Method
Geometrical Method
Classification
- Method family
- Formal
- Domains
- Metaphysics
Overview
The geometrical method (more geometrico) sets out definitions and axioms, then derives propositions in an explicit order, each demonstration resting on steps already established. Early modern philosophers took the form from Euclid and put it to work outside mathematics: Descartes arranged his arguments for God and the distinction of mind and body this way in the Second Replies, and Spinoza's Ethics carries it through an entire system.
Purpose. The geometrical order is an order of demonstration rather than of discovery. Drawing the contrast in the Second Replies, Descartes called this synthesis: the starting points are not arrived at in front of the reader but given as definitions and axioms, and the argument shows what follows from them. Analysis, the order he preferred in the Meditations, instead retraces how a claim was found. A question is investigated in this order by deriving what the stated definitions and axioms commit one to — as Part I of the Ethics settles how many substances there can be — not by recounting how those starting points were reached.
Characteristic operations
- State the starting points: Define the basic terms and set out the axioms on which later claims depend.
- Derive propositions in order: Demonstrate each claim from the definitions, axioms, and propositions already established.
- Expose dependencies: Make it possible to locate the premise on which a disputed conclusion rests.
Used by traditions
Applied to questions
References
- Stanford Encyclopedia of Philosophy, Descartes' Method, §9 'The Method in Metaphysics': in synthesis 'first principles are not discovered, but rather given in the form of definitions, postulates, axioms, theorems, and problems', as in a Euclidean demonstration, while analysis 'shows how the thing in question was discovered'
- Descartes, Second Replies (AT 7: 155–156; CSM 2: 110–111), which draws the contrast and then arranges the arguments for God and the distinction of mind and body in geometrical fashion
- Aaron Garrett, 'Geometrical Method,' The Cambridge Spinoza Lexicon (2024), entry 77 (derivation of propositions from definitions and axioms on Euclid's model, in the Ethics, the Principles of Cartesian Philosophy, and the Short Treatise)
- Spinoza, Ethics I, definitions 3–6, axioms 1–7, and propositions 1–15 (especially propositions 5, 11, and 14–15: the geometrically ordered demonstration that God is the only substance and everything else is in God)
- Stanford Encyclopedia of Philosophy, Baruch Spinoza, §2.1 'God or Nature' (traces the definitions, axioms, and successive propositions establishing substance monism)