Abstract
There are several interpretations of what mathesis universalis could have been. Some interpreters claim that mathesis universalis is a universal method of discovery; others understand it as coextensive with algebra as a fundamental discipline of mathematics; while still others identify it with the mathematical articulation of material reality. Each of these three interpretations is safe. We can find traces of all of them in Descartes’ work. However, each of them captures only one aspect of Descartes’ mathematical work, ignoring others. Also, they allow us to understand neither the unity of these three aspects of Descartes’ mathematical work nor the relationship between mathesis universalis and his method. The method, as Descartes presented it in the Regulae, was geometrical rather than algebraic, its aim was certainty rather than discovery, and it did not discriminate between material and non-material reality. I will show how Descartes gradually expanded and radicalized his mathesis universalis.
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Notes
- 1.
Even though I admit that in the Regulae we have the only example of Descartes’ use of that term.
- 2.
This was, for instance, the view of Chikara Sasaki, who writes, “For Descartes, mathesis universalis seems to have connoted some kind of algebra” (Sasaki 2003, p. 7).
- 3.
The relationship between mathesis vulgaris and mathesis universalis cannot be seen to mean that there was a general, neutral mathematics which subsequently split into these two areas. Rather, alongside the classical mathematics of antiquity and the Middle Ages, symbolic algebra developed as a new discipline. In the early modern period Descartes realised the novelty of this discipline, chiefly its universality, and realised that he could remodel the whole of mathematics on the model of algebra into a mathesis universalis. In retrospect, from the viewpoint of the new mathesis universalis, the whole of ancient mathematics appears as mathesis vulgaris.
- 4.
Turning to Frege for help in interpreting Descartes may seem anachronistic. However, in the spirit of Sect. 2.3 above, Frege was (together with Euclid, Newton, Cantor and Turing) a mathematician of the same kind as Descartes, viewing mathematics with the same degree of visionary radicalism.
- 5.
That means, on the one hand that symbols like x or b can stand for any number (in algebra), in contrast to symbols like 5 or 17, which stand always for the same number; yet on the other hand, x and b can stand for arbitrary quantities such as the length of an arbitrary line segment (in analytic geometry), an arbitrary time interval (in astronomy), an arbitrary angle of reflection (in optics), or an arbitrary velocity (in mechanics). Therefore, these disciplines belong to mathesis universalis, understood as the discipline able to express universality.
- 6.
In Funktion und Begriff, Frege interpreted the history of mathematics as a process of the gradual increase of generality. In doing so, he showed that mathematics achieved higher generality by creating representational tools enabling it to prove more general theorems and construct more general objects. That is, there is not one kind of generality in mathematics, but several. Each new representational tool, such as symbolic algebra, differential and integral calculus, and the predicate calculus opens access to a new kind of generality, which includes the previous ones as special cases (see Kvasz 2008, pp. 14–84).
- 7.
“I began my investigation by inquiring what exactly is generally meant by the term mathematics and why it is that, in addition to arithmetic and geometry, sciences such as astronomy, music, optics, mechanics, among others, are called branches of mathematics. ... When I considered the matter more closely, I came to see that the exclusive concern of mathematics is with questions of order and measure and that it is irrelevant whether the measure in question involves numbers, shapes, stars, sounds, or any other objects whatever. This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis” (Descartes 1701, p. 19; AT10, pp. 377 and 378).
- 8.
For more detail concerning the relation between Descartes and Proclus see (Rabouin 2018, pp. 4756–4761).
- 9.
More precisely, on the certainty of an instrumentally founded geometrical intuition. Since the time of Euclid, the steps of a geometric construction have been based on postulates that sanction the acts of construction by means of a ruler and compass (for details see Kvasz 2019).
- 10.
Symbolic geometry, the first steps of which were developed in the Regulae, is in my view a further example of Descartes’ ability to do mathematics in a completely new way. The fact that at approximately the same time Stevin introduced infinite decimal expansions, which led to the creation of the notion of real numbers and which were later used as the foundation of arithmetic and algebra, only shows that Descartes’ geometrical approach was an attempt to solve a real, existing and pressing problem. That his geometrical approach to the foundations was replaced by the arithmetical does not mean that his idea of a symbolic geometry would be wrong or unfeasible. It is simply a further example of an abandoned mathematics, just like Bolzano’s set theory or Euler’s calculus based on infinitesimals. For Euler’s calculus, Abraham Robinson offered its new formulation in the form of non-standard analysis (see Robinson 1965) and for Bolzano’s set theory, Petr Vopěnka made an attempt at its revival in the form of alternative set theory (see Vopěnka 1979). Who knows, maybe some mathematician will become interested in reviving Descartes’ symbolic geometry in some new form? It is possible that in Hartry Field’s project of a science without numbers we have an example of such an attempt (see Field 1980).
- 11.
Today, under the influence of Turing and von Neumann, we understand ‘thinking’ as symbolic manipulation. However, with equal justification we can interpret thinking as the manipulation of ideas in the imagination, that is, as the manipulation of the figures of symbolic geometry. Thus, the Regulae symbolic geometry project can be seen as one of the first attempts to naturalize epistemology, to transform it into a scientific discipline.
- 12.
To discriminate these two roles—developing the foundations of mathesis universalis versus offering a foundation for all knowledge—or at least to indicate their difference, I have used in this section’s title two adjectives. By symbolic geometry I mean primarily an attempt to found algebra on rules for manipulating plane geometric figures, while by epistemic geometry I mean a further extension of this approach to all (visual) knowledge; but of course, symbolic geometry is epistemic in the sense that it develops the foundations for algebraic knowledge, and epistemic geometry is symbolic in the sense that the rules of transformation of the geometric figures in the imagination do not belong to simple plane geometry, but express the rules of epistemology in geometric language, which is thus used in a symbolic way. Thus, the difference is rather a difference of emphasis.
- 13.
Thus, I propose that the mathesis universalis project did not end in failure, but instead split into two distinct projects. The first focused on the cogito and culminated in Meditations on First Philosophy (Descartes 1641). The second, turned towards nature, aimed at a mathematical knowledge of extended substance, and its outcome was Le Monde (Descartes 1664).
- 14.
For instance, in the letter to Mersenne of 27th July 1638 Descartes writes: “For if he will please consider what I have written on salt, on snow, on the rainbow, etc., he will understand that my entire physics is nothing other than mathematics” (AT 2, p. 268).
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Kvasz, L. (2024). What is Mathesis Universalis?. In: Descartes on Mathematics, Method and Motion. SpringerBriefs in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-031-57061-2_3
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