


‘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 shortlived 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 bookform 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 threedimensional 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 threedimensional
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 metalogic and a metaimage,
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 ‘VISÀVIS’:
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
halfremembered, 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.
VISÀVIS 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 zerosetting 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 quasialgebraic formulae and quasigeometric
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 languageworld,
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 nicknamed ‘the mitre’ or
crosscap (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]
Intext 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
 Gaspard Monge, Géométrie
descriptive (Paris: Baudouin, 1799). [back to text]
 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]
 Ludwig Wittgenstein, Notebooks
1914–16 (Oxford: Basil
Blackwell, 1961). [back
to text]
 Francis Picabia, Poèmes et dessins
de la fille née sans mère (Lausanne:
Imprimeries Réunies, 1918). [back to text]
 Tristan
Tzara, ‘Manifeste dada’, Dada,
3, 1918. [back
to text]
 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]
 Jacques Lacan, ‘L’étourdit’,
Scilicet, 4, 1973. [back to text]
 ‘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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