The intelligibility of motion and construction: Descartes’ early mathematics and metaphysics, 1619–1637

Re‐Examining Descartes’ Algebra and Geometry: An Account Based on theReguale

“… My view is also not entirely unfriendly to commentators such as Domski () who considers her approach opposed to a methodological one. For example, one might use the method to explain why the focus on intelligibility and constructability that Domski sees in Descartes’ classification of curves in the Geometrie and later in Le Monde even arises and in what sense they may or may not be related.…”

“…I recognize that this is a deviation from the typical practice of focusing primarily on Descartes’ Geometrie when engaged in a discussion of his mathematics. But I am engaged in a different project than commentators such as Domski ; who are looking to see what his mathematical problem‐solving techniques can tell us about his subsequent philosophical views. Even commentators such as Grosholz ; Grosholz and Hintikka who explicitly look at Descartes’ “method” are starting with mathematical methods.…”

The debate between Peletier and Clavius on superposition

“…168 [Bos, 2001, 160-165]. 169 On this issue, see [Molland, 1976], [Mancosu, 1996, 71-79], [Bos, 2001, 335-342], [Mancosu, 2008], [Domski, 2009] and [Crippa, 2014, 68-72, 100-103, 221-228, 238-247 and 255-263]. See also [Panza, 2011, 74-91].…”

Rethinking geometrical exactness

“…Two examples are enough to explain why. Domski (Domski (2009), 123) emphasises the role played in Descartes' geometry by "a standard of intelligibility grounded on simple and clearly conceivable motions", and suggests that geometrical curves are those that can be obtained through an "intelligible motion". Arana (Arana (Forthcoming)) suggests, instead, that for Descartes "constructed objects are known best when the construction is carried out in a way that is fully present to the attentive constructing mind".…”

“…The case of Villalpando's locus and compass provides an example. Descartes seems to think of a situation like this when he alleges to have "furnished a way to describe" a curve, by "having explained the way of finding an infinite number of points though which" it passes, and adds that "this way of finding a curve by finding several of its points at random applies only to those curves which can also be described by a regular and continuous motion" (Descartes (1637), 339-340; Descartes (AT), VI, 411-412;Bos (2001), 344-345;Domski (2009), 125-129). One could then conclude that generic point-wise constructions of a curve are admitted by Descartes if and only if the curve is also traceable by a geometrical linkage suggested by the very construction.…”