Abstract
Descartes’s doctrine of clarity and distinctness states that whatever is clearly and distinctly perceived is true. This paper looks at his early doctrine from Rules for the Direction of the Mind, and its application to the demarcation problem of curves in Descartes’s Geometry. This paper offers and defends a novel account of the demarcation criterion of curves: a curve is geometrical just in case it is clearly and distinctly perceivable. This account connects Descartes’s rationalist epistemological programme with his ontological views about mathematics.
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Notes
This, as I show, is the key to the demarcation of curves in his Geometry.
I only focus on the previously known manuscript of the Regulae in this paper, primarily from the Adam-Tannery edition (Descartes, 1996).
By ‘reduction’ here, I mean that deduction can be shortened to be as instantaneous as clear and distinct perception. This does not mean the latter is inferior to the former.
Throughout this paper, I will abbreviate the English translation of the Geometry (Descartes, 1637) as ‘S &L’, referring to the translators Smith and Latham. The original page number from the 1637 publication will be cited after ‘Geometry’. The Adam-Tannery edition (Descartes, 1996) will be cited as ‘AT’ followed by the volume and the page numbers.
Even in this remark, Descartes suggests that this demarcation criterion is suitable because exact knowledge can be obtained.
However, Kempe’s original ‘proof’ was flawed, and a correct proof was only offered in 2002 (Kapovich & Millson, 2002).
I will abbreviate the two volumes of The philosophical writings of Descartes (Descartes, 1985) as ‘CSM’, referring to the translators, followed by the volume and page numbers.
One could question how ‘true belief’ could be possible for Descartes as opposed to knowledge. For Descartes, it is possible to arrive at truth beliefs from experience, but what makes it belief rather than knowledge is that its truth comes from the bodily experience rather than the pure intellect. There is a difference between cognition and scientia in Descartes, which are both translated as knowledge in English. In that sense, a true belief is a cognition rather than scientia, as it is only the latter that can be known by the pure intellect. Thus, any certainty or knowledge [scientia] is obtained from intuition or deduction alone.
By the ‘embodied mind’, I do not mean to suggest that every faculty depends on the body.
As I will show later, deduction is such method in which we make use of the faculties.
See Simmons (2003, p. 564ff) for a discussion on the different perception of the faculties.
As Descartes develops this further, he seems to distinguish the picture or image in one’s mind from one in imagination. For example, in a letter to Mersenne, he says that there is a difference between the ideas in imagination and in the mind. The latter can be conceived without an ‘image’ (i.e. a picture), while the other cannot be (AT 3:395). See Simmons (2003) for more on this. In the earlier work, however, this distinction is not as clear. In Book II of the Regulae, Descartes talks in detail about abstracting and picturing bare figures in imagination for intuiting facts about mathematics. But when it comes to the non-mathematical examples, it is unclear how that ‘he exists’ or that ‘he is thinking’ can be pictured as such. This is perhaps why Descartes distinguishes the ideas in the mind and in the imagination in his later doctrine of clarity and distinctness, so the doctrine generalises to more abstract truths beyond mathematics. According to Hatfield (2017, p. 444), Descartes could not accept the view that the pure intellect relied on imagination, as that would suggest God could not be known clearly and distinctly. While this is true, this is no longer a problem when we focus on the mathematical examples. This might even be the reason why Descartes could not finish the Regulae – Book III, which was never written, was to consider non-mathematical certainties. Thus, narrowing our domain of certainty to be that of mathematics, I claim that if we can intuit a truth p, then we could form a picture of p in imagination, and conversely, if we can form a picture of p, then we could intuit an appropriate truth.
Although whether the perception of pure intellect and of imagination ought to be treated the same for Descartes is not entirely clear, certain passages and letters from Descartes indicate that they should be treated differently – see Simmons (2003, p. 565) for more discussion on this. The usual contrast between them seem to be whether the figures are imagined or understood, rather than whether the ideas are the same. In that sense, having the image alone does not consist in understanding, i.e. intuiting the truth by the pure intellect. So we can contrast intuition, i.e. the understanding and seeing the truth, from just seeing the mere picture.
Another problem is that Descartes is not clear himself how exactly deduction, which relies on other faculties, guarantees that the ideas are indeed of the pure intellect. This problem is beyond the scope of this paper.
It is not even clear whether Descartes would intuit \(52 + 73 =125\), but only equations that are simpler.
For Descartes, the line itself would have been the magnitude. That is, the line A is the magnitude |A|. However, the modern notation differentiates the line from the ‘magnitude’ of the line. I am using the modern notation here for clarification.
It is possible that Descartes was criticising the ancients for only using deduction by imagination. I do not discuss this further here.
Note that, Mancosu does not explain whether he means construction in imagination instead of physical/pencil-paper construction. But since Descartes discusses how the tracing of curves is in his imagination, I assume that Mancosu would agree.
I omit the figure for the operation here, since we do not need it for this paper. For those interested, see Bos (2001, pp. 293–296).
There are several other ways to find this equation. One way is to use Thales’ theorem: the triangle drawn from the diameter and a point on the circle is a right triangle at the point. Then we can use the properties of similar triangles, as Descartes does for solving geometrical problems.
The way we obtained the equation of the circle shows that it satisfies the second and the third conditions of geometricality given by Descartes. Namely, that there is a definite relation between the point P and the point X (ii). This relation is then expressed as an equation F(x, y) in two variables (iii).
In order to really show that the curve is mechanical by its construction, we must evaluate all possible constructions of the curve. Since this is not humanly possible, one ought to turn to algebraic equations to explicate the curve’s mechanicality.
One possible historical reason (other than Descartes’s recognition that the construction of quadratrix cannot be geometrical) for rejecting that such ratio can be known might be due to Viète’s equation, which shows that there is an infinite product expression of \(\pi \) as follows (see Boyer (1968, pp. 352–353)):
$$\frac{2}{\pi } = \sqrt{\frac{1}{2}}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}}}\cdots .$$Unlike the Archimedian approximation which is given geometrically, this equation is given algebraically. Descartes, possibly having known that there is an infinite product expression of \(\pi \), and so that the ratios between a curve and a straight line can only be expressed in infinite products (or sums), which suggests that it would take infinitely long chains of deduction to know the ratio exactly. Thus the ratio cannot be known to the finite human mind.
In fact, Descartes allowed certain constructions using strings even if they were inexact, as long as they were used ‘only to determine lines whose lengths are known’ (S &L p. 92, Geometry, p. 341, AT 6:412). For example, as Panza (2011, pp. 83ff) shows, Descartes accepts the gardener’s constructions of ellipses and hyperbolas using strings, despite the inexactness of the equipment.
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Acknowledgements
I would like to thank Marco Panza, Jeremy Gray, Jeremy Heis, and Charles Leitz, the audience at the Orange County Inland Empire Seminar in History and Philosophy of Mathematics and Logic, as well as the members of the University of California, Irvine, for their valuable feedback and discussions on earlier versions of this paper. I would also like to thank the anonymous reviewers for taking the time and effort necessary to review the manuscript. I sincerely appreciate all valuable comments and suggestions, which helped us to improve the quality of the paper.
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Moon, S.S. Demarcating Descartes’s geometry with clarity and distinctness. Synthese 202, 123 (2023). https://doi.org/10.1007/s11229-023-04356-3
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DOI: https://doi.org/10.1007/s11229-023-04356-3