Plato’s Mathematical Psychophysics of Color

Abstract

Aristotle is often regarded as providing a potentially appropriate model for a naturalistic human psychology that is able to reconcile the commonly opposed normative or “manifest” and factual or “scientific” images of the world and restore to the world the qualities that constitute its value. Such Aristotelian features were taken up after Newton by Goethe in his Theory of Color in his attempt to restore the actual color to the world that had seemingly been drained of it by Newtonian science. Here, I argue that beneath the “modificationalist” elements that Goethe took from Aristotle lies a mathematical approach to color originating in Plato that exploits similarities between color and tonal consonances and dissonances. The logical structure of Goethe’s color theory has recently been investigated by proponents of “universal logic”, but only when this theory is viewed against the background of Plato’s appropriation of Pythagorean harmonic theory does its full explanatory potential become apparent.

1. Introduction

Many practicing natural scientists would no doubt be puzzled by the continuing interest among psychologists and philosophers in the psychological studies of Aristotle1. Modern science is generally conceived to have, in the seventeenth century, liberated itself from Aristotelian natural philosophy; why, then, four centuries later, should his writings still be taken seriously? One particular reason has to do with the contrast between what Wilfrid Sellars described as the “manifest” and “scientific” images of the world and the role of humans within it [3]. While modern Western culture since the 17th century has generally been marked by the rise of a naturalistic view of the world, this has left significant problems for our view of ourselves and our roles within it. Among other things, naturalism faces problems in accounting for those aspects of ourselves that we take for granted in intellectual activities, such as pursuing the naturalistic point of view itself.

From a naturalistic point of view, we are simply complex causal mechanisms, but in the context of our conceptions of ourselves as investigators of the world, we think of and treat ourselves and others as free and rational, for example, as accountable to the rational norms presupposed in all scientific inquiry and debate. Our assertions, then, belong not simply to the “space of causes” but also to the normative “space of reasons”. However, from the scientific viewpoint itself, norms and reasons are ontologically peculiar entities that have no discernible place in the world of scientific facts.

To this well-known aspect of Sellar’s distinction could be added the issue of the qualities found by perceivers in the manifest world—qualities with the peculiar “what it is like”, such as the taste of coffee or the distinct color of lemons [4]. Like norms, these qualities have no obvious place in the world of facts. One might understand all the physics and physiology behind human color perception without knowing what it is like to experience, for example, the color of a lemon [5].

The loss of color within the view of the world introduced in particular by Newton in his Opticks of 1704 [6]—one aspect of what would become a greater “disenchantment” of the world—was noted a hundred years after Newton by Johann Wolfgang von Goethe:

My worthy friend, gray are all theories,

And green alone Life’s golden tree

[7], (57).

However, Goethe would also challenge Newton’s science by attempting, in his Theory of Color [Zur Farbenlehre] of 1820 [8], to restore color to the world as scientifically understood by reviving what was essentially Aristotle’s account of color that Newton’s had, Goethe thought, wrongly displaced.

Goethe would in no way succeed in displacing Newton as the genuine scientist of color; nevertheless, many since, including Ludwig Wittgenstein in the twentieth century [9] and, more recently, proponents of “universal logic”2, have acknowledged the importance of Goethe’s qualitative treatment of color within science generally. Here, I will argue that to appreciate Goethe’s contribution we should look beyond Aristotle’s peculiar theory of color to another theory from which Aristotle clearly drew. This approach was more overtly mathematical than Aristotle’s own and combined qualitative and quantitative features. This approach was used by Plato and his followers. In this Platonic approach, the qualitative relations among colors, and much more besides, were conceived on the basis of the model of those existing between qualitative tones in music, which, in turn, had already been given a quantitative representation by the Pythagorean mathematicians. Aristotle attempted to utilize this approach but struggled, I suggest, with the mathematics of its form, a mathematics that must be unraveled in order to show the rational dimensions of Goethe’s theory of color.

In Section 2 and Section 3, I start with those features of Aristotle’s approach that led him to attempt to articulate the field of color using the analogy of the differentiation of sound into musical consonances and dissonances. It is this aspect of his approach, I suggest, that draws upon the same Pythagorean considerations from which Plato had drawn in his late dialogues and which he applied in the dialogue Timaeus to cosmology and his related account of the “world-soul”, and by implication, to the souls of humans constructed on the model of the world-soul itself. In Section 4, the relevance of the harmonic structure of the world-soul for color is explored further, specifically in relation to the levels of intelligence found in the cosmos in Plato’s Timaeus—at the highest level, nous, followed by that of the world-soul, and finally that of individually embodied humans. Then, in Section 5, building on approaches to Goethe’s color theory by recent “universal logicians”, the Platonic infrastructure of Goethe’s approach is suggested to be the “manifest” equivalent of modern color science.

2. Aristotle and the Manifest View of Color

The modern scientific view of the world is typically contrasted with that of Aristotle, who allowed both norms and qualities into the world as scientifically understood, through which he became the “great defender of the manifest image” in the classical world [10] (110). The scientific loss of such qualities can, in turn, be extended to the loss of values in the sense captured by what Bernard Williams once described as the evaluatively “thick”, “agent-guiding” predicates found in everyday discourse [11] (ch. 8). Such predicates are “action-guiding” in the sense of relative to the needs and desires of perceivers qua organisms, that is, relative to what they consider their good. This now-lost view of qualities and values found in Aristotle is summed up in the words of Alan Code: “For Aristotle visual information is the most useful sensory information for an animal when it comes to coping with the needs of life and survival. Physical objects are colored, and by seeing their colors together with the shapes and sizes and motions that accompany them animals are able to keep track of objects, and to distinguish by visual appearance food, water, predators and prey. Of course, in intelligent creatures the information acquired through sight and the other senses is the indispensable starting point for various sorts of knowledge, culminating ultimately in scientific understanding” [12] (237). Modern scientific understanding, however, typically sees the acquisition of knowledge itself as requiring complete abstraction from the particularity and contingency of specific natural needs, such as the need for food or the avoidance of predators.

For our purposes, we will focus here on qualities themselves, and in particular, colors, the experience of which Aristotle conceived of realistically, with the sensuous quality of experience directly resembling the quality inherent in the object itself [13,14]. That is, for him, colors were what came to be described in the seventeenth century as primary qualities, rather than secondary qualities as conceived after Newton, qualities somehow dependent on the subjective responses of the perceiver to the causal impact of light on their retinas, much like William’s subjectively constituted values. Aristotle’s realist attitude toward color was, in turn, linked to another aspect of his color theory—what has been called its modificationalism [15,16].

In relation to the former, in On the Soul [17], Aristotle focused mainly on the epistemological issue of how the perceived color of a substance represented an actual quality of that substance, a quality understood as actually being the way it was represented in perceptual experience. In contrast, Newton treated color subjectively as an effect produced in the psyche by reflected light rays, which were themselves essentially colorless. For Newton, perceived colors could represent objective features of reflected light without resembling them. Given his realism, however, Aristotle had to explain what it was about perceived colored things that allowed them to be resembled by the sensory qualities experienced in perception. It was thus in works such as Sense and Sensibilia [18], Meteorology [19], and On Generation and Corruption [20] that he attempted to explain this with the so-called modificationalist side of his position. Thus, he would address the ontological question of specifying what “color is, or sound, or odour, or savour … the point of our present discussion is to determine what each sensible object must be in itself, in order to produce actual sensation” [18] (439a10–17).

His theory here was modificationalist in the sense that different colors were thought of as “modifications” of two basic colors, white and black, which, when combined in different proportions, resulted in the array of all the other colors. Mark Kalderon has summarized Aristotle’s theory and the history on which he built: “That white and black, or light and dark, are the primary colors, the colors in terms of which all other colors are explained, is an ancient doctrine, arguably of Homeric roots, that Parmenides and Empedocles share. Aristotle follows them in this. Moreover, Aristotle takes over from Parmenides and Empedocles the idea that light and dark are contraries that constitute the extreme ends of an ordered range of sensible qualities. Moreover, he emphasizes Empedocles’ contribution to this tradition in claiming that it is the ratio of light and dark when combined that determines an intermediary color” [10] (120).

As can be inferred from Kalderon’s description, when viewed from a modern perspective, Aristotle seems to have conflated two different color-related concepts—black and white as qualitatively understood colors and the contrasting quantitative features of brightness and darkness, or degrees of ambient illumination. In relation to this, Richard Sorabji had pointed out that such a conflation may have been rooted in the semantics of Greek color words themselves, noting that “a number of Greek color words did double duty. They were used as much to denote the brilliance of a color as to denote its hue” [21] (294). Thus, in ancient Greek, “leukon means bright, or light-colored, as much as it means white. And melan means dark-colored, as much as it means black” [21] (294). In fact, in modern color theory, black and white are not regarded as having hue [22] (ch. 2), [23] (159–160). However, presumably, Sorabji’s use of this word reflects a more “manifest-image” sense of the word qua, that manifest quality through which a perceiving subject can discriminate among various colors, including black and white. In any case, his point is clear: Aristotle did not distinguish between the brightness of colors and the differentiating qualities they possess (that is, including white and black) as colors.

It seems that Aristotle’s conflation of manifest color and brightness for the chromatic and non-chromatic colors alike allowed him to think of white/brightness and black/darkness as the extremes of a continuous gradation of different chromatic colors, conceiving of those chromatic colors as determined by the different ratios of white and black within them. We might think of this arrangement as analogous to the way in which bodies of different comparative felt warmth might be arrayed with extremes of maximally cold and hot. In the eighteenth and nineteenth centuries, some scientists would think of such an arrangement as due to the bodies containing different amounts of “heat”, conceived as a type of fluid—“caloric”—flowing from hot to cold bodies. Similarly, Aristotle seems to have thought of differently colored objects as containing different proportions of white and its “lack” (steresis), black. In Sense and Sensibilia, Aristotle discusses three hypotheses concerning how colors may be generated from the combination of white and black: first, as juxtaposed, next, as superposed, and then, as mixed. After critically examining the first two, he settled on this third option, such a combination by mixture, he said, having been dealt with “generally in its most comprehensive aspect” in his earlier “treatise on mixture” [18] (440b4), presumably the work On Generation and Corruption [20].

There, blurring the quality-brightness distinction, the opposed qualities of white/brightness and black/darkness are, like those of hot–cold, dry–moist, soft–hard, and so on, treated as opposing properties of a substance, each capable of affecting and being affected by the contrary state of another substance [20] (314b13–30)3. Thus, white and black are considered to be at the extremes of a sensory continuum in the way that hot and cold are similarly located at the ends of a continuum of intermediary degrees of warmth. Similarly, just as a hot body will warm a cold one, contact with a white-bright body will brighten a dark one4.

Many obvious problems present themselves, not the least of which is one raised by many interpreters (e.g., [21] (293); [10] (92)): why is it that mixtures of black and white do not simply produce an array of shades of gray? Where does the distinctive array of chromatic colors actually come from? Moreover, in connection with the indeterminacy of the semantics of color words noted above, what are we to make of the odd status of black as a privation [steresis] of white? The language of mixing suggests something substantive, something like black dye mixed with white5, but steresis suggests a simple absence of white. In much the same way, Goethe, in his arguments against Newton, would bizarrely suggest the existence of something like black light, rays of darkness that could interact with the opposed rays of white light [25].

Leaving such problems to one side, I want to focus on a more specific problem: even if one accepts that the intermediaries stretching from white to black are somehow colored rather than grey, how does one find some appropriate order for them? Aristotle’s idea of the different secondary colors resulting from specific ratios of white and black within the mixture suggests that the colors should be ordered in terms of their degree of brightness, with purple, for example, being closer to black and yellow closer to white. However, reflecting the independence of hue and brightness, the colors themselves can surely vary in terms of brightness or darkness while still retaining their distinctive hue. The chromatic colors, it would seem, need an independent ordering principle distinct from the issue of brightness, leading to the need for a model of greater dimensionality than that provided by the linear continuum of Aristotle’s brightness-darkness model. This need for a multidimensional ordering of color would be addressed by Goethe in Zur Farbenlehre, where he would present his own account of the intermediate colors in a hexagonal arrangement of six colors arrayed in a color wheel, in which each color faces its reciprocal or complement across the three major diagonals of the hexagon: red facing green; purple facing yellow; and orange facing blue, as in Figure 16.

3. The Implicit Harmonic Ordering of Colors by Aristotle and Goethe

Goethe’s color circle/hexagon was part of the tradition of color wheels used by artists in the eighteenth century as guides for color mixing and aesthetic juxtaposition7, and he would appeal to various phenomena from “daily life” to justify this arrangement. The opposition across the diagonal of a color and its complement is, for example, revealed in afterimages, where intense focus on a scarlet color patch results in a similarly shaped green afterimage [8] (176). While he treats this as a subjectively based relation, he nevertheless, like Aristotle, thinks of this as mirroring an objective relation found in actual color mixing in nature, as expressed diagrammatically in the mixing relationship represented between a color and its two immediate neighbors. As represented on the hexagon, mixing blue and yellow will produce green, which lies between them; mixing purple and orange will produce red, and so on [8] (165). Moreover, he believed that the objectivity of these principles was manifest in the evaluative aesthetic pleasure we take in the colors of the world. In relation to the coherence of colored elements into wholes, he says, “we may rightly speak of … a harmony” [8] (178)8.

An allusion to such a music-like harmony existing among colors can be found in Sense and Sensibilia, in which Aristotle offers a suggestion regarding the differentiation of specific intermediate colors when he invokes the actual ratios of black and white constituents of color, which “may be juxtaposed in the ratio of 3 to 2, or of 3 to 4, or in ratios expressible by other numbers; while some may be juxtaposed according to no numerically expressible ratio, but according to some incommensurable relation of excess or defect …” [18] (439b20–30). These quantitative ratios, as he makes clear, derive from Pythagorean music theory:

“Accordingly, we may regard all these colors as analogous to concords, and suppose that those involving numerical ratios, like the concords in music, may be those generally regarded as most agreeable; as, for example, purple, crimson, and some few such colors, their fewness being due to the same causes which render the concords few. The other compound colors may be those which are not based on numbers. Or it may be that, while all colors whatever are based on numbers, some are regular in this respect, others irregular; and that the latter, whenever they are not pure, owe this character to a corresponding impurity in their numerical ratios. This then is one way to explain the genesis of intermediate colors”

[18] (439b30–440a6).

Here, Aristotle raises a suite of issues that go beyond the limits of the simple hot–cold model of the mixing of opposite qualities, and as Richard Sorabji has pointed out, the origin of this extension of the model of musical concords [symphoniai] to color was clearly not Aristotle himself:

How much of the foregoing scheme is Aristotle’s and how much did he inherit? Oskar Becker has suggested that the rational/irrational division stems from Archytas and Eudoxus, while the alternative regular/irregular division is due to Philolaus, Plato, and the Old Academy. Konrad Gaiser thinks that … the mathematical ideas were already being worked on in the Academy before Aristotle wrote about them. A. E. Taylor detects a Pythagorean source. In fact, it is hard to say how much is due to Aristotle. He certainly learnt from others the theories that the remaining colors are produced from black and white by juxtaposition or by superimposition, while the substitution of chemical mixture for juxtaposition and superimposition is his own

[21] (297).

It is not difficult to grasp how Pythagorean harmonic theory might be thought to be applied to colors, nor is it difficult to appreciate why one might invoke some sort of analogy between the respective fields of visually experienced colors and auditorily experienced tones. As Goethe would point out, colors are typically perceived as capable of harmonious juxtapositions analogous to those of sounds, and in cases of both colors and sounds, we are dealing with qualities arranged on some sort of continuum. Pythagorean theory had already come up with a way of identifying points on the qualitative tone continuum that linked the tones correlated with those points as concordant as opposed to dissonant. The principal intervals within the range of the octave, which is itself represented by the interval 2:1, are those mentioned by Aristotle: 3:2 and 4:3. These represent the intervals of the diapente, or “perfect fifth”, of the modern scale, as in C to G, and the diatessaron, the modern “perfect fourth”, as in C to F. However, we need to delve further into the mathematics of this triad of ratios, 2:1, 3:2, and 4:3, to understand how they may have been extended beyond the rather narrow sphere of music theory to play the role they did in Pythagorean and, later, Platonic thought.

That the ratios 3:2 and 4:3 represented two intra-octaval concords in Pythagorean harmonic theory had been established well before Aristotle’s time. While the earliest recorded version of this mathematical structure is found in Philolaus of Croton [28] (ch. 1), [29] (ch. 10), a rough contemporary of Socrates, the first advances in the science of harmonics are thought to have been made sometime earlier in the fifth century BCE through experiments on a “monochord”—a single-stringed instrument in which a bridge dividing the string could be moved, its position being measured by some type of tape measure9. It had been found that when the length of a vibrating string was halved, the pitch of the tone emitted was raised one octave producing a note heard as being higher than but nevertheless, in some sense, identical to or “in unison with” the first. Dividing the string again in half, produced a note another octave above the original. The octave was thus identified as the interval between the numbers 1 and 2 when these were conceived as the first two terms of a sequence that increased by terms that doubled “geometrically”, as in the sequence 1, 2, 4, 8, 16, … In such a continuous geometric sequence, the middle term of any triad is the “geometric mean” of the other two extremes.

A follower of Philolaus, Archytas of Tarentum, a major fourth-century mathematician and contemporary and friend of Plato [29] (ch. 11), went on to define the three means that coincide with the major musical consonances. The geometric mean is defined as “when the first term is to the second one, the second one is to the third one” [30] (247). We might picture this proportion as existing between three lines drawn on a plane, l₁, l_g, and l₂. Line l_g will be the geometric mean of l₁ and l₂ when, in terms of their lengths, l₁:l_g = l_g:l₂.

The geometric sequence and geometric mean are known to have held a particularly ideal status for the Greeks, presumably because the unison of the octave was considered to be the purest of musical intervals. By the time of Philolaus, however, it had become apparent that the geometric mean could not be specified numerically for non-square numbers such as 2, an example of the earlier discovered phenomenon of the incommensurability of continuous magnitudes (measurable distances) and discrete multitudes (numbers). For an octaval interval represented by the ratio 2:1, there was, in the Greek sense, simply no number or ratio of numbers that, like the modern number √2, could “divide” that interval geometrically. Indeed, Archytas would produce a formal proof of this impossibility [31] (65–75; 79–80).

For consonances within the octave, experiments with the monochord had shown that the octave could be divided at two points to produce two inter-octaval concords: the diatessaron, or modern fourth, and the diapente, or modern fifth, both consonant with the tone emitted by the undivided string. Archytas defined the proportions correlated with these concords. The diapente was sounded when the string was divided at the arithmetic mean, defined as “when three terms are in proportion (analogon) according to the following kind of excess: that by which the first exceeds the second is the same as that by which the second exceeds the third”. For its part, the diatessaron was sounded when the string was divided at “the subcontrary”, which, he added, “we call harmonic”. This is formed when “by that part of itself <by which> the first term exceeds the second one, by this part of the third term the middle term exceeds the third one” [30] (247–249). The earlier name, “subcontrary”, for the harmonic mean, a term deriving from geometry, betrays the fact that the harmonic mean is, in fact, the mathematical inverse of the arithmetic mean10. This correlates with the phenomenon of harmonic inversion, when two notes, say C and F, that can be considered a chord resulting from a fourth built on C, can also be considered an inversion of a C built on F, which is a fifth11.

In the twentieth century, it was not uncommon to stress the importance of the discovery of the incommensurability of continuous and discrete magnitudes for the Greeks, specifically, the line lengths of geometry and the numbers of arithmetic [32] (part I). Traditionally, discovery of such incommensurability had been thought to have been a consequence of Pythagoras’s theorem; however, more recently, some have argued that it was more likely discovered in a musical context [33] (intro. and ch. 1), [28] (42–44)). However, if music had been the context for this discovery, it also seems to have been that in which a type of solution to this problem was first formulated. The three musical means may invoke incommensurable quantities, but the three means had been given a form of expression that demonstrated their mediated unity12.

First, the fundamental intervals of Pythagorean musical intonation can be arranged in the sequence 1, 4/3, 3/2, 2, or (by multiplying all terms by 6, the more convenient sequence 6, 8, 9, 12) a sequence representing the root note, its fourth, fifth, and octave, and known as the musical tetraktys. By cross-pairing the tonic with the arithmetic mean and the octave with the harmonic mean, reflecting the inverse relationship between arithmetic and harmonic means, this sequence of numbers can be arranged as an identity between two ratios: 6:9 = 8:12.

This mathematical object seems to have been afforded great significance in Plato’s time [34,35] (95–96) and is described in the late “Platonic” work, Epinomis (probably written not by Plato but by his follower, Philip of Opus), as having been “granted to the human race by the blessed choir of the Muses and [that] has bestowed upon us the use of concord and symmetry to promote play in the form of rhythm and harmony” [36] (991b). The importance that this held for the author is clarified by the discussion of the role of mathematics in astronomy and its teaching in the passage that precedes this statement.

Number theory, “the study of numbers in their own right, as opposed to numbers that possess bodies”—that is, numbers used in counting and measuring worldly objects—is described as the first topic to be taught, followed by planar geometry and three-dimensional geometry or stereometry [36] (990c–d). Both geometry and stereometry, however, involve “the assimilation by reference to plane surfaces of numbers that are not by nature similar to one another”. What are not “similar” here are surely those “incommensurable” discrete multitudes of arithmetic and the continuous magnitudes of geometry and stereometry. “What people who look into these matters and understand them find divine and miraculous”, notes the author, “is how nature as a whole molds sorts and kinds according to each proportion, with reference to the power that is always based on the double and the power opposite to this [the half]” [36] (990e–991a). The “reference” here is to the ideal geometric proportion, a:m_g :: m_g:c. However, the passage continues, “the sequence that gives the mean of the double [i.e., the geometric mean] involves both the mean that exceeds the smaller and is exceeded by the larger by an equal amount, and the mean that exceeds one of the extremes by the same fraction of that extreme as the fraction of the other extreme by which it is exceeded by that extreme” [36] (991a). These are the arithmetic and harmonic means, respectively. Plato had used the same three bonds in his account of the structure of the world-soul in the Timaeus [37] (36a–b), and accordingly, many late neo-Platonist and neo-Pythagorean philosophers, such as Iamblichus [38] (49–50) and Nicomachus of Gerasa [39] (284–285), had taken this inverse double ratio to be the ”best bond” that, in the Timaeus, Plato had described as responsible for the unity of the parts of the cosmos itself [37] (31b–32a).

After Plato’s death, Speusippus and Xenocrates, the first and second directors of the Academy, like Philip of Opus, attempted to develop this Pythagorianizing mathematical direction taken by Plato in his later years. However, this tradition seems to have soon faded, only to be revived in the first century BCE, from whence it would become a feature of late neo-Platonic thought [31] (392–396). In this later tradition, the earlier Pythagorean mathematics became intertwined with religious and mystical doctrines, and any scientific status attributed to such neo-Platonic “arithmology” is often discounted because of such associations. However, these need to be kept distinct from the actual mathematical issues involved, mathematical issues more directly seen in earlier attempts to unify the three “musical” means in the century after Plato.

Here, the works of two figures important for the development of Greek mathematical astronomy stand out. These are Eratosthenes of Cyrene (c. 276–c. 195 BCE), most famous for his calculation of the diameter of the earth [31] (236–238), and Apollonius of Perga (c. 240 BCE–c. 190 BCE), the author of the major work On Conics [40], which focuses on the properties of the conic sections: circle, ellipse, parabola, and hyperbola. Eratosthenes had written a work, now lost, called Loci with Respect to Means, which seems to have been concerned with diagrammatic solutions to the unification of the geometric mean with its contrasting incommensurable arithmetic and harmonic means13. However, perhaps the most successful example of such a solution is found in Apollonius’s On Conics. There, in Book III, Proposition 37, the following construction is described (illustrated as Figure 2 below)14: “Let there be the section of a cone AB and tangents AC and CB and let AB be joined, and let CDEF be drawn across. I say that CF:CD :: FE:ED” [40] (236–237). Apollonius’s construction would be significant for the development of what would become “projective geometry”, which was seen briefly in seventeenth-century Europe but later blossomed in the nineteenth century, as the “harmonic cross-ratio” involved would become the principal “invariant” of projective geometry. It is clear, however, that the harmonic cross-ratio is a generalization, conceived on the basis of diagrams, of the musical tetraktys. Were the points C, D, E, and F to correspond to the values 12, 9, 8, and 6, then CF:CD would equal 6:2, while FE:ED would equal 3:1 (which is equal to 6:2). That is, E will be the harmonic mean of CF when D is its arithmetic mean.

Figure 2. Apollonius’s “Harmonic” division of a line15.

Figure 2. Apollonius’s “Harmonic” division of a line15.

While generally critical of Pythagorean metaphysical claims and the philosophical importance Plato attributed to Pythagorean mathematics, Aristotle used the results of Pythagorean mathematical approaches to natural science in various places throughout his work (c.f., [45] (ch. 12)). Nevertheless, his understanding of mathematics seems not to have been particularly advanced for the time16, and his understanding of the details of Pythagorean harmonic theory especially seems to have been limited [47] (24–26). Specifically in relation to the application of harmonic intervals, Kalderon describes Aristotle’s approach as “less an account than the beginnings of one. Aristotle never specifies which intermediate secondary color goes with which simple ratio, such as 3 to 2 or 3 to 4. Nor does he explore to what extent ancient acoustical theory may be extended to the colors in the way that he envisions” [10] (127).

While alluding here to the relationship between harmonic tones and the issue of incommensurability [18] (439b20–30), Aristotle does not seem to have been as challenged by this finding as Plato had been. Here, the influence of Eudoxus of Cnidus, who joined Plato’s Academy around the same time as Aristotle himself, may have been significant, as Eudoxus developed an approach to ratios and proportions that seemed to anticipate the modern solution in terms of the expansion of the number system to include irrationals [48]. If Aristotle’s approach was based on some extant model circulating in Plato’s Academy during the years in which he was a member, in his hands it seems to have been deprived of the features that had given it such significance for Philip of Opus and, we assume, Plato. In contrast to Aristotle’s failing attempts, in the following sections, we will focus on the possibility of extending to color theory the role played by the musical tetraktys, in virtue of the role it played in Plato’s “harmonic” treatment of the structure of the world-soul in the Timaeus. This extension will lead us to Goethe’s color theory.

4. The Harmonic Structure of the World-Soul in the Timaeus and Its Relevance to Color

The early Pythagoreans had an arithmetical view of the universe, and the discovery of the incommensurability of discrete and continuous quantities clearly challenged this view. It was shown that not all ratios of measurable distances could be expressed as a ratio of two numbers. The musical tetraktys, understood as a unified arithmetico-geometric structure as in Apollonius, offered a type of mediated unification of these otherwise incommensurable quantities. In his late work, Plato apparently flirted with this model to address problems in his own theory of Ideas that became apparent in the dialogue Parmenides17.

For the Pythagoreans, the musical tetraktys seems to have provided a way of finding, for practical purposes, an approximate solution to the problem of numerically specifying an “irrational” number—in this case, the geometric mean of the interval between 1 and 2, which can be seen to fall roughly equally between the harmonic and arithmetic means as in Figure 3 below. In the Greek number system, the geometric mean of this interval, while definable, cannot be numerically specified; however, the arithmetic and harmonic means can be specified, giving an approximation of the geometric mean.

Even more striking, however, is the fact that this process of finding the arithmetic and harmonic means of an interval can be iterated to find the harmonic and arithmetic means of the original harmonic and arithmetic means. This, of course, results in a smaller range of possible values, giving a much closer approximation to the value needed in this example (1.41176… < √2 < 1.41666…). The idea of the process being repeated indefinitely now allows for a conception of the value of √2 as the limit of a converging pair of potentially infinite series, in much the same way that irrational numbers would come to be understood later in modern European mathematics18.

Such an approach to mathematics, with an emphasis on its application to the empirical world as reflected in the Epinomis, is characteristic of Plato’s late dialogues [51,52], expressing a different emphasis than that found earlier in the Republic. In the Republic Book VII, Socrates outlined the mathematical curriculum for future leaders that would eventually lead to the study of dialectic [55] (525b–531c)19. While the sequence is much the same as in the Epinomis, the focus in this classic middle-period work is much more on mathematics as understood as a “pure” rather than “applied” discipline, with the main consideration being the role of mathematics in leading “the soul forcibly upward and compel[ling] it to discuss the numbers themselves, never permitting anyone to propose for discussion numbers attached to visible or tangible bodies” [55] (525d). For the most part, in the Republic, calculation [logismos] is generally dismissed as relevant only to “tradesmen and retailers, for the sake of buying and selling” [55] (525c). In short, Plato’s attitude toward mathematics in the Republic is focused on the conceptual definition of numbers—magnitudes that can be “grasped only in thought and can’t be dealt with in any other way”—rather than their roles in calculation. Evidence that in his later dialogues Plato used the Pythagorean idea of the mediated unity of otherwise incommensurable magnitudes as a model for the unification of similarly incommensurable ideas can be found in his appeal to the Pythagorean cosmologist Philolaus in the dialogue Philebus.

In the opening sentences of Philolaus’s book, On Nature, Philolaus describes nature as “fitted together [harmozein] out of unlimited things (apeiron) and limiting ones (perainonton), both the whole world and everything in it” [56] (155)20. Such harmonizing, explains Philolaus, is only needed where there exists some otherwise incommensurable difference, such as that, in the mathematical context, existing between numbers and continuous magnitudes, here designated as limiting things and “the unlimited”21. In introducing a new “Philolaic” method of dialectic in the Philebus [57], Plato has Socrates suggest making “a division of everything that actually exists now in the universe into two kinds, or if this seems preferable, into three” and reintroduces an earlier reference to the Philolaic pair of “the unlimited and the limit” [49] (23c). “Let us now take these as two of the kinds, while treating the one that results from the mixture of these two as our third kind”. He then adds a fourth to Philolaus’s three, the two archai and their mixture: “Look at the cause of this combination of those two together, and posit it as my fourth kind in addition to those three” [49] (23c–d). By positing the mixture as the result of a cause, a role traditionally assigned to Ideas, Socrates thus links this mixture to Ideas as more traditionally conceived. It is in this sense that the mixture of the limit and the unlimited is posited as the worldly imperfect equivalent or paradigm (paradigma) of the Idea that is unable to be realized in the actual world.

In describing Plato’s late-period assimilation of Pythagorean ideas, Aristotle noted that while the Pythagoreans identified the ultimate constituents of the world as numbers, Plato identified these ultimates as Ideas, treating numbers as intermediaries between empirical things and Ideas [24] (987b13–18). In accordance with this, harmonic ratios in the Timaeus are not tightly bound to their musical origins as they had been for Philolaus. As Huffman notes [28] (150), while Timaeus (Plato) closely follows the ratios of Philolaus’s musical scale in the Timaeus, he never mentions their musical origins. This clearly signals that for Plato, musical harmonies themselves were merely the contingent forms in which these abstract mathematical proportions were discovered. The proportions themselves are in no sense essentially musical. Their application to the cosmos does not imply, as it would later for Eratosthenes, for example [58] (199), that the planets actually emit harmonious heavenly chords in their movement about the earth. The relevance of the proportions resides in their role in mitigating the effects of incommensurability.

We might understand the significance of numbers for Ideas for Plato via the relationship between the conceptual definitions of numbers and the actual numbers used in counting, measurement, and computation. An “irrational” number, such as that coinciding with the length of the diagonal of a unit square or the geometric mean of the extremes of an octave, can be easily defined as the number that, when multiplied by itself, results in the non-square number 222. That is, with such a definition, we can have the Idea of a number without possessing an actual number that can be applied to the world. However, if we can have such an indirect instantiation of the Idea of a number, might we not similarly have approximate representations of uninstantiated Ideas of other things as well—the idea of “the good”, for example, for which no actual exemplars need to be found in the world? In his later dialogues after the Parmenides, Plato seems to have searched for ways in which such Ideas could nevertheless be acted upon to produce ultimately imperfect but increasingly better substitutions for those Ideas. That is, might one not search for something akin to Philolaus’s method for finding, for practical purposes, close numerical approximations to rationally defined magnitudes, such as the square root of 223?

In accord with this interpretation, in the Philebus, within the context of a debate regarding whether the pursuit of knowledge or pleasure constitutes the good life, Socrates introduces the idea of a form of life worthy of “second prize” in that, while this form does not perfectly instantiate the good (and thus is not worthy of the first prize), it comes closer to it than its rival [49] (22d). If the good life is “closer to knowledge, then knowledge wins over pleasure and pleasure loses” [49] (11e–12a)24.

In his actual discussion of color in the Timaeus [37] (67d–68d), Plato is revealed to be not a perceptual realist like Aristotle. The qualitative “feel” of colors is not a direct disclosure of reality itself. It concerns the way that color is known by individuals constituted as they are. As one recent interpreter has summarized Plato’s position, “we may ask whether perceptible qualia, according to the theory put forward in Plato’s Timaeus, are objective, i.e., subject-independent properties of the things perceived. We must answer this question in the negative […]. Unlike geometrical characteristics of the things perceived, they do not constitute intrinsic properties of these things. In this respect, Plato’s account of what is perceptible is in agreement with that of Democritus rather than with that of Aristotle” [59] (226), and, we might add, Newton. Nevertheless, qualitative experience in the realm of color might, with Plato’s adoption of Philolaus’s approach, still be eligible to play the role of second-best. To understand how this is the case, it is necessary to examine the different levels of intelligence implicit in the Timaeus.

After the discussion of the cosmic soul in the Timaeus, we learn that the Demiurge left it to his/her progeny to fashion the souls of mortal beings [37] (69c–71e). This means that there are at least three levels of intelligence to be found in the Timaeus: the highest intelligence (nous), here represented by the Demiurge, who has the capacity to cognize and act upon transcendent pure Ideas; next, the immortal but nevertheless embodied world-soul itself; and thirdly, that exercised by individual humans. In comparison, this second level of intelligence possessed by the world-soul was missing from Plato’s early and middle dialogues [60] (120–121). The intelligence of the world-soul, like that of individual humans, is embodied. However, for the world-soul, its body simply is the cosmos, while the souls of mortals are each embodied in particular bits of the cosmos and are thereby located within some specific region within it. Again, unlike the world-soul, which has no surroundings (there is nothing, not even a void, beyond the cosmos), individually embodied souls of humans are in need of external senses like eyes and ears, as well as external organs like hands and feet, to interact in rational ways with their surroundings. As emphasized by Klaus Corcilius, while the world-soul is a type of “ideal cognitive agent” that serves as a role model for individualized embodied souls and is able to cognize “every single thing that there is and that happens in the cosmos”, it has “in a certain sense only a limited access to reality”. “Not only is perceptual experience in particular unavailable to it, but other forms of immediate cognition as well: everything the world soul cognizes is the outcome of its active recognition of the sameness and difference between things in the cosmos by actively comparing them with each other. Its worldly science of every single thing in the cosmos, therefore, is of an essentially contrastive character. There is no direct ‘vision’ of the intrinsic being of things in the world soul, but only a propositional grasp of the features that things have in relation to each other” [61] (52–53). One could say that its equivalent of “perceptual knowledge” is limited to “knowing that”, in the way that one might know, after being told, that lemons are yellow, without, as in the case of Frank Jackson’s famed “Mary” [5], even knowing what it is for something to look yellow. As Concilius puts it, “the world soul knows (and has true and firm opinions about) reality, but does not feel it and is not affected by it” [61] (96).

At the summit of Plato’s three levels of knowledge of color will thus be the Idea of color, as known by and guiding the world-forming action of the Demiurge25. Below this is Plato’s equivalent of the modern scientific knowledge of color—that of the world-soul—the quality-free sort possessed by Jackson’s Mary. Below that is the sensuously perceived color that the rest of us tend to think of as unproblematically revealing the actual properties of the objects of our vision. At the second and third levels, the determination of color given may then represent the “second prize” in relation to a grasp of the level above it. The scientists’ propositional account of color is not truly philosophical, but it relates to that philosophical grasp as the best out of a series of approximations to the idea, and the everyday qualitative grasp is not the scientific one, but it is clearly better than, for example, that found in a blind or color-blind person.

At the physical level, perception for Plato clearly relies on the causal interaction between the body of the perceiver and the object perceived. However, the perceptible characteristic is not the sensation (aisthesis) qua modification of the body, but rather an “aistheton” grasped via a cognitive act of “opinion accompanied by irrational sensation [doxa met’ aistheseos alogou]” [59] (215)26. The rational structure of color—the subject of a properly human psychology of color—should thus be understandable in relative abstraction from the specifics of the underlying physics necessary for its occurrence, a physics known indirectly and comprehensively by the world-soul. In this sense, beyond setting the boundary conditions for color experience to take place, Plato’s account of the specific underlying physical mechanisms—an account known by the world-soul—is not really relevant for our purposes27. Rather, I suggest that color experience, in the psychological sense, will thus be conceived from a holistic or relational point of view. That is, we might think of color “space” as internally partitioned, such that the qualities of distinct colors can be thought of as determined by principles of similarity and difference operating within such a closed sensory domain. In a sense, this mirrors the entirely structural and contrastive nature of the propositional knowledge that constitutes the world-soul’s understanding of the underlying physics of this aspect of human psychology.

Thus, Plato suggests that the principles of similarity and difference operating within different sensory modalities exhibit a type of structural uniformity, such that the fundamental color opposition between light and dark is “‘cousin’ to what is cold or hot in the case of the flesh, and, in the case of the tongue, with what is sour, or with all those things that generate heat and that we have therefore called ‘pungent’” [37] (67d–e). Plato’s distinction between the levels of physics (level 2) and human psychology (level 3) in turn allows him to differentiate that which is conflated by Aristotle between the degree of ambient light—brightness and darkness—and the fundamentally experienced colors, including white and black. The former belongs to the knowledge of color possessed by the world soul, whereas the latter pertains to the way color is known to individual humans, but this does not mean that there is not something “objective” at this lowest level. When considered as regions of subjective color space, black and white can be “really the same as these other properties” [37] (67e), that is, cold and hot, sour and sweet, and so on, because they occupy similar regions within their similarly partitioned sensory spaces. It is something about the nature of these different spaces themselves that gives, say, white and hot a different “appearance” [37] (67e). However, at a structural level, a particular region within one space might stand in the same characteristic “consonances and dissonances” to regions of other spaces, with some distinctions of taste being “cousin” to black and white, cold and hot, and so on.

Recall the fundamental problems facing Aristotle’s application of musical harmonies to the array of colors in finding an intelligible order among the hues, in contrast to that found on the linear continuum stretching from white to black or light and its steresis. With no equivalent distinction between the physical and the rational—“aesthesis” and “aistheton”—Aristotle was forced to think of colors as somehow correlating with the comparative ratios of black to white within the mixture. It would seem, then, that like the mixtures of black and white in grey, all colors will be conceived as simple arithmetical proportions. Within the properly Platonic framework, however, we should expect there to be two means implicated, the harmonic and arithmetic means, which stand as the inversely related intra-worldly surrogates of an ideal geometric structure, which is possessed at the level of the scientific intelligence of the world-soul.

The general principle I have suggested here is a generalization of the way that within an octave, the harmonic and arithmetic means serve as opposed surrogates for the more ideal geometric mean that defines the perfect unison between octaves. When applied to the model of a color continuum, somehow conceived as stretching from white to black, one must now search for a structure involving pairs of inversely related ratios that reproduce something analogous to the ideal meta-opposition of black and white itself—that is, structured by multiply colored analogues to this primordial opposition. This is the broad context within which something like the musical tetraktys seems to have been intended to function for the later Plato and his followers. Let us now relate this approach to Goethe’s diagrammatic presentation of color.

We have noted some of the parallels between colors and audible tones that may have suggested themselves to Aristotle, but what Aristotle does not seem to have appreciated is the parallel between the inverse relation holding between the fourth and fifth and the case of color complements of the sort recognized by Goethe. What impeded Aristotle’s attempt to arrange the chromatic colors on a continuum between white and black was the one-dimensionality of this array. However, in the case of tone, the inverse relation between arithmetic and harmonic means gives a bi-directionality to the line, making it, we might say, implicitly two-dimensional28.

The field of color as portrayed in Goethe’s hexagon is, of course, more complex than the Pythagoreans’ portrayal of tone, but here too is found the underlying idea of bi-directional linear continua with complementary or “inverse” colors located at the extremes. We can interpret edges like that joining blue and green in the hexagon, as representing a continuous transition from paradigmatic blue to paradigmatic green, passing through various intermediate grades of blue-green and green-blue. However, across the major diagonals of opposed “complementary” colors that face each other as inverted, as with red and green, for example, there can be no similar unbroken continuity.

Moreover, Goethe resists Aristotle’s simple conflation of white and black qua colors with lightness and darkness qua degrees of illumination. He states that the colors represent the states of illumination, although what he means by this is far from clear: “Black, as a representative of darkness, leaves the eye in a state of repose; while, representing light, stirs it to activity” [8] (170). Nevertheless, as we have seen, in the hexagon, the colors of white and black themselves are not displayed with the others, although lightness and darkness are indirectly represented in the diagram by the fact that he conceives of the hexagon as constructed from two overlapping inverted triangles, representing the colors of a light spectrum (red, blue, yellow) on the one hand and a dark spectrum (purple, orange, green) on the other [62] (145–149). It would seem, then, that white and black themselves, directly representing the principles underlying the mechanisms of color production, cannot be represented on the same level as the other colors. We might suggest, then, that while the qualitative colors of white and black imperfectly represent the scientifically known states of illumination, the complementary polar opposites found among the chromatic colors imperfectly represent, at a yet lower level, the purer polarity of white and black itself29. In short, the plural polarities of the chromatic colors stand to that of black and white somewhat like the way the intra-octaval concords stand to the unison relation of octaval extremes.

There are structural differences, however; for instance, in the case of color, there are three such lowest-level oppositions, while in the intra-octaval, there is just one. Thus, for the former, a further dimension is required, as is evident in the explicitly two-dimensional color hexagon, which features three inverse pairs of chromatic colors. Recently, Goethe’s color hexagon has been interpreted in ways that conform with recent developments in logic, and this may allow us to further pursue Plato’s approach.

5. Graphical Versions of Goethe’s Color Hexagon: From Quantitative to Qualitative Interpretation

In the 1960s, the French logician Robert Blanché [63] extended a modal interpretation of the traditional “square of opposition” into a hexagon—a diagrammatic representation of Aristotle’s judgment forms from On Interpretation [64]30. In the traditional square of opposition, the edges of the diagram represent the logical relations holding between propositions located at the corners or “vertices” of the graph. Thus, as shown in Figure 4a below, the judgment “All Fs are G” at the top left or “A” corner of the square is seen to imply that “Some Fs are G” at the “I” corner below it, just as “No Fs are G” at the E corner implies that “Some Fs are not G” at the O corner. In the modal interpretation, the equivalent inferences would be from the proposition necessarily Fs are G to possibly Fs are G, down the AI edge, or from “Impossibly Fs are G” to the weaker “Not necessarily Fs are G”, down the EO edge. In both cases, the inferences are one-way, and thus the direction of the inferences can be represented by arrows directed downward.

Blanché added two new vertices to the square to represent six modal judgment types. To the traditional vertices (A, necessarily p; E, impossibly p; I, possibly p; O, not necessarily p), he added Y (contingently p) and U (not-contingently p), as shown in Figure 5a below. Six inference patterns were now established: from A to U; from E to U, from A to I; from E to O; from Y to I; and from Y to O. Recently, Blanché’s hexagon was extended from modal logic to Goethe’s color hexagon by Jean-Yves Béziau [65] (Section 4), as shown in Figure 5b31.

The labeling of the vertices of Béziau’s diagram is a close approximation of Goethe’s, and, like Goethe, he conceives of the hexagon as constructed from two triangles. However, in Béziau’s case, the triangles represent the three primary and three secondary color triads of modern color theory. In this color context, two converging arrows represent the mixing of the colors from which the arrows are directed. Thus, blue and green are represented as mixing to form cyan. From this modern perspective, Goethe can be considered to have misconceived the status of his color theory by thinking of it as a physical theory operating at the same level as Newton’s. However, when reinterpreted in a modern way, Goethe’s color hexagon can be thought to conform to what is known about the eye’s processing of the light falling on the retina32.

It is not clear, however, that Béziau’s diagram is meant to capture the qualitative relations that are so important to Goethe. Béziau represents the vertices not, as in the diagram above, with color words, but rather with color samples. I suggest that just as with the appeal to modern color theory, the “knowledge” represented at each of the vertices represents something like the quality-less abstract concepts of these colors. That is, in its compatibility with the “scientific image”, it represents a type of knowledge to be found at the level of the world-soul itself, rather than at the level of individuals. In contrast, a Platonic interpretation of Goethe’s diagram might more closely represent the human “space” of color contrasts and relations.

A related diagrammatic presentation of the color hexagon can be found in the work of Dany Jaspers [68], who draws upon a different type of logical diagram known as a “Hasse” diagram, as shown in Figure 6a,b33.

Hasse diagrams are typically understood as geometric presentations of relations that can be represented arithmetically in terms of set theory. For example, in Figure 5a, there are four levels of vertices representing sets. At the bottom is the empty set, represented by the symbol “{}”. At the next level, there are three single-membered sets, {1}, {2}, and {3}, and at the next highest level, there are three more sets, each with two members, {1,2}, {1,3}, and {2,3}. At the highest level, there is another single set, now with three members, {1,2,3}. The arrows, which consistently point upward, represent the inclusion of the member(s) of the lower set in the set immediately above it, so converging edges can represent the combinations of the contents of the sets at the lower nodes. For example, members of both sets {1} and {2} are included in the set {1,2}. In the case of the relation of the bottom-most set to the sets above it, the empty set is treated in set theory as being included in all other sets.

In the color equivalent, it can be appreciated how blue can be considered a component of both cyan and magenta, as, regarding color mixing, blue mixed with green results in cyan, while blue mixed with red results in magenta. However, just as the empty set is included in the sets {1}, {2}, and {3}, in Jaspers’s color Hasse diagram, black is represented as “mixed” with all the three colors immediately above it, blue, green, and red. Clearly, Jasper’s diagram clashes with the intent of Goethe’s hexagon to represent colors in terms of their sensed quality of mixing. For Jaspers, adding black to another color is not like not adding anything at all—that is, it “adds” the absence of color. In short, the role of black in Jasper’s diagram only makes sense when it is considered as the absence of white or, more correctly, brightness (light).

From a Platonic perspective, white and black can represent the “purest” binary distinction between qualitative colors, which provide models of which the other chromatic color oppositions provide imperfect realizations. Thus, we can now appreciate that there exist a total of 10 “mini-spectra” (black–blue; black–red; black–green; blue–cyan; blue–magenta; green–cyan; green–yellow; cyan–white; yellow–white; and magenta–white), in which there exists a continuous qualitative transition (edge) between any two neighboring distinct colors (directly connected vertices). Each of these ten continua may now be thought to be isomorphic, in its own way, to the underlying Aristotelian one-dimensional graph, which joins white to black through a continuum of greys. Furthermore, it can be appreciated that in the graph, these edges of continuous color transition can be arranged end-to-end to provide a variety of paths through which one could journey between two non-directly contiguous colors, like black and white, which are considered contrasting regions of a color space that includes all the chromatic colors. For example, one can travel a mediated path from black to white via the intermediaries of blue and magenta, or green and yellow34. It can be appreciated from a Platonic perspective that the major purport of the graph is to represent possible color transitions along a continuum. The edges of the graph are the main focus of interest rather than the contents of the individual vertices, and with this focus, a new meaning is assigned to the triads of “primary” (red, green, blue) and “secondary” colors (cyan, magenta, and yellow) at the vertices, as these colors now represent the points on the network of color continua where paths split or converge35.

All in all, there will be six alternate paths that can be constructed, passing from black to, ultimately, white: black-green-yellow-white; black-green-cyan-white; black-blue-cyan-white; black-blue-magenta-white; black-red-magenta-white; and black-red-yellow-white. Here, the existence of the “two means” between the ultimate extremes should give us pause, as this is exactly what Plato described in his account of the relations of three-dimensional actual space, as well as those among the four elements of the cosmos: fire, air, water, and earth when understood on the harmonic model [37] (31b–32c). The “symphony of proportion” structuring these domains was reflected in the structure of the world-soul itself and, from there, into the particularized souls of mortals. We should not be surprised to find it cropping up in the psychophysics of color perception among those mortals, because the souls of mortals were created as imperfect replicas of the world-soul itself. For this reason, it is surely significant that the Hasse diagram, while depicted as planar, can equally be understood as instantiating a cube in three-dimensional space36.

6. Conclusions

Aristotle clearly considered the Pythagoreans’ tendency to associate some distinctive numbers with everything in the universe ridiculous—justice, for example [24], (985b29), to which they assigned the number 4. In accordance with this, apart from his attempted pursuit to apply numbers in the context of color theory, he was generally skeptical of the role played by quantity in knowledge. The known world was basically the manifest world, and it was firmly qualitative. Plato, I have suggested, seems to have had an approach to color in which he was able to exploit the parallels between colors and tones with which Aristotle had experimented in his attempts to pair specific chromatic colors with specifically proportioned mixtures of black and white. For Plato, however, “mixtures” were not mixtures of anything like substances; they were, following Philolaus, mixtures of “limit” and “the unlimited”, modeled on patterned mixtures of discrete and continuous magnitudes as found in the approaches of mathematicians of his time.

In the modern sciences of color, numbers would be assigned to colors; for example, red is described as a property of electromagnetic radiation with a certain wavelength (between 620 and 750 nanometers). This, however, is just the type of quantitative approach to such a topic to which both Aristotle and Goethe would surely have objected. No numbers could possibly explain what it means for roses or tomatoes to be red, just as no numbers could help in understanding the nature of justice. In relation to color, Plato ultimately agreed with the positions common to Democritus and Newton against Aristotle and Goethe—there is nothing about color to which numbers are not truly adequate. There are no objective qualitative properties in the world like the redness commonly associated with a human’s experience of roses and tomatoes—properties that simply resist numerical analysis. For Plato, there existed a level of knowledge, characteristic of the world-soul, that was abstract and non-qualitative. The modern equivalent would be that possessed by a color scientist who, like Newton or Jackson’s Mary, could comprehensively understand the realm of color in entirely non-qualitative and particularly numerical ways. For Plato, while not revealing ultimate reality, such knowledge was still closer to it than our efforts to juggle relations of similarity and difference among qualia in describing the world. Numbers stood between empirical phenomena and Ideas, but this was compatible with the existence of an objective order within the patterns of subjective color experience. Relations among the most subjective features of our experience could still be articulated by patterns of numbers—patterns that were, in many ways, the closest thing to Ideas that we could know.