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‘Mathesis Universalis’ in the Sixteenth Century

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Descartes’s Mathematical Thought

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 237))

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Abstract

It was through Proclus’s Commentary on the First Book of Euclid’s Elements, among others, that a novel mathematical idea entered the intellectual world of the sixteenth century. It is not too much to say that there has been no impact greater than Proclus regarding the status of mathematical sciences among the philosophies of mathematics in early modern Europe. We have seen in Chapter 2, § 2 that Clavius was greatly inspired by Proclus’s philosophy of mathematics. By attempting to graft, as it were, the mathematical thought of Proclus on the trunk of the scholastic-Aristotelian scheme of learning, the Jesuit mathematician promoted the study of mathematics. Descartes is conjectured to have acquired the thought of Proclus through Clavius or others. Proclus also played a crucial role in spreading the idea that there is a substantial mathematical discipline prior and superior to arithmetic and geometry. This idea was useful to convince thinkers of the sixteenth century that there existed a mathematical discipline of ‘mathesis universalis’.

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References

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  18. See I. Mueller, “Forward to the 1992 Edition” (n. 1), p. xxix. Mueller uses “universal mathematics” deliberately for our “common mathematics.” This use is, I think, legitimate.

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  19. Aristotle states in Book I of Topics (101a37–101b4): “It [the treatise] is a further use in relation to the principles used in the several sciences. For it is impossible to discuss them at all from the principles proper to the particular science in hand, seeing that the principles are primitive in relation to everything else: it is through reputable opinions about them that these have to be discussed, and this task belongs properly, or most appropriately, to dialectic; for dialectic is a process of criticism wherein lies the path to the principles of all inquiries.” For Plato’s influence on Aristotle, see H. D. P. Lee, “Geometrical Method and Aristotle’s Account of First Principles,” The Classical Quarterly, XXIX (1935), pp. 113–124, esp. pp. 119–124.

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  20. See J. D. G. Evans, Aristotle’s Concept of Dialectic (Cambridge, 1977), esp. “Summary: Aristotle and Plato on Dialectic and Science,” pp. 49–52. “Aristotle preserves the Platonic idea of dialectic as unrestricted in its scope. On the other hand this lack of restriction is for Aristotle an indication of the unscientific character of dialectic, whereas for Plato it had been an indication that dialectic was the only true science.” (p. 50.) Similarities and dissimilarities in general between Plato’s and Aristotle’s philosophy are discussed by G. E. L. Owen in his excellent paper “The Platonism of Aristotle,” in Logic, Science and Dialectic: Collected Papers in Greek Philosophy, ed. Martha Nussbaum (Ithaca, 1986), pp. 200–220. On mathematics and dialectic, consult especially pp. 211–216.

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  22. See Simon Grynaeus’s “Praefatio” to his Greek editio princeps of Euclid’s Elements (n. Ch. 2, n. 51). An interesting discussion can be seen in E. Kessler, “Clavius entre Proclus et Descartes” (Ch. 2, n. 16), pp. 290–294. Cf. H. Schilling, Die Geschichte der axiomatischen Methode im 16. and beginnenden 17. Jahrhundert (Hildesheim, 1969), p. 37.

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  23. See Stanislas Breton, Philosophie et mathématique chez Proclus (Paris, 1969), p. 89; Annick Charles-Saget, L’Architecture du divin: Mathématique et philosophie chez Plotin et Proclus (Paris, 1982), pp. 195, 205, and 295. On the other hand, Nicolai Hartmann doesn’s seem to see the concept of `mathesis universalis’ in Proclus’s Commentary, for, according to him even the theory of proportion is not common to both geometry and arithmetic. Nicolai Hartmann, Des Proklus Diadochus philosophische Anfangsgründe der Mathematik nach den ersten zwei Büchern des Euklidkommentars (Gießen, 1909), p. 15: “Die Geometrie stellt den Grundsatz der Proportion für dei Größen, die Arithmetik für die Zahlen auf, jede aus ihren eigenen Mitteln und für ihre eigenen Zwecke.” Cf. Geneviève de Pesloüan’s French translation in Breton, Ibid., p. 198.

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  30. Ibid., p. 106: “[…] in sola mathematica sunt demonstrationes potissimae propter per causam necessariam.” Cf. Burke’s translation, p. 124. See also Communia mathematica, p. 15: “[…] cum sola mathematica sit vere demonstrativa secundum demonstrationem potissimam

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  31. Communia mathematica, p. 63: “Methaphysica est prior mathematica et magis abstracta

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  32. Johannes Kepler, Gesammelte Werke, Bd. VI: Harmonise mundi (München, 1940). In “Personenregister” “Proklus” appears 34 times.

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  33. A. Crapulli, Mathesis universalis (Roma, 1969), p. 146. The quoted Latin passage is as follows (A. Romanus, Apologia pro Archimede, pp. 22–23): “Geometriae and Arithmeticae communem esse scientiam, gum quantitatem generaliter vti mensurabilem considerat […] ad quam spectarent affectiones communes omnibus quantitatibus […] non abstractis tantum vt numeris and magnitudinibus, sed concretis etiam, vti temporibus, sonis, vocibus, locis, motibus, potentiis […] propositiones eas quæ spectant ad analogias […] ad scientiam aliquam vniuersalem iure merito pertinere existimandum est.” We have referred to Van Roomen’s original rather than what Crapulli cited.

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  34. See in detail Crapulli, p. 150.

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  35. Piccolomini, Op. cit. (Ch. 2, n. 18), f. LXXXXVIIv: “cuiusdam facultas communis ad Geometriam, and Arithmeticam.”

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  36. Dasypodius, Analyseis Geometricee sex librorum Euclidis (Argentinensi, 1566), f. LXXXVIv: “[S]unt etiam quæ cum sint communia ex geometria in arithmeticam: et ex hac in illam transferuntur: uel etiam utrinque se habent: et peræque in geometria, atque in arithmetica considerantur: imo in cæteris quoque disciplinis adhiberi possunt: quia ex illa uniuersali µaBr)µareni) desumuntur: ut sunt ?vaaaaryal,aal AvavTpocpaì T(DI/ Aory’uiv: Q4Jv0É’QELç,bcacp€vECç et quæ huius sunt generis.”

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  37. /bid., f. XCVIIr: “His quintus liber peculiarem dectrinam instituit de proportionibus: qu diffusæ sunt per uniuersam rerum naturam, non ad geometriam tantum aut arithmeti-cam pertinent, sed ad alias scientias atque idcirco potius referri debet, ad uniuersalem illam disciplinam mathematicam, qu reliquas omnes tanquam species sub se comprehendit.”

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  38. Benedictus Pererius, De Cornmunibus omnium rerum naturalium principijs f4 affectionibus, libri gvindecim (Rome, 1576), p. 34: “Quaemadmodum non est dubium quin sit aliqua scientia Mathematica communis, quae debeat speculari affectiones communes magnitudini and numero, quae tamen scientia, à Mathematicis non numeratur distincta à Geometria and Arithmetica.” Crapulli mistakenly refers to this statement on p. 37 instead of p. 34. See Crapulli, p. 95. Notice that Pereira admitted `scientia Mathematica communis’ while still claiming that “the mathematical doctrine is not properly a science” (dottrina Mathematica non est propriè scientia) (Pereira, p. 24).

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  39. A. Romanus, Apologia, p. 23. See our discussion on pp. 354–356.

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  40. Crapulli, pp. 125–143.

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  41. A. Romanus, Apologia, p. 23. Cf. Philippe Gilbert, “Les sciences exactes dans l’ancienne université de Louvain,” Revue des questions scientifiques, XVI (1884), pp. 438–453, esp. p. 449. Figure 7.1 shows the Latin original. The Komaba Library of the University of Tokyo possesses a copy of this rare book.

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  42. This is the point Crapulli emphasizes. See Crapulli, p. 8.

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  44. Romanus, Vniversee mathesis idea (Würzburg, 1602), p. 3.

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  45. /bid.,pp. 6–7: “Demonstrationes Mathematica an sint potissimæ, apud nonnullos controversum est; Benedictus Pererius sentit potissimas non esse, cùm non progrediantur ex principijs, qva sint causa conclusionis, uti pluribus ostendit exemplis ex elementis Geometri-cis Euclidis desumptis. Verùm qvicqvid sit, (hoc est sive potissimæ sint, sive non) hoc unum indubitatum est apud omnes, demonstrationes mathematicas esse certissimas inter omnes demonstrationes qvarumcumqve facultatum. In rebus sanè naturalibus and sensibilibus dicere ferè semper cogimur (Aristotele teste) hoc vel illud fieri 2ç Éirì rò 7roÀv, vel etiam g àv µr1 Tì.p roöI(et; at contrà (2. Metaphys. 3) Mathematicarum scientiarum summa est certitudo excedens omnem aliarum facultatum certitudinem.” Pererius, Op. cit. (n. 38), Lib. I, Cap. XII, “Scientia speculativam non dici vniuovè de Mathematicis disciplinis and alijs, quoniam doctrina Mathematica non est propriè scientia,” pp. 24–26. Aristotle’s statement in the Metaphysics, II, 3 (995a14–17) can be seen in Ch. 2, n. 19.

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  46. /bid., p. 8: “Vnde à Metaphysica superatur, Physicam verò superat.”

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  47. /bid., p. 12: “Metaphysicae non modo vtilis sed and necessaria videtur.” Van Roomen continues to state the reason: “For the entrance to metaphysics opens to nobody but through the mathematical disciplines. Certainly if we should try to raise the power and sharpenness of our intellect without any mediation from sensible things, which the physicist considers, to things divided and separated from every sensible matter, which the metaphysician contemplates, we ourselves will be made blind just as it happens to one who is sent out of a certain dark cave in which he has been hidden long into the clearest light of the sun. And for this reason, before the intellect ascends from physical things, to which matter subjected to senses is joined, to metaphysical things, which are separated greatly from matter, in order not to be filled with the brightness of the latter it is necessary, earlier than that, to be accustomed with less abstract things as are considered by mathematicians in order that one can comprehend them easier. And therefore the Divine Plato rightly maintains that the mathematical disciplines raise the soul to the contemplation of divine things and sharpen the mind.” (p. 12.) “Nulli etenim ad Metaphysicam pater aditus, nisi per mathematicas disciplinas. Nam si à rebus sensibilibus, qvas Physicus considerat, ad res ab omni materia sensibili secretas, seiunctasqve qvas contemplatur Metaphysicus, vires, aciemqve nostri intellectus attollere absque ullo medio tentemus: nos metipsos excæcabimus, non secus, ac ei contingit, qvi è carcere aliqvo tenebricoso, in qvo diu latuit, in lucem Solis clarissimam emittitur. Qvamobrem, anteqvam à rebus physicis, qvæ materiæ sensibus obnoxiæ, sunt conjunctæ, ad res metaphysicas, qv æ sunt ab eadem maxime avulsæ, intellectus ascendat; necesse est, ne harum claritate offundatur, prius eum assvefieri rebus minùs abstractis, qvales à Mathematicis considerantur, ut faciliùs illas possit comprehendere. Qvocirca rectè Divinus Plato mathematicas disciplinas erigere animum, ad divinarum rerum contemplationum exacuere mentis aciem affirmat.”

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  48. Romanus, Ibid., p. 16: “Vniuersalis continet elementa motus violenti; diciturqve propriè Mechanica, Pappo Manganaria.” The term `Manganaria’ is used in Pappus’s Collection, VIII: “7) TÉxv71 T Jv µaryryavaplwv” (Hultsch, III, p. 1024, I. 14); “ars manganorum” (Commandino, f. 305v); “l’art des artificiers” (Ver Eecke, II, p. 810). Incidentally, Van Roomen wrote the title of Chapter XVI as “De Manganaria sive mechanaria proprie dicta,” probably by mistake. “Mechanaria” should be corrected to “mechanica”, of course.

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  49. P. 56: “Evthymetria est scientia quæ per radios visuales aptè directos, lineas rectas finitas metitur.”

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  50. See Heath, A History of Greek Mathematics, Vol. 1, p. 18.

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  51. P 17: “Svpputatrix Grzecis aoryuvruci dicta, est quæ beneficio canonum universalium, ex datis numeris rebus accommodatis quæsitum elicit.”

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  52. P. 17: “Posset non incommodè si vox recepta esset Arithmopraxia vocari. Supputatricem hanc à reliquis Mathematicis scientijs separare placuit, quod ea non peculiaris obietti tradat proprietates, verùm omnem numerum respiciat concretum; hoc est, rem quam vis numerabilem sine ulla numerorum proprietate, ita ut omnibus servire queat Matheseos partibus, imò omni ciuili usui. Quodque Logica est in universa Philosophia, idipsum Supputatrix censeri potest in Mathesi.”

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  53. /bid “Differt porrò Supputatrix ab Arithmetica, quod h2ec numerorum exhibeat proprietates demonstratione confirmatas, ilia verò sine ulla demonstratione, sine ulla numerorum proprietate, canones paucissimos exhibeat inveniendi and iudicandi, eosque generalissimos longè à Theorematibus and Problematibus Arithmetic2e diversos.”

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  54. P. 18: “Principia præter terminorum scienti propriorum explicationem vis habet alia; usus tarnen supputatricis requirit cognitionem proprietatum earum rerum quibus numeri applicantur.”

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  55. Ibid “Supputatrix ergo meritò omnibus mathematicis præmitti potest. Nam Supputatrix doceri sine ulla reliquarum Mathematicarum præexistenti cognitione.”

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  57. /bid., p. 53.

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  60. The first twenty chapters constituting the first part of De principiis ex mathesi desumendis may correspond to the previous Vnivers x mathesis idea. See Henri Bosmans, “Romain (Adrien)” (Ch. 5, n. 189), cols. 879–880.

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  62. ee “Principia praeter propria, etiam desumit, turn ex prima Mathesi, turn ex Arithmetic”.“ (p. 23) and ”Proximum in Mathesi locum post Arithmeticam obtinet.“ (p. 24)

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Sasaki, C. (2003). ‘Mathesis Universalis’ in the Sixteenth Century. In: Descartes’s Mathematical Thought. Boston Studies in the Philosophy of Science, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1225-5_9

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