1 - Spinoza’s Ontology Geometrically Illustrated: A Reading of Ethics IIP8S

Summary

The Ethics is probably the most famous and finest example of a philosophical treatise written in the synthetic geometrical style in which propositions are derived from basic definitions and axioms. This alone leaves little doubt that, for Spinoza, geometry provides the pivotal model for philosophy. However, we should not rush into thinking that a certain form of exposition is the only, or even the most important, sense in which geometry informs his philosophy; there are reasons to think that geometricity is ingrained deeper than that in his thought. Most notably, Spinoza's penchant for geometrical examples to illustrate key points of his system signals this. Perhaps the best-known instance of this is the following:

I think I have shown clearly enough (see P16) that from God's supreme power, or infinite nature, infinitely many things in infinitely many modes, i.e., all things, have necessarily flowed, or always follow, by the same necessity and in the same way as from the nature of a triangle it follows, from eternity and to eternity, that its three angles are equal to two right angles. (E IP17S)

In other words, all things as modifications of the single substance follow from the nature (or essence) of that substance, precisely in the way that certain necessary properties follow from the essence of a geometrical figure such as a triangle. This arresting claim is in line with and, I think, the source of Spinoza's no less striking necessitarianism, according to which nothing could have been otherwise since everything takes place with absolute necessity.

In addition to the example concerning (the essence of) a triangle and its properties, I would like to draw attention to three especially prominent illustrations. To take the earliest first, in the Treatise on the Emendation of the Intellect Spinoza sets two requirements for a proper definition (of the essence of a finite thing). Here is the first:

The definition … will have to include the proximate cause. E.g., according to this law, a circle would have to be defined as follows: it is the figure that is described by any line of which one end is fixed and the other movable. The definition clearly includes the proximate cause. (TIE 96)