‘Grapheus Was Here’ by Anthony Auerbach, in Drawing A Hypothesis: Figures of Thought ed. by Nikolaus Gansterer, Vienna/New York: Springer, 2011, pp. 65–76. The book publication includes drawings by Nikolaus Gansterer.

Untangling drawing and theory reveals a knot which cannot be undone historically. While cutting it is considered the sign of progress in mathematics, the knot doesn’t relinquish its primordial status all that easily. Euclid’s first postulate, ‘To draw a straight line from any point to any point,’ is the graphic hypothesis on which is founded the notion that there are drawings which may be considered to all intents and purposes equivalent to abstract thought. By literally drawing a hypothesis, the postulate at once recruits drawing to the cause of deductive reasoning and furnishes quod erat demonstrandum with an image; it warrants a line to draw a conclusion (a theorem) and the a priori to compel reality as surely as a geometer constructs figures.

Drawing thus enacted the isomorphism of geometry and its image as a law of nature, and signed the expedients — that is to say, authorised the departures from the strict domain of mathematics — that we associate with the names, for instance, Alberti, Galileo, Newton.

The point is not to insist on the purity of mathematics, which would be bound to anachronism: the truths mathematics claims are timeless remain so, but, whereas the antique fell short of its ideal only by modern standards, the modern refuses to realise antique expectations. Better to note that while mathematics admits no contradiction, in history, contradictions abound. The period when pure mathematics came to be defined by the elaboration of arbitrary hypotheses, free from intuitive and realistic content or meaning, was also the period of accelerating expansion of the domain of applied mathematics. The types of mathematics that were applied and the fields of knowledge to which they were applied multiplied, along with the number and variety of drawings imagined as embodying demonstration on the Euclidean model (construction in the Kantian version). Let us call such drawings diagrams. The burgeoning of the scope of mathematics along with its graphic counterparts perhaps also prompted the revival of interest in philosophising ad more geometrico, albeit not according to the old method.

My approach may be called pragmatic because it is concerned with the meaning produced, transferred and transmitted by the use of diagrams: content not reducible to the abstractions in which diagrams purport to deal, nor necessarily derivable from the hypotheses on which diagrams rest, more or less explicitly, more or less consistently. The metaphorical economy of diagrams is a web of exchange in which drawings function not only as tokens but also as agents.

Highlighting some transactions in that network — drawing attention to some of its threads and crossings — is probably as much as can be expected from a short text like this one. While the approach doesn’t promise a fundamental theory of diagrams, nor the format a comprehensive survey, at least I examine the tangle intact.

Gaspard Monge: Géométrie descriptive

Monge’s method wasn’t published under the ancien régime because it was a military secret. It came out first in the Séances of the shambolic and short-lived École normale de l’an III, the institution hastily contrived for the formation of a revolutionary curriculum and corps of teachers. Monge’s lectures were collected in book-form four years later in 1799, and became a cornerstone of the polytechnic tradition that Monge himself helped establish. [note 1]

Descriptive geometry proposes a universal method of engineering drawing with a double aim: to represent exactly any three-dimensional object — provided it is ‘susceptible of rigorous definition’ — by means of drawings, and furthermore, to derive, from an exact description of any object, everything that follows necessarily from its form. In that sense, Monge adds, it is ‘a means of searching out the truth’. It would therefore be ‘necessary,’ he concludes, that descriptive geometry be part of a national plan of education, not just for the intellectual benefit of a great people and thereby of mankind, but for the practical benefit of French industry and, by implication, the military capability of the Republic. Monge envisages, on the one hand, the power of nature harnessed by machines, determined graphically part by part, and on the other hand, the knowledge of nature, described by geometry, turned to the profit of the arts.

When Monge mentions the construction of perspectives and of shadows as notable applications of the method, it is as if, at once to align descriptive geometry with the Vitruvian canon, and to distinguish it from it, indeed, to assert the priority of descriptive geometry — both logically and pedagogically — over the methods taught in the academies, that is to say, Renaissance methods invented in honour of the antique.

Descriptive geometry distinguishes itself from the perspective inherited from Alberti in several important ways. Above all, descriptive geometry is not optical: its notional rays do not converge in an eye. Instead of a bundle, descriptive geometry supposes parallel projectors — like the sun’s rays that project shadows, although Monge himself avoids such metaphors. Descriptive geometry does not produce a picture: not one, because more than one drawing is required to describe an object, and no picture, as long as Alberti’s intersection of the visual pyramid defines the very idea of a picture. Whereas perspective rests on the theory of proportion (expressed geometrically by the similar triangles that encompass base and intersection of the pyramid) and pays homage to Euclid, descriptive geometry is conceptualised in terms of three-dimensional Cartesian analytic geometry, with which, Monge says, it has ‘the most intimate relations’.

The correspondance between geometry and algebra (which Monge inherits from Descartes) means that ‘every analytical operation can be regarded as the script for a play (l'écriture d'un spectacle) in geometry’, and reciprocally, there is ‘no construction in descriptive geometry which cannot be translated into analytic form,’ the évidence of drawing complementing the généralité of algebraic expression.

Monge conceived of drawing as a rational machine, responding efficiently to the tasks appointed by mathematics as well as to practical engineering problems. Although descriptive geometry’s mathematical credentials lent it prestige and supported institutional hierarchies of knowledge, the method was mainly transmitted as technical drawing: as a repertoire of graphic gestures taught and learned mechanically, producing signs standing not only for their intended objects but for rational form as such — just as perspective constructions had earlier come to signify the authentic trace of appearance, and schoolbook geometry stood for reason itself.

C. S. Peirce: Existential Graphs

Come on, my Reader, and let us construct a diagram to illustrate the general course of thought.

This opening sentence of the ‘Prolegomena to an Apology for Pragmaticism’ [note 2] is the gambit Peirce offered at the outset of the philosophy he distinguished for himself. Peirce’s enthusiasm for diagrams goes, on the one hand, with his insistence on the purity of mathematics, and on the other hand, with his receptivity to metaphor, but it has still one more preliminary: a semeiotic adequate to diagrams, that will state clearly Peirce’s hypothesis. ‘Not that the particular signs employed are themselves the thought! Oh no; no whit more than the skins of an onion are the onion. (About as much so, however.)’ (4.6).

While Peirce’s semiotics is widely accepted (like Monge’s geometry, usually in simplistic form), his system of diagrammatic logic, the Existential Graphs, as he called them, are regarded as something of an eccentric curiosity. For Peirce, the effort he devoted to elaborating the system promised ‘moving pictures of thought’ (4.8), ‘rendering literally visible before one’s very eyes the operation of thinking in actu’ (4.6). Peirce’s imagination, however, wasn’t quite the same as that of Kant, for whom necessary reasoning was performed by constructing geometric figures; nor quite like that of Monge, whose geometry traced the spectacle mouvant predicted by analysis. Existential Graphs were to be understood, under the strictest regime of abstraction, as expressions concerning a hypothetical universe, ‘perfectly definite and entirely determinate, but the arbitrary creation of an imaginary mind’ (4.432).

Peirce’s notion of Existential Graphs suggests both a meta-logic and a meta-image, since he already regarded deduction, exemplified by mathematical reasoning — which set the standard for exact logic — as none other than ‘diagrammatical, or, iconic, thought’ (3.429). Whereas Peirce often cites algebra as iconic thinking par excellence (3.364) — since algebraic formulae display relations, and further, open them to experiment and observation — he was convinced Existential Graphs would amount to a ‘far more powerful method of diagrammatisation’ (3.418).

Wittgenstein’s reflections on logic (Wittgenstein once imagined his own book ‘might well be equipped with diagrams’ [note 3] — though it was not) could suggest why Peirce’s diagrammatic system did not, after all, catch on. The problem with Existential Graphs isn’t that there are more efficient methods of notation. Peirce preferred a more complex (not to say unwieldy) style in so far as it disclosed the structure of thought in more detail. Still — despite Peirce’s pains to generalise the relations between ‘the grapheus’, out of whose imaginary mind a hypothetical universe is continuously developed, ‘the graphist’, whose graphs, or rather, whose successive modifications of ‘the entire graph’, assert something about that universe, and ‘the interpreter’ who is to make sense of the graphs — despite all that, the system is probably too heavily burdened with (sometimes bizarre) metaphors (like the heraldic ‘tinctures’ applied to the figures) for most logicians’ taste, and remains much too abstract for most literary philosophers.

The problem is, Existential Graphs don’t elucidate the question that mobilised Peirce’s effort and would justify it, namely, ‘how the diagram is to be connected with nature’ (3.423), in short: the question of representation. Peirce doesn’t need to be told that this isn’t a question for logic, but he insists that thought isn’t just a mental thing. That ‘there cannot be thought without Signs’ (4.551) is not more evident from Existential Graphs than from any other notation. The boldest hypothesis of Semeiotic, however, goes beyond what logic can demonstrate, for the means by which ‘Reality ... contrives to determine the Sign to its Representation’ (4.536) is not the force of reason. In other words, if we agree with Peirce that all Signs are ultimately Indices — as typified by physical traces, pointers and interpellations — then no line can be drawn between thought and matter.

Francis Picabia: Ce qui défigure la mesure

Picabia’s drawings would certainly qualify as diagrams on Peirce’s criteria: as experimental devices for investigating and demonstrating the structure of reason. The title imprinted on the first drawing in Poèmes et dessins de la fille née sans mère [note 4] reads ‘VIS-À-VIS’: an icon of relations, as Peirce would say. The combination of more or less abstract graphic gestures with more or less abstract verbal labels gives Picabia’s drawings all the appearance of diagrams familiar, or half-remembered, from a variety of didactic contexts: perhaps mechanics, biology, geography, philosophy. Yet they have no such context to lend continuity to what Peirce would call ‘the sheet of assertion’, nor any hypothetical grid to map each gesture — each discrete sign — to a field of knowledge. These ‘Witticism Machines’ feed on ardour, Madagascar, hermaphroditism, truth, error, madman’s hands, limpidity, vernal vagina, to cite only a few of Picabia’s indices. They are vivid in the context in which they appear: a book of dull poems exhibiting the Dada strategy in its pure form: sabotage meaning! (It’s not that Dada has no cargo of meaning, only that it’s going to explode.) Half image, half sentence, the drawings by ‘the girl born without a mother’ are no image and no sentence. The blanks which reason does not leap gape for association, the tentative and anxious web spun by the interpreter who exists to make sense of signs.

VIS-À-VIS is inscribed, ‘That which disfigures measurement’. Even as it appears to discredit and deform reason, Picabia’s drawing hints at a discipline. The line of reasoning which can be traced through projective geometry (the science of properties and relations preserved under projective deformations), and which finds its most general expression under the term topology, could be called geometry without measurement. Topology stands for thinking from which all constraints of measure and matter have been rigorously subtracted, and hence preserves (in altered form) the promise of necessity that had made Euclidean geometry so compelling. While Picabia’s drawing, in a book dedicated to ‘tous les docteurs neurologues en général’ and to his own psychotherapists in particular, is a comic play on the script of analysis (to distort Monge’s terms a little), it is Lacan’s affectation for diagrams which draws the consequences, in all seriousness, of Dada logic.

Jacques Lacan: La logique du fantasme

It is as if the headline ‘Dada signifies nothing’ which interrupted Tristan Tzara’s manifesto [note 5] with a typographic pointing finger were condensed into twenty years of weekly seminars in front of the blackboard of the École normale. Lacan posits his geometric origin at a double crossing: a hybridisation and a crossed purpose. His zero-setting of subjectivity identifies a supposed Freudian subject with a subject he claims originates with Descartes. ‘What does that imply?’ Lacan asks rhetorically, ‘if not that we are going to be able to start playing with the little letters of algebra, which transform geometry into analysis [...] — that we can allow ourselves everything as hypothesis of truth’. [note 6] The geometrisation of psychoanalysis, Lacan believes, will secure its constitution as the ‘science of the unconscious’. From a likeness of the structure of the unconscious, he professes to have ‘deduced a topology whose aim is to account for the constitution of the subject’ (27 May 1964). A repertoire of quasi-algebraic formulae and quasi-geometric diagrams will therefore prove indispensable to a teaching in which such figures are assigned the duty of demonstration, despite being deprived of any consistent premise or rule of transformation that would allow anything to be deduced independently — any premise or rule, that is, other than that language-world, that law of semeiosis, in which everything is permitted — Lacan’s diagrams are no autonomous machines.

When there is no difference between metaphor and theory, apodeixis is reduced to a didactic gesture that would command reality like an abracadabra. The performer’s flourish, not to say sleight of hand, masks a schoolmaster’s charisma with ecclesiastical authority. Lacan did not fail to remark of one of his favourite objets trouvés — a drawing sometimes used to illustrate a kind of surface that crops up in topology and nick-named ‘the mitre’ or cross-cap (although topologically speaking, the figure has no particular form) — that it is worn by bishops. [note 7]

The ambition of constructing the science of the unconscious after Descartes (E. F. P., the school Lacan founded and dissolved, stood briefly for the French School of Psychoanalysis before it was altered to the Freudian School of Paris) stumbled on no obstacles among its empirical data (supposedly the practice of psychoanalysis) nor amid the abstractions it borrowed from philosophy and mathematics. It turned out, indeed, very like a language. The knots in which the project finally became embroiled were the result of tangling with drawing. [note 8]

Images

Grapheus Was Here, photograph by Anthony Auerbach, 2010 [back to image]

In-text images:

Existential Graph, after C. S. Peirce

Francis Picabia: Ce qui défigure la mesure

Tristan Tzara: Dada Manifesto

Topology and Time, transcript of Jacques Lacan'’s seminar, 16 January 1979.

Notes

1. Gaspard Monge, Géométrie descriptive (Paris: Baudouin, 1799). [back to text]
2. Charles S. Peirce, Collected papers of Charles Sanders Peirce (Cambridge: Harvard University Press, 1931), 4.530. Additional references from this and other published and unpublished papers are given by volume and paragraph numbers in the text. [back to text]
3. Ludwig Wittgenstein, Notebooks 1914–16 (Oxford: Basil Blackwell, 1961). [back to text]
4. Francis Picabia, Poèmes et dessins de la fille née sans mère (Lausanne: Imprimeries Réunies, 1918). [back to text]
5. Tristan Tzara, ‘Manifeste dada’, Dada, 3, 1918. [back to text]
6. Jaques Lacan, Les quatre concepts fondamentaux de la psychanalyse (Paris: Seuil, 1973), 29 January 1964. Further references to the seminars collected in this book are given by date. [back to text]
7. Jacques Lacan, ‘L’étourdit’, Scilicet, 4, 1973. [back to text]
8. ‘Bon c’est ennuyeux que je m’embrouille, mais je dois dire que je dois avouer que je m’embrouille. Bien. Ça sera assez pour aujourd’hui.’ (It’s annoying, but I’m confused. I have to say that I have to admit that I’m confused. Well, that will be enough for today.) Jacques Lacan, La topologie et le temps, 16 January 1979, unpublished transcript. [back to text]

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