Abstract
Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge.
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Notes
This paper is the outcome of a talk given at the conference “Cartesian Epistemology” organized by Jean-Baptiste Rauzy, Xiaoxing Zhang and Stefano Cossara at the Université Paris IV-Sorbonne in June 2015. I was asked to compare the development of “Cartesian epistemology” and its possible echoes in philosophy of mathematics with development coming from a more historically minded approach of Descartes’ philosophy and scientific practice. I thank the organizers for having given me the opportunity of presenting my ideas and for their insightful comments on a previous version of this paper. I also thank the anonymous referees for their help in improving this paper.
See Mancosu (1999).
Especially since a new copy of the Regulae has been discovered by R. Serjeantson in Cambridge in 2011. This copy is significantly different in many respects and brings up new questions about the date of this treatise or the possible evolution it might display. For a survey of some debates concerning the place of the Regulae in Descartes’ work, see for example Israel (1998) or Van de Pitte (1991).
See in particular Hintikka (1981) and Parsons (1992). In fact, as recalled by Carson (2009), Hintikka’s strategy was more likely to get rid of imagination. One can state, however, that Hintikka’s reading launched a renewal of interest in Kant’s philosophy of mathematics, of which the reappraisal of Kant’s mathematical imagination (as proposed by Carson) was an essential part.
See Sect. 2 for references.
I refer here to the Elementa nova matheseos universalis from 1683 (Leibniz 1999, pp. 513–524).
Imaginatio generaliter circa duo versatur, Qualitatem et Quantitatem, sive magnitudinem et formam; secundum quae res dicuntur, similes aut dissimiles, aequales aut inaequales. Et vero similitudinis considerationem pertinere ad Mathesin generalem non minus quam aequalitatis (Leibniz 1999, p. 514).
The beginning of the text is a direct reference to the work done by Descartes and Vieta, which Leibniz intends to pursue and to overtake (Leibniz 1999, p. 513).
My emphasis. Mathesis universalis tradere debet Methodum aliquid exacte determinandi per ea quae sub imaginationem cadunt, sive, ut ita dicam, Logicam imaginationis (Leibniz 1999, p. 513).
This occurs, once again, in the context of a description of Universal Mathematics: Interim Algebra cum Mathesi universali non videtur confundenda. Equidem si Mathesis de sola quantitate ageret sive de aequali et inaequali, ratione et proportione, Algebram (quae tractat quantitatem in universum) pro parte ejus generali haberi nihil prohiberet. Verum mathesi subesse videtur quicquid imaginationi subest, quatenus distincte concipitur, et proinde non tantum de quantitate, sed et de dipositione rerum in ea tractari (De ortu, progressu et natura algebrae, GM VII, 205). See also the letter to Sophie-Charlotte “On What is Independent of Sense and Matter” (1702): “these clear and distinct ideas which are subject to the imagination are the objects of the mathematical sciences” (G VI 501, my emphasis; translation in Leibniz 1986, p. 548).
See, amongst many others, the opening section of Arndt (1971). More recently, this narrative has been taken up by Macbeth (2014) in her chapters 2 and 3, where she considers a corpus of texts very close to the one dealt with in this paper (and also in the context of the philosophy of mathematical practice).
See Macbeth (2014, p. 15) : “Descartes’ new mathematical practice, which reveals, so he thinks, the pure intellect as a power of knowing, enables in turn a radically new understanding of inanimate nature in terms of exceptionless physical laws governing the motions of matter” (This view is then developed in chapter 3 whose second section is entitled “Mathesis universalis”).
“They [scil. Mathematical objects such as circles] are instead (so Descartes thinks) purely intelligible objects, objects whose essences are given by equations that are grasped by the pure intellect independent of any images, and indeed of our sense organs generally” (Macbeth 2014, p. 140).
See Rabouin (2009), especially Introduction and Annexe II.
Contrary to a widespread myth, neither Stevin nor Vieta tackled the issue of a universal mathematics. Van Roomen was an important and innovative algebraist of his time. However, in the book in which he mentions the mathesis universalis (Van Roomen 1597), he does not identify it with the arithmetica universalis (inherited from Stevin), to which a separate chapter is dedicated and in which algebra makes a brief appearance. In later texts, Van Roomen sometimes talk of a mathematica universalis including the logistica (with reference to Viète) and the prima mathesis. But once again he does not seem to confuse these various notions (one being an organon to mathematics the other a genuine scientia). See Rabouin (2009), chap. IV.
On these two aspects, see Rabouin (2009), chap. III.
Picccolomini (1547).
“Hence Proclus concludes from Plato that these mathematical beings, on which we give demonstrations, are neither completely in a sensible subject, nor completely freed from it, but that mathematical figures are to be related to imagination” (Picccolomini 1547, fol. 97).
“One thing of great weight is to be noted: when we have shown the imagined quantity to be the matter or subject (materiam sive subjectum) of mathematicians, this was said to be the subject not of geometry or arithmetic, which are the first genera of mathematics, but of a certain faculty common to geometry and arithmetic. In effect, as Proclus indicates in many passages from his first and second books, there is a certain science common to those two, which recognizes a subject with properties and principles proper to itself” (Picccolomini 1547, fol. 97 r).
Other scholars have emphasized the parallel between Proclus and Descartes’ thoughts, but through an exclusive focus on the role of construction, see Lachterman (1989), Sepper (1996) and Nikulin (2002). This is in line with the emphasis on the continuity between the views expressed in La Géométrie and some features of the Greek constructivist point of view (see already Molland 1976). As will appear more clearly in what follows, I doubt that this is the real place for a continuity between the two authors considering that the constructivist point of view on mathematical objects plays no role in the Regulae (in which, moreover, there is no question of geometrical curves). This continuity seems, however, very important for other authors from the Early Modern period, see Domski (2012) on Newton.
See Rule XIV, AT X, 441; CSM I, 58.
The doctrine of “ratios and proportions” was also what Piccolimini had in mind, like all the other authors of our corpus, when referring to the general science of mathematics, compare fol. 97 r.
The French says : “sans les supposer que dans les sujets qui serviraient à m’en rendre la connaissance plus aisée” (my emphasis).
“Next I observed that in order to know these proportions I would need sometimes to consider them separately, and sometimes merely to keep them in mind or understand many together. And I thought that in order the better to consider them separately I should suppose them to hold between lines, because I did not find anything simpler, nor anything that I could represent more distinctly to my imagination and senses. But in order to keep them in mind or understand several together, I thought it necessary to designate them by the briefest possible symbols. In this way I would take over all that is best in geometrical analysis and in algebra, using the one to correct all the defects of the other” (AT VI, 20, CSM I, 121. My emphasis).
On this continuity, see Araujo Silva (2008) (in particular p. 78, n. 30 about the fact that the passage from le Discours indifferently applies to the Regulae and to La Géométrie).
See Rule XII, AT X, 416; CSM I, 42: “It is one and the same power: when applying itself along with imagination to the ’common’ sense, it is said to see, touch etc.; when addressing itself to the imagination alone, in so far as the latter is invested with various figures, it is said to remember; when applying itself to the imagination in order to form new figures, it is said to imagine or conceive; and lastly, when it acts on its own, it is said to understand”. The relationship between the use of symbols, and more generally the role of writings, and memory is presented in Rule XVI, AT X, 454–455; CSM I, 66–67.
One could even say that they open the door to some form of ‘symbolic’ imagination. As is well known, Leibniz coined the term cognitio symbolica (or caeca) for this kind of knowledge. What is of particular interest is that he introduced it publicly in a reference to Descartes’ famous example of the chiliagon (see note 63). Leibniz’ point was that when thinking of a chiliagon, one was not necessarily relying on pure ideas, as opposed to images, but on symbols, which are just another kind of images. Symbols in this case are words referring to notions which required further analysis in order to lead to a genuine “idea”. Indeed, the mere combination of notions cannot suffice to attest for the possibility of ‘ideas’, since it can entail a hidden contradiction (as when we talk of a regular solid constituted of ten equal faces or “regular decahedron”, see Nouveaux Essais sur l’entendement humain III, 3, § 15). On this occasion, as is well known, Leibniz stated that “one such cognition I am accustomed to calling blind or also symbolic; we make use of the same in algebra and arithmetic, indeed virtually everywhere” (Meditationes de Cognitione, veritate et ideis, A VI, 4, 587 ; Engl. Transl. M. Wertz. My emphasis). On the fact that symbols or ‘caracters’ “represent ideas in imagination”, see De mente, de universe, de Deo (1675; A VI, 3, 463).
On Proclus’ singular position regarding mathematical imagination, see O’Meara (1989).
Barozzi (1560).
On Proclus’ projectionism see O’Meara (1989) and Mueller (1987). The locus classicus is the passage from the Second Prologue stating that “what projects the images is the understanding; the source of what is projected is the form in the understanding; and what they are projected in is the ‘passive nous’ [scil. imagination]” (in Eucl. 56).
On the material aspect of imagination, see In Eucl. 51: “If we assume two classes of things that participate in the universal, namely, sense objects and objects that have existence in the imagination (for matter likewise is twofold, as Aristotle somewhere says: the matter of things tied to sensation and the matter of imagined objects)”.
“The imagination, occupying the central position in the scale of knowing, is moved by itself to put forth what it knows, but because it is not outside the body, when it draws its objects out of the undivided center of its life, it expresses them in the medium of division, extension, and figure” (In Eucl. 52. My emphasis).
See in particular In Eucl. 12–15.
The Greek term for ‘medium’ here is hupodochè, a term playing a central role in platonic philosophy since the Timaeus and often times translated by ‘receptacle’.
Rule XII: “Finally we must make use of all the aids which intellect, imagination, sense-perception, and memory afford in order, firstly, to intuit simple propositions distinctly; secondly, to combine correctly the matters under investigation with what we already know, so that they too may be known; and thirdly, to find out what things should be compared with each other so that we make the most thorough use of all our human powers.” (CSM I, 39).
AT X, 416; CSM I, 42 (translation modified).
See the quote from Rule XIV below where Descartes says that imagination is “nothing but a real body with a real extension and shape” (verum corpus reale extensum et figuratum, AT X 441).
One could object that there is no ambivalence here and that imaginatio and phantasia are simply two different things (one spiritual, the other corporeal). This would not touch the heart of the difficulty which is to understand how the “purely spiritual” power can “form” ideas in the corporeal medium. But, at any rate, the distinction is not as strong as it seems. The two terms are introduced as equivalent (in phantasia vel imaginatione veluti in cera fomandas, AT X, 414, 18–19) and Descartes talks indifferently of ingenium as forming images in fantasy (in phantasia) or in imagination (imaginatio occupatur). For passages in which Descartes uses the two terms equivalently, see AT X, 416, 19–20; 441, 8; 449, 19. On the fact that “imagination” is in Descartes an ambivalent term, because it has an “intellectual” and a “corporeal” side, see AT III, 361.
Compare In Eucl. 52 and AT X, 415, 25–27.
AT X, 378–379; CSM I, 20 : “Aware how slender my powers are, I have resolved in my search for knowledge of things to adhere unswervingly to a definite order, always starting with the simplest and easiest things and never going beyond them till there seems to be nothing further which is worth achieving where they are concerned. Up to now, therefore, I have devoted all my energies to this universal mathematics, so that I think I shall be able in due course to tackle the somewhat more advanced sciences, without my efforts being premature. But before I embark on this task I shall try to bring together and arrange in an orderly manner whatever I thought noteworthy in my previous studies, so that when old age dims my memory I can readily recall it hereafter, if I need to, by consulting this book, and so that, having disburdened my memory, I can henceforth devote my mind more freely to what remains”. One could note in passing that mathesis universalis is not here a program, or a reform of mathematics, but one of these “simplest and easiest things” which has to be studied at first. This is in accordance with the description in the previous part of the Rule where Descartes says that this science should not be described with a foreign name, but with a venerable and widely received one (non ascititio vocabulo, sed jam inveterato atque usu recepto).
In the previous development, Descartes has explained how any problem could be expressed as a relation between magnitudes.
“Toute ma physique n’est autre que géométrie” (A Mersenne, 27 juillet 1638, AT II, 268).
Amongst others, this view was strongly expressed by Brunschvicg in “ Mathématique et Métaphysique chez Descartes ” [1927], in Brunschvicg (1951, pp. 11–54).
For a more detailed description of Descartes mathematical practice at the time, see Rabouin (2016).
On the fact that Proclus often time criticizes the use of imagination, see Claessens (2012). Descartes criticized the role of imagination already in the Regulae, just before mentioning mathesis universalis: “there is nothing more futile than devoting our energies to those superficial proofs which are discovered more through chance than method and which have more to do with our eyes and imagination than our intellect; for the outcome of this is that, in a way, we get out of the habit of using our reason” (AT X, 375).
The last three rules (XIX–XXI) are only known by their titles.
This is the point made by Macbeth (2014), for example p. 141.
I am referring here, in particular, to the seminal study by Bos (2001). On the auxiliary role of algebra and the importance of geometric construction, see in particular, Bos (2001, p. 397): “Each of the special algebraic techniques he explained in the Geometry had its purpose within the geometrical rationale of the book and was not developed further than necessary for that purpose. We may therefore characterize Descartes’ algebra as subservient to geometry, more precisely to the canon of construction that Descartes elaborated in order to solve ‘all the problems of geometry”’.
Descartes’ expression in AT VI, 390.
AT VI, 389–390.
On these two examples, see Maronne (2007, pp. 164–165).
This point is emphasized in Bos (2001). Although Book III of La Géométrie deals with the theory of algebraic equations, it is entitled: “Façon générale pour construire tous les problèmes solides, réduits à une équation de trois ou quatre dimensions” (AT VI, 464).
See, for an early formulation of this interpretation, the influential paper by Léon Brunschvicg quoted in note 49.
For a more detailed analysis on the role of imagination in Descartes’ Géométrie, see Araujo Silva (2008).
Let me recall, for example, a famous passage in which Descartes considers that mathematical objects are products of the thought although they are not fabricated by us, because they have “their own and immutable natures”: “I find within me countless ideas of certain things, that, even if perhaps they do not exist anywhere outside me, still cannot be said to be nothing. And although, in a sense, I think them at will, nevertheless they are not something I have fabricated; rather they have their own true and immutable natures. For example, when I imagine a triangle...” (triangulum imaginor, AT VII, 64; transl. Ariew 64).
See AT VII, 325 and 445.
Letter to Elisabeth, June 28 1643 (AT III, 692; transl. CSM III, 227). My emphasis.
See also AT II, 622: “la partie de l’esprit qui aide le plus aux mathématiques, à savoir l’imagination, nuit plus qu’elle ne sert pour les spéculations métaphysiques” (To Mersenne, 13 November 1639. My emphasis).
For a parallel discussion in Proclus, see Rabouin (2015, pp. 108–111).
“If I want to think about a chiliagon, I certainly understand that it is a figure consisting of a thousand sides, just as well as I understand that a triangle is a figure consisting of three sides, yet I do not imagine those thousand sides in the same way, or envisage them as if they were present. And although in that case, because of force of habit I always imagine something whenever I think about a corporeal thing, I may perchance represent to myself some figure in a confused fashion, nevertheless this figure is obviously not a chiliagon. For this figure is really no different from the figure I would represent to myself, were I thinking of a myriagon or any other figure with a large number of sides. Nor is this figure of any help in knowing the properties that differentiate a chiliagon from other polygons. But if the figure in question is a pentagon, I surely can understand its figure, just as was the case with the chiliagon, without the help of my imagination. But I can also imagine a pentagon by turning the mind’s eye both to its five sides and at the same time to the area bounded by those sides” (AT VII, 72; transl. R. Ariew in Descartes 2000, p. 132). See also Descartes and Burman (1648), AT V, 162–163.
“I observe one great difference between these three kinds of notions. The soul is conceived only by the pure intellect; body (i.e. extension, shapes and motions) can likewise be known by the intellect alone, but much better by the intellect aided by the imagination; and finally what belongs to the union of the soul and the body is known only obscurely by the intellect alone or even by the intellect aided by the imagination, but it is known very clearly by the senses” (To Elisabeth, 28 June 1643, AT III, 691–692; CSM III, 226–227).
AT VII, 73; transl. Ariew (2000, p. 133).
There is also no evolution as regards the other part of the alternative (imagination receiving figures from the senses), see Rule XII AT X, 414.
See La Géométrie II, AT VI, 416–418.
This is the very core of Leibniz’ point in his Elementa nova matheseos universalis, where the comparatio aequationum is presented as a perfect example of the fact that we need an analysis of “forms” and of “similarity” in algebra as we do in geometry (A VI, 4, 516).
Grosholz and Breger (2000).
Mancosu et al. (2005).
J.-P. Van Bendengem, “The Impact of the Philosophy of Mathematical Practice on the Philosophy of Mathematics”, in Soler, Zwart, Lynch and Israel-Jost (eds.) (2014, pp. 215–226).
Netz (1999).
On could add here that, according to experts, there are very few errors in ancient Greek Geometry in general, and certainly less than in Bourbaki for example.
See Netz (1999, pp. 23–24).
The proposition asks “to construct an equilateral triangle on a given finite straight line” and make use of two circles, with the center taken as one extremity of the given segment and the radius taken as the given segment.
Miller (2001), which led to the book Miller (2007). Recently, Miller has critized Mumma’s formulation for being unsound (Miller 2012). It is not yet clear whether or not the problems raised by Miller can be fixed in a revised version of Mumma’s original system (“EU”), but in any case, the criticism does not apply to the system Avigad et al. (2009), which also relies on Manders’ distinction between exact and coexact attribution (whereas Miller’s system relies on a topological formulation).
Interestingly enough, it does not apply so immediately to the Kantian version of the role of imagination because it adds a strong constructive requisite in the use of geometrical “schemes”. Although Descartes and Proclus also emphasize the role of construction, their attention to mathematical proofs, and in particular to geometrical analyses, make them more aware of cases in which diagrams do not correspond to constructions. On this point, see Rabouin (2015). On the fact that a too strict constructivism is incompatible with a faithful formalization of Euclid’s proof, see the very interesting paper by Mumma (2012).
Similar attitude in Netz about ‘ontology’, but from an historical perspective: “The lettered diagram supplies a universe of discourse. Speaking of their diagrams, Greek mathematicians need not speak about their ontological principles. This is a characteristic feature of Greek mathematics. Proofs were done at an object-level, other questions being pushed aside. One went directly to diagrams, did the dirty work, and, when asked what the ontology behind it was, one mumbled something about the weather and went back to work. (...) There is a certain singlemindedness about Greek mathematics, a deliberate choice to do mathematics and nothing else. That this was at all possible is partly explicable through the role of the diagram, which acted, effectively, as a substitute for ontology” Netz (1999, p. 57).
“La façon de démontrer qui réduit à l’impossible, et qui est la moins estimée et la moins ingénieuse de toutes” (To Mersenne, January 1638, AT I, 490).
For a more detailed argument on that matter, see Rabouin (2015). One simple example of this kind of reasoning with case distinctions has been presented above with Descartes’ method for normals. One draws a general situation with a circle intersecting a curve, which can be expressed by a quadratic equation, and one then examines how the roots behave when one imagines the points of intersection getting closer and closer until joining in one.
Hilbert (1950, theorem 10, p. 11).
For a more detail account of the argument, I refer to Rabouin (2017).
It is no coincidence that Manders puts as epigraph to his paper the following quote by Leibniz: “[Geometrical] figures must also be regarded as characters, for the circle described on paper is not a true circle and need not be; it is enough that we take it for a circle”. One could also think here of the similar claim made by Hilbert: “arithmetical signs are written diagrams, and geometrical diagrams are drawn formulas” (Hilbert 1935).
On this question, see also Grosholz (2007).
Hutchins (2005).
Descartes did not talk explicitly about ‘projection’, but, as we have seen, he emphasized the ‘formative’ activity of the mind on the material proxy, as opposed to standard theory of abstraction (which he extensively discusses and criticizes in Rule XIV).
Frake (1985).
“The wind rose is an ancient schema that, for most of its history and in most places, had nothing in particular to do with representing knowledge of the tides. It was used, first and foremost, as a way to express direction” (Frake 1985, p. 264).
Hutchins (2005, p. 1571).
This conclusive section (6.4) is entitled by Hutchins: Imaginary material anchors (p. 1575).
His section 4.1 is entitled: “4.1 Euclidean diagrams: artifacts of control or semantics?”
See, for example, the paper by Marco Panza in which he argues in detail that Descartes’ Geometry can be seen as a conservative extension of Euclid’s geometry (Panza 2011). See also the work undertaken by Grosholz (2007) which, beginning with Early modern times, also deals with examples taken from contemporary mathematics.
To take up on another beautiful Leibnizian expression: “ Ce qui a fait qu’il a été plus aisé de raisonner démonstrativement en mathématiques, c’est en bonne partie parce que l’expérience peut y garantir le raisonnement à tout moment, comme il arrive aussi dans les figures du syllogisme. Mais dans la métaphysique et dans la morale ce parallélisme des raisons et des expériences ne se trouve plus ” (Nouveaux Essais sur l’entendement humain, IV, 2, 12; A VI, 6, 371, my emphasis).
There are, however, other good reasons to call contemporary mathematical practice ‘conceptual’, especially if this view is related to a new coupling between imagination and concepts. This point is the central thesis of Macbeth (2014), based on a close study of Frege’s idea of a Begriffsschrift.
I cannot resist here quoting the warning which Descartes expressed to the same Elisabeth in his letter from June 28th 1643: “I believe that it is very necessary to have properly understood, once in a lifetime, the principles of metaphysics, since they are what gives us the knowledge of God and of our soul. But I think also that it would be very harmful to occupy one’s intellect frequently in meditating upon them, since this would impede it from devoting itself to the functions of the imagination and the senses. I think the best thing is to content oneself with keeping in one’s memory and one’s belief the conclusions which one has once drawn from them, and then employ the rest of one’s study time to thoughts in which the intellect co-operates with the imagination and the senses” (AT III, 694; transl. CSM III, 228. My emphasis).
References
Arana, A. (2016). Imagination in mathematics. In A. Kind (Ed.), Routledge handbook of the philosophy of imagination (pp. 463–476). Routledge: Taylor & Francis Group.
Araujo Silva, M. (2008). L’imagination dans la géométrie de Descartes. In D. Descotes et & M. Serfati (Eds.), Mathématiciens français du XVIIe siècle : Descartes, Fermat, Pascal (pp. 69–128). Clermont-Ferrand: Presses Universitaires Blaise Pascal.
Arndt, H. W. (1971). Methodo scientifica pertractatum. Berlin: W. de Gruyter.
Avigad, J., Mumma, J., & Dean, R. (2009). A formal system for Euclid’s elements. Review of Symbolic Logic, 2(4), 700–768.
Barozzi, F. (1560). Procli Diadochi Lycii philosophi platonici ac mathematici probatissimi in primum Euclidis Elementorum librum Commentariorum ad universam mathematicam disciplinam principium eruditionis tradentium libri III. Padova: G. Perchacinus.
Bos, H. J. (2001). Redefining geometrical exactness: Descartes’ transformation of the early modern concept of construction. New York: Springer.
Brunschvicg, L. (1951). Écrits Philosophiques. Paris: PUF.
Carson, E. (2009). Hintikka on Kant’s mathematical method. Revue internationale de philosophie, 250, 435–449.
Claessens, G. (2012). Proclus: Imagination as a symptom. Ancient Philosophy, 32(2), 394–395.
Corfield, D. (2003). Towards a philosophy of real mathematics. Cambridge: Cambridge University Press.
Crapulli, G. (1969). Mathesis universalis. Genesi di una idea nel XVI secolo. Rome: Dell’Ateneo.
De Pace, A. (1993). Le Matematiche e il mondo. Milan: Francoangeli.
Descartes, R. (1964–1974). Oeuvres de Descartes. publiées par C. Adam et P. Tannery, 11 vol., nouvelle présentation en coédition avec le CNRS, Paris, Vrin.
Descartes, R. (1966). Regulae ad directionem ingenii (edited by Crapulli). La Haye: M. Nijhoff.
Descartes, R. (1988). The philosophical writings of Descartes (J. Cottingham, R. Stoothoff, & D. Murdoch, Trans.). Cambridge: Cambridge University Press.
Descartes, R. (2007). The correspondence between Princess Elisabeth of Bohemia and René Descartes (L. Shapiro, Trans.). University of Chicago Press.
Descartes, R. (2000). René Descartes. Philosophical essays and correspondence. Edited with an introduction by R. Ariew. Indianapolis: Hackett Publishing Company.
Domski, M. (2012). Newton and Proclus: Geometry, imagination, and knowing space. The Southern Journal of Philosophy, 50/3, 389–413.
Fauconnier, G. (1997). Mappings in thought and language. Cambridge: Cambridge University Press.
Fauconnier, G., & Turner, M. (2002). The way we think. New York: Basic Books.
Frake, C. (1985). Cognitive maps of time and tide among medieval seafarers. Man, 20, 254–270.
Grosholz, E. (2007). Representation and productive ambiguity in mathematics and the sciences. Oxford: Oxford University Press.
Grosholz, E., & Breger, H. (Eds.). (2000). The growth of mathematical knowledge. Dordrecht: Kluwer Academic Publishers.
Hilbert, D. (1935). “Mathematische Probleme”, Gesammelte Abhandlungen (Vol. III, pp. 290–329). Berlin: Springer.
Hilbert, D. (1950). The Foundations of Geometry. La Salle: Open Court.
Hintikka, J. (1981). Kant’s theory of mathematics revisited. Philosophical Topics, 12(2), 201–215.
Husserl, E. (1969). Formal and transcendental logic (D. Cairns, Trans.). Amsterdam: M. Nijhoff.
Husserl, E. (1970). The crisis of European sciences and transcendental phenomenology (D. Carr, Trans.). Northwestern University Press.
Hutchins, E. (2005). Material anchors for conceptual blends. Journal of Pragmatics, 37, 1555–1577.
Israel, G. (1998). Des Regulae à la Géométrie. Revue d’histoire des Sciences, 51(2/3), 183–236.
Jullien, V. (1996). Descartes : La Géométrie de 1637. Paris: PUF.
Lachterman, D. (1989). The ethics of geometry: A genealogy of modernity. New York: Routledge.
Leibniz, G. W. (1986). Philosophical papers and letters (2nd ed.), (L. E. Loemker, Trans.). Dordrecht: D. Reidel.
Leibniz, G. W. (1999). Sämtliche Schriften und Briefe, herausgegeben von der deutschen Akademie der Wissenschaften zu Berlin, series VI, tome 4.
Macbeth, D. (2014). Realizing reason: A narrative of truth and knowing. Oxford: Oxford University Press.
Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford: Oxford University Press.
Mancosu, P. (1999). Literature survey: Recent publications in the history and philosophy of mathematics from the Renaissance to Berkeley. Metascience, 8(1), 102–124.
Mancosu, P. (Ed.). (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.
Mancosu, P., Jørgensen, K., & Pedersen, S. A. (Eds.). (2005). Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer.
Manders, K. (2008a). Diagram-based geometric practice. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 65–79). Oxford: Oxford University Press.
Manders, K. (2008b). The Euclidean diagram. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 80–133). Oxford: Oxford University Press.
Maronne, S. (2007). La théorie des courbes et des équations dans la Géométrie cartésienne: 1637–1661. Philosophie: Université Paris-Diderot - Paris VII. https://tel.archives-ouvertes.fr/tel-00203094. Accessed September 2016.
Miller, N. (2001). A diagrammatic formal system for Euclidean geometry. Ph.D. Dissertation, Cornell University.
Miller, N. (2007). Euclid and his twentieth century rivals: Diagrams in the logic of Euclidean geometry. Stanford: Centre for the Study of Language and Information Press.
Miller, N. (2012). On the Inconsistency of Mumma’s Eu. Notre Dame Journal of Formal Logic, 53(1), 27–52.
Molland, G. (1976). Shifting the foundations: Descartes’ transformation of ancient geometry. Historia Mathematica, 3, 21–49.
Mueller, I. (1987). Mathematics and philosophy in Proclus’ commentary on book I of Euclid’s elements. In J. Pépin & H. D. Saffrey (Eds.), Proclus lecteur et interprète des anciens (pp. 305–318). Paris: Edition du CNRS.
Mumma, J. (2006). Intuition formalized: Ancient and modern methods of proof in elementary Euclidean geometry. Ph.D. Dissertation, Carnegie Mellon University.
Mumma, J. (2010). Proofs, pictures, and Euclid. Synthese, 175(2), 255–287.
Mumma, J. (2012). Constructive geometrical reasoning and diagrams. Synthese, 186, 103–119.
Netz, R. (1999). The shaping of deduction in Greek mathematics: A study in cognitive history. Cambridge: Cambridge University Press.
Nikulin, D. (2002). Imagination and geometry: Ontology, natural philosophy and mathematics in Plotinus, Proclus and Descartes. Aldershot: Ashgate.
O’Meara, D. J. (1989). Pythagoras revived. Oxford: Clarendon Press.
Panza, M. (2011). Rethinking geometrical exactness. Historia Mathematica, 38(1), 42–95.
Parsons, C. (1992). The transcendental aesthetic. In P. Guyer (Ed.), The Cambridge companion to Kant. Cambridge: Cambridge University Press.
Perera, B. (Benedictus Pereirus) (1576). De communibus omnium rerum naturalium principiis et affectionibus. Rome: F. Zanettus et B. Tosius.
Piccolomini, A. (1547). In Mechanicas Quaestiones Aristotelis, Paraphrasis paulo quidem plenior. Rome: A. Bladus Asulanus.
Proclus. (1533). Commentarium Procli editio prima quae Simonis Grynaei opera addita est Euclidis elementis graece editis. Bâle: J. Hervagius.
Proclus. (1873). Procli Diadochi in primum Euclidis elementorum librum commentarii ex recognitione Godofredi Friedlein. Teubner: Leipzig. (reprint Olms, 1992).
Rabouin, D. (2009). Mathesis universalis. L’idée de ‘mathématique universelle’ d’Aristote à Descartes. Paris: PUF.
Rabouin, D. (2015). Proclus’ conception of geometric space and its actuality. In V. De Risi (Ed.), Mathematizing space. The objects of geometry from antiquity to the early modern age, Basel, Birkhäuser (pp. 105–142). Series: Trends in the History of Science.
Rabouin, D. (2016). Mathesis universalis et algèbre générale dans les Regulae ad directionem ingenii de Descartes. Revue d’histoire des sciences, 69/2, 259–309.
Rabouin, D. (2017). Les mathématiques comme logique de l’imagination. Une proposition leibnizienne et son actualité. Bulletin d’analyse phénoménologique, XIII(2), 222–251. (Actes 10 : L’acte d’imagination : Approches phénoménologiques).
Sepper, D. L. (1996). Descartes’ s imagination : Proportion, images, and the activity of thinking. Berkeley: University of California Press.
Sepper, D. L. (2016). Descartes. In A. Kind (Ed.), The Routledge handbook of philosophy of imagination. London: Routledge.
Tymoczko, T. (1998). New directions in the philosophy of mathematics. Princeton: Princeton University Press.
Van Bendengem, J.-P. (2014). The impact of the philosophy of mathematical practice on the philosophy of mathematics. In L. Soler, S. Zwart, M. Lynch, & V. Israel-Jost (Eds.), Science after the practice turn in the philosophy, history, and social studies of science (pp. 215–226). London: Routledge.
Van de Pitte, F. (1991). Descartes’ mathesis universalis. In G. J. D. Moyal (Ed.), René Descartes : Critical assesments (pp. 61–79). London: Routledge.
Van Kerkhove, B., & Van Bendegem, J.-P. (ed.) (2002). Perspectives on mathematical practices. Special issue of Logique et Analyse 45, 179–180.
Van Kerkhove, B., & Van Bendegem, J.-P. (Eds.). (2007). Perspectives on mathematical practices: Bringing together philosophy of mathematics, sociology of mathematics, and mathematics education. Dordrecht: Springer.
Van Kerkhove, B., & Van Bendegem, J.-P. (Eds.). (2009). New perspectives on mathematical practices. Essays in philosophy and history of mathematics. Singapore: World Scientific.
Van Kerkhove, B., Van Bendegem, J.-P., & De Vuyst, J. (ed.) (2010). Philosophical perspectives on mathematical practice. College Publications.
Van Roomen, A. (1597). In Archimedis Circuli dimensionem expositio et analysis. Apologia pro Archimede. Genève.
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Rabouin, D. Logic of imagination. Echoes of Cartesian epistemology in contemporary philosophy of mathematics and beyond. Synthese 195, 4751–4783 (2018). https://doi.org/10.1007/s11229-017-1562-1
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DOI: https://doi.org/10.1007/s11229-017-1562-1