Abstract
The aim of this paper is to explore some aspects of the connection between mathematics and philosophy in Leibniz’s thought, and in particular the role that a certain model of logical analysis played in it. In a first section, I will briefly recall the central role ascribed very early by Leibniz to analysis of notions (analysis notionum) and to the constitution of an “alphabet of human thoughts”, from which all true knowledge was to be recovered by some form of “combinations”. I will then give some testimonies of the doubts raised by Leibniz himself against this program as early as 1675–1676. In order to understand better how these doubts arose, and the change that they induced in Leibniz’s philosophical orientations, I will consider the influence played in this evolution by his mathematical practice. In particular, I will emphasize the role played by some demonstrations of impossibility elaborated at the beginning of the stay in Paris and mentioned in later philosophical texts. As a conclusion, I will sketch how this evolution of Leibniz’s philosophical ideas, which was provoked by mathematics, had a ricochet effect on his mathematical practice. This will provide evidence in a simple case of how mathematics and philosophy really did interact in Leibniz’s thought. I will claim that this form of interaction is quite different from the one generally reconstructed by commentators in the past.
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- 1.
Unless otherwise stated, all the translations are mine.
- 2.
“Mais pour moy je ne cherissois les Mathematiques, que par ce que j’y trouvois les traces de l’art d’inventer en général, et il me semble que je découvris à la fin que Monsieur des Cartes luy même n’avoit pas encor penetré le mystère de cette grande science” (A II, 1, 662).
- 3.
“Car j’ay reconnu que la Metaphysique n’est gueres differente de la vraye Logique, c’est à dire de l’art d’inventer en general” (A II, 1, 662).
- 4.
To De Volder, September 6th 1700, GP II, 213.
- 5.
“Il faut avouer aussi que la preuve de Mons. des Cartes qu’il apporte à fin d’establir l’idée de Dieu est imparfaite. Comment dira-il pourroit on parler de Dieu sans y penser, et pourroit on penser à Dieu sans en avoir l’idée. Ouy sans doute, on pense quelques fois à des choses impossibles, et mêmes on en fait des demonstrations. Par exemple Mons. des Cartes tient que la quadrature du cercle est impossible, et on ne laisse pas d’y penser, et de tirer des consequences de ce qui arriveroit si elle estoit donnée. Le mouvement de la derniere vistesse est impossible dans quelque corps que ce soit (…). De même le plus grand de tous les Cercles, est une chose impossible, et le nombre de toutes les unités possibles ne l’est pas moins: il y en a démonstration.” (A II, 664).
- 6.
“I (…) arrived at this remarkable thought, namely that a kind of alphabet of human thoughts can be worked out and that everything can be discovered and judged by a comparison of the letters of this alphabet and an analysis of the words made from them. This discovery gave me great joy though it was childish of course, for I had not grasped the true importance of the matter” ( De numeris characteristicis ad linguam universalem constituendam (1679), A VI, 4, 265; transl. Loemker (1989b), 222).
- 7.
This program was of prime importance in the picture of Leibniz’s thought drawn by Couturat (1901)—especially chap. VI: “La Science Générale”, 180–184, where Couturat summarizes his interpretation by commenting on a fragment on which I shall also put some emphasis: De la Sagesse [(1676); A VI, 3, 671]. On the relation between this program and that of a Characteristica Universalis, its origins and sources, see also Rossi (2006) and Pombo (1987).
- 8.
“We can make the analysis of thoughts perceptible and conduct it as if by some mechanical guide, since the analysis of characters is something that is somewhat perceptible. Indeed Analysis of characters occurs when we substitute certain characters for others, which are equivalent to the former in their use.” ( Analysis Linguarum (September1678); A VI, 4, 102).
- 9.
“This is the origin of the ingenious specious analysis which Descartes was the first to work out, and which Frans van Schooten and Erasmus Bartholinus later organized into precepts; the latter in what he calls the elements of a universal mathematics. Analysis is thus the science of ratios and proportions, or of unknown quantity, while arithmetic is the science of known quantity, or numbers” (GP IV, 35; transl. Sasaki (2003, 298)).
- 10.
Novas complures meditationes non poenitendas, quibus semina artis inveniendi sparguntur, (…) atque inter caeteras palmariam illam de Analysi cogitationum humanarum in Alphabetum quasi quoddam notionum primitivum (A VI, 2, 549).
- 11.
See Picon (2003).
- 12.
More than targeting at Descartes himself, the model of “Cartesian” logical analysis aimed at by Leibniz here is the one provided by Arnauld and Nicole in their Logique (1662).
- 13.
Meditationes de Cognitione, veritate et ideis (A VI, 4, 587).
- 14.
“Two and two are four is not quite an immediate truth. Assume that four signifies three and one. Then we can demonstrate it, and here is how.
Definitions:
(1) Two is one and one
(2) Three is two and one
(3) Four is three and one
Axiom: If equals be substituted for equals, equal remains.
Demonstration:
2 and 2 is 2 and 1 and 1 (def. 1) 2 + 2
2 and 1 and 1 is 3 and 1 (def. 2) (2 + 1) + 1
3 and 1 is 4 (def. 3) 3 + 1
Therefore (by the axiom) 2 and 2 is 4, which is what was to be demonstrated
- 15.
As is well known, Leibniz never uses the expression “analytical proposition” which would come to have such an importance in later philosophy. Nonetheless, he talks of “analytical truths” (analyticae veritates) in order to designate truths which can be analyzed into simple terms and therefore expressed by natural numbers (A VI, 4, 715).
- 16.
The role of the reduction to “identicals” in Leibniz’s mathematical practice is the subject of another paper of mine, which could be considered as a continuation of the present one cf. Rabouin (2013).
- 17.
A VI, 3, 504; my emphasis. Note that Leibniz explicitly mentions a case in which the notions are not analyzed in contrast to a case in which they would be assumed to be already analyzed, i.e. free of any potential contradiction.
- 18.
“Il est très difficile de venir à bout de l’analyse des choses, mais il n’est pas si difficile d’achever l’analyse des vérités dont on a besoin. Parce que l’analyse d’une vérité est achevée quand on en a trouvé la démonstration: et il n’est pas toujours nécessaire d’achever l’analyse du sujet ou du prédicat pour trouver la démonstration de la proposition. Le plus souvent le commencement de l’analyse de la chose suffit à l’analyse ou connaissance parfaite de la vérité qu’on connaît de la chose” (De la Sagesse (1676); A VI, 3, 671; my emphasis).
- 19.
A II, 1, 677; see below note 46.
- 20.
On the methods used by Leibniz see Hofmann (1974, 15–20).
- 21.
This was of course before his first journey to London where he came to realize just how limited his knowledge of mathematics in general, and of series in particular, actually was. See S. Probst’s paper in this volume.
- 22.
A II, 1, 342–356. Part of the story is told by Leibniz himself at the beginning of the text. In fact, the paper was not submitted because the journal ceased publication for two years after December 1672.
- 23.
As Pascal already put it, the rule of formation for the triangle does not hold that the first line be generated by unity. Leibniz therefore gives a more general result concerning any series of fractions formed with an arbitrary generator of a Pascal Triangle as numerator and a line of the triangle as denominator: “Regula Universalis haec est: Summa seriei fractionum, quarum numerator est generator, nominatores sunt termini cujusdam progressionis Arithmeticae Replicatae est fractio seu ratio cujus numerator seu antecedens (…) est exponens seriei proximae praecedentis seu penultimae (data scilicet supposita ultima) nominator vero seu consequens est exponens seriei proxime praecedentis praecedentem, seu antepenultimate” (A II, 1, 346).
- 24.
“Numerum istum infinitum sive Numerum maximum seu omnium Unitatum possibilium summam, quam et infinitissimum appellare possis, sive numerum omnium numerorum esse 0 seu Nihil” (A II, 1, 352).
- 25.
This “paradox” is modeled on the fact that one can establish a one to one correspondence between natural numbers and their squares (or their cubes), where it seems that there are “more” of the former than the latter (not all natural numbers are squares). See S. Levey’s paper in this volume.
- 26.
I thank Marco Panza for having pressed me on this issue when I first presented this study.
- 27.
As is well known, Euclid’s proof states that whatever prime number is considered to be the largest, it is always possible to construct a larger one.
- 28.
See, for example, the letter of late August 1698 to Johann Bernoulli: “Many years ago I proved beyond any doubt that the number or multitude of all numbers implies a contradiction, if taken as a unitary whole. I think that the same is true of the largest number, and of the smallest number, or the lowest of all fractions. The same has to be said about these, as about the fastest motion and the such-like” (GM III, 535, transl. MacDonald Ross (1990, 129)). On the impossibility of an actual infinite number, see also Essais de Théodicée (GP VI, 90).
- 29.
“exceptis scilicet ipsis definitionibus, quae ut toties in suis scriptis inculcat restaurator philosophiae Galilaeus, arbitrariae sunt, nec falsitatis, sed ineptiae obscuritatisque tantum arguendae” (A II, 1, 351).
- 30.
Multa videmur nobis cogitare (confuse scilicet) quae tamen implicant. Exempli causa: Numerus omnium numerorum. Unde valde suspecta esse debet nobis notio infiniti, et minimi et maximi, et perfectissimi, et ipsius omnitatis. Neque fidendum his notionibus, antequam ad illud criterion exigantur, quod mihi agnoscere videor, et quod velut Mechanica ratione, fixam et visibilem, et ut ita dicam irresistibilem reddit veritatem (A II, 1, 393).
- 31.
Even if it is dubious that this conception was genuinely Cartesian, it is certainly very close to the kind of representation of knowledge presented in the Regulae ad directionem ingenii. In the Regulae Descartes reduced the entirety of knowledge, at least such as is accessible to certitude, to two basic operations: intuition and deduction (these being considered as chains of directly evident inferences). Although Leibniz read the Regulae only at the beginning of 1676, this model was clearly presented by the Logique de Port-Royal (1662) as the paradigm of the new, that is “Cartesian”, logical “analysis”. The comparison of the fragment De la Sagesse (1676) with this Cartesian concept is very interesting. Leibniz takes up the same model, but with the important distinction between “analysis of things” and “analysis of truth” mentioned in the previous section, i.e. he points to a possible gap between analysis of notions and truth based on simple notions.
- 32.
The last demonstration of Euclid’s Elements, in Book XIII, establishes precisely the impossibility of constructing another one. Note that the passage devoted to the chiliogon in the Meditationes is in the same vein. Leibniz turns Descartes’s example back on him, by noticing that the problem is not linked to the use of sensible imagination, but to the use of symbolic thinking in general (be it through words or through diagrams). Because we don’t proceed to the analysis of the notions involve to its end, we therefore have no guarantee that this notion does not imply a contradiction. This becomes obvious if we replace polygons by regular polyhedra in the example.
- 33.
In contrast, the arguments taken from the new analysis simply indicate that Descartes’s conception of “geometrical curves” was too narrow, without establishing clearly that his general “method” was responsible for this.
- 34.
They constitute a complete volume of the Academy Edition. See volume VII, 6: Arithmetische Kreisquadratur 1672−1676.
- 35.
It is the last of 51 proposition and its last words are: “Impossibilis est ergo quadratura generalis sive constructio serviens pro data qualibet parte Hyperbolae aut Circuli adeoque et Ellipseos, quae magis geometrica sit, quam nostra est. Q.E.D.” (Parmentier 2004, 354).
- 36.
“Geometrical” should be understood here in the Cartesian sense of “geometrical curve”, i.e. one which can be expressed through a finite algebraic formula. Leibniz also has other arguments, which I shall not go into here, by means of which he is able to defend the fact that his particular series is better than others.
- 37.
This latter claim is not substantiated by Leibniz and one might doubt whether Vieta could be said to have produced a demonstration of this fact.
- 38.
Leibniz would agree with them on this point (which is why he carefully distinguishes between having a “notion” and having an “idea” of something).
- 39.
GP V, 301; Nouveaux Essais III, 6, § 28. The continuation of the sentence being precisely: “autrement on auroit droit de parler des Decaedres reguliers” (“otherwise, one would have the right to speak about regular decahedra”).
- 40.
See Ouverture nouvelle des nombres multiples, et des diviseurs des puissances, January 1676 (A VII, 1, 576–578); Figuram numerorum ordine dispositorum et punctatorum ut appareant qui multipli qui primitivi (A VII, 1, 579–581); De numeris figuratis divisoribusque potestatum (A VII, 1, 583–586); De natura numerorum primorum et in genere multiplorum (A VII, 1, 594–598).
- 41.
On the history of the study of prime numbers in the seventeenth century, see Bullynck (2010).
- 42.
- 43.
This last problem occupied Leibniz so intensely that he made no less than thirty attempts at solving it—around four hundred pages in the Academy Edition. See Hofmann (1969).
- 44.
See also this striking passage from the Nouveaux Essais about prime numbers: “C’est la multitude des considérations aussi qui fait que dans la science des nombres même il y a des difficultés très grandes, car on y cherche des abregés et on ne scait pas quelquesfois, si la nature en a dans ses replis pour le cas dont il s’agit. Par exemple, qu’y a t-il de plus simple en apparence que la notion du nombre primitif? c’est à dire du nombre entier indivisible par tout autre excepté par l’unité et par luy même. Cependant on cherche encor une marque positive et facile pour les reconnoistre certainement sans essayer tous les diviseurs primitifs, moindres que la racine quarrée du primitif donné. Il y a quantité de marques qui font connoistre sans beaucoup de calcul, que tel nombre n’est point primitif, mais on en demande une qui soit facile et qui fasse connoistre certainement qu’il est primitif quand il l’est” (Nouveaux Essais IV, 17, § 9, GP V, 470).
- 45.
The demonstration of Fermat’s last theorem was certainly the most spectacular example of this fact in recent years, but there are many other examples in present day mathematics. One could mention that this is also something which was emphasized by the Bourbaki group in their structuralist manifesto, L’architecture des mathématiques: “[…] in certain theories (for example in the theory of Numbers), there exist many isolated results that up till now no one has been able to classify, nor connect in a satisfactory way with known structures” (Bourbaki 1950).
- 46.
“Je voudrois sçavoir si vostre M. Prestet continue à travailler dans l’analyse. Je le souhaite, parce qu’il y paroist propre. Je reconnois de plus en plus l’imperfection de celle que nous avons. Par exemple, elle ne donne pas un moyen seur pour resoudre les problemes de l’Arithmetique de Diophante […] Enfin, je pourrois faire un livre des recherches où elle n’arrive point, et où quelque Cartesien que ce soit ne sçauroit arriver sans inventer quelque methode au delà de la methode de des Cartes. (Letter to Malebranche, January 1679; A II, 1, 677). See also A VI, 4, 2047 (1689) and Nouveaux Essais IV, 2, § 7: “On n’a pas encore trouvé l’analyse des nombres: Il arrive aussi que l’induction nous presente des verités dans les nombres et dans les figures dont on n’a pas encor decouvert la raison generale. Car il s’en faut beaucoup, qu’on soit parvenu à la perfection de l’Analyse en Geometrie et en nombres, comme plusieurs se sont imaginés sur les Gasconnades de quelques hommes excellens d’ailleurs, mais un peu trop prompts ou trop ambitieux” (GP V, 349).
- 47.
This situation has to be contrasted with the fact that Leibniz was also very explicit about some connections which he resisted making—although modern commentators tend to put a lot of emphasis on them. One famous example is given by the provocative declaration made to Masson in 1716: “The infinitesimal calculus is useful with respect to the application of mathematics to physics; however, that is not how I claim to account for the nature of things. For I consider infinitesimal quantities to be useful fictions” (GP VI, 629; transl. in Ariew & Garber, 230).
- 48.
A VI, 4, 489.
- 49.
One can remember here the striking formulation of “La vraie méthode” (1677): “certain experiments are always necessary to serve as a basis for reasoning” (A VI, 4, 3. My emphasis). One could object that, according to the Meditationes, experience can only serve as basis for an a posteriori proof. But the fact that mathematical truths are a priori does not exclude in and of itself the validity of a posteriori proof of their possibility (especially if we don’t have access to complete analysis of these notions).
- 50.
See the list of “identicals” given to Conring in the letter from March 1678 (A II, 1, 599) or the one given ten years after at the beginning of the Principa logico-metaphysica (A VI, 4, 1645; 1689).
- 51.
On this issue, see Rabouin (2013), “Analytica Generalissima Humanorum Cognitionum. Some reflections on the relationship between logical and mathematical analysis in Leibniz”.
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Acknowledgements
I would like to thank Philip Beeley, Norma Goethe , and an anonymous referee for their helpful comments and corrections on this paper.
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Rabouin, D. (2015). The Difficulty of Being Simple: On Some Interactions Between Mathematics and Philosophy in Leibniz’s Analysis of Notions. In: Goethe, N., Beeley, P., Rabouin, D. (eds) G.W. Leibniz, Interrelations between Mathematics and Philosophy. Archimedes, vol 41. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9664-4_3
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