Dividing Lines. Four Images, Somewhat Fewer Ideas.

Lapo Lappin, 2021

A Different Space

Pythagorean symmetries, Euclidean architecture, marmorean glaciations of Apollonic ratios.

The Parthenon, for instance, or the fountain-square on Rhodes. But before the dust of ages had settled; before the Corinthians capitella were required to hold up the weight of civilization. When they were still new: polished marbles, untarnished in their adamantine lustre and opalescent twinklings. And because they were new, they could be eternal, rippled mirror-images of hyperouranian spaces.

It is enframed within these geometries that Pliny the Elder traces an anecdote about the legendary painter Apelles. While visiting the rival artist Protogenes in his home on Rhodes, Apelles walks up to the frescoed walls of the villa and paints a line so thin it can barely be seen by the human eye. Upon his return, Protogenes picks up the gauntlet and, in a different colour, places an even thinner line on top of the previous one. Thereupon, Apelles, with preternatural precision, plots a third line that cleaves the first two perfectly in half.

 A Space of Difference

The Aegean sun dapples an inscription in the marble, percolating through the olive grove of Athena. The epigraph, etched into the congealed geometry of the Academy, reads: ”Let no one ungeometrical enter under my roof”. 

Eleni Ieremia, ”Vessels”, 2020

The verbal adjective picked by Plato — ageômetrètos — denotes someone who is not ”geometrical”, one who has not yet actualized the possibility of thinking geometrically. Whether or not this tradition records an accurate historical inscription, or is just a hyperstitious mytheme of the later Platonic tradition, the sentiment finds plenty of support in Plato’s dialogues: one must first wrestle with geometry before the mind is fit to rise up to things divine. One would assume, as is commonly done, that this is because geometry is based on logic. So too is philosophy, which is what makes geometry into the perfect proemial activity to the love of wisdom.

But that assumption would be wrong. It is not geometry that is ”logical”; it is logic that is geometrical. Logic is principally a matter of drawing distinctions, of dividing, of dividing these distinctions into new divisions. It is, as much as we would want to resist it, a spatial matter, a geometrical matter. What Plato looks for in an aspiring lover of wisdom is the capacity to draw segments, trace circumferences, split lines.

Plotted along these coordinates, the fable of Apelles is about philosophy. Philosophy is a matter of finding differences, of splitting hairs. Apelles picks out a difference so subtle that no onlooker could have imagined it. At heart, it is a fable about spatial imagination: through a geometrical intuition Apelles gleans the difference nested within another difference. It is this same spatial imagination that Plato requires, the lack of which bars entrance to the Academy.

Spaces of difference

A female figure splashed on parchment, shimmering with cobalt paste and gold-leaf. Beautiful, yet asymmetrical: one side is lithe, dexterous; the other skeletal, spidery, almost sinister. On the one hand, she clutches a phial of the sweetest honey; on the other, a scorpion scuttles over her knuckles, while a viper enwreaths itself around her forearm.

The medieval personifications of Dialectica betray one fundamental point: the natural model of difference is a spatial one. Lady Dialectic collocates differences along a spatial axis, dividing left from right: poison from delicacy, punishment from dessert. The other images floating around ancient philosophical iconography belabour the same point: the groves of ”trees of Porphyry”, which bind together definitions in leafy crowns of differentiae. Or the ”columns” of syllogisms, which inscribe the diverging shapes of argument within architectural complexes.

And so on to this day, except that we seldom think much of it. We speak profusely of ”the logical space of reasons”, of modal-logical ”frames”, of ”grounds” and ”grounding”, of ”supervenience” — all spatial imagery, upon which we calque our philosophical concepts. 

But — someone may object — we should not make mountains out of mole-hills: these are, after all, only metaphors! That, however, is precisely the point.  These are indeed metaphors, but metaphors without which philosophy would not function. No philosophy to date, no matter its commitment to the deserted landscapes of analyticity, has eluded metaphor. And metaphors are, after all, images. And images are ineluctably spatial, which means philosophy, as much as we would like to burst free, is nested in a cocoon of two things it cannot moult: images and spaces.

Different Spaces

Among all the spatial imagery that defines ancient philosophy, the most central is Cicero’s picture of the art of argument as the exploration of the topoi — the ”places”, or ”spaces” — of thought.

Philosophy, according to this image, is really a kind of travel. The philosopher explores the landscapes of thought, foraging in the hinterlands, staking out the forest paths from the all-engulfing undergrowth. This topographical image, which veered the ancient philosophers onto their path, differs from our contemporary philosophical practices in one fundamental way. The space within which philosophy takes place is the locus of our spatial fantasy — the human imagination. It is precisely this faculty that Plato believes that geometry embodies.

It is due to an ever so slight misunderstanding about the nature of geometry that modern philosophy lost its spatial imagination. While modern philosophy was from its inception Euclidean (think of the ”geometrical” methods of Descartes, Spinoza, Hobbes), it remained fundamentally ungeometrical in the proper, Platonic sense. In these modern methods geometry becomes a matter of proofs, rather than a matter of geometry proper — that is to say, of space, of lines, of shapes. Modern philosophy became the philosophy of ageômetrètos

To escape this predicament philosophy must rediscover its dependence on spatial imagery. It must become a philosophy whose relation to images is neither pedagogical (that does not reduce then to didactic exemplifications of this or that doctrine) nor simply erudite flexing (through which the philosophist cashes in cultural capital through cultivated allusions to Cezanne or Vasari), nor, still, a specialized sub-culture of philosophy that just happens to adopt art as its subject, in the same way as one could apply philosophical tool-kits to snorkeling or bus-driving.

Which leads us to the final image: 

A philosophy that is iconographic, that dwells among the images from which it was born. A philosophy that not only sustains itself on spatial imagery, but is also attentive to it, reflecting on it. It is both informed by its foundational images, and at the same time informs those very images through its reflection. Only when philosophy can do this will it shed its chrysalidal trappings, and become properly imaginal.


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