Abstract
The notion of mathesis universalis appears in many of Edmund Husserl’s works, where it corresponds essentially to “a universal a priori ontology”. This paper has two purposes; one, largely exegetical, of clarifying how Husserl elaborates on Leibniz’ concept of mathesis universalis and associated notions like symbolic thinking and symbolic knowledge filtering them through the lesson of the so called “bohemian Leibniz”, Bernard Bolzano; another, more properly philosophical, of examining the role that the universal mathesis is allowed to play, and the space it occupies in Husserl’s intuition-based epistemology.
[L]a bonne caractéristique est.
une des plus grandes aides
de l’esprit humain ([A] good symbolism
is one the greatest aids to the human mind
Leibniz, N.E. IV, Ch.7, § 6)
[O]hne die Möglichkeit symbolischer […] Vorstellungen
gäbe es kein höheres Geistesleben, geschweige denn eine Wissenschaft
([W]ithout the possibility of symbolic representations […]
there would not exist a higher spiritual life, and even less science
Husserl, Zur Logik der Zeichen, PdA 349)
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- 1.
Cp. Piccolomini’s Commentarium de certitudine mathematicarum (1547), ch. 7; Dasypodius’ mathematical writings (1564a, b, 1571, 1593); van Roomen’s Apologia pro Archimede (1597). All these Authors, except for van Roomen, refer to Proclus’ Commentary on the First Book of Euclid’s Elements, especially to the first prologue of the Commentary, which deals with the mathematical sciences in general. In Proclo’s commentary there are many hints at a common mathematical discipline that shall precede all other mathematical disciplines and has, therefore, a more general character. Crapulli 1969 offers a very interesting investigation about the development of the idea of the mathesis in the XVI century.
- 2.
Before Descartes and Leibniz there were mainly two criteria for gathering together mathematical disciplines, namely, that they all had, in some way, quantity as their object (think of the disciplines traditionally unified in the Quadrivium) and an higher degree of certitude in proofs with respect to non-mathematical disciplines.
- 3.
Descartes Rule IV, AT.X, 372–379, here 374f. Actually it is a recurrent theme by Descartes that “all sciences are concatenated”, “that it is much more easier to apprehend them all together, then to separate one of them from the others” and focus solely on a unique one (Rule I, AT.X, 361). Hereto also cp. AT.X, 255: “[one single science] cannot be brought to perfection, without doing the same with the others”. Responsibility for translations from German, French and Latin is ours, even when we refer to, benefit from, or simply echo published translations.
- 4.
AT.X, 376.
- 5.
Pappus of Alexandria (290–350 c. AD) and Diophantus of Alexandria (201/215(?)-285–299(?) AD) were Alexandrian Greek Mathematicians.
- 6.
AT.X, 378.
- 7.
loc. cit.
- 8.
loc. cit.
- 9.
AT.X, 378–379.
- 10.
Descartes 1954.
- 11.
Gerolamo Cardano (1501–1576) was an Italian mathematician, physicians and astrologers. He occupies an important place in the history of Renaissance philosophy. He wrote more than 200 works in the most disparate fields, but especially, mathematics, medicine, philosophy and astrology. Among his works we remember his Artis magnae sive de regulis algebraicis liber unus (1545), commonly known as Ars magna, that draws the anger of Nicolò Tartaglia for having published the solution of third degree equations revealed to him by Tartaglia 6 years earlier, though under oath, not to reveal it.
- 12.
Scipione del Ferro (1465–1596) was an Italian mathematician who first discovered a method to solve the depressed cubic equations.
- 13.
Niccolo Fontana Tartaglia (1499/1500–1557) was an Italian mathematician famous for having been the first to translate Euclid and Archimedes into Italian as well as for his controversy with Cardano as to the solution of cubic equations.
- 14.
Raphael Bombelli (1526–1572) was an Italian mathematician. He is Author on a treatise on algebra (1572) and gave important contributions in the understanding of imaginary numbers.
- 15.
Cardano 2007.
- 16.
Descartes 1954.
- 17.
See Nahin 1998.
- 18.
Leibniz 1684.
- 19.
Berkeley 1948–57.
- 20.
Bonaventura Cavalieri (1598–1647) was a pupil of Galileo. He developed Galileo’s thoughts into a geometrical method and published in 1635 a work on the subject: Geometria Indivisilibus Continuorum Nova quadam Ratione Promota.
- 21.
Johannes Kepler (1571–1630) is one of the most representative figures in the Renaissance. He was strongly influenced by Copernicus and endorsed the Platonic conception that universe is ordered according to a pre-established mathematical plan. He his Author of many works, among them a Mysterium Cosmographicum (1596) and his Astronomia Nova (1609), in which we find him saying: “The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics” quoted after Pearcey & Thaxton 1994, 126.
- 22.
See Mancosu 1996.
- 23.
This comes out clearly in Leibniz’s practice of the calculus he invented, see Leibniz 1684 note 18.
- 24.
Leibniz 1678, quoted after Ariew & Gaber 1989, 236.
- 25.
See Belaval 1960.
- 26.
Leibniz 1684. Henceforth quoted as MED. MED is Leibniz’s first mature philosophical publication. It appeared in November 1684 in the Leipzig Journal Acta Eruditorum as Leibniz’s contribution to the famous Arnauld-Malebranche controversy, triggered by the publication of Arnauld’s Des vrais et des fausses idées in 1683, an attack on Malebranches’s philosophy.
- 27.
Henrich Scholz hypothesizes Trendelenburg as source. See Husserl’s and Scholz’s Anmerkungen in Frege’s 1964.
- 28.
Husserl 1891a.
- 29.
Husserl 1891b.
- 30.
Hereto cp. Chap. 6 in this volume.
- 31.
Husserl, Philosophie der Arithmetik. Mit ergänzenden Texten (1890–1901), HGW XII, 1–283. Henceforth cited as PdA. English translation cited as PoA.
- 32.
Husserl, Logische Untersuchungen II/1 (Tübingen: Max Niemeyer, 19937). Henceforth cited as LU. English translation cited as LI.
- 33.
Formal and Transcendental Logic [ed. 1929] (henceforth cited as FTL), 49 ff; Engl. transl. 56 ff.
- 34.
Hereto see Centrone 2010b.
- 35.
MED 585–586.
- 36.
Talk of “notio” as well as “nominal definition” below indicates that Leibniz conceives of “notiones” as abstract objects. The question is controversial, for this is against his official position presented in the short essay Quid sit Idea? (Leibniz 1677a), in which he maintains that an idea is a certain power of the mind. With “idea“, he writes there, we mean “something that is in our mind (aliquid, quod in mente nostra est)”: “The idea is [...] not a certain mental act, but rather a power, and thus we say that we have the idea of an object even if we don’t actually think of it but have the ability to do so on any given occasion. (Idea [...] non in quodam cogitandi actu, sed facul- tate consistit, et ideam rei habere dicimur, etsi de ea non cogitemus, modo data occasione de ea cogitare possimus.)” (Ak VI.4B, 1.370).
- 37.
MED 585.
- 38.
The bearer has changed! It is no longer a cognitio!
- 39.
The bearer has changed again!
- 40.
MED 585.
- 41.
MED 585–586 (our emphasis).
- 42.
See e.g. LU I, §20; LI 304–306 (Thought without intuition and the ‘surrogative function’ of signs).
- 43.
NE II, Ch. 9, §§ 8–10; and Ch. 29 that bears the title “Of Clear and Obscure, Distinct and Confused Ideas.”
- 44.
PdA 339; PoA 357.
- 45.
See also Husserl, Zur Logik der Zeichen (Semiotik), in: PdA 340–373. Here PdA p. 349f.
- 46.
Cp. Willard 1984, 26.
- 47.
“To [Brentano] I owe the deeper understanding of the vast significance of inauthentic presentations for our whole mental life; this is something which, so far as I can see, no one before him had grasped” (PdA 193; PoA 205). See also Husserl, Zur Logik der Zeichen (Semiotik): PdA, p. 340.
- 48.
PdA 193; PoA 201–202.
- 49.
LU IV, §45, 144; LI 785–786.
- 50.
PdA 194; PoA 205;
- 51.
PdA 193–194, 237–239; PoA 205–206, 247–240. See also Husserl, Zum Begriff der Operation (On the Concept of Operation), in: PdA 408–429. PoA, 385–408. Here: PdA, p. 418; PoA, p. 395.
- 52.
Leibniz 1666, 5.
- 53.
Leibniz 1677b, in: Ak. IV.4.A, 23–24. Also cp. Leibniz 1688?: “Characterem voco, notam visibilem cogitationes repraesentantem. Ars characteristica est ars ita formandi atque ordinandi characteres, ut referant cogitations, seu ut eam inter se habeant relationem, quam cogitationes inter se habent. Expressio est. aggregatum characterum rem quae exprimitur repraesentantium.”
- 54.
Trendelenburg 1856, 39f.
- 55.
Schröder 1890, 48; 49. Hereto cp. ch. 6, section 4 in this volume.
- 56.
See da Silva 201x.
- 57.
PdA 258; PoA 273 (italics in the original).
- 58.
Husserl, Aufsätze und Rezensionen, 393.
- 59.
loc. cit.
- 60.
Cp. Aristotle, Metaphysics.∆ 5, 1015a 34–36: “We say that that which cannot be otherwise is necessarily as it is. And from this sense of ‘necessary’ all the others are somehow derived.” Cp. Leibniz, Confessio philosophi (Fall 1672-Winter 1672/73): “Impossibile, quod possibile non est. Necessarium cuius oppositum impossibile est., Contingens cuius oppositum possibile est” (Ak VI.3127).
- 61.
Cp. e.g. Generales Inquisitiones de analysi notionum et veritatum (1686): “Necessaria autem proposition est […] cujus oppositam assumendo per resolutionem devenitur in contradictionem” (Ak VI.4A, 761).
- 62.
In De Interpretation 13 he defines “possible” (“endekomenon”, “dunaton”) as “that which is not impossible”, In An. Pr. 13bis and 22 he takes “endekomenon” to mean “neither possible nor impossible”, that is the characterization we usually give nowadays of “contingent”.
- 63.
Cp. Leibniz, Definitiones: ens, possibile, existens (1687–96?) (Ak VI.4A, 867). Also cp. Definitiones: terminus vel aliquid, nihil (1688–89) (Ak VI.4A, 936).
- 64.
See Couturat 1901, 283–322.
- 65.
PoA 277; PdA 262. Hereto cp. Centrone 2010a, 45ff.
- 66.
PR 219–222 (§ 60 Anknüpfungen an Leibniz/ Links with Leibniz), Pre 138–141. Henceforth: PR = Husserl, Logische Untersuchungen I, Prolegomena zur reinen Logik, Tübingen 1993; PRe = English translation thereof, in: Logical Investigations, London 1970, Vol. I, 51–247.
- 67.
loc. cit.
- 68.
loc. cit.
- 69.
LV ‘06/07, 54–55.
- 70.
LV ‘06/07, 91.
- 71.
LV ‘06/07, 104.
- 72.
LV ‘06/07, 376.
- 73.
Ideen I, §7, 16ff. (o.p.).
- 74.
EU 1–2; Engl. transl. 11.
- 75.
FTL 65; Engl. transl. 73–74.
- 76.
FTL 122; Engl. transl. 138.
- 77.
FTL 127; Engl. transl. 143.
- 78.
FTL 123; Engl. transl. 138.
- 79.
Krisis §9 g.
- 80.
Husserl, Das Imaginäre in der Mathematik. I: Zu einem Vortrag in der mathematischen Gesellschaft in Göttingen 1901; in: PdA 430–451; PoA 409–452.
- 81.
See also Drei Studien zur Definitheit und Erweiterung eines Axiomensystems, in: PdA 452–469, PoA 432–438 & 453–464.
- 82.
PdA 430; PoA 409–410.
- 83.
We borrow here terminology from Casari 2004.
- 84.
Webb 1980, xii.
- 85.
Hereto cp. Casari 2004.
- 86.
LV 06/07, 55.
- 87.
loc.cit. (quotation marks added).
- 88.
loc.cit.
- 89.
LV 06/07, 55 (quotation marks added).
- 90.
Cp. LV 06/07, 59 ff.
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Centrone, S., Da Silva, J.J. (2017). Husserl and Leibniz: Notes on the Mathesis Universalis . In: Centrone, S. (eds) Essays on Husserl's Logic and Philosophy of Mathematics. Synthese Library, vol 384. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1132-4_1
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