Neher, How Perspective Could Be a Symbolic Form

How Perspective Could Be a Symbolic FormAuthor(s): Allister NeherSource: The Journal of Aesthetics and Art Criticism, Vol. 63, No. 4 (Autumn, 2005), pp. 359-373Published by: Wiley on behalf of The American Society for AestheticsStable URL: http://www.jstor.org/stable/3700512 .

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ALLISTER NEHER

How Perspective Could be a Symbolic Form

Erwin Panofsky's Perspective as Symbolic Form remains one of the most important works on the history and nature of pictorial space.' As Hubert Damisch put it in the first line of The Origin ofPerspective:

Anyone investigating perspective today, in its double capacities as object of knowledge and object of reflection, must negotiate territory that remains under the jurisdiction of a text that has attained classic sta- tus, one that still constitutes, more than a half a century after its appearance, the inescapable horizon line and reference point for all inquiry concerning this object of study and all related matters, to say nothing of its theoretical and philosophical implications.2

The importance of Perspective as Symbolic Form is not confined to the history of art. W. J. T. Mitchell has suggested that "[t]he current revival of interest in Panofsky is surely a symp- tom of the pictorial turn, Panofsky's magisterial range, his ability to move with authority from ancient to modern art, to borrow provocative and telling insights from philosophy, optics, theology, psychology, and philology, make him an inevitable model and starting point for any general account of what is now called 'visual culture.' "3 The resurgence of interest in Panof- sky's early theoretical works that we saw in the 1980s and early 1990s has engendered an impressive body of publications that refer to the ideas and arguments in Perspective as Symbolic Form, but what has struck me as unusual in all of the disputes and deliberations is the dearth of discussion on what Panofsky meant by claiming that perspective is a symbolic form; it is after all, as the title announces, the defining claim of the work, and thus should be fundamental to any evaluation or exposition of it.

It is not that no one has commented on his use of the term 'symbolic form,' or that no one has ventured to reflect on why he might have chosen to employ it, but the reflections that have been offered have been slight and no con- vincing effort has been made to determine how the idea of a symbolic form, which was taken over from Ernst Cassirer's philosophy of culture, underlies Panofsky's study. Without a thorough understanding of Cassirer's philosophy, nothing can be said about Panofsky's claim. One cannot enter the text.

In this paper I shall provide an explanation of what it could mean to call perspective a symbolic form, and in doing so I shall show how the central arguments of Perspective as Symbolic Form are connected to the doctrines of Cassirer's philosophy.4 I am not claiming that Panofsky would have given an exposition of his arguments in the language I employing, but I do want to maintain that my explanation and his arguments are entirely compatible.

I shall begin then with a brief exposition of those aspects of Cassirer's philosophy that are most relevant to understanding Panofsky's text. Cassirer's idea of symbolic form is derived from the philosophical doctrines of both Immanuel Kant and G. W. F. Hegel. Kant was the philosopher who most effectively broke the hold of the idea that our knowledge of the empirical world is simply a case of consciousness mirroring a preestablished reality that exists independently of our apprehension of it. In the Critique of Pure Reason, Kant developed this thesis principally in relation to the question

The Journal of Aesthetics and Art Criticism 63:4 Fall 2005



360 The Journal of Aesthetics and Art Criticism

of how and why our claims to empirical knowledge, especially in the sciences, are secure. He argued that they are, and indeed that they are possible, only because consciousness brings to experience certain forms that structure it and constitute it in a way that allows for such knowledge. These forms are of two types: there are what Kant calls the a priori forms of intuition, which are space and time, and there are the a priori concepts that he calls the categories. 'Intuition' is Kant's word for sensation formed by space and time. Categories are concepts that, formally speaking, are prior to experience and, in their application, make empirical knowledge possible. There are twelve categories in total-- the two most familiar are substance and attribute and cause and effect-and together they constitute and demarcate the realm of possible experience for us, the boundaries of sense. It is important to note that Kant is not making a psychological claim about human experience; rather it is a claim about the conditions of possibility for empirical knowledge in general. This is Kant's singular achievement; he provided philosophy with a new standpoint from which to approach the question of knowledge, a standpoint that he dubbed critical or transcendental philosophy.

What Cassirer attempts to do is extend the Kantian critical analysis of empirical knowledge to, as he would put it, all of the other ways human beings can have a world. In other words, he investigates the conditions of possibility for such greater cultural forms as myth, natural language, art, and, of course, contemporary science. In its most general sense, the term 'symbolic form' refers to these greater cultural forms. As organs rather than mirrors of reality, these structures are not static; unlike Kant, whose analysis of empirical knowledge assumed that Euclidean geometry, Aristotelian logic, and Newtonian mechanics were com- pleted achievements, Cassirer builds transformation and development into his analyses of the various symbolic forms. The developmental model is borrowed from Hegel, but unlike Hegel, philosophy and reason are not given the place of privilege.

Within this general framework, Cassirer con- ceives of symbolic forms structurally and gives them an inner articulation that is based on a semiotic that is peculiar to his philosophy. In

turning to these subjects, it must be remem- bered that Cassirer developed his structuralist approach and his semiotics well before the elaborate systems advanced in the latter half of the twentieth century, and thus what he offers is sketchy in comparison.

Let us consider first Cassirer's conception of the symbol, or what is now more commonly referred to as the sign. In one of his more explicit statements, he defines the symbol as covering "the totality of all phenomena which-in whatever form-exhibit 'sense in the senses' (Sinnerfiillung im Sinnlichen) and in which something 'sensuous' (ein Sinnliches) is represented as a particular embodiment of 'sense' (Bedeutung, meaning)."'5 As this passage indicates, Cassirer is working with a tripar- tite model of the sign: there is something sensuous, a particular sense, and a relation in which the former is taken as representing the latter.6 The fourth element of this model, assumed but not mentioned in the above passage, is the symbolic form-the greater signifying structure in which the relation of representation takes place. Figure 1 represents Cassirer's model of signification graphically.

One of the most readily noticeable features of Cassirer's definition of the symbol is its general- ity. This is deliberate, as he wants it to be as inclu- sive as possible and not privilege a particular Symbolic form. Not surprisingly, then, Cassirer criticizes Ferdinand de Saussure for the central place that he assigns to language in his semiology.

"Apart from language," says a distinguished modem linguist, Ferdinand de Saussure, in his Cours de linguistique gnderale, "our thought is only an amor- phous mass.... Taken in itself, thought is like a misty veil. There are no pre-established ideas and nothing is distinct before the appearance of language."

sensuous element sense

symbolic form

representation

Figure 1. Cassirer's model of signification.



Neher How Perspective Could be a Symbolic Form 361

I do by no means wish to contradict this statement that is in perfect accordance with my own position. But I do not think that from a comprehensive philosophical point of view we can accept it without any restriction. ... When giving such a judgment we should [not] forget that beside the world of language there is another human world which has a meaning and a structure of its own. There is, as it were, another symbolic universe beyond the universe of speech, of verbal symbols. This universe is the world of arts---of music and poetry, of painting, of sculpture or architecture.7

This is not a remark specific to art, for Cassirer makes essentially the same point in relation to myth-his greater concern is that a particular symbolic form (probably language or science) will be given primacy and that all other forms of sense making will be reduced to a single model of meaning.

There is, all the same, much that converges in the structuralist conceptions of linguistics promoted by Saussure and Cassirer (though Saus- sure never employed the term 'structuralism' and Cassirer did so only later in his career). For example, Cassirer holds to a three-fold distinction that is similar to the parole, langue, and langage distinction made by Saussure, though Cassirer's has its roots in the work of Wilhelm von Humboldt, whose writings in fact predate Saussure's by a century. Like Saussure, von Humboldt distinguishes between language as a formal system, ergon (langue), and as a living, formative force, energia (parole).8 Langage for Saussure is the general phenomenon of language itself, encompassing both langue and parole as its complementary aspects. This idea finds its counterpart in Cassirer's claim that language is a symbolic form, a structurally conceived semiotic system under the pressures of historical transformation.

When Cassirer considers language as ergon, his approach is in essential harmony with the major tenets of the Prague Linguistic Circle, as is revealed in the following quotation in which he discusses the interpenetration of formal and factual considerations that typify structural analyses.

If the adherents and defenders of the program of linguistic structuralism are right, then we must say that in the realm of language there is no opposition between what is "formal" and what is merely "factual."

"In a given state of language," says Viggo Bron- dal, "everything is systematic; any language is con- stituted by a general unity in which everything is related: systems of sounds (or phonemes), systems of forms and of words (morphemes and semantemes). To speak of a system is to speak of a coherent whole: if everything is related, each term must depend upon another. Now, one would like to know the modalities of this coherence, the possible degrees and variables of this mutual dependence. In other words, it is necessary to study the conditions of the linguistic structure, to distinguish in the phonological and mor- phological systems what is possible and what is impossible, what is contingent and what is necessary."

The same conviction appears in Saussure's Cours de linguistique gindrale (2nd ed., Geneva, 1922), in the works of Trubetzkoy, of Roman Jakobson, and the other members of the "Cercle Linguistique de Prague." Obviously, the necessity claimed here for a linguistic system has no metaphysical connotation. It is no absolute but a relative or hypothetical necessity. Roman Jakobson has expressed the character of this necessity by the formulae:

1. if a exists, b exists also 2. if a exists, b is absent 3. if a exists, b is also absent

These relations, which certainly have the value of a law, constitute one of the most important factors for phonetic change.9

What distinguishes Cassirer's approach from more static structuralist models that employ the above principles is that when he considers language as energia, his neo-Hegelian vision becomes preeminent and the structural analysis is set into motion and historicized. For Cassirer, the distinction between ergon and energia can only, however, be a methodological one, for in actuality they are a dialectical unity that must be conceived of in terms fitting an organic whole, that is, they must be conceived of as the symbolic form they jointly constitute.10 This sketch of Cassirer' s structuralist approach can be deepened and strengthened by considering it in relation to group theory. I will, however, postpone this discussion until I take up the question of how perspective could be a symbolic form."11

Cassirer grounds his concept of the symbol, and the philosophy of symbolic forms itself, on his doctrine of "symbolic pregnance" (symbolische




Priignanz). Symbolic pregnance is the

Urphiinomen that is the transcendental condition upon which the phenomenon of meaning depends.

By symbolic pregnance we mean the way in which a perception as a sensory experience contains at the same time a certain non-intuitive meaning which it immediately and concretely represents. Here we are not dealing with bare perceptive data, on which some sort of apperceptive acts are later grafted, through which they are interpreted, judged, transformed. Rather, it is the perception itself which by virtue of its own immanent organization, takes on a kind of spiritual articulation-which being ordered in itself, also belongs to a determinate order of meaning. In its full actuality, its living totality, it is at the same time a life "in" meaning. It is not only subsequently received into this sphere but is, one might say, born into it. It is this ideal interwoven- ness, this relatedness of the single perceptive phenomenon, given here and now, to a characteristic total meaning that the term "pregnance" is meant to designate.12

Cassirer makes a distinction between conventional meaning and natural meaning. Conven- tional meaning depends on the production of meaning through the giving of signs (Zeichengebung), as takes place in language, mathematics, art, and other cultural creations. Natural meaning refers to the essential mean- ingfulness of perception itself. We give meaning to our experience with signs, but the basic function of signification is already in our experience, and it is the basis for all signification, whether natural or conventional; both rest on symbolic pregnance, "the imprinting of sense in the senses," which is a condition of possibility for all experience. An example that Cassirer provides will make this less abstract.

We can consider an optical structure, a simple line, for example, according to its purely expressive meaning. As we immerse ourselves in the design and construct it for ourselves, we become aware of a distinct physiognomic character in it. A peculiar mood is expressed in the purely spatial determination ... And here we do not merely read our own inner states subjectively and arbitrarily into the spatial form; rather, the form gives itself to us as an animated totality, an independent manifestation of life.... But these qualities recede and vanish as soon as we take

the line in another sense-as soon as we understand it as a mathematical structure, a geometrical figure. Now it becomes a mere schema, a means of representing a universal law. Whatever does not serve to represent this law, what merely appears as an individual factor in the line, now becomes utterly insignificant ... The spatial form is nothing but a paradigm for the formula; it remains the mere out- ward cloak of an essentially unintuitive mathematical idea.... And once again we stand in an entirely different sphere of vision when we take the line as a mythical symbol or as an aesthetic ornament. The mythical symbol as such embraces the fundamental mythical opposition between the sacred and the profane.... Viewed as an ornament, the drawing seems remote both from signification in the logical- conceptual sense and from the magical-mythical warning symbol. Its meaning lies in itself and discloses itself only to pure artistic vision, to the aesthetic eye.13

As an optical structure, the line is, then, as Cassirer states in the passage on symbolic pregnance, the "perception" that each symbolic form can take up into its own symbolizing activity. Each symbolic form is "an independent energy of the human spirit through which [the line] ... assumes a definite 'meaning."'14 With this brief outline of the central elements of Cassirer's philosophy in hand, let us turn to Perspective as Symbolic Form.

Panofsky's central contention, in its most general statement, is that the system of linear perspective that was developed during the Italian Renaissance, and that dominated Western art until the late nineteenth century, is not the only and natural way of representing three-dimensional space pictorially, but an artifice suited to a certain Western manner of apprehending the world. Panofsky also maintains that the forms of spatial organization inherent in works of art, whether they are of the Renaissance or another era, are direct correlates of the cosmologies and modes of perception of the ages to which they belong. The two eras that Panofsky singles out for special consideration in support of these claims are antiquity and the Renaissance. He argues that we can discern in the art of these




times two very different and fundamentally opposed methods of perspective construction that are derived from distinct reckonings of visual experience, from distinct emphases on the elements of visual experience.

Panofsky supports this assertion by appealing to the writings of Guido Hauck, a nineteenth- century mathematician with an interest in optical theory and the practical problems of

painting.5 Hauck discusses two dissimilar ways in which we determine relative visual size. The first way, which he names "collinearity," refers to what could be commonly called focal vision, seeing things in clear focus because one is attending to an object and light rays are entering the eye in such a way that they predominantly strike the area around the fovea centralis. In such circumstances, straight lines are perceived as straight lines. According to Hauck (and Panofsky), the kind of method most in harmony with representing this visual situation on a two- dimensional surface is that embodied by Renaissance linear perspective. The other way of determining relative visual size, which Hauck names "conformity," refers to the perception of features peripheral to the visual field. In this situation, perception of relative size accords with the size of angle subtended when straight lines representing visual rays are drawn from the outer edges of the object of vision to the eye. In such circumstances, Hauck maintains, straight lines appear to be curved and become increasingly curved the further they are removed from the center of focus. Hauck believes that we alternate between collinearity and conformity as modes of vision, but because we attempt to achieve unified vision, collinearity gets the upper hand, as it corresponds to our belief that straight lines appear straight.

Panofsky goes on to argue that an awareness of the curvatures in vision, and an attempt to incorporate them in representation, can be discerned in certain telling efforts at two-dimensional depiction that have survived antiquity. This is supported, we are told, by an examina- tion of the relevant philosophical and theoretical texts. From Euclid's Optics, for instance, we discover not only that the ancients measured relative size according to visual angles, but also that they lacked the Renaissance conception of an infinite, homogeneous space, and, hence,

that they could not have conceived of a pictorial surface as a plane intersecting a visual cone. The implied conclusion is that different peoples have apprehended the world around them in differing terms, which we see in the forms of spatial representation developed by them, and in what we could loosely call their scientific and cosmological texts; thus, as Panofsky says: "This is why it is essential to ask of artistic periods and regions not only whether they have perspective, but also which perspective they have."'16

Immediately preceding this passage, and in partial explanation of it, Panofsky states that "[perspective] may even be characterized as (to extend Ernst Cassirer's felicitous term to the history of art) one of those 'symbolic forms' in which 'spiritual meaning is attached to a concrete, material sign and intrinsically given to this sign.' "'7 For some commentators, the gen- erality of this statement does little to help us understand how perspective could be a symbolic form. As Damisch states, "the definition he uses, which holds that perspective is one of those 'symbolic forms' by means of which 'intellectual meaning becomes so closely linked to a concrete sign as to be indistinguishable from it,' is sufficiently vague and generalized to justify any interpretation one would like."18 In Damisch's estimation, Panofsky's study does not in any consequential way engage the questions to which Cassirer's philosophy is a response: "In truth, these questions do not seem to have interested Panofsky."'19

III

Obviously, I do not share Damisch's evaluation, but I can understand why he came to that conclusion and my account will be an answer to it. However, there are other impediments to understanding how perspective could be given a Cas- sirerian characterization. A major one results from Cassirer's use of the term, for he never provides a complete definition of 'symbolic form.' Typically, when Cassirer speaks of symbolic forms he refers, as I have already noted, to the greater cultural forms of myth, language, science, and art, but he never provides a straightforward principle that could be used to identify and individuate symbolic forms.20 In




The Myth of the State, Cassirer does make the following statement: "It is a common characteristic of all symbolic forms that they are applicable to any object whatsoever. There is nothing that is inaccessible or impermeable to them: the particular character of an object does not affect their activity."'21 This is less helpful than it appears. Cassirer's statement about the universal range of symbolic forms is too vague to provide us with a principle of individuation and, furthermore, it needs to be qualified so that it does not conflict with established doctrines of his philosophy. Symbolic forms do not apply to everything, to any object whatsoever. They are different modes of meaning that apprehend different objects. Consider science and myth. Myth does not address the objects of scientific inquiry, and science has nothing to say about the defining concerns of myth. Cassirer says on many occasions that science and myth have fundamentally different attitudes toward the world, which can be summed up by saying that science sees the world as an "it" and myth as a "thou." Myth does not have science's conception of law-governed causal connections, it does not share science's understanding of things and their properties, it does not experience time and space in any way that is comparable, and so on. Strictly speaking, then, they cannot be said to apply to the same "everything."22 The Myth of the State is a book that is intended for a general audience, and it is not aimed at answering foun- dational questions about Cassirer's philosophy. It is likely that Cassirer made the statement about universal applicability as a way of con- veying the general idea of symbolic form to an audience of nonphilosophers.

One of the reasons Cassirer did not provide a precise principle of individuation for symbolic forms was that he did not want to limit his philosophical approach; he did not want to overly determine its range of application: "The 'Philosophy of Symbolic Forms' cannot and does not try to be a philosophical system in the traditional sense of this word. All that it attempted to furnish were the 'Prolegomena' to a future philosophy of culture.... Only from a continued collaboration between philosophy and the special disciplines of the 'Humanities' (Geisteswissenschaften) may one hope for a solution of this task."23 This is why Panofsky believed that he could write an

essay on perspective as a symbolic form; he saw a way to continue this collaboration and "extend Ernst Cassirer's felicitous term to the history of art."24

A more useful passage on the nature of symbolic forms that allows us to see how the idea might be extended to perspective is this one: "We have so far tried to show how the individual symbolic forms ... are aspects in the intelli- gent organization of reality. Each of them presented us with an independent, architectonic principle, an ideal 'structure,' or, better-since we are here never dealing with describing purely static relationships, but rather with exposing dynamic processes--a characteristic way of 'structuring' itself."'25 Symbolic forms are symbol systems that structure an aspect of reality according to certain organizing principles, and this is what Panofsky argues perspective does for the representation of the visible world.26

Panofsky does not explicitly set out how his claims fit the tenets of Cassirer's philosophy and this has led some authors, Damisch can again serve as the spokesman, to maintain that Cassirer's doctrines have no real role in Panofsky's study: "The fact is, however, that far from making reference to [the idea of symbolic form] from the start, he only introduces it after extended developments, supposedly based on psychophysiology, that directly contradict Cassirer's arguments because they take the retinal image, which has nothing to do with the symbolic order, to be the touchstone of perspective construction."27 I shall argue, of course, that this evaluation is mistaken. Panofsky implicitly relies on Cassirer's doctrines throughout his study, and his conclusions are entirely compatible with the philosophy of symbolic forms. Besides, it would not be charitable to suggest that Panofsky chose his title care- lessly or without due consideration of what it implies-Panofsky was not a careless scholar. Furthermore, it is well known that he and Cassirer were colleagues at the University of Hamburg and that Panofsky attended Cassirer's lectures. Plus, we should not forget that they worked together at the Warburg Institute and produced publications that addressed common topics.28 It would be a great surprise, indeed, if Panofsky did not understand the philosophy of symbolic forms.




IV

Let us consider a simple system of perspective that fits the basic tenets of Cassirer's approach and reveals how perspective can be a symbol system that structures an aspect of reality according to certain organizing principles. As Panofsky's antique angle perspective exists only as a hypothesis, we will have to use linear perspective as our representative case.

At the most basic level, in order to have a signifying system one needs to have elements, rules of combination, and an interpretation (a syntax/semantics distinction is not going to be delineated here, in part because giving in to this distinction in its familiar form would be to prematurely surrender to standard semiotic models that privilege language, and we know that Cassirer does not want one symbolic form to be the model for others; he insists that symbolic forms be investigated according to the principles of formation that are particular to them, whatever those should be). What could be the elements of a simple system of linear perspective that involves a vanishing point and a distance point?29 Speaking broadly, and not very exactly, we could suggest lines and shapes. This is a rather large and open class, but I think it can be used to at least indicate the direction in which we might go in conceiving of perspective semiotically. The next step will make this clearer. Consider the idea of a semantic field. By semantic field I mean an arrangement in space or time or both in which a sign, or to use Cassirer's word, a symbol, takes on a different meaning depending on its location in the field, depending on its place in the arrangement. One example of a semantic field is a musical staff, where a circle, which stands for a whole note, represents a different tone depending on its placement on the staff (or above or below it). This seems to have an immediate application to perspective constructions. In musical notation, the staff is the structure that determines the value of the note; similarly, the determining structure in a painting in linear perspective results from the interrelation of the bounded painting surface, the placement of the horizon line, the vanishing point, the distance point, and the vertical that intersects the visual pyramid. The choice of configuration involving these five basic conceptual components constitutes the

structure that will give value to an element placed on the flat painting surface (and will transform the painting surface into a picture plane). The same line placed in different loca- tions can assume very different meanings, for example, in being lower or higher than the horizon line it is either above or below the beholder's eye level and can be interpreted as closer to or further away from the ground plane in the image. Obviously, the same could be said for the employment of other lines: a line starting at the top right-hand corner and reaching the vanishing point will be taken as being above the beholder's head and extending to ground level, whereas the same line starting in the lower left- hand corner and reaching the vanishing point will be taken as running parallel to the ground surface, and so on. Likewise for the placement of a shape, which will be evaluated as nearer to or further from the beholder depending on its location on the painting surface within the structure that has been set up. However, the five basic conceptual components are not all that there is to the determining structure, for they are the basis of a constructional procedure that gen- erates a grid that maps the painting surface, and allows for more exact estimations of how far shapes are from each other and from the beholder.

In this brief explanation I have run together structure and sense, as I said I would. Even so, we can see how the elements, the rules of combination, and their interpretation can be distinguished: the elements, as already suggested, are lines and shapes; the rules of combination spec- ify relations such as how lines are combined to form plane figures; and the interpretation concerns the establishment of relations between elements and rules of combination that will serve as conventions for the representation of three dimensions on a two-dimensional surface, for example, that any two parallel lines drawn within the bounded surface will have to converge (unless they run at right angles to the picture plane).

Some might think that this is too simple to be a signifying system. True, one could have this impression if one's paradigm of semiotics was the work of someone such as, say, Algirdas Greimas. But semiotics is defined by its aims, not the elaborateness of its theoretical machin- ery. This system is, however, more complex




than it appears, and it can be given a sophisti- cated theoretical foundation, as I would like now to demonstrate.

What are the principles we could use to interpret the placement of lines and shapes as signifying spatial relations? The answer can be found through group theory. Appealing to group theory is particularly appropriate in relation to the question of how perspective could be a symbolic form, for the theoretical orientation offered by group theory was fundamental to shaping Cassirer's thought.30 Although the concept of group had its origins in the development of modern mathematics, it is applicable to a much wider range of studies. Cassirer thought that "[t]here must be a reason for this universal applicability of the concept to so many fields of inquiry that, at first sight, seem to have little in common, or even to deal with quite disparate subjects."31 He never formally articulated what the reason is, but the entirety of the philosophy of symbolic forms stands as a testimony to his faith in the explanatory power of this orientation.

The basic ideas of group theory are most readily graspable when they are presented through mathematics, and thus that is how I would like to introduce them. The most fundamental idea is that of a group of transformations. What are studied as a group are not mathematical objects or elements, but trans- formative operations to which elements are sub- jected. A group then is defined as the totality of unique operations a, b, c, and so on, so that from the combination of any two operations a and b there results an operation c that also belongs to the totality, that is, a group is a set of operations that has the property that when one executes two operations in succession what is created is something that could be obtained by a single operation of the set. For example, if a circle is projected onto a flat surface at an angle of 10? as an ellipse, and then this figure is itself projected at an angle of 20, a new figure will be created that will be equivalent to the single operation of projecting a circle at 30 as an ellipse. The consequences of group theory for our concerns can be made more concrete if we look at how the concept of group affected the geometrical study of space.

With the creation of non-Euclidean geometries by figures such as N. I. Lobachevsky and G. F. B. Riemann, space became a highly problematic

concept, for if all of these geometries were equally valid, how could it be decided which one was the true description of reality? Felix Klein offered a way out by arguing that what distinguishes geometrical concerns from, say, topological ones is that geometry studies properties of space that remain invariant under certain groups of operations; every geometry studies a specific manifold in relation to a definite group of transformations, which Klein calls its Hauptgruppe.32 From this point of view, different geometries are not only possible but also inevitable, and geometry is relieved of the task of describing reality. No geometry then is truer than another, though some will be more powerful or more general and will include others as special cases.

To make all this even more concrete, and more directly related to our topic, consider the case of projective geometry, to which linear perspective has deep and strong familial ties. Take a Euclidean object such as a circle and imagine it drawn on a sheet of rubber, which can then be deformed in any number of ways. In Euclidean terms, any such deformation results in a new figure (for example, an ellipse) because Euclidean geometry recognizes simi- larity only in terms of congruence. Under such deformations, however, some properties are not destroyed, for on another level, the level of projective geometry, one can continue to consider the two figures as the same. This is possible because the Hauptgruppe of operations in projective geometry is larger than that of Euclidean geometry, and it accepts these kinds of central projections as invariant.

Such deformations are also familiar to us from linear perspective. As a simple example think of a receding tile floor in a Quattrocento painting. From a Euclidean point of view, each tile is a different object because it has a unique shape that is not congruent with any of the others. Nevertheless, we take each tile as being the same as all the others because we view the represented figures through a transformation group within which they remain invariant. Thus we can appeal to the principles of group theory to explain how lines and shapes can be interpreted as indicating spatial relations.

Even though linear perspective is not projective geometry, it can nevertheless be presented as a consistent system of spatial representation




that is based on a subset of the principles that govern projective geometry; indeed, it is worth remembering that projective geometry came into existence in part out of the desire to provide a geometrical account of linear perspective. Their substantial commonality can be used to make a powerful argument by analogy for Pan- ofsky's thesis, for projective geometry is a clear example of what Cassirer would consider to be a structurally conceived semiotic system.33 This is central to what it means to call perspective a symbolic form.

Damisch would hold that nevertheless there are reasons why such an argument is not sufficient.

At first glance the "strong" hypothesis [that perspective is a bona fide symbolic form] ...has against it, since we are dealing here with perpectiva artificialis, that it is difficult to see how the latter would be accessible to an analysis using the notion of the sign, rigorously construed. Although it managed, over an extended historical time frame, to appropriate for its graphic ends a set of objects essentially borrowed from architecture-aediculae, architectonic elements, colonnades, coffered ceilings, and vaults in recession, not to forget the pavement divided into squares that was the foundation of costruzione legittima, study of whose trace lines would eventually lead to the discovery of the vanishing point, according to Panofsky's argument-perspective is not a language, in Saussure's sense of a system of distinct signs that correspond to distinct ideas.34

Damisch is without doubt right that perspective is not a language in Saussure's sense, but then that is not a concern for the philosophy of symbolic forms. As has been stressed before, natural language does not provide the model for Cassirer's semiotics, and consequently does not determine what can be thought of as a symbolic form. (But even if language did supply the model, what would we say of art itself, which is always cited in the canonical list of symbolic forms? Is it, too, not a symbolic form? Or are we to understand that art is a language in Saus- sure's sense?)

The theory of the symbol upon which Cassirer builds his semiotics is based on the Kantian

notion of schemata. Schemata are mediating representations produced by the faculty of imagination to match the intuitions of sensibility with the categories, the a priori concepts of the understanding. A schema is, in Kant's words, "a rule for the synthesis of the imagination," the "representation of a universal procedure of imagination in providing an image for a concept."35 Even more, "[t]he schema is, prop- erly, only the phenomenon, or sensible concept, of an object in agreement with the category."36 It is the idea that schemata are sensible concepts that Cassirer found attractive, for it is wide enough to encompass all of our symbolizing activities. It is especially fitting for the proper consideration of art, given that art, as Cassirer often says, provides the most striking example of a type of symbolic creation that completely and concretely unites sense and understanding in singular symbols.

Near the end of an important discussion of invariants in perception in "The Concept of Group and the Theory of Perception," Cassirer poses this pregnant question: "Is there any logical connection between the subject of our discussion and the question discussed by Kant in the chapter on the Schematism, in his Critique of Pure Reason?"37 Cassirer immediately adds in qualification: "To be sure the two problems cannot really be identified with each other, since they belong, methodologically speaking, to different dimensions.... The fact, however, that there is nonetheless a point of contact between Kant and modern psychology has been noted and commented upon by investigators interested in the philosophical foundations of a theory of perceptual constancy."38 What then is their relation?

We may venture to answer this question, reminding ourselves of Kant's characterization of the schemata as "monogrammata of pure imagination."... The schemata are monogrammata because they express an original function of unification. The "images" which we receive from objects, the "impressions" which sensationalism tried to reduce perception to, exhibit no such unity. Each and every one of these images possesses a particularity of its own; they are and remain discrete as far as their contents are concerned. But the analysis of perception discloses a formal factor that supersedes this particularity and disparity. Perception unifies and, as it were, concentrates the




manifold of particular images with which we are sup- plied at every moment. Perception fits this stream of images into definite channels. It cannot be reduced to a mere manifold of impressions, the "polygrammata" of sensibility, in any more satisfactory manner than to a mere reproductive function in terms of "engrammata" of memory. Beyond these "polygrammata" and "engrammata" there appears a specific function of perception: the "monogram of imagination." Each invariant of perception is in fact such a "monogram," a schema toward which the particular sense-experiences are oriented and with reference to which they are interpreted.39

One of the connections amongst diverse phenomena that group theory makes apparent to us is that everyday visual experience, like geometry, depends very much on invariants. As we can see in the passage above, Cassirer is a fierce opponent of sensationalism and the kind of empiricist epistemology with which it is associ- ated. Cassirer never tires of pointing out that the experiments of Gestalt psychologists have dealt a fatal blow to this atomistic approach to experience. What the Gestalt psychologists revealed is that perception depends on the apprehension of patterns that can be treated as invariants. This is perhaps most obvious in questions of perceptual constancy, where, for example, a circle can be moved closer to a viewer, further away from the viewer, or even tilted so that it looks like an ellipse, and still be perceived by the viewer to be the same circle. We attribute to the objects in our environment certain comparatively consistent qualities that are not only psychologically important but epistemologically important as well, for without the constancy that invariants deliver to experience "it would be impossible to segregate 'things' or 'properties' from the stream of becoming. To use Heraclitus' meta- phor, we would be unable 'to step twice in the same river.' "40

Apart from reaffirming these important epis- temological points, the passage above makes the interesting suggestion that invariants in perception can be understood as similar in function to schemata. This is an idea that can help us clarify how linear perspective can be a semiotic system, for not only is Cassirer's concept of the symbol based on a reinterpretation of the doctrine of schemata, but also the symbols that perspective uses to create a two-dimensional

illusion of three-dimensional space are derived from the invariants in perception. To see how, let us return to our model of the Cassirerian symbol (see Figure 1).

Consider how this model might apply to a drawing whose spatial effect is based on a receding floor composed of square tiles that are of equal size. Let each tile be a symbol. The sensuous element will be the bounded shape of the tile; the sense will be the group of transformations that determine that this figure is equivalent to a square; the relation of representation will be the viewer's understanding that this figure, familiar from deformations in the objects of everyday experience, is to be taken as square; and the symbolic form will be the set of formative principles through which the viewer sees the figure, along with the others, as representing three- dimensional spatial properties. The invariants familiar from perception will be built into this complex at the level of sense because the group of geometrical transformations that will have been chosen will have been those that are con- sonant with the deformations encountered in visual experience. But it is at the level of symbolic form that the figure becomes a symbol, for it is only then that its two-dimensional configuration is read as indicating something beyond itself, in other words, as representing recession and depth. The square tile is a straightforward example of a Cassirerian symbol, for, like the concept of schema from which it is derived, it is a form in which sensibility and understanding are united through the greater formative principles of a symbolic order.

vi

We are now in a position to understand the place that Panofsky's hypothesized antique angle perspective has in his claim that perspective is a symbolic form. There is a passage that is of singular importance for understanding this hypothesis, but to my knowledge it has been overlooked by commentators.

For, just as for Aristotle there is no "quantum contin- uum" in which the quiddity of individual things would be dissolved, so there is for him also no energeiai apeiron (actual infinite) which would extend beyond the Dasein of individual objects (for, in modern




terms, even the sphere of fixed stars would be an "individual object"). And precisely here it becomes quite clear that "aesthetic space" and "theoretical space" recast perceptual space in the guise of one and the same sensation: in one case that sensation is visually symbolized, in the other it appears in logical form.41

For anyone reading through Panofsky's text, looking for material that indicates that he knew what was involved in calling perspective a symbolic form, this last sentence is an important one. Recall the doctrine of symbolic pregnance and the example of the line. The distinction that Cassirer makes between "the simple presence of the phenomenon" given in perception and the symbolic orders through which it is apprehended is clearly at work in the above passage: what is given in the perception of space can be symbolized and comprehended as aesthetic space or theoretical space in the unified world- view of antiquity. But it is what is given in perceptual space that will help us understand how both antique angle perspective and linear perspective can be said to capture what we per- ceive. Let us consider another important, though misunderstood, passage.

[Linear perspective] takes no account of the enor- mous difference between the psychologically conditioned "visual image" through which the visible world is brought to our consciousness, and the mechanically conditioned "retinal image" which paints itself upon our physical eye.... Finally, per- spectival construction ignores the crucial circum- stance that this retinal image--entirely apart from its subsequent psychological "interpretation," and even apart from the fact that the eyes move-is a projection not on a flat but on a concave surface. Thus already on this lowest, still pre-psychological level of facts there is a fundamental discrepancy between "reality" and its construction.42

The "retinal image" is the phenomenon that remains the same for both angle and linear perspective, or, to speak more precisely, it is the retinal image acting as the causal basis of what I will call the "optical image," a relation that will be specified shortly. At the basis of vision is the retinal image that, as Panofsky remarks in an aptly ambiguous turn of phrase, "paints itself upon our physical eye." When I look at the

world, what I see, for lack of a more precise way of stating it, is derived from the retinal image (this is what Panofsky refers to as the prepsychological level). If I pay careful atten- tion to the experience of seeing, what I will notice is that there are curvatures in my field of vision. Supposedly, these curvatures exist-in what we will now call the "optical image"- because the retinal image from which the optical image is derived is a projection onto a curved surface. This is why, if I am standing perpendicular to a rectangular wall, with my line of sight directed to its approximate center, I will see its upper edge as slightly convex and its foundation line as slightly concave, and I might also notice that the outer edges have a convex curvature. It seems, though, that what is given by the retinal image in perception-in the optical image-can be overridden, for this is apparently what happened with linear perspective: as the mode for the visual representation of space in the extraordinarily powerful world view of modern science, linear perspective has gradu- ally habituated us to ignoring the subjective curvatures in the optical image.43 This is why Panofsky says of linear perspective that "already on this lowest, still pre-psychological level of facts there is a fundamental discrepancy between 'reality' and its construction." Reality, which he here places in quotation marks, does not refer to some pre-Kantian absolute reality, but to what we earlier referred to as "the simple presence of the phenomenon," in other words, to the retinal image as the causal basis of the optical image. The retinal image is what is registered on the retina by the outside world; the optical image is the physiologically created, unmediated experience of sight; and the visual image is the mediated experience of sight-- what we would report, if we were asked to describe what we were seeing.44 Panofsky implicitly employs these distinctions but does not explain them. That does not detract from the fact that his account is in complete harmony with Cassirer's doctrine of symbolic pregnance.45 As a result, we have to question Damisch' s claim, remarked on earlier, that Pan- ofsky's appeal to the retinal image is a clear indication that he does not fully understand the philosophy of symbolic forms: "The fact is, however, that far from making reference to [the concept of symbolic form] from the start, he




only introduces it after extended developments, supposedly based on psychophysiology, that directly contradict Cassirer's arguments because they take the retinal image, which has nothing to do with the symbolic order, to be the touchstone of perspective construction."46

Let us now consider how these distinctions underpin Panofsky's speculations about the existence of an antique angle perspective. There is, evidently, a great discrepancy between the curvatures in an image projected onto the retina and the curvatures that we are aware of in vision. This is accounted for, Panofsky tells us, by "a peculiar stabilizing tendency within our consciousness-promoted by the cooperation of vision and the tactile sense-[that] ascribes to perceived objects a definite and proper size and form, and thus tends not to take notice, at least not full notice, of the distortions which these sizes and forms suffer on the retina."47 The full distortions are not transferred, if you will, to the visual cortex and registered in sight, though they persist to a certain degree and are noticeable in our field of vision. Panofsky's unstated assumption is that the curvatures in the optical image are ultimately derived from the retinal image. Most people would claim that they do not see any curvatures in vision-that is the truth of their visual experience, and that is what Panofsky would refer to as their visual image. However, we can always try to aid them in noticing the curvatures in the visual field, and if we succeed then we have brought that person to an awareness of the optical image, that is, to the unmediated, pretheoretical visual experience offered by the human apparatus of sight. Of course, one can also come to the discovery by oneself, as Panofsky suggests Johannes Kepler and other scientists did.

If what I have said is correct, we can now more clearly state what distinguishes Panof- sky's two types of perspective. Even though we all have the same retinal image and the same optical image when looking at the same rectangular wall, the visual image of a modern citizen of the Western world will be distinctly different from what would have been experienced by an ancient Greek or Roman. Although Renaissance linear perspective and the modern conception of science has led us to overlook curvatures in the visual field, the ancients noted them, and, according to Panofsky, attempted to

incorporate them into representations of the world. We deny what they stress, though our visual images are derived from the same retinal image.

The hypothesis of antique angle perspective can be brought into even closer accord with the philosophy of symbolic forms. In "Reflections on the Concept of Group and the Theory of Per- ception," Cassirer argues that, just as we can choose our transformation group in geometry, "the same free choice appears already- although much less distinctly-in the act of sense perception. Even here we constantly make selections from among the vast manifold of utterly heterogeneous sense data. We give pref- erence to certain phenomena which hence assume a privileged position."48 If Cassirer believes that in visual perception it is possible to have a different transformation group for the construction of reality, drawn from the vast manifold that is given to us, even "[i]mpres- sions received by peripheral parts of the retina," then in advancing the theory of antique angle perspective, Panofsky is yet again in complete agreement with the basics of Cassirer's epistemology and his doctrine of symbolic pregnance. In stressing the curvatures in visual experience, the ancients came to have a different, but equally valid, transformation group. They came to it because, or so Panofsky argues, their greater common culture, as is revealed to us in their philosophical and scientific texts, pos- sessed a different understanding of the spatial world, one incompatible in its fundamentals with our own. If Cassirer is right that we can have other transformation groups, then we can also say that these are probably not the only two possibilities, as we can imagine that there could be other coherent symbolic articulations of what is given to us in sensibility. Just as there are different languages in the symbolic form of language, so there are different perspectives in the symbolic form of perspective.

To state that perspective is a symbolic form is to maintain that it is a signifying system founded on the tenets of Cassirer's theory of meaning, grounded in his doctrine of symbolic pregnance. It is to claim that the elements of this system of signification conform to Cassirer's model of the symbol, and they are united by the structural principles that are constitutive of symbolic forms. It is also to hold




that, as an organ rather than a mirror of reality, a symbolic form can be instantiated in different ways depending on how its principles are applied to what is given in sense. All of this fits with Panofsky's discussion of perspective, which is why he thought he could call it a symbolic form.49

ALLISTER NEHER

Humanities Department Dawson College Montreal, Quebec Canada

INTERNET: aneher@ dawsoncollege.qc.ca

1. Erwin Panofsky, Perspective as Symbolic Form, trans. C. S. Wood (New York: Zone Books, 1991).

2. Hubert Damisch, The Origin of Perspective, trans. J. Goodman (MIT Press, 1995), p. 3.

3. W. J. T. Mitchell, Picture Theory (University of Chicago Press, 1994), p. 16.

4. Ernst Cassirer, The Philosophy of Symbolic Forms, vol. 1, Language, vol. 2, Mythic Thought, vol. 3, The Phenomenology of Knowledge, trans. R. Mannheim (Yale University Press, 1953-1957). The fourth volume was published posthumously as The Philosophy of Symbolic Forms, vol. 4, The Metaphysics of Symbolic Forms, ed. J. Krois and D. Verene, trans. J. Krois (Yale University Press, 1996).

5. Cassirer, The Phenomenology of Knowledge, p. 93. The translation I have chosen, however, is not Mannheim's, but C. H. Hamburg's from "Cassirer's Conception of Philosophy," in The Philosophy of Ernst Cassirer, ed. P. Schilpp (Lasalle: Open Court, 1949), p. 78.

6. The component "sense" is a concept. Cassirer's theory of concept formation is based on the logic of relations. Although Cassirer's theory was developed in relation to discursive symbolizing activities, particularly those of the natural sciences, he believes that this model can be applied to all other types of symbol relation: "the same is true in principle of the simplest intuitive concepts. They never form a mere conglomerate of sense impressions and remem- bered images but contain a peculiar articulation of these impressions and images, a form of organization. In them the separate is 'seen together'-not in the sense that its components are mingled but in the sense that their connection in regard to some linking factor is retained." Cassirer, The Phenomenology of Knowledge, p. 312.

7. Ernst Cassirer, "Language and Art II," in Symbol, Myth, and Culture: Essays and Lectures of Ernst Cassirer, 1935- 1945, ed. D. Verne (Yale University Press, 1979), p. 183.

8. See Wilhelm von Humboldt, "Linguistic Variability and Intellectual Development," in Miami Linguistics Series No. 9, trans. G. C. Guck and F. A. Raven (University of Miami Press, 1971).

9. Ernst Cassirer, "Structuralism in Modern Linguistics," Word 1 (1945): 105-106. I have translated the passages that Cassirer quotes from Viggo Brondal and Roman Jakobson.

In the original French, the passage from Brondal is: "Dans un 6tat de langue donn6 ... tout est syst6matique; une langue quelconque est constitu6e par des ensembles oi tout se tient: systhmes des sons (ou phonhmes), systhmes de formes et de mots (morphhmes et s6manthmes). Qui dit systhme dit ensemble coh6rent: si tout se tient, chaque terme doit d6pen- dre de l'autre. Or on voudrait connaitre les modalitds de cette cohdrence, les degr6s possibles et variables de cette d6pendance mutuelle, en d'autres termes, il faudrait 6tudier les conditions de la structure linguistique, distinguer dans les systhmes phonologiques et morphologiques ce qui est possible de ce qui est impossible, le contingent du n6ces- saire." In the original French, the passage from Jakobson is:

1. si a existe, b existe aussi 2. si a existe, b manque 3. si a manque, b manque aussi

Ces rapports qui ont infailliblement valeur de loi constituent un des facteurs les plus importants des changements phonologiques.

10. As organic wholes, symbolic forms have at least three characteristics: First, all of the symbols of the structure are irreducible unities of the sensible and the intellectual. See Cassirer, Substance and Function and Einstein's Theory of Relativity, trans. William Curtis Swabey and Marie Collins Swabey (New York: Dover Publications, 1953), pp. 266-268, and the discussion surrounding p. 91 in Cassirer's The Philosophy of Symbolic Forms, vol. 1. Second, the structure discloses a thoroughgoing organization according to law. See Cassirer, The Philosophy of Symbolic Forms, vol. 3, pp 118-121, and the discussion surrounding p. 31 of Cassirer's The Philosophy of the Enlightenment, trans. F. C. A. Koelln and J. P. Pettegrove (Princeton University Press, 1951). Third, the objects studied through the structure are not things existing independently of them, but constructs that get their meaning and objective reality through their functional relation to the other constructs with which they form an organic whole. See the discussion surrounding p. 145 of Cassirer's The Philosophy of Symbolic Forms, vol. 3. Ultimately, these principles are derived from Kant's Critique of Judgment.

11. Group theory and its extension to nonmathematical topics is the subject of Cassirer's paper "Reflections on the Concept of Group and the Theory of Perception," in Symbol, Myth, and Culture, ed. D. Verene (Yale University Press, 1979), pp. 271-291. This subject, as Cassirer tells us, was one of the first questions to arouse his philosophical interest, and it was one that he thought about for the entire course of his career-he was working on this paper the day he died.

12. Cassirer, The Phenomenology of Knowledge, p. 202. 13. Cassirer, The Phenomenology of Knowledge,

pp. 200-201. See also the discussions in "The Problem of the Symbol and Its Place in the System of Philosophy," trans. J. Krois, Man and World 11 (1978): 414.

14. Cassirer, Language, p. 78. 15. Guido Hauck, Die subjektive Perspektive und die

horizontalen Curvaturen des Dorischen Styls (Stuttgart, 1879). Hauck achieved a certain distinction during his life- time for his mathematically consistent system of curvilinear perspective.

16. Panofsky, Perspective as Symbolic Form, p. 41. 17. Panofsky, Perspective as Symbolic Form, pp. 40-41. 18. Damisch, The Origin of Perspective, pp. 11-12. 19. Damisch, The Origin of Perspective, p. 11.




20. Cassirer's use of the term is, however, more varied than is usually supposed. Carl Hamburg's studies have revealed that Cassirer uses 'symbolic form' in at least three distinct, though related, senses: "(1) It covers what is more frequently referred to as the 'symbolic relation,' the 'symbol-concept,' the 'symbolic function,' or, simply, the 'symbolic' (das Symbolische). (2) It denotes the variety of cultural forms which-as myth, art, religion, language, and science--exemplify the realms of application for the symbol-concept. (3) It is applied to space, time, cause, number, etc. which-as the most pervasive symbol-relations-are said to constitute, with characteristic modifications, such domains of objectivity as listed under (2)." Carl Hamburg, "Cassirer's Conception of Philosophy," p. 77. See also Carl Hamburg, Symbol and Reality: Studies in the Philosophy of Ernst Cassirer (The Hague: Martinus Nijhoff, 1956).

21. Cassirer, The Myth of the State (Yale University Press, 1946), p. 34.

22. In fact, the idea of a common "everything" to which all symbolic forms apply is just a useful fiction. It is, Cas- sirer tells us, "no psychological datum, but rather a liminal notion ... What we call the 'matter' of perception is not a certain sum-total of impressions, a concrete substratum at the basis of artistic, mythical or theoretical representation. It is rather a line towards which the various formal modes converge." Ernst Cassirer, "Zur Logik des Symbolbegriffs," Theoria 4 (1938): 173. Hamburg's translation, "Cassirer's Conception of Philosophy," p. 119.

23. Cassirer, "Zur Logik des Symbolbegriffs," p. 173. Hamburg's translation, "Cassirer's Conception of Philosophy," p. 119.

24. Panofsky, Perspective as Symbolic Form, pp. 40-41. 25. Cassirer, The Metaphysics of Symbolic Forms,

pp. 50-51. 26. Michael Podro has suggested (in conversation, Feb-

ruary 2002) that Panofsky could have seen his investigation into perspective as an integral part of a larger inquiry into the symbolic form of art, for if art has its own architectonic principle or principles, perspective would be an embodiment of them, just as physics would be representative of the symbolic form science.

27. Damisch, The Origin of Perspective, p. 11. 28. Panofsky's Idea: A Concept in Art Theory was origi-

nally written to complement Cassirer's The Idea of the Beautiful in Plato's Dialogues. Both were publications in the Studien der Bibliothek Warburg. Furthermore, in The Individual and the Cosmos in Renaissance Philosophy, Cas- sirer refers positively to Perspective as Symbolic Form in footnote 84 on p. 182. My understanding of the professional and social ties that linked Panofsky and Cassirer has bene- fited from conversations with their colleague, Professor Raymond Klibansky.

29. For an example of the kind of system I am referring to, see J. V. Field's discussion of Alberti in the second chapter of The Invention of Infinity: Mathematics and Art in the Renaissance (Oxford University Press, 1997).

30. Ernst Cassirer, "The Concept of Group and the The- ory of Perception," trans. A. Gurwitsch Philosophy and Phenomenological Research 5 (1944): 1-35. "The Concept of Group and the Theory of Perception" is Cassirer's longer, more philosophically elaborate, presentation of the theses presented in "Reflections on the Concept of Group and the Theory of Perception." That latter work was originally

published in French as "Le concept de groupe et la th6orie de la perception," Journal de Psychologie July-December (1938): 368-414. On the first page of "Reflections on the Concept of Group and the Theory of Perception," Cassirer says "the subject of this paper was one of the first to arouse my philosophical interest; I began to ponder on it when still an undergraduate in philosophy. I took, however, a very long time before I found the courage to publish the results to which I had been led." It apparently was also one of the last subjects that Cassirer pondered. He delivered the paper to the Philosophy Club in April 1945 while he was a visiting professor at Columbia. He died from a heart attack on the campus of Columbia (while talking to a student) on April 13, 1945. Toni Cassirer, his wife, reports in her book Aus meinem Leben mit Ernst Cassirer (New York, 1950), p. 308, that Cassirer was working on the introduction to the paper the morning of the day of his death.

31. Cassirer, "Reflections on the Concept of Group and the Theory of Perception," p. 274.

32. Felix Klein, "Vergleichende Betrachtungen fiber neuere geometrische Forschungen," in Gesammelte mathe- matische Abhandlungen, vol. 1 (Berlin: Springer, 1921). Klein's text was called the Erlanger Programm because it was Klein's inaugural address as professor of mathematics in Erlangen in 1872.

33. Both The Phenomenology of Knowledge and Substance and Function and Einstein's Theory of Relativity address this matter in detail.

34. Damisch, The Origin of Perspective, p. 23. 35. Kant, Critique of Pure Reason, trans. Norman Kemp

Smith (New York: St. Martin's Press, 1963), ?? B 180 and 181.

36. Kant, Critique of Pure Reason, ? B 186. 37. Cassirer, "The Concept of Group and the Theory of

Perception," p. 31. 38. Ibid. 39. Cassirer, "The Concept of Group and the Theory of

Perception," p. 32. 40. Cassirer, "Reflections on the Concept of Group and

the Theory of Perception," p. 288. 41. Panofsky, Perspective as Symbolic Form, pp. 44-45

(emphasis added). See also Cassirer's discussion in "Mythic, Aesthetic, and Theoretical Space," trans. D. Verene and L. Foster, Man and World 2 (1969): 3-17.

42. Cassirer, "Mythic, Aesthetic, and Theoretical Space," p. 31.

43. Panofsky goes to some length to try to demonstrate that historical figures in the dawning of the modern era of science were aware of the discrepancies between the curvatures in the optical image and the symbolically mediated visual image. He cites Kepler, for example, who noted "the possibility that the objectively straight tail of a comet or the objectively straight trajectory of a meteor is subjectively perceived as a curve" (Perspective as Symbolic Form, p. 34). "What is most interesting," Panofsky continues, "is that Kepler fully recognized that he had originally overlooked or even denied these illusory curves only because he had been schooled in linear perspective."

44. James Elkins is alert to these distinctions, The Poetics of Perspective (Cornell University Press, 1994), pp. 191-195, but does nothing with them.

45. Panofsky's account of the movement from one conception of space to another is also in accord with Cassirer's




theory of the internal development of symbolic forms, which was presented in its final form in the third volume of The Philosophy of Symbolic Forms, The Phenomenology of Knowledge. In this work, Cassirer argues that each symbolic form passes through three (ideal) stages of development: a mimetic, an analogical, and a purely symbolic stage. Although the third volume was not published until 1929, it seems to have been available to Panofsky, perhaps through conversations with Cassirer when they were colleagues at the University of Hamburg. I say this because although Per- spective as Symbolic Form was published in 1925, and thus could not have made full use of this theory in its final form, it nevertheless appears to employ the three general developmental stages that distinguish Cassirer's doctrine, especially if they are referred to Cassirer's discussions of the history of the concept of space in Western civilization. This can be easily discerned in Panofsky's text (see pp. 43-45 for a discussion of the mimetic qualities of antique space, pp. 53- 56 for a discussion of the analogical nature of medieval

space, and for the symbolic character of modem space, see the long quote from Cassirer on p. 30). Parallel to the kind of neo-Hegelian account that Cassirer gives of the development of individual symbolic forms, Panofsky connects these different conceptions of space through relations of dialectical presupposition: in the movement from antique to modemrn perspective a series of stages have to be obtained and worked through before a later one, which is built out of the preserved but transformed elements of earlier ones, can be achieved.

46. Damisch, The Origin of Perspective, p. 11. 47. Panofsky, Perspective as Symbolic Form, p. 31. 48. Cassirer, "Reflections on the Concept of Group and

the Theory of Perception," p. 289. 49. I would like to thank Margaret Iversen, Johanne

Lamoureux, Michael Podro, and Vladimir Zeman for their comments on earlier versions of this paper. I would like to thank the Department of Art History and Theory of the Uni- versity of Essex for providing me with the ideal environment in which to write it.



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