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THESIS: The road goes from WPU to NYC.
ANTITHESIS: The road goes from NYC to WPU.
SYNTHESIS: The road is the same no matter which direction you go.
THESIS: The sequence ABCDE; A is to the left of B, etc.
ANTITHESIS: The sequence EDCBA; A is to the right of B, etc.
SYNTHESIS: All that matters is that the letters have the same order relative to one another: A is beside B, B is beside C; the arrangement of the letters, the structure of the sequence, is the same no matter whether it goes from right to left or left to right.
THESIS: You wear your clothes normally.
ANTITHESIS: You wear your clothes turned inside-out.
SYNTHESIS: You can wear your clothes either way, because either way they have the same form. Sameness of form is isomorphism.
THESIS: This photograph.
ANTITHESIS: The photographic negative of this picture.
SYNTHESIS: Both the photo and its negative show the same form; the colors are inverted, but the form (the arrangement of things) remains unchanged.
Colors and words are not the same. It's entirely possible that you and your neighbor do not associate the same color with the same word. Suppose that your color perceptions are inverted or switched. For instance, green and red, yellow and blue, black and white, are switched.
What you see and say
Roses are red.
Grass is green.
Sun is yellow.
Sky is blue.
What your neighbor sees and says
Roses are red.
Grass is green.
Sun is yellow.
Sky is blue.
Since you both learned the words "red", "green", "yellow", "blue", "black", "white" by people pointing to the relevant objects and telling you their color-names, you and your neighbor always use those color-names in exactly the same ways. You'll never detect any difference by means of speech or behavior. If you ask for the red cup, your neighbor will get the one you expected.
The British philosopher John Locke was the first to deal with the Inverted Spectrum problem. Here's what he says:
Though one man's idea of blue [could] be different from another's. Neither would it carry any imputation of falsehood to our simple ideas, if by the different structure of our organs it were so ordered, that the same object should produce in several men's minds different ideas at the same time; v.g. if the idea that a violet produced in one man's mind by his eyes were the same that a marigold produced in another man's, and vice versa. For, since this could never be known, because one man's mind could not pass into another man's body, to perceive what appearances were produced by those organs; neither the ideas hereby, nor the names, would be at all confounded, or any falsehood be in either. For all things that had the texture of a violet, producing constantly the idea that he called blue, and those which had the texture of a marigold, producing constantly the idea which he as constantly called yellow, whatever those appearances were in his mind; he would be able as regularly to distinguish things for his use by those appearances, and understand and signify those distinctions marked by the name blue and yellow, as if the appearances or ideas in his mind received from those two flowers were exactly the same with the ideas in other men's minds. 
Here's the dialectic:
THESIS: Color-names assigned to spectrum one way.
ANTITHESIS: Color-names assigned to spectrum the opposite way.
SYNTHESIS: All that matters is that the arrangement (the sequential ordering) of color-names and the arrangement (the sequential ordering) of colors are the same. It doesn't matter how names are paired with colors, so long as the pairing preserves the relative order of words and the relative order of colors.
Having advanced beyond the stages of mere sense-certainty and perception, Hegel says that consciousness now finds itself faced with a world of appearances that it aims to understand (¶132-165). The dialectical development of the understanding begins with a thesis, a positive world of appearances experienced as the spatio-temporal play of opposed forces. We model this thesis using a computational structure known asthe game of life.
The game of life is played on a grid like a chess board. This grid is called the life grid. Since the life grid is divided into square regions, known as cells, the game of life is a cellular game. Each cell on the life grid is a unit of force. Using the science of his day, Hegel speaks of force as divided into two opposed aspects (like positive and negative electricity); he refers to these two aspects as soliciting and solicited (¶136-141). We denote these two aspects of force simply as LIVE and DEAD. Each cell is either LIVE or DEAD, so that at any moment the life grid is occupied by a pattern of living cells amidst dead ones. The pattern on the life grid at any moment is how we model the appearance of the world to consciousness. Since life and death are mutual opposites, we model them using opposed colors: we display LIVE cells as black and DEAD cells as white.
The game of life consists of setting up the life grid with some initial pattern of LIVE cells, and then watching the changes in that pattern from one tick of the clock to the next. Since these changes take place automatically, the game of life is a cellular automaton. As Hegel reminds us, if consciousness were still at the level of sense-certainty, it would simply be entertained by the patterns it sees: the world for sense-certainty contains only what is immediately here and now. If consciousness were still at the level of perception, it would see that the patterns are internally structured, but it would not see that there are causal relations among them. It would not see that the set of all possible patterns on the life grid is a system. But consciousness is growing up, and now it understands that the set of all possible patterns is a totality organized by causal laws (¶142).
Over time, the patterns of forces on the life grid change. Many change in irregular ways, or rapidly die off. Other patterns, however, exhibit regular behaviors. Experiencing such changes at the level of the understanding, Hegel says that consciousness apprehends their regularities and posits an inner world of laws (¶144-146) beyond or behind the play of forces, behind the patterns on the life grid.
Initially, this inner world is empty (¶146), for consciousness has posited it as the negation of appearance, which has been the only kind of object it knows. But since the intentionality of consciousness demands an object, consciousness initially fills this void by projecting into it images produced by consciousness itself (¶146). The images thus projected are not, however, objects produced by the lower cognitive activities of sense-certainty or perception (¶147); rather, the images that consciousness as understanding projects behind the appearances on the life grid are abstractions. Consciousness fills the inner world it projects behind the life grid with the causal structures it abstracts from the play of forces. The simplest of these structures is the mutual opposition between the two types of force, which enables each to solicit the other (¶148). In the game of life, this reciprocal solicitation is illustrated by simple cyclical patterns in which LIVE and DEAD cells seem to alternatively create and destroy one another. Figure 1 shows several simple cyclical patterns.
Figure 1. Oscillators with period two.
Consciousness quickly advances beyond the mere apprehension of symmetry and cyclical alternation to the apprehension of more complex spatio-temporal regularities among forces, and it grasps all these regularities as the laws of force (¶148). It consequently grasps the inner world as a realm of laws (¶149), which it has projected behind appearances by means of its own abstractive activity. These laws of force are the spatio-temporal invariants of patterns.
A spatio-temporal invariant correlates a shift in space with a shift in time; such a correlation reconciles the superficial opposition of space and time by uniting them in a single abstract form. Laws of motion, such as Newton's law of gravitational attraction, are spatio-temporal invariants (¶150). The game of life contains many patterns that exhibit simple laws of motion. One such pattern is called the glider. The glider moves diagonally across the life grid, shifting itself 1 square vertically and 1 square horizontally every 5 moments. Figure 2 illustrates glider motion. Importantly, while the glider moves, the cells over which it moves do not. Glider motion is the repetition of the same pattern on different cells; the glider is not a thing, but a pattern supervening on things.
Figure 2. The motion of a glider.
As consciousness reflects further on the laws of motion of the life world, it realizes that these laws are just empirical regularities; they are abstracted directly from appearances, and such abstraction amounts to no more than mere generalization. But Hegel insists that the understanding wants to grasp the set of all possible appearances of the world as a unified whole. It therefore projects behind the multiplicity of empirical regularities a single, theoretical law from which all the empirical regularities can be deduced (¶150). This projection is not mere imagination, but the scientific construction of a hypothesis able to serve the needs of explanation (¶154).
Consciousness projects this theoretical law when it realizes that each cell changes its state according to a rule it computes. It first encounters this rule when it sees that whether a cell is LIVE or DEAD in the next moment depends on just two local factors: (1) the cell's current state; and (2) the total number of its LIVE neighbors. To abstract this rule, consciousness proceeds by the scientific method, testing various hypotheses concerning the conditions in which a cell changes its state. What consciousness discovers is that if a cell is LIVE now, then it will be LIVE in the next moment only if it has either 2 or 3 LIVE neighbors; otherwise it dies; but if a cell is DEAD now, then it will become LIVE in the next moment only if it has exactly 3 LIVE neighbors; otherwise it stays DEAD. This local transition rule is a program computed simultaneously and independently by each cell, it is the single theoretical law behind the empirical reality of the life world. It is the Concept of the game of life, the program driving the evolution of the life world as a whole, and the understanding is the first shape of consciousness able to see that dynamical rational patterns -- programs or concepts -- really exist in things.
Although consciousness now understands the spatio-temporal structure of the world theoretically, Hegel points out that it does so only accidentally. So far consciousness has focused only on one side of the opposition between LIVE and DEAD. Although it sees patterns as LIVE cells on a DEAD background, this is arbitrary: the DEAD cells form patterns too. By focusing just on the LIVE cells, consciousness does not see that any pattern of LIVE cells presupposes a pattern of DEAD cells, and vice versa. For consciousness, the life world is like a coin with just one side (¶153). Consequently, consciousness does not see the other side of this world; it does not see that this world is like a shirt that can be turned inside-out and still worn. Because consciousness does not see the other side of the life world, does not see that DEAD cells are also able to form patterns, the explanations it gives by applying theoretical laws to patterns of LIVE cells are just empty syntactic procedures, mere tautologies (¶154-155).
To really understand the world of appearances, consciousness must grasp all oppositions in this world as reflections of one another, and in so doing it must grasp the axis of symmetry around which they are in fact reflected. It is around this axis of symmetry that, as Hegel puts it, what is self-same repels itself from itself and what is not self-same attracts itself to itself (¶156-157). By grasping such an axis as the unity on which every mutual opposition is based, the understanding grasps the oppositions essentially as two sides of the same coin. What consciousness grasps is that all oppositions are systematically invertible (¶157-159). Seeing that these oppositions are invertible, consciousness now sets out to invert them.
There is only one kind of opposition in the game of life: that between the concrete values LIVE and DEAD. When LIVE and DEAD are grasped together as opposed aspects of the same quality, they are grasped as symmetric: what is grasped is the mutuality of their opposition. The opposition between LIVE and DEAD is invertible in both patterns and laws. For each pattern in the game of life, there is an inverted pattern derived by replacing all the LIVE cells with DEAD cells, and vice versa. This anti-pattern is the mirror-image of its original, something like a photographic negative. For both the local and global transition rules, there are inverted rules derived by replacing all references to LIVE with DEAD, and vice versa. These two anti-transition rules are also mirror-images of their originals. Consciousness at the earlier levels of sense-certainty and perception cannot see that an original and its mirror-image are related, that they are two sides of the same coin. Such unification requires the understanding, which grasps the symmetry behind opposition.
By exchanging LIVE and DEAD in both patterns and transition rules, the understanding has turned the game of life inside-out. It thus derives from Life another game, which we call AntiLife. The global anti-transition rule is the Concept of AntiLife. AntiLife really is the mirror-image of Life, but on the theoretical level only. On the empirical level, Life and AntiLife are the same: in both games, the color white indicates a DEAD cell and the color black indicates a LIVE cell. What Hegel calls the sensuous determinations, which in our models are the colors, are not inverted. What is inverted are the roles these sensuous determinations play in patterns and theoretical laws: what is LIVE in the patterns and laws of Life is DEAD in those of AntiLife, and vice versa. For example, while the initial state of Life is all DEAD cells, the initial state of AntiLife is all LIVE cells. Patterns in Life are LIVE cells on a DEAD background; anti-patterns in AntiLife are DEAD cells on a LIVE background. For instance, the anti-glider, shown in Figure 3, looks just like a glider formed of DEAD cells rather than LIVE ones. What is surprising is not that pairs of patterns and anti-patterns exist, but that the spatio-temporal invariants of patterns in Life and their anti-patterns in AntiLife are exactly the same. This is striking because the theoretical laws from which these empirical regularities are deduced are exactly the opposite.
Figure 3. Motion of the anti-glider.
Consciousness has just realized that the inner world of laws is like a shirt that can be turned inside-out and still worn; it has in fact turned the game of Life inside-out to produce the game of AntiLife. What consciousness needs to understand next, Hegel tells us, is that the reason an inside-out shirt can be worn is just because there is a single abstract form of the shirt that is common to both its normal and its inside-out instantiations.
For example, if a photograph is a thesis, then its photographic negative is its antithesis; but their synthesis is the recognition that both display the same pictorial structure.
Consciousness has just grasped that the opposites it used to separate are two sides of the same coin; it now needs to grasp that coin itself. It needs to see how each of the two games "is itself and its opposite in one unity" (¶160).
Simple perception of the motion of the glider and anti-glider shows that the very same law of motion applies equally to each: both the glider and the anti-glider move diagonally across their respective grids. For perception, such observations serve as the basis for an inductive argument with a universal conclusion: every law of motion that is true of any pattern is also true of its anti-pattern. For the understanding, however, such inductive arguments are not adequate: the understanding requires a mathematical proof that the two games share a common spatio-temporal relational structure. Such a proof is straightforward, but far too technical to present here. We present it in an Appendix. The proof reveals a correspondence between Life and AntiLife that preserves their causal structures. It shows that any physical theory that is empirically adequate for Life has an empirically equivalent complement that is adequate for AntiLife.
The correspondence between the causal structures of Life and AntiLife permits consciousness to unify their physical theories: at a deep theoretical level, Life and AntiLife are the same. These two games are not just mirror-images of one another: both are really specialized versions of a more abstract game. Life and AntiLife are both realizations or implementations of a deeper synthesis, the definition of which replaces all concrete determinations with formal parameters whose different settings yield different games. The definition of this abstract game is entirely free of any concrete content; it is purely formal, purely functional.
While the two games Life and AntiLife would simply have been opposed
by sense-certainty and perception, the understanding sees that they are
really just opposite realizations of the same basic theory. In grasping
that a single theory has opposite realizations, the understanding realizes
that there is no limit to the set of possible concrete realizations of any
given theory; it learns that the set of concrete things able to realize
a particular function is potentially infinite (¶161), and that what
matters for science is form, not content. This conceptual leap from the
constraining particularity of content to the liberating universality of
form prepares consciouness for its advance to self-consciousness (¶162-165).
The synthesis of Life and AntiLife is the mirror-image of itself, but self-consciousness
is also the mirror-image of itself, and Hegel will argue that this double
mirroring is the basis for all social and cultural relations.
1. John Locke, Essay Concerning Human Understanding, Bk. II, Ch. 32, Sec. 15.
William Paterson University Philosophy Department