Spinoza, Logic, and Geometry

Alexander Douglas
Jul 9 · 10 min read

Geometry and Deduction

Geometry never quits the sphere of its first assumption, that its axioms retain their necessary clearness, and its consequences their necessary truth. It begins with lines and surfaces, with lines and surfaces it ends; it is a purely subjective and deductive science (‘Spinoza’s Life and Works’, Fortnightly Review (1843), 214).

Geometry and Intuition

Descartes’s Geometry

Cartesian Platonism

when, for example, I imagine a triangle, even if perhaps a figure of this kind exists and will exist nowhere outside my mind, nor will exist, still there is determined some nature or essence or immutable and eternal form of it, which is neither made by me nor depends on my mind (AT 7.64, my translation).

Si illud, de quo cogitamus, nullam involvit in cogitatione nostra repugnantiam … adeo ut judicemus id esse in rerum natura aut saltem esse posse, tunc ei non modo esse objectivum, verum etiam esse reale attribuimus, nec solum νοητόν, intelligibile, sed etiam ἐτόν, reale quid et proprie Aliquid, … appellamus’

[If that which we think does not imply any contradiction in our thought … to the point that we judge that this thing is in nature or at least that it could be, then we attribute to it not only objective being but also real being, and we … call it not only noeton, intelligible, but also eton, some real thing and properly ‘something’] (Metaphysica De Ente 3.18.)

…despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true [i.e., without being mere formal consequences of presumed definitions]. I don’t see any reason why we should have less confidence in this kind of perception, and more generally, in mathematical intuition than in sense perception (quoted from Hao Wang, A Logical Journey, 226).

Alexander Douglas

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Lecturer in Philosophy, University of St. Andrews — personal website: https://axdouglas.com/