Abstract
The purpose of this paper is to give the key-principles which are involved in the ideal of the mathematization and of the ‘more geometrico’ — idea in Descartes and Leibniz. This is done in the following way. In Chapter 1 some of the main-principles of the philosophical doctrine of rationalism are given. These principles it is thought underly the philosophy of both thinkers Descartes and Leibniz. And so they underly also their doctrines concerning the ideal of more geometico and of mathematization of all sciences. In Chapter 2 the method ‘more geometrico’ will be analyzed. In Chapter 3 the ideal of mathematization will be dealt with. In respect to ‘more geometrico’ and to the ideal of mathematization Descartes and Leibniz differ considerably and it is Leibniz who has to offer the main principles here.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
The problem of the different meanings of ‘proposition’,’ statement’ and ‘judgement’ cannot be discussed here. It would lead too far away from the topics of this paper. It may however be mentioned that the English term ‘judgement’ would be probably more appropriate to Descartes — cf. MD 3, 8-12-where he gives his division of psychic phenomena. On the other hand the term ‘proposition’ will be more appropriate to Leibniz. Here the term ‘proposition’ is used because it is more neutral in respect to an interpretation which would be too close to some psychologistic view for symbolization I use the usual first order predicate logic and quantification for propositions.
RD 3; AT 10, 368.
RD 3; AT 10, 370. Cf. Röd (DEP) p. 49ff.
Cf. RD 4; AT 10, 371-2.
GP 5, 21.
GP 7, 194-195. Cf. GP 7, 296.
RD 3; AT 10, 370.
GP 5, 343; NE 4, 2, 1.
RD 3; AT 10, 368.
GP 5, 343; NE 4, 2, 1. On some passages Leibniz says that the principle of noncontradiction asserts that every proposition is true or false (GP 5, 343). This is a rather restricted formulation which rules out any many-valued logic. Cf. Rescher (MVL) p. 143ff. But Leibniz knows also forms of the principle which are different and apply to an axiomatic system. Cf. Chapter 2.2. and 2.4.
GP 5, 347; NE 4, 2, 1.
Copleston (HPh) Vol. IV, p. 73.
Russell (CEP), p. 167.
Cf. GP 4, 423-424.
DM 4, 3; AT 6, 33. Cf. RD 3 and Descartes’ commentary to that rule.
DM 2; AT 6, 18.
GP 4, 422-424. Cf. the citation of Russell, note 13.
GP 4, 422-426 and GP 1, 381-385.
GP 1, 384.
GP 1, 384. Cf. GP 7, 194 (footnote) where Leibniz gives a definition of truth (of reason) with the help of identical propositions. Cf. Chapter 2.3. note 63.
RD 3; AT 10, 368.
RD 3; AT 10, 369.
RD 3; AT 10, 370. Cf. RD 12, 23.
Cf. especially his commentary to Rule 3 in RD.
GP 5, 461.
GP 7, 199-200.
RD 4; AT 10, 371-372.
DM 2; AT 6, 18.
Cf. Röd (DEP) p. 13f. There is a discussion in historical works about Descartes whether’ simple natures’ can only have the form of propositions or whether they may be understood as certain simple ideas as well. It would go beyond the scope of this paper to deal with this problem. It seems that Descartes himself was not entirely clear about that matter and that he used’ simple natures’ for both, propositions and ideas in such a way that simple natures as ideas like “existence”, “thinking”, “three lines” are disengaged from first principles known by intuition like “I think, therefore I exist” or “a triangle is bounded by three lines”. Cf. Copleston (HPh), Vol. IV, p. 77f.
RD 12, 13; AT 10, 418.
Cf. PP 1, 10.
GP 4, 64-65 (= De arte combinatoria, 64). Cf. Kneale (DLg), p. 333.
GP 4, 450.
Rescher (LIP) p. 22. Cf. Röd (RAM). Concerning Leibniz’ views on the subject-predicate form of propositions cf. §8 of his Discourse on Metaphysics (= GP 4, 427-463) and Rescher (LIP) p. 21. It should be mentioned that the characterization of a proposition of subject-predicate-form as true if the predicate is contained in the subject — which can be found also in Thomas Aquinas (cf. (STh) I, 2, 1) — requires an intensional interpretation: In the true proposition “men (subject) is animal (predicate)” the subject (class of men) is extensionally included in the predicate (class of animals). But animal (the properties of being an animal) is intensionally included in man (the properties of being a man). Elsewhere I have given an interpretation of intensional inclusion which is able to explain the traditional doctrines concerning intensional inclu sion in Aristotle, the Scholastics, Leibniz and De Morgan and which can be introduced in the modern logic of classes (or into first order predicate logic). Cf. Weingartner (IIS), (BIG), and (SPI).
MD 3, 13.
DM 6; AT 6, 64. PP 2, 3; AT 8,42.
RD 12, 14.
NE 1,1,18.
Russell (CEP), p. 161. Cf. GP 5, 66 and 79. Cf. Rescher (PhL), p. 128; (LIP), p. 122.
Cf. RD 12, 14; cf. note 37.
NE 1, 1, 18.
NE 1, 3, 3.
NE 2, 1, 2.
Cf. Copleston (HPh), Vol. IV, p. 83: “AU clear and distinct ideas are innate. And all scientific knowledge is knowledge of or by means of innate ideas”. Cf. the citations in Chapters 1.2.2. and 1.2.3.
NE 1, 2, 12.
Cf. Tarski (LSM), p. 185, Def. 18. α is a set of sentences (propositions). It is further assumed that α ⊂S where S is the set of meaningful sentences (propositions). This set of meaningful sentences has to be understood relative to a language which has been prepared and made precise in some sense to discuss scientific or philosophical problems. It would be unnecessary to require a symbolic language with a recursive definition of well formed formula. For Descartes and Leibniz a language in which books of geometry (for instance Euclids elements) are written would be ideal. But observe that not only the symbols of the deductive system are meant here but the whole language in which the axioms and theorems are discussed. Descartes would certainly view his Meditations as a case in point, and Leibniz his Nouveaux Essays.
RD 5. AT 10, 379.
PP, Letter to C. Picot, AT 9-2 (19). Cf. Röd (DEP), p. 47 and Scholz (PZM), p. 29.
GP 7, 195. Cf. GP 7, 295.
GP 4, 437.
NE 4, 2, 1. Cf. GP 7, 299: “Ante omnia assumo Enuntiationem omnem (hoc est affîationem aut negationem) autveram aut falsam esse, et quidem si vera sit affirmatio, falsam esse negationem; si vera sit negatio, falsam esse affîationem.” Cf. Kauppi (LLg), p. 82f.
There are some extravagant many valued calculi, in which a proposition p can take both the value T and F. Rescher calls them quasi-truth functional. Cf. Rescher (MVL), p. 166ff. Calculi of that sort would certainly be ruled out by Leibniz’ weaker consequence of his principle of non-contradiction.
At least not in this and related passages. — For different versions of the principles of contradiction and tertium non datur see Rescher (MVL), p. 143-154.
NE 4, 2, 1.
Observe that the expression ‘many valued’ has two different meanings: a system can be many-valued in the sense that its propositions can take values which are different (for instance: indefinite) from true and false. On the other hand it can be many-valued in the sense that there is no value between or outside true or false but a proposition can take more than one value of true (or: designated value) and more than one value of false (or: antidesignated value) such that the antidesignated values coincide with the non-designated ones. The passage in Leibniz’s Nouveaux Essays is formulated in such a way that I think one can assume that also the second form of many-valued systems are ruled out; i.e. it is assumed that there is only a single truth-value true and a single truth-value false.
That Leibniz defends the subject-predicate form of propositions is clear from many passages-cf. GP 4, 432, GP 7, 295/96. But as Brentano already noticed he was not of the opinion that this was the only possible form. Thus he says for instance: “On peut tousjours dire que cette proposition: j’existe est de la dernière evidence, estant une proposition, qui ne sauroit estre prouvée par aucune autre, ou bien une verité immediate.” (NE 4, 7, 7). Cf. NE 4, 5, 3.
v. Fraassen (STT) and Kripke (OTT).
RD 4, 2; AT 10, 371-2.
MD 3, AT 7, 36f.
MD 3, AT 7, 37.
The definition used here is due to Tarski: A deductive system a is finitely axioma-tizable (FAX) iff there is at least one axiom system (i.e. iff there is a finite set β of sentences) such that β is identical with the consequence class of α. Cf. Tarski (LSM), p.355.
RD 3, 8; AT 10, 369.
RD 5; AT 10, 379.
GP 7, 57. Cf. GP 7, 25, where he describes his new invention as “certaines vues toutes nouvelles, pour reduire tous les raisonnemens humaines à un espece de calcul, qui serviroit à decouvrir la verité.” Cf. further Courturat (OF), p. 221.
GP 7, 194. Cf. the passage cited in note 49 (GP 7, 195) and GP 7, 295.
Cf. GP 7, 195.
GP 7, 194. Cf. Chapter 1.1.1., note 6.
GP 7, 355. But it is very probable that he includes here also the principle of contradiction as an inference rule, i.e. as the reductio ad absurdum principle. Cf. Chapter 2.4.
NE 4,7, 10.
GP 7, 195. Cf. the passage cited in Chapter 1.2.1., note 32.
Copleston (HPh), Vol. IV, p. 277, Rescher (PhL), p. 26 and (LIP) p. 24.
GP 5, 343. NE 4, 2,1.
GP 7, 355. Cf. Chapter 2.3., note 66. This passage is as a matter of fact also cited by Copleston, p. 276.
Cf. Monadology 31 (= GP 6, 612)
For such definitions see the example cited in note 67 above. In special cases the definitions have to be real, not only nominal according to Leibniz. This is so when the conclusion should be stated absolutely not only hypothetically. Cf. note 65. Concerning inference rules one should observe that Leibniz held that firstly the syllogisms of the first figure can be proved by the law of non-contradiction (in the sense of indirect proof or reductio ad absurdum) and secondly the syllogisms of the second and third figure can be derived from the first, also by using only the principle of non-contradiction. Cf. NE 4, 2, 1.
GP 7, 295-296. Cf. GP 7, 194-195.
RD 4, AT 10, 371. Cf. the fifth rule which was cited in note 61.
Cf. Russell, (CEP), p. 30ff.
GP 2, 62.
GP 7; 300, 301.
GP 7, 199-200.
GP 6, 612 (Monadology 31-36).
This seems to be also the opinion of Couturat — cf. his (LgL), p. 214f. — and of Kauppi in her detailed discussion of the principle of sufficient reason, cf. (LLg), p. 87-94.
Rescher (PhL), p. 23, cf. (LIP), p. 23, cf. Leibniz’s description which was given in Chapter 1.2.1., note 32.
GP 6, 602.
GP 1, 57. Cf. GP 7, 301, where he lists metaphysics, physics, the moral sciences and proofs of the existence of God for the application of the principle of sufficient reason.
That this is so for logic and mathematics is evident, but it is also true for metaphysics according to Leibniz though metaphysics deals with different kinds of truths of reason (hypothetical truths about possible substances, categorical truths about abstract concepts, analytic truths and last but not least verités innées). Cf. Rescher, (PhL), p. 127; (LIP), p. 121. Cf. Kauppi, (LLg), p. 31, note 3.
Cf. Rescher, (PhL), p. 17.
GP 3, 400.
See the citation about method, Chapter 1.2.1., note 25.
RD 4, AT 10, 372.
RD 7, AT 10, 387.
Kauppi is therefore right to say that the principle of contradiction and that of sufficient reason are in the first place not axioms but metatheoretical principles of proof in general. Cf. Kauppi, (LLg), p. 80 and (OF), p. 184.
GP 7, 295. Cf. NE 4, 2,1.
For a discussion of normative principles in the methodology of all or specific scientific disciplines cf. my (NCS).
RD 2, AT 10, 366.
DM 2, 11; AT 6, 19.
OF 175.
Cf. Frege (BGS), Preface and (WBB).
GP 7, 205.
This expression is not used unambiguously in the literature on Leibniz. Very often it is used in a broad sense for some or all of the different parts listed subsequently.
GP 7, 184.
GP 7, 189.
Cf. OF p. 49 and 50. This is also the opinion of Kauppi, cf. her (LLg), p. 146.
Cf. OF p. 78 and 79.
Observe that according to Leibniz a categorical proposition is true iff the predicate is contained in the subject: “Propositio vera est cujus praedicatum continetur in subjecto seu ei inest”. OF, p. 68. Cf. GP 7, 199-200; GP 7, 208. I have shown elsewhere that the traditional expression “the predicate is contained in the subject” (predicatum inest subjecto) can be interpreted as a kind of intensional inclusion which can be defined as P⊑S↔(M) [P⊆M ↓S⊆M], where ⊆ is the usual class-inclusion. When this definition plus two additional ones for intensional intersection and intensional union are introduced into the usual theory of classes (with out the need of any set-theoretical axiom) one gets a dual theory which is an intensional class-theory. The theorems of this theory-as for instance the theorem P⊑S ↔ S ⊆ P which expresses the antagonistic relation between extensional and intensional inclusion — give suitable interpretations of doctrines of traditional logic especially of those of Aristotle, the Scholastics, Leibniz and De Morgan. Cf. my (IIS) and (SPI).
OF, p. 42.
Cf. OF, p. 40-84.
Cf. OF, p. 77-84. The discovery that this calculation method is successful for syllogistics is due to Lukasiewicz (ASy). Cf. D. Marshall (LLA). I am indebted to P. Simons for drawing my attention to this discovery and to the paper of Marshall.
OF, p. 78.
OF, p. 79. Only a selection of Leibniz’s description has been cited. Under (III) Leibniz describes the particular negative proposition and under (V) the particular affirmative proposition. He also states 8 theorems concerning the validity of different syllogistic laws like that of conversion and of square of opposition. (Translation by G. H. R. Parkinson, in: Parkinson (LLP).)
cf Lukasiewicz (ASy), p. 126-129. Kauppi and Rescher seem not to be aware of the success of Leibniz’s last proposal. They discuss only earlier proposals where Leibniz represents terms by simple numbers (not pairs). In this respect they were correct with their doubts whether it works with all the syllogistic modes (though the simple method works for some, for example BARBARA, as they observe.)
Marshall (LLA), p. 241. For the historically difficult question whether Leibniz had doubts about the correctness of his last proposal see Marshall, ibid., p. 241f.
GP 1, 57.
OF, p. 161.
Cf. the citations I have given for the principle of completeness in Chapter 2.5.3., note 89 and 90.
Hermes (ILG), p. 98.
Hermes (ILG), p. 99.
That Leibniz called the principle so is a claim by Rescher — cf. his (PhL), p. 41, note 15: “He termed it ‘la conspiration universelle’”. But Rescher doesn’t give a reference. The Tentamen Anagogicum (GP 7, 270-279) which contains some passages concerning that principle does not however contain the name ‘conspiration universelle’. It may be that Rescher has the term from Couturat.
GP 7, 303.
GP 7, 274.
GP 7, 303.
GP 2, 58. The conspiration universelle is stated by Couturat and Rescher almost only in connection with some of the other principles listed above, especially with the principle of perfection. Cf. Couturat, (LgL), p. 230ff. and Rescher, (LIP), p. 38-39. But its importance as a basic principle underlying all others seems not to be seen clearly. Rescher views the principle of perfection as the source-principle of those of plenitude, continuity, and harmony, cf. (PhL), p. 57. For Leibniz’ principle of plenitude cf. Hintikka (LPR).
GP 7, 275.
Cf. GM 6, 130 (About the principle of continuity). The conception of the universe as a universe of moving things goes back to Aristotle. Cf. his Physics, 2, 1. Similarly the view that physics is concerned with moving being. This was accepted also by the great philosophers of scholastics. Cf. Thomas Aquinas (APC), 1, 1, 3: “… It follows that mobile being is the subject of natural philosophy. For natural philosophy is about natural things, and natural things are those whose principle is nature. But nature is a principle of motion and rest in that in which it is. Therefore natural science deals with those things which have in them a principle of motion.”
GP 7, 303.
Grua (LTD, 1,393.
GP 7, 303.
Cf. Rescher (LIP), p. 39 and (PhL), p. 30.
Cf. Rescher (LIP), p. 39, (PhL), p. 38.
GP 4, 438.
GP 3, 582.
GP 7, 272-273.
GP 7, 309-310.
Monadology §41.
OF, p. 360.
GP 7, 309-310.
Cf. Friedmann (RLM), p. 345f.
Cf. Friedmann (RLM), p. 346f. Friedmann discusses then a second analogy between Leibniz’s principle that every monad represents the entire universe from its point of view with the greatest possible series of perceptions it may have on one hand and the generalized maximization principle that every local universe is maximized (which is a generalization of 3.2.4.1.) on the other. He again states three similar analogies which are illuminating in respect to the conception of a monad as a series of perceptions, in respect to the hierarchical order of the monads and to the logical relation of the two maximization principles of Leibniz. Cf. (RLM), p. 348f.
GM 6, 129.
GP 3, 52. These formulations of the principle of continuity support at the same time principle 3.2.6.
Russell (CEP), p. 64-65.
Cf. the discussion of principle 2.2.
Cf. Rescher (LIP), p. 63f. (PhL), p. 51f.
GP 6, 107. Leibniz presupposed of course that a process originating from a place can have an unlimited extension. This is unlike the theory of relativity, where a process originating from a point can have at any time only limited extensions.
GP 2, 450.
GP 4, 501.
GP 2, 58.
I have shown in (AnS) that one can give different definitions of analogy with the help of the mathematical concepts of homomorphism and isomorphism which can be applied both to traditional doctrines of analogy and to analogies in contemporary science (for instance physics).
Friedmann (RLM), p. 336-337. Friedmann’s conjecture is not without reason. He showed in (GCH) that the generalized continuum hypothesis is equivalent to the generalized maximization principle.
GP 7, 200.
References
Aristotle Physics, Intr. Comm. W. D. Ross, Oxford, 1955.
Copleston, F. ‘A History of Philosophy’, Vol. IV, Descartes to Leibniz, Burns & Oates, London, 1960.
Couturat, L. La Logique de Leibniz d’ apres des documents inédits, Paris, 1901.
Descartes Oeuvres de Descartes, C. Adam and P. Tannery (eds.), 13 Vols., Paris, 1897-1913. Librairie Philosophique J. Vrin, Paris.
Descartes Discourse on Method
Descartes Meditations
Descartes Principles of Philosophy
Descartes Rules for the Direction of the Mind
van Fraassen, Bas ‘Singular Terms, Truth-value Gaps and Free Logic’, in The Journal of Philosophy 63 (1966), 481–495.
Frege, G. Begriffsschrift, Halle 1879, in I. Angelelli (ed.), Begriffsschrift und andere Aufsätze, Wiss. Buchgesellschaft, Darmstadt, 1964.
Frege, G. ‘Über die wissenschaftliche Berechtigung einer Begriffsschrift’, in Zeitschrift für Philosophie und philosophische Kritik 81 (1882), 48–56. Reprinted in, I. Angelelli (ed.), Begriffsschrift und andere Aufsätze, p. 106-114.
Friedmann, J. I. ‘The Generalized Continuum Hypothesis is Equivalent to the Generalized Maximization Principle’, in Journal of Symbolic Logic 36 (1971), 39–54.
Friedmann, J. I. ‘On some Relations between Leibniz’ Monadology and Transfînite Set Theory (A Complement to the Russell Thesis)’, in Studia Leibnitiana Supplemente, Vol. 15-3, K. Müller, H. Schepers, and W. Totok (eds.), Steiner, Wiesbaden, 1975.
Grua, G. Leibniz, Textes inédits, G. Grua (ed.), 2 Vols., Paris, 1948.
Hermes, H. ‘Ideen von Leibniz zur Grundlagenforschung: Die ars inveniendi und die ars iudicandi’, in Studia Leibnitiana Supplementa, Vol. 15-4, K. Müller, H. Schepers, and W. Totok (eds.), Steiner, Wiesbaden, 1975.
Hintikka, K. J. J. Leibniz on Plenitude, Relations, and the ‘Reign of Law’, in H. Frankfurt (ed.), Leibniz. A Collection of Critical Essays, University of Notre Dame Press 1976.
Kauppi, R. ‘Über die Leibnizsche Logik. Mit besonderer Berücksichtigung des Problems der Intension und der Extension’, Acta Philosophica Fennica XII, Helsinki, 1960.
Kneale, W. and Kneale, M. Development of Logic, Oxford, 1962.
Kripke, S. ‘Outline of a Theory of Truth’, in The Journal of Philosophy (1975), 690–716.
Leibniz Die mathematischen Schriften von G. W. Leibniz, C. I. Gerhardt (ed.), 7 Vols., Berlin, 1849-63.
Leibniz Die philosophischen Schriften von G. W. Leibniz, C. I. Gerhardt (ed.), 7 Vols., Berlin, 1875-90. Georg Olms Verlag, Hildesheim.
Leibniz Nouveaux essais sur l’entendement humain. Wissenschaftliche Buchgesellschaft, Darmstadt.
Leibniz Opuscules et fragments inédits de Leibniz, L. Couturat (ed.), Paris, 1903. Georg Olms Verlag, Hildesheim.
Lukasiewicz, J. Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic. London (Oxford UP) 21957.
Marshall, D. Jr. ‘Lukasiewicz, Leibniz and the Arithmetization of the Syllogism’, in Notre Dame Journal of Formal Logic, 18 (1977), 235–242.
Parkinson, G. H. R. Leibniz, Logical Papers, A Selection Translated and Edited with an Introduction by G. H. R. Parkinson. Oxford (Clarendon) 1966.
Rescher, N. Leibniz. An Introduction to his Philosophy, Oxford, 1979.
Rescher, N. Many-valued Logic, McGraw Hill, New York, 1969.
Rescher, N. The Philosophy of Leibniz, Englewood Cliffs, 1967.
Röd, W. Descartes’ Erste Philosophie. Versuch einer Analyse mit besonderer Berücksichtigung der Cartesianischen Methodologie, Bouvier, Bonn, 1971.
Röd, W. ‘Die Rolle der analytischen Methode in der Grundlegung der Leibnizschen Metaphysik’, in Studia Leibnitiana Supplementa, Vol. 15-4, K. Müller, H. Schepers, and W. Totok (eds.), Steiner, Wiesbaden, 1975.
Russell, B. A Critical Exposition of the Philosophy of Leibniz, London, 1900, new edition 1937.
Scholz, H. Die Philosophie im Zeitalter der Mathesis Universalis, Descartes, Pascal, Leibniz (Lecture at the University of Münster, 1933-34), ed. by Mathematische Fachschaft of the University of Münster (mimeographed).
Tarski, A. Logic, Semantics, Metamathematics, Oxford, 1956.
Thomas Aquinas In Octo Libros Physicorum Aristotelis Expositio, P. M. Maggiolo (ed.), Marietti, Roma, 1954.
Thomas Aquinas Summa Theologica, Latin Text and English Translation, Blackfriars, Cambridge, 1964.
Weingartner, P. ‘Bemerkungen zum Intensionsbegriff in der Geschichte der Logik’, in Zeitschrift für Philosophische Forschung 30 (1976), 51–68.
Weingartner, P. ‘On the Introduction of Intensions into Set Theory’, in M. Przelecki, K. Szaniawski, and R. Wójcicki (eds.), Proceedings of the Conference for Formal Methods in the Methodology of Empirical Sciences, Warszaw, 1974, (1976), D. Reidel Publ. Co., Dordrecht, Holland, pp. 287–296.
Weingartner, P. ‘Normative Characteristics of Scientific Activity’, in R. Hilpinen (ed.), Proceedings of the Conference on Ethics and Science, Dubrovnik, 1978, 1980, D. Reidel Publ. Co., Dordrecht, Holland, pp. 209–230.
Weingartner, P. ‘A Simple Proposal for Intruducing Intensions into Science’, in: Archives de l’Institut International des Sciences Théoriques, Actes du Colloque “La Sémantique dans les Sciences” de l’Academie Internationale de Philosophie des Sciences Rixensaart, 1974, Bruxelles, 1978, pp. 193–209.
Weingartner, P. ‘Analogy among Systems’, in Dialectica 33 (1979), p. 1–23.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 D. Reidel Publishing Company, Dordrecht, Holland
About this chapter
Cite this chapter
Weingartner, P. (1983). The Ideal of the Mathematization of All Sciences and of ‘More Geometrico’ in Descartes and Leibniz. In: Shea, W.R. (eds) Nature Mathematized. The University of Western Ontario Series in Philosophy of Science, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6957-5_7
Download citation
DOI: https://doi.org/10.1007/978-94-009-6957-5_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-6959-9
Online ISBN: 978-94-009-6957-5
eBook Packages: Springer Book Archive