Abstract
The history of mathematics can be read in two ways. On the one hand, unlike the history of physics, it does not proceed by conjectures and refutations. New theories rarely refute old theories, but give them new foundations, generalize them, and reinterpret them through new concepts. This reading is unifying, highlighting the unity of the history of mathematics from its origins, through the permanence of its truths. On the other hand, many contemporary historians of mathematics have insisted on the diversity of methods, objects, and practices throughout history. They have shown that the objects of the past mathematics are not the same as those of today. That second reading shatters the illusion of unity. It disjoins what the first reading unifies.
Rather than deciding between these two readings, this chapter examines what makes them both possible and legitimate, based on an analysis of the acts of reading that we have to perform when confronted with the pages of a mathematical text, its signs, diagrams and images, and the different language modalities it employs. Starting with a discussion of Husserl's theses in The Origin of Geometry on the role of writing in the history of mathematics, and using a number of historical examples, the chapter shows that any reading of a mathematical text involves at least three divisions, three delimitations made by the reader’s gaze: divisions between the true and the false, between language and image, and between what is inside mathematics and what is outside. For the same text, these divisions are traced by the gaze in different ways throughout history.
The chapter develops an example in greater detail, concerning the status of the equation in Descartes’ Geometry. It discusses the contrasting recent readings of the equation by two historians of science, Henk Bos and Enrico Giusti. It shows the concepts of The Geometry and the ideas the philosopher develops on reading and the nature of the sign. It shows how these ideas can be used to reconstruct how Descartes might have read a mathematical text, and it explains how successive readings of The Geometry can at the same time insert it into the mathematics of later centuries.
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Notes
- 1.
Leonhard Euler (1751) On the controversy between Messrs. Leibniz and Bernoulli concerning the logarithms of negative and imaginary numbers. Berlin Acad Sci 5:139–179, 1749. The text is reproduced in the Opera omnia, XVII, series 1, pp 195–232. Euler obtained this result by giving x the value −2 in the series \( \frac{1}{1+x}=1-x+{x}^2-{x}^3+{x}^4-{x}^5+\dots \) Whose validity, formally demonstrated, is for him necessarily universal (Euler 1996).
- 2.
Jacques Derrida, Edmund Husserl’s Origin of Geometry: An Introduction, translated with a preface and afterword by John P. Leavey, University of Nebraska Press, Lincoln/London (Derrida 1962, p. 158).
- 3.
In his letter to Florimond de Beaune dated 20 February 1639 (AT, III, 409).
- 4.
Husserl, op. cit., p. 165.
- 5.
Ibid., pp. 169–170.
- 6.
In his Mathematics as Sign (Stanford University Press, 2000), Brian Rotman proposes just such a semiotic, drawing in part on Peirce’s theory, and expressly criticizing Husserl’s theory in The Origin of Geometry (cf. p. 47). Brian Rotman is undoubtedly unfair to Husserl in considering that the latter merely grants the sign the role of recording an ideality born originally in the solitary consciousness of a mathematician. Actually, the German philosopher defends the idea that the object or the mathematical truth only becomes truly objective, and thus mathematical, properly speaking, through the mediation of writing, its “linguistic flesh.” The sign thus has a constitutive role for him. Nevertheless, Brian Rotman is right to point out that a reflection on the signs’ nature and form, and on how they influence and transform the nature and form of mathematical objects, is utterly foreign to Husserl’s thinking.
- 7.
Op. cit., p. 40.
- 8.
Gaston Bachelard (1965) L’activité rationaliste dans la physique contemporaine. PUF, p. 25.
- 9.
Jean Cavaillès (1962) Remarques sur la formation de la théorie abstraite des ensembles. Philosophie Mathématique, éd Hermann, p. 27.
- 10.
On this question, see Jean-Michel Salanskis, Le Constructivisme non standard, Septentrion, 1999, which presents Robinson’s conceptions in “Robinson” while discussing the question of historical recurrence in the introduction (Salanskis 1999).
- 11.
On this issue, see Quine (1999). Word and object. MIT Press.
- 12.
Donald Davidson (1984) Inquiries into truth and interpretation. Oxford University Press, “On saying that,” pp. 100–1.
- 13.
Ibid., p. 101. In “Radical interpretation”, Davidson discusses “the trade-off between the beliefs we attribute to a speaker and the interpretations we give his words” (Ibid., p. 207).
- 14.
On the relation between these general analyses of beliefs and meanings and some problems in history of science, see “On the Very Idea of a Conceptual Schema,” Ibid., pp. 183–98.
- 15.
- 16.
For example, we read: “In any arithmetic triangle, the sum of the cells of each base is the double those of the preceding base” (Pascal, op. cit., p. 1292).
- 17.
Gerolamo Cardano (1974) The great art of the rules of algebra, trad: Richard Witmer T. MIT Press.
- 18.
Jacques Peletier (1554) L’Algèbre… departie en deus livres. J. de Tournes, Lion.
- 19.
Ibid., p. 24.
- 20.
This instrument generalizes another one, sometimes called “mesolabum,” known before Descartes. It is used by Descartes for the first time in the Cogitationes privates of 1619.
- 21.
Descartes (1954) The Geometry, trad: Smith DE, Latham M. Dover Publications, p. 47 (AT, VI, 442).
- 22.
“Mathematics goes towards its purity as much and more than it comes from it,” Michel Serres writes in “La querelle des anciens et des modernes”, Critique, 1963, reprinted in La communication. Seuil, 1969, p. 73 (translation ours). Suzanne Bachelard comments: “Mathematics never stops moving toward purity. The principle of purity remains a regulating principle. The principle of purity is a principle native to mathematics. Recognizing it does not prevent us from recognizing that impure conceptions, when they are born with the great mathematicians, can be catalysts” (La Représentation géométrique des quantités imaginaires au début du XIXe siècle, Palais de la Découverte, D 113, translation ours) (Bachelard 1966).
- 23.
In a recent book, I tried to describe what we can call a fourth limit, the limit between the subjective position of the mathematician in the text and the space of its objects. On this question, see Lucien Vinciguerra, Celui qui parle. Science et roman (Hermann 2019).
- 24.
See Gottlob Frege (1960) Function and concept. In: Geach P, Black M (eds). Translations from the philosophical writings of Gottlob Frege. Basil Blackwell.
- 25.
See Enrico Giusti (1990) Numeri, grandezze e Géométrie. In: Descartes, il methodo e i saggi. Armando Paoletti, pp. 419–431.
- 26.
“A figure is that which is contained by some boundary or boundaries.” Euclid, Elements, Book I, Def. 14 (Euclid 1956).
- 27.
Enrico Giusti (1998) La Révolution cartésienne en géométrie. In: Descartes et son œuvre aujourd’hui. Pierre Mardaga, pp. 47–62
- 28.
See Henk JM Bos (1990) The structure of Descartes’ Géométrie. In: Descartes, il methodo e i saggi. Armando Paoletti, pp. 349–69.
- 29.
Henk JM Bos (2001) redefining geometrical exactness. Descartes’ transformation of the early modern concept of construction. Springer.
- 30.
See rule 7: “Si igitur, ex.gr., per diversas operationes cognoverim primo, qualis sit habitudo inter magnitudines A and B, deinde inter B and C, tum inter C and D: non idcirco video qualis sit inter A and E, nec possum intelligere praecise ex jam cognitis nisi omnium recorder. Quamobrem illas continuo quodam imaginationis motu singula intuentis simul and ad allia transeuntis aliquoties percurram, donec a prima ad ultimam tam celeriter transire didicerim, ut fere nullas memoriae partes relinquendo, rem totam simul videar intueri” (AT, X, 388). “If, for example, by way of separate operations, I have come to know first what the relation between the magnitudes A and B is, and then between B and C, and between C and D, and finally between D and E, that does not entail my seeing what the relation is between A and E; and I cannot grasp what the relation is just from those I already know, unless I recall all of them. So I shall run through them several times in a continuous movement of the imagination, simultaneously intuiting one relation and passing on to the next, until I have learnt to pass from the first to the last so swiftly that memory is left with practically no role to play, and I seem to intuit the whole thing at once” (Philosophical writings, vol. 1, p. 42). On the relationship between the compass device and the Rules, see Vuillemin J (1960) Mathématiques et métaphysique chez Descartes. PUF and Serfati M (1993) Les compas cartésiens. Arch Philos 56:197–230.
- 31.
Henk JM Bos, op. cit., p. 361.
- 32.
Descartes (1954), The Geometry, op.cit., p. 8 (AT, VI, 372).
- 33.
Ibid. (translation modified).
- 34.
Ibid., 374. This metaphor is not translated in the English version. And at the next paragraph, “lorsque la dernière équation aura été entièrement demêlée” is translated by “when the last equation shall have been entierely solved” (op. cit., p. 13).
- 35.
Descartes, Principles, I, 45.
- 36.
Descartes (2009), The Philosophical Writings, vol. 1, p. 107 (AT, XI, 4).
- 37.
Descartes (1991), The Philosophical Writings, p. 378 (AT, V, 357).
- 38.
Descartes, Principes, IV, 197, in The Philosophical Writings, vol. 1, p. 329 (AT, IX, 316).
- 39.
On this issue, we may of course also refer to the fourth discourse of Dioptrics, op. cit., p. 202 (AT, VI, 112).
- 40.
In the fourth discourse of Dioptrics, it is indeed what is sent “inside our brain,” such as the vibration of the nerve fiber or the image on the back of the eye, which receives the qualification of sign likely to excite thoughts in the soul, “as for example signs and words, which do not resemble in any way the things they signify” (Ibid.).
- 41.
Descartes, Principes, IV, 198 (French translation of Claude Picot revised by Descartes).
- 42.
AT, IX, 58. In this two passages, the french text dated 1647 is very different from the latin text dated 1644. It was probably modified by Descartes himself.
- 43.
Dioptric, discourse four, in Descartes (2009), The Philosophical Writings, vol. 1, p. 202 (AT, VI, 114).
- 44.
It might be argued that The Geometry never presents the problem in this light, and that it never evokes the theory of the sign of The World or of the Dioptrics. However, it should be remembered that The Geometry is the third essay of a series of works which began with the Discourse of Method, Meteors and the Dioptrics; and that Descartes prefaces his table of contents at the end of the work with the following warning, which remains relevant to this day: “Those who visit the tables of books only in order to choose the subjects they wish to see, and to exempt themselves from the trouble of reading the rest, will derive no satisfaction from this table: for the explanation of the questions mentioned therein almost always depends so expressly on what precedes them, and often also on what follows them, that it cannot be understood perfectly unless the whole book is read with attention” (AT, VI, 486; this passage is not translated in the English version).
- 45.
This idea is at the heart of the fifth and sixth Rules for the direction of the mind.
- 46.
Descartes evokes “the fact that all points of those curves which we may call ‘geometric’, that is those which admit of precise and exact mesurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by means of a single equation, The Geometry, op. cit., p. 48, livre second (AT VI, 392).”
- 47.
Ibid., p. 91 (AT VI, 411).
- 48.
Ibid., p. 43 (AT VI, 389).
- 49.
Michel Fichant, in “La géométrisation du regard. Réflexion sur la Dioptrique de Descartes” (re-edited in Science et métaphysique dans Descartes et Leibniz, PUF, 1998), has emphasized the role of movement in the Cartesian analysis of vision: “It is the ordered set of all these movements (and not the more or less resembling traces they leave in their wake) which signifies to the soul what it interprets as a thought of seeing” (p. 48; translation ours) (Fichant 1998).
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Vinciguerra, L. (2023). What Happens, from a Historical Point of View, When We Read a Mathematical Text?. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_80-1
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