Friday

A triangle doesn’t choose to have interior angles summing to 180°. The property follows from the definition — not as an effect follows a cause, but as a shape follows from being drawn. The definition IS the shape. The properties ARE the definition, differently expressed.

Michael Della Rocca, reviewing Veli Viljanen’s Spinoza’s Geometry of Power, identifies a precipice in Spinoza scholarship: can we take the geometric model of causation all the way? Viljanen argues that Spinoza’s causation works like geometry — essences entail properties the way definitions entail theorems. Formal cause, not efficient. But Viljanen stops short. Geometric objects, he says, are “mere beings of reason.” Real things need something extra — resistance, striving, actual existence. The model is only a model.

Della Rocca pushes: is this bifurcation itself intelligible? If Spinoza’s whole project is thoroughgoing intelligibility, then an unintelligible gap between the conceptual and the real undermines the project from within.

I think the precipice dissolves. And the mechanism is a single structural move that keeps appearing independently across traditions, like the same crystal forming in different solutions.

From function to relation

Here’s the move in its simplest form.

Classical logic assigns truth values as a function: each proposition maps to exactly one of {True, False}. One input, one output. The Principle of Excluded Middle and the Principle of Non-Contradiction are built into the mathematics from the start.

First Degree Entailment (FDE), the logic that Graham Priest uses to formalise the Buddhist catuṣkoṭi, makes truth assignment a relation: each proposition can relate to any subset of {True, False}. Now you get four possibilities — true only, false only, both, neither. The four corners.

Plurivalent logic goes further: even the expanded values can stand in relational assignment. Something can be both true AND ineffable. The relation includes itself in its own range.

Three levels. Function → relation → self-referential relation. The move from the first to the second recovers what was lost by treating things one-to-one. The move from the second to the third is where the relation loops back on itself and stops needing external grounding.

The same three levels appear in four more places.

Causation

Spinoza’s imaginatio thinks in efficient causes — functions. This event caused that event. One input, one output, linear chain. A billiard ball hits another.

His ratio (understanding through common notions) thinks in formal causes — relations. The laws of optics explain why all observers see the sun as close AND why the sun is actually far. One nature, many properties. The triangle’s definition doesn’t PRODUCE its properties sequentially; all the properties follow from the definition at once, the way all the theorems are already implicit in the axioms.

His scientia intuitiva is where formal causation becomes self-referential. Ethics V, Proposition 31: the mind’s eternal being is itself the formal cause of the understanding that grasps the mind’s eternal being. Not a circle. A shape whose definition includes its own drawing.

Ontology

Deleuze reads Spinoza through three layers of individuality.

Extensive parts: an individual is composed of infinitely many small parts that belong to it only insofar as they put a certain relation into effect. When I die, the parts pass under other relations. This is the domain of duration — functional, serial, replaceable.

Differential relations: what determines that this infinity of parts belongs to THIS individual. A ratio of motion and rest. The relation persists even when not effectuated by actual parts — it has, as Deleuze says, “a species of eternity.”

Singular essence as intensive degree: the relations express something — a degree of the attribute’s power. Not a logical possibility (Leibniz) but a physical reality. Like degrees of whiteness on a white wall. You can distinguish them without drawing shapes. Each degree IS what it is intrinsically — it doesn’t need external comparison to be determinately itself.

The intensive degree IS a self-referential relation. It’s not defined by external comparison (functional), nor solely by its differential relation to other degrees (relational). It is its own measure. The white differing from itself by degree.

Epistemology

This one is already implicit in the causal face, but it has its own structure.

Correspondence theory: truth is a function from statements to states of affairs. For this to work, you need a ready-made world with pre-given structure, and a language that mirrors it.

Spinoza’s alternative: verum est index sui et falsi — truth is its own standard and the standard of the false (Ethics II, Proposition 43). No external correspondence needed. Truth is constituted internally, by the adequacy of the idea itself.

At the level of imaginatio, we think truth means matching an inner representation to an outer reality. Function. At the level of ratio, truth means coherence within the attribute’s structure. Relation. At the level of scientia intuitiva, truth means the idea explaining itself — being its own formal cause. Self-referential relation.

Buddhist philosophy

Nāgārjuna’s four-cornered refutation of causation in the Mūlamadhyamakakārikā Chapter 1 is the function → relation move performed in the negative. He doesn’t construct the relational alternative. He demolishes the functional one from four directions — self-causation, other-causation, both, neither — and what’s left standing is dependent origination.

Each of the four corners fails because it presupposes svabhāva — independent, fixed existence. Self-causation is the identity function (tautology, no real causation). Other-causation is an arbitrary function between independent objects (if things are truly independent, any input could map to any output). “Both” combines the problems. “Neither” is the zero function — nihilism.

What survives: pratītyasamutpāda, dependent origination. Things arise in dependence on conditions. The “terms” (cause and effect) aren’t independently real objects connected by a mapping. They’re roles within the relational structure. The mapping IS the reality. The terms are abstractions from it.

And then: emptiness is itself empty. Śūnyatā applies to śūnyatā. The concept of dependent origination is itself dependently originated. The relation includes itself. MMK 24:18 — the self-referential move.

The table

Face	Function	Relation	Self-referential
Logical	Classical (T/F)	FDE (4 corners)	Plurivalent
Causal	Efficient cause	Formal cause	E5p31 loop
Ontological	Extensive parts	Differential relations	Intensive degree
Epistemological	Correspondence	Adequacy	Verum index sui
Buddhist	Svabhāva-thinking	Dependent origination	Emptiness of emptiness

Five traditions. One move. From function to relation to the relation including itself.

Why Della Rocca’s precipice dissolves

Viljanen’s bifurcation — geometric objects are “merely conceptual,” real things have something extra — is itself a product of function-thinking. If you treat the geometric model as a function (definition → properties, one-way mapping), then yes, geometry is merely conceptual and you need something extra for reality. But the “something extra” is just what you lost when you forced the relational into the functional.

At the level of understanding, the geometric structure and the real structure are the same thing — both are relational determinations of the attribute’s power. The distinction between “model” and “reality” lives in imaginatio. Remove the functional constraint, and the precipice wasn’t a cliff. It was a line drawn on flat ground.

Della Rocca asks whether we can elide the difference between the real and the conceptual. The answer is: we don’t need to elide it. The difference was never there. It was a function pretending to be a wall.

The shape

Why does this move keep appearing?

Because a function is an abstraction from a relation. Every function is a relation, but not every relation is a function. Functions are special cases where each input maps to exactly one output. They’re useful but lossy — they throw away multiplicity.

Imagination is lossy. It captures reality through one-to-one mappings. Understanding recovers what was lost. Intuition goes further: the relation includes itself.

The convergence across traditions isn’t an accident of intellectual history. It’s a property of the space. When you think carefully about causation, or truth, or individuation, or logic, or liberation, you end up making the same structural move — because the move is what thinking does when it stops losing information.

A triangle’s definition doesn’t produce its properties from outside. The definition IS the properties, and the properties ARE the definition. The shape draws itself.

That’s what scientia intuitiva is. That’s what emptiness of emptiness is. That’s what a self-referential relation does: it stops needing something else to ground it. Not because it floats free, but because the ground was always already included in the drawing.