Abstract
Philosophers before Kant assumed that all a priori (non-empirical) judgments must be analytic, and also that all synthetic judgments must be a posteriori (empirical). Applied to the classical example of the axioms of geometry, these assumptions produced two opposing schools of thought—some philosophers said that geometry had to be empirical because its axioms were obviously synthetic; the other said that geometry had to be analytic because its axioms were obviously a priori. Kant’s discovery that the two distinctions were not identical allowed for a middle ground position in which the axioms of geometry (as well as many other propositions) had to be synthetic a priori.
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Notes
- 1.
See Critique of Pure Reason, A25, B39.
- 2.
This seems to have been the position of David Hilbert, who was Nelson’s mentor in mathematics. It also was undoubtedly Frege’s position, who, in contrast to Russell, did distinguish between the analytic truths of arithmetic and the synthetic truths of geometry (see Frege 1884). Even at the end of his life, when, as a consequence of Russell’s paradox, Frege had given up the project of founding arithmetic on logic, he believed that arithmetic must be as synthetic as geometry (see Hermes et al. 1983). Poincaré’s position is discussed later in Chapters “Lecture X” and “Lecture XI”.
As we know, the logical empiricists and positivists, whose star had just begun to rise at the time this course was delivered, would defend the analytic character of geometry in terms that Nelson could no longer subject to scrutiny due to his early death. In any case, it may be opportune to say here that Nelson did not ignore or deny the existence of non-Euclidean geometries. In fact, he dedicated to them no less than two lengthy papers, one technical (Nelson 1905) and the other popular (Nelson 1906), as well as two oral expositions (Nelson 1914, 1928). See also Chapters “Lecture IX”, “Lecture X” and “Lecture XI”.
- 3.
The phrase ‘complete disjunction’ (and its opposite ‘incomplete disjunction’) refers in Nelson’s German to a division of a concept (genus) into two sub-concepts (species) according to traditional logic, i.e. so that these two are mutually exclusive and jointly exhaust the superordinate concept. It is a very important term for Nelson’s analysis in that the history of philosophy is marked by the presence of innumerable false dilemmas, all of which depend on a previous disjunction that appears to be complete yet is in fact incomplete. Figure 2 of this chapter, Fig. 1 in Chapter “Lecture X”, Fig. 1 in Chapter “Lecture XIII”, Fig. 1 in Chapter “Lecture XIX”, Fig. 1 in Chapter “Lecture XXI” and Fig. 1 in Chapter “Lecture XXII” will give examples of several incomplete disjunctions and the false dilemmas that rest on them.
- 4.
The reader should be reminded that after Kripke (1980) we have become used to draw this table of judgments (or propositions) differently, i.e. combining the distinction between a priori and a posteriori with that between necessary and contingent (rather than with that between analytic and synthetic). Viewed in that way, it can be argued that none of the four cells is actually empty; in particular there may be necessary a posteriori propositions as well as contingent a priori ones. This certainly appears to be quite non-Kantian; but this is not the place to take up the issue.
- 5.
All quotations refer to Hume (1748, Sect. IV, Part I).
- 6.
Figure 2 contains the first of several diagrams of the kind Nelson used to draw in his lectures and publications. As noted in the introduction, these diagrams were highly praised by Popper at the time (see Popper 1979). They consist of boxes connected by lines (mathematically speaking, they are directed graphs).
Each box contains a proposition, usually expressed fully by means of a sentence, although in some cases the proposition is just indicated by a noun phrase; see Fig. 1 in Chapter “Lecture XIX” for an example. Each proposition is metalogical in character, i.e. it does not say things about the world but about our concepts of the world. As for Fig. 2 in this chapter, note that of the four propositions in the middle of the diagram the two on top directly contradict the diagonally opposite ones.
The lines represent a logical derivation that is always directed from top to bottom. For the purposes of this book I have replaced those lines by arrows to make Nelson’s convention perfectly explicit. Incidentally, whenever two (or more) lines reach the same box, it is always understood that they are not independent premises, each one leading to the conclusion, but rather joint premises within one and the same inference. In spite of this obvious formal defect of his diagrams, Nelson was clearly a pioneer in argument mapping, a recent and currently flourishing area within the theory of argumentation (see e.g. Monk and van Gelder 2004; van Gelder 2005; Rowe et al. 2006). His diagrams usually contain six, seven or eight boxes, although on one occasion he uses up to ten boxes (so in Nelson 1917, 629).
- 7.
I remind the reader that the German word is Denkfehler, literally ‘error in (or of) thinking’. My justification for translating it by ‘fallacy’ is given in Chapter “Introduction”.
References
Frege, Gottlob. 1884. Die Grundlagen der Arithmetik: Eine logisch-arithmetische Untersuchung über den Begriff der Zahl. Wrocław: Wilhelm Koebner. [English translation by J.L. Austin: The foundations of arithmetic: A logico-mathematical enquiry into the concept of number. Oxford: Blackwell, 1950. New translation by Dale Jacquette: The foundations of arithmetic, New York: Pearson, 2007].
van Gelder, Tim. 2005. Teaching critical thinking: Some lessons from cognitive science. College Teaching 53(1): 41–46.
Hermes, Hans, Friedrich Kambartel, and Friedrich Kaulbach (eds.). 1983. Gottlob Frege: Nachgelassene Schriften. Hamburg: Felix Meiner.
Hume, David. 1748. Philosophical essays concerning human understanding. London: A. Millar [This is now widely known as An enquiry concerning human understanding, according to the second edition of 1772].
Kripke, Saul. 1980. Naming and necessity. Oxford: Blackwell.
Monk, Paul, and Tim van Gelder. 2004. Enhancing our grasp of complex arguments. Plenary address to the 2004 Fenner conference on the environment, Australian Academy of Science. [Revised version available at www.austhinkconsulting.com].
Nelson, Leonard. 1905. Bemerkungen über die nicht-euklidische Geometrie und den Ursprung der mathematischen Gewißheit [Remarks on non-Euclidean geometry and the origin of mathematical certainty]. Abhandlungen der Fries’schen Schule (N.F.) 1(3): 373–430 [Reprinted in Nelson (1971–1977), vol. III, 3–52].
Nelson, Leonard. 1906. Kant und die nicht-euklidische Geometrie [Kant and non-Euclidean Geometry]. Das Weltall 6(10–12): 147–155, 174–182, 187–193 [Reprinted in Nelson (1971–1977), vol. III, 53–94].
Nelson, Leonard. 1914. Des fondements de la géométrie [On the foundations of geometry]. Paper at the Paris conference in which the Société internationale de philosophie mathématique was founded [Reprinted with German translation in Nelson (1971–1977), vol. III, pp. 129–185].
Nelson, Leonard. 1917. Kritik der praktischen Vernunft [Critique of practical reason]. Göttingen: Vandenhoeck & Ruprecht [Reprinted in Nelson (1971–1977), vol. IV].
Nelson, Leonard. 1928. Kritische Philosophy und axiomatische Mathematik [Critical philosophy and axiomatic mathematics]. Unterrichtsblätter für Mathematik und Naturwissenschaften, 35(4). [Reprinted in Nelson (1971–1977), vol. III, 187–220. The posthumous publication of this address was preceded by a prologue by David Hilbert and followed by Nelson’s replies to objections by Richard Courant und Paul Bernays].
Nelson, Leonard. 1971–1977. Gesammelte Schriften, 9 vols, ed. Paul Bernays, Willy Eichler, Arnold Gysin, Gustav Heckmann, Grete Henry-Hermann, Fritz von Hippel, Stephan Körner, Werner Kroebel, and Gerhard Weisser. Hamburg: Felix Meiner.
Popper, Karl. 1979. Die beiden Grundprobleme der Erkenntnistheorie. Tübingen: Mohr. English translation: 2008. The two fundamental problems of the theory of knowledge. London: Routledge. [The original German text was written by Popper in the late 1920s or early 1930s].
Rowe, Glenn, Fabrizio Macagno, Chris Reed, and Douglas Walton. 2006. Araucaria as a tool for diagramming argument in teaching and studying philosophy. Teaching Philosophy 29(2): 111–124.
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Nelson, L. (2016). Lecture VIII. In: A Theory of Philosophical Fallacies. Argumentation Library, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-20783-4_9
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