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If Descartes wanted to found philosophy on the certainty of mathematics, it seems he must have considered arguments for Euclid's parallel postulate cogent, or at least not doubted them.

Gerolamo Saccheri (1667—1733), one of the early forerunners of non-Euclidean geometry, postdated Descartes (1596-1650), so perhaps Descartes never doubted the foundations of geometry.

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  • "But geometric space, for Descartes, had to be Euclidean. This is because the theory of parallel lines is crucial for Descartes’ analytic geometry—not for Cartesian coordinates, which Descartes did not have, but because he needed the theory of similar figures in order to give meaning to expressions of arbitrary powers of x", Grabiner, Why Did Lagrange “Prove” the Parallel Postulate? He required indefinite extendability of straight lines as well.
    – Conifold
    Dec 21, 2019 at 1:25
  • @Conifold Re: Descartes's "method of analysis or resolution," Copleston vol. 4 p. 75 claims "he considered [where?] that Euclidean geometry, for example, has a serious drawback, namely, that the axioms and first principles are not ‘justified’. That is to say, the geometer does not show how his first principles are reached. The method of analysis or resolution, however, ‘justifies’ the first principles of a science by making it clear in a systematic manner how they are reached and why they are asserted."
    – Geremia
    Dec 21, 2019 at 3:57
  • There is nothing of the sort in Discourse on the Method. According to De Risi's book:"Arnauld and the Cartesian school claimed that the self-evidence of the axioms makes the proof of them useless... Arnauld thinks that it is not wise to “lose time and rack one’s brain” to prove the Parallel Postulate, which has “clarity enough” to be accepted without demonstration." This seems to be more applicable to Leibniz, but even he did not doubt the PP, only sought to prove it "from definitions".
    – Conifold
    Dec 21, 2019 at 4:47
  • @Conifold Discourse on Method pt. 4: "I perceived that there was nothing at all in these [geometry] demonstrations which could assure me of the existence of their object […] [I]t is at least as certain that God, who is this Perfect Being, is, or exists, as any demonstration of geometry can be." (In fact, he considers it more certain, as his previous ¶ shows, where he gives a sort of Anselm ontological argument and shows that only his idea of God cannot be doubted.)
    – Geremia
    Dec 21, 2019 at 5:21
  • The reasoning goes the other way, from "the great certitude which by common consent is accorded to these demonstrations" to "at least as certain that God... is". As for the "great certitude", he is apparently with Arnauld. If any further justification is needed, it is of the "God is not a deceiver" sort, not any cogent arguments for what is already très-claire.
    – Conifold
    Dec 21, 2019 at 5:32

1 Answer 1

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Descartes considered that he needed Euclid's parallel postulate:

Descartes identified space and the extension of matter, so geometry was, for him, about real physical space. But *geometric space, for Descartes, had to be Euclidean. This is because the theory of parallel lines is crucial for Descartes' analytic geometry - not for Cartesian coordinates, which Descartes did not have, but because he needed the theory of similar figures in order to give meaning to expressions of arbitrary powers of x [23, p. 197]. Descartes was the first person to justify using such powers. But an expression like x4 for Descartes is not the volume of a 4-dimensional figure, but a line, which can be defined as the fourth proportional to the unit line, x, and x3. That is, l/x = x3/x4. They are all lines, and since all powers of x are lines, they can all be constructed geometrically - but only if we have the theory of similar triangles, for which we need the theory of parallels.

Judith V. Grabiner, 'Why Did Lagrange "Prove" the Parallel Postulate?', The American Mathematical Monthly, Vol. 116, No. 1 (Jan., 2009), pp. 3-18: 7.

[23, p. 197] = L. Hodgkin, A History of Mathematics, Oxford: OUP, 205: 197.

Edit - Ram - added superscripts e.g. x3

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  • Hi, I edited the answer by adding superscripts.
    – Ram Tobolski
    Dec 21, 2019 at 23:02
  • Much appreciated. I really must master the technique :)- Best - Geoff
    – Geoffrey Thomas ♦
    Dec 21, 2019 at 23:47

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