspective P, there is another monad y in that set such that x has perspective P on y. A perspectivally closed set of monads is a complete world space.
If a set of monads is perspectivally closed, then those monads can be given locations in a single common objective world space. This world space is structured by its own neighbor relations, which are public and objective. These relations supervene on the monads, and relative to them, are just virtual. Only God actually is able to know these relations, though monads can construct them.
Every natural transformation is a movement in space. The geometry of physical space is defined by the natural transforms that are permitted, since each natural transform defines a neighbor. Suppose some world permits only shifts. Its geometry is thus Euclidean (with a discrete metric). For example, on a 4 by 4 grid, using only the shift operators, there are just 4*4 = 16 ways to shift any appearance. A set of 16 monads, each of which is transformable by a shift into some other one in that set, is perspectivally closed.
Two monads are comparable according to how many shifts it takes to transform the VR detail in the one into the VR detail in the other. For instance, in Figure 4 appearance W is transformed into Z through a right shift R and then two down shifts D and D. The transformation of W by R is written R(W); application of shift D to R(W) is written D(R(W)). So, with all 3 transforms, Z = D(D(R(W))). The monad X is more similar to monad W than Z is to W because it only takes 1 shift to go from W to X, but 3 to go from W to Z. Similarity is distance! Each transformation is a step in space. It takes 1 step to go from W to X, but 3 steps to go from W to Z, so W is more similar to X than it is to Z, and therefore W is closer to X than W is to Z.
Figure 4. Transformations of VR detail.
Given a perspectivally closed set of monads, it is possible to arrange them all by comparing and contrasting their VR details so that they are distributed in space. We can now identify any monad we want as the origin of space, and give it null coordinates. In 2-dimensions, these are (0,0). We can then assign coordinates to all other monads by letting a shift in one direction be +1 and in the opposite direction be -1, for all dimensions of that space. The result is a system of coordinates that are purely relational. Figure 5 shows a system closed under shifting.
Figure 5. A set of monads closed under shiftings.
58. "And by this means there is obtained as great variety as possible, along with the greatest possible order; that is to say, it is the way to get as much perfection as possible. (Theod. 120, 124, 241 sqq., 214, 243, 275.)"
59. "Besides, no hypothesis but this (which I venture to call proved) fittingly exalts the greatness of God; and this Monsieur Bayle recognized when, in his Dictionary (article Rorarius), he raised objections to it, in which indeed he was inclined to think that I was attributing too much to God- more than it is possible to attribute. But he was unable to give any reason which could show the impossibility of this universal harmony, according to which every substance exactly expresses all others through the relations it has with them."
59. Universal harmony. For each cell in every monad's life grid, there exists (outside of it) some other monad corresponding to that cell. The analogies between life grids in monads are isomorphisms, so the monads are coordinated spatially. Because the monads are synchronized, they are coordinated temporally. Thus they are spatio-temporally harmonized.
60. "Further, in what I have just said there may be seen the reasons a priori why things could not be otherwise than they are. For God in regulating the whole has had regard to each part, and in particular to each Monad, whose nature being to represent, nothing can confine it to the representing of only one part of things; though it is true that this representation is merely confused as regards the variety of particular things [le detail] in the whole universe, and can be distinct only as regards a small part of things, namely, those which are either nearest or greatest in relation to each of the Monads; otherwise each Monad would be a deity. It is not as regards their object, but as regards the different ways in which they have knowledge of their object, that the Monads are limited. In a confused way they all strive after [vont a] the infinite, the whole; but they are limited and differentiated through the degrees of their distinct perceptions."
60. So far we have treated the VR and AI subprograms of monads as if they were entirely separate; now we must integrate them. This is the mind/body integration. All monads perceive the whole (thus all are running VR detail that includes the entirity of space, hence the same space); but all are focuses only on a part. What is nearer to a monad (i.e. to the center of its body) is clearer; what is farther away is confused. But clarity and confusion are features of the monad's AI subprogram, which represents the whole world from the monad's particular point of view. Each monad's AI program is harmonized with its VR program. Harmony is maintained by transforming the output of the VR program into input to the AI program. The input into the AI program is the sensory field (one for each modality of the monad). Our monads only have visual experience, so they just have a visual field. The visual field of each monad's AI subprogram is much like the VR detail, except that it realizes the decreasing clarity as distance from its center. The AI visual field contains cells (like retinal cells). To implement loss of clarity with distance from the center, we make the cells in the middle larger and have them get smaller as they get farther away (their size decreases vertically & horizontally; indeed, one could thus place an infinite series of cells in a finite bounded square by having their sizes decrease by 1/2 every step). The VR details are objective perspectives; the AI visual field is a subjective perspective. Figure 6 shows the grid as it appears to the AI program (the AI visual field).
Figure 6. The visual field.
61. "And compounds are in this respect analogous with [symbolisent avec] simple substances. For all is a plenum (and thus all matter is connected together) and in the plenum every motion has an effect upon distant bodies in proportion to their distance, so that each body not only is affected by those which are in contact with it and in some way feels the effect of everything that happens to them, but also is mediately affected by bodies adjoining those with which it itself is in immediate contact. Wherefore it follows that this inter-communication of things extends to any distance, however great. And consequently every body feels the effect of all that takes place in the universe, so that he who sees all might read in each what is happening everywhere, and even what has happened or shall happen, observing in the present that which is far off as well in time as in place: sympnoia panta, as Hippocrates said. But a soul can read in itself only that which is there represented distinctly; it cannot all at once unroll everything that is enfolded in it, for its complexity is infinite."
61. We have already mentioned that we can construct an objective life grid over the totaliity of monads by extracting the focal cell of each monad and placing it at the row and column indicated by its internal coordinates. This life grid is the material world, it is the world as a complex substance. The material world supervenes on monads, whose internal life grids are ideal. It might seem as if we've put a copy of the material world in each monad, but we haven't. The reason is that cells in the material world are parts of it, but cells in the internal life grid of a monad are not parts of it but functional aspects of it. Material part-whole relations supervene on the ideal functional aspects internal to monads. Materiality is just a way of looking at the whole world as if relations among things were external rather than internal. This has scientific utility but not metaphysical truth. All action in the material world is apparent, not real, but there is a real structure to these appearances. For instance, there are laws of glider motion and spaceship motion, as well as more complex spatio-temporal invariants. Physics is the study of all the spatio-temporal regularities in the life grid; matter is just the totality of appearances (patterns) with physical regularities. Material motion is transmitted locally from one neighbor to another. God alone is able to read in each thing all that happens to all, for God alone has the external perspective on the whole universe. Souls are able to see clearly only what is happening near their focal cells.
62. "Thus, although each created Monad represents the whole universe, it represents more distinctly the body which specially pertains to it, and of which it is the entelechy; and as this body expresses the whole universe through the connexion of all matter in the plenum, the soul also represents the whole universe in representing this body, which belongs to it in a special way. (Theod. 400.)"
62. Each monad's AI program remains focused on the point in the center of its body. As the body of the monad moves, the monad computes a natural transformation of the VR detail onto the visual field such that the body stays in the center of the visual field. Thus the body remains in the center of clarity and attention, and the world grows fuzzy around it. Suppose the body of a monad is a glider and that there is a transformation T of the monad's VR detail onto its visual field, so that what the monad sees is T(VR). After 2 clock ticks the glider has moved 1 unit vertically (e.g. down), so the AI-VR mapping must shift the life grid up to keep the center of the glider in focus in the center of the visual field; the result is the transformation U(T(VR)). In another 2 clock ticks the glider has moved horizontally (say to the right), so the AI-VR mapping must shift the VR detail left to keep the center of the glider in focus; the result is the transformation L(U(T(VR))). As the VR detail goes through its changes, the AI-VR mapping also changes. In this case, after n clock ticks the AI-VR mapping is (L°U)n(T). Thus the center of the glider is always in the center of the visual field.
In Figure 7, the visual field shifts so as to keep the center of the glider on the center cell. This is analogous to our own situation as we walk through space: our visual relation to our own walking body is constant; if the origin of my gaze is the center of my phenomenal world, then my body surrounds this origin, and as I walk towards a thing, it appears to come towards me (the distance between it and my body decreases, but the distance between myself and my body always remains the same, namely, zero).
Figure 7. The convergence of the body and an object.
Each monad represents the whole universe, but more particularly its body. The AI program of each monad is focused on the focal cell in the life grid, which it brings into the center according to the law that everything around it diminishes in clarity proportional to the distance. Thus the VR detail is just the life grid; but the AI program perceives a transformed version of that detail. The transformation is like this: (1) the life grid is dynamically shifted so the monad's body is in the center of the visual field (the main monadic program, the larger program of which the VR and AI programs are subprograms, performs this shift so that it stays focused on the center of its body); (2) cells shrink proportional to the distance from the center. The transformation is structure-preserving, so the internal AI program's representation is isomorphic with and thus harmonized with that of the VR program.
Having a multiplicity of subjective and objective perspectives seems massively redundant. Why not just have one objective perspective (one life grid) and many subjective perspectives (natural transforms) in it, all of which change to maintain mind/body coordination. The objective perspective would just be God. This is not Leibniz's theory, but it seems to have been that of Malebranche, and it might also be that of Berkeley. In such a system, there are perspectives that are unoccupied (by souls); such a world is not complete, and perhaps Leibniz (with his desire for algebraic perfection and spatial plena) prizes completeness above all else. But in a Malebranche-Berkeley world, no information is missing even though some computations are not being performed: such a world is complete with missing parts.
63. "The body belonging to a Monad (which is its entelechy or its soul) constitutes along with the entelechy what may be called a living being, and along with the soul what is called an animal. Now this body of living being or of an animal is always organic; for, as every Monad is, in its own way, a mirror of the universe, and as the universe is ruled according to a perfect order, there must also be order in that which represents it, i.e. in the perceptions of the soul, and consequently there must be order in the body, through which the universe is represented in the soul. (Theod. 403.)"
63. The body of a monad (the pattern in the center of the grid), along with the AI program, is an organism; if the AI program is a soul, then this pair is an animal.
64. "Thus the organic body of each living being is a kind of divine machine or natural automaton, which infinitely surpasses all artificial automata. For a machine made by the skill of man is not a machine in each of its parts. For instance, the tooth of a brass wheel has parts or fragments which for us are not artificial products, and which do not have the special characteristics of the machine, for they give no indication of the use for which the wheel was intended. But the machines of nature, namely, living bodies, are still machines in their smallest parts ad infinitum. It is this that constitutes the difference between nature and art, that is to say, between the divine art and ours. (Theod. 134, 146, 194, 403.)"
65. "And the Author of nature has been able to employ this divine and infinitely wonderful power of art, because each portion of matter is not only infinitely divisible, as the ancients observed, but is also actually subdivided without end, each part into further parts, of which each has some motion of its own; otherwise it would be impossible for each portion of matter to express the whole universe. (Theod. Prelim., Disc. de la Conform. 70, and 195.)"
66. "Whence it appears that in the smallest particle of matter there is a world of creatures, living beings, animals, entelechies, souls."
67. "Each portion of matter may be conceived as like a garden full of plants and like a pond full of fishes. But each branch of every plant, each member of every animal, each drop of its liquid parts is also some such garden or pond."
68. "And though the earth and the air which are between the plants of the garden, or the water which is between the fish of the pond, be neither plant nor fish; yet they also contain plants and fishes, but mostly so minute as to be imperceptible to us."
69. "Thus there is nothing fallow, nothing sterile, nothing dead in the universe, no chaos, no confusion save in appearance, somewhat as it might appear to be in a pond at a distance, in which one would see a confused movement and, as it were, a swarming of fish in the pond, without separately distinguishing the fish themselves. (Theod. Pref. [E. 475 b; 477 b; G. vi. 40, 44].)"
64-69. Natural bodies are divine machines that are machines in all their parts; they are infinitely complex. Every bit of matter is not merely infinitely divisible, but actually infinitely divided. The infinite external complexity of the material world parallels the infinite internal complexity of every monad. To make an actually infinitely divided grid, we would have to subdivide the grid infinitely many times. Figure 1 shows a series of subdivided grids, named G(1), G(2), and G(3), containing 4, 9, and 25 monads at grid sites. The series {G(i)} of such grids is recursive: grid G(i+1) is obtained by dividing each square in G(i) into 4 equal squares and putting monads at the new sites. If G(i) is the i-th iteration of this subdivision, then we obtain an actually infinitely divided grid by taking the union of G(i) for i = 1 to infinity. But the actually divided grid contains only countably many monads due to its recursive construction. But classical continuity requires uncountably many monads. Such a structure is not recursively constructible: it is not definable by any computation. Even if it were, it is not clear how the grid would compute, because the neighbor relation would remain undefined (by arguments like those employed by Zeno). So we can't use such a construction: our model fails drastically in this respect. But the failure is revelatory: there is no procedure for computation on an infinitely subdivided network. Leibniz's remarks that artificial machines (made by humans) are not machines in all their parts is profound: it anticipates the Church-Turing thesis. God's art may well be superior to ours, but it is also incomprehensible. The life grid in each of our model monads is only finitely complex, because it is only finitely divided. Therefore, the material world supervening on the totality of monads is finitely divided. Thus every bit of matter is finitely divisible and divided. There is nothing sterile or dead in the world, but the hierarchical constructions of bodies are limited.
70. "Hence it appears that each living body has a dominant entelechy, which in an animal is the soul; but the members of this living body are full of other living beings, plants, animals, each of which has also its dominant entelechy or soul."
71. "But it must not be imagined, as has been done by some who have misunderstood my thought, that each soul has a quantity or portion of matter belonging exclusively to itself or attached to it for ever, and that it consequently owns other inferior living beings, which are devoted for ever to its service. For all bodies are in a perpetual flux like rivers, and parts are entering into them and passing out of them continually."
72. "Thus the soul changes its body only by degrees, little by little, so that it is never all at once deprived of all its organs; and there is often metamorphosis in animals, but never metempsychosis or transmigration of souls; nor are there souls entirely separate [from bodies] nor unembodied spirits [genies sans corps]. God alone is completely without body. (Theod. 90, 124.)"
73. "It also follows from this that there never is absolute birth [generation] nor complete death, in the strict sense, consisting in the separation of the soul from the body. What we call births [generations] are developments and growths, while what we call deaths are envelopments and diminutions."
74. "Philosophers have been much perplexed about the origin of forms, entelechies, or souls; but nowadays it has become known, through careful studies of plants, insects, and animals, that the organic bodies of nature are never products of chaos or putrefaction, but always come from seeds, in which there was undoubtedly some preformation; and it is held that not only the organic body was already there before conception, but also a soul in this body, and, in short, the animal itself; and that by means of conception this animal has merely been prepared for the great transformation involved in its becoming an animal of another kind. Something like this is indeed seen apart from birth [generation], as when worms become flies and caterpillars become butterflies. (Theod. 86, 89. Pref. [E. 475 b; G. vi. 40 sqq.]; 90, 187, 188, 403, 86, 397.)"
75. "The animals, of which some are raised by means of conception to the rank of larger animals, may be called spermatic, but those among them which are not so raised but remain in their own kind (that is, the majority) are born, multiply, and are destroyed like the large animals, and it is only a few chosen ones [elus] that pass to a greater theatre."
76. "But this is only half of the truth, and accordingly I hold that if an animal never comes into being by natural means [naturellement], no more does it come to an end by natural means; and that not only will there be no birth [generation], but also no complete destruction or death in the strict sense. And these reasonings, made a posteriori and drawn from experience are in perfect agreement with my principles deduced a priori, as above. (Theod. 90.)"
77. "Thus it may be said that not only the soul (mirror of an indestructible universe) is indestructible, but also the animal itself, though its mechanism [machine] may often perish in part and take off or put on an organic slough [des depouilles organiques]."
70-77. Every animal's body is a pattern centered on its focal cell. So this pattern is distributed over the material world. These bodies grow and decay, unfolding and enfolding their complexities. These bodies are infintely complex. They vary in size from what we see to microscopic (even infinitesimal). The arrangements of parts of bodies change over time, but bodies never entirely perish; at worst, they simply become infinitesimal microscopic seeds. While our finitary construction of the life grid cannot account for the changes in animals, our infinitary construction can. Even so, our finite construction suffices for much (but certainly not all) of Leibniz's doctrine. On our version, animals lose their bodies at death, though this does not stop the internal actions of their monads. Resurrection is permitted since God is able to restore the body (the pattern), though God must do this over all monads.
78. "These principles have given me a way of explaining naturally the union or rather the mutual agreement [conformite] of the soul and the organic body. The soul follows its own laws, and the body likewise follows its own laws; and they agree with each other in virtue of the pre-established harmony between all substances, since they are all representations of one and the same universe. (Pref. [E. 475 a; G. vi. 39]; Theod. 340, 352, 353, 358.)"
79. "Souls act according to the laws of final causes through appetitions, ends, and means. Bodies act according to the laws of efficient causes or motions. And the two realms, that of efficient causes and that of final causes, are in harmony with one another."
80. "Descartes recognized that souls cannot impart any force to bodies, because there is always the same quantity of force in matter. Nevertheless he was of opinion that the soul could change the direction of bodies. But that is because in his time it was not known that there is a law of nature which affirms also the conservation of the same total direction in matter. Had Descartes noticed this he would have come upon my system of pre-established harmony. (Pref. [E. 477 a; G. vi. 44]; Theod. 22, 59, 60, 61, 63, 66, 345, 346 sqq., 354, 355.)"
81. "According to this system bodies act as if (to suppose the impossible) there were no souls, and souls act as if there were no bodies, and both act as if each influenced the other."
78-81. There are no causal interactions betweeen body and mind. The body does not cause effects in the mind during perception, nor does the mind cause effects in the body during action. Rather, both perception and action are (inverse) isomorphisms between body and mind. Our model clearly demonstrates the impossibility of mind-body interaction (or at least action of the mind on the body). Since every monad contains a copy of the whole world, if the mind acted on its body it would have to change some pattern on the life grid in every other monad. Its action would have to extend over the whole world. But that's impossible. The mental functionality of each monad is just its AI program; the bodily functionality of each monad is a pattern centered on the focal cell of its internal life grid (run by its VR program). The AI program is a cellular automaton whose state-transitions mirror those of the body. The body associated with every monad contains a nervous system, including sense organs and a brain. Neurology establishes correspondences between the mind and the nervous system. Beginning with sensation, there is a formal structure (a program) whose states correspond to states of the nervous system. For example, every retinal event corresponds to a visual sensation. Every visual sensation has a vertical and horizontal location in an ideal phenomenal grid isomorphic to the retina; but every visual sensation has three other dimensions: hue, saturation, and brightness. Every visual sensation is a 5-tuple of qualities: (vertical, horizontal, hue, saturation, brightness). The totality of these 5-tuples is a 5-dimensional visual space in which every visual sensation is a point. There is one such 5D space for each retina; the 2 5D space are integrated via a stereoscopic synthesis into a single 3D visual space whose points are visible things. These visible things are intentional objects; each of which corresponds to a complex pattern of impulses in the nervous system. Analogous spaces exist for all the other senses. The mind has thoughts and images that supervene on the brain as the 5-dimensional visual space supervenes on the retina. The mind is an AI program whose algorithmic activity is the logical construction of the world, like that carried out by Carnap. The mind is a cellular automaton whose states are just these formal mental structures. Both states and transitions of the mental automaton correspond to those of the bodily automaton. This is the pre-established harmony between body and mind. Mind and body are coordinated, but they do not interact.