Kant (vs. Leibniz, Wolff and Lambert) on real definitions in geometry

Abstract

This paper gives a contextualized reading of Kant's theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid's definitions. I argue that Kant accepted Euclid's definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions.

Keywords::

• Kant
• Lambert
• Leibniz
• Wolff
• Euclid
• geometry
• definition

Notes

^1. Of the many passages where Kant aligns himself with Euclid, see for instance, Reflexion 11, where Kant criticizes Wolff for not conducting his proof of Euclid I.29 ‘in the Euclidean way’ (Reflexion 11, 14:52). Wolff aligns himself with Euclid (against his early modern critics, like Ramus) at Preliminary Discourse (§131) (in Wolff 1740). Lambert aligns himself with Euclid (against Wolff, whom he thinks distorts Euclid) in Lambert (1895a, 1915, §§78–§§82). Of course, none of these thinkers hold Euclid above criticism: each rejects Euclid's axiom of parallels, for example, and Wolff, Leibniz and Kant all reject Euclid's definition of < parallel lines>.

Works by Wolff and Lambert will be cited by paragraph number (‘§’). Citations of works of Kant besides the Critique of Pure Reason are according to the German Academy (‘Ak’) edition pagination in Kant (1902). I also cite Kant's reflexions by number. For the Critique of Pure Reason, I follow the common practice of citing the original page numbers in the first (‘A’) or second (‘B’) edition of 1781 and 1787. Passages from Kant's JL (edited by Kant's student Jäsche and published under Kant's name in 1800) are also cited by paragraph number (‘§’). I use the translation in Kant (1992) both for JL and for Kant's other logic lectures. Passages from Kant's correspondence are cited by Ak page number and by date; translations are from Kant (1967).On Euclid's influence on Kant, see Shabel (2003) and Friedman (1992); on Lambert's self-conscious Euclideanness, see Laywine (2001, 2010) and Dunlop (2009).

^2. There are notable exceptions: on Kant's notes on parallels, see Adickes (1911) and Webb (2006); on Leibniz, see De Risi (2007); on Adickes's and Webb's readings of Kant, see note 17.

^3. I discuss Kant's theory of real definitions also in Heis (forthcoming, §I). Two paragraphs in this section – and one paragraph from §2 – appear also slightly modified in that paper.

^4. I follow the common practice of referring to concepts in angled brackets and words in quotation marks.

^5. Kant is speaking loosely when he says that one cannot think a circle without describing it. Indeed, he is at pains elsewhere to insist that it is possible to think of a mathematical concept without carrying out its construction, as for instance a philosopher would if she were trying to prove Euclid I.32 in a merely discursive way: A716/B744. Kant is being more precise when, in a passage parallel to B154, he says: ‘in order to cognize something in space, e.g., a line, I must draw it’ (B138). Kant's point is that one cannot employ the concept < circle> in the uniquely mathematical way – that is, in the way that leads to rational cognition – without carrying out the construction and describing a circle.

^6. Kant characterizes ‘real definitions’ in multiple ways, all of which turn out to be equivalent. In the first Critique (A241-2), Kant says real definitions ‘make distinct [the concept's] objective reality’, which is equivalent to the ‘real possibility’ of the concept (see e.g. A220/B268). Elsewhere, he says that real definitions contain the ‘essence of the thing’ (Vienna Logic, Ak 24:918), which is equivalent to the ‘first inner principle of all that belongs to the possibility of a thing’ (Kant 2004, Ak 4:467, 2002, Ak 8:229). That mathematical definitions are real, see A242; Blomberg Logic, Ak 24:268; Dohna-Wundlacken Logic, Ak 24:760; JL §106, note 2, Ak 9:144; Reflexion 3000, Ak 16:609.

^7. On genetic definitions, see also Reflexion 3001, Ak 16:609. That mathematical definitions are genetic: Reflexion 3002, Ak 16:609.

^8. Dunlop (2012) makes the same point in her discussion of real definitions in geometry.

^9. 26 May 1789, Letter to Herz, Ak 11:53. I quote and discuss this letter in Section 2. In the letter, Kant is referring specifically to the concept < circle>. I take Kant to be further committed to the immediacy condition for all geometrical definitions, not just for geometrical definitions of basic concepts such as < circle>. This further commitment is quite strong, and my reading will require some defense. I return to this issue in the closing section of the paper.

^10. On postulates, see also Reflexion 3133, Ak 16:673; Heschel Logic 87 (Kant 1992, 381); ‘Über Kästner's Abhandlungen’, Ak 20:410-1; Letter to Schultz, 25 November 1788, Ak 10:556.

^11. A24, A47/B65, A163/B204, A239-40/B299, A300/B356. Though Kant does not say explicitly in JL §35 that axioms are theoretical and never practical, I believe that this was his view. See Heis (forthcoming, §3).

^12. By ‘basic’, I mean those concepts whose definitions have postulates as corollaries. Definitions of complex concepts that are composed from basic concepts are genetic, but their corollaries are not postulates, but problems. The concept < triangle> is complex in this sense, since it is defined as ‘a figure enclosed in three straight lines’ (Wolff 1716, 1417) and is therefore composed from the basic concept < straight line>.

^13. For similar definitions of circle, see Wolff (1741, Geometriae, §37, 1710, Introduction §4, 1710, Geometrie, Definition 5). That all radii in a circle are equal is Axiom 3 in Wolff (1710).

^14. It is obvious that Leibniz, Wolff and Lambert each rejected Euclid's axiom because each tried to prove it. The argument that Kant also would have rejected Euclid's axiom would require a fuller defense, which I hope to present on another occasion.

^15. For a more extended interpretation and discussion of these notes, see (Heis forthcoming).

^16. ‘Rectae, quae se invicem ubique habent eodem modo’ (Leibniz 1858, 201, Definition XXXIV, §3). This definition actually is more abstract than the definition suggested in New Essays. Literally, it says that parallel lines are ‘straight lines that everywhere have the same situations with respect to one another’. In §9, he notes that this is equivalent to saying that parallels are equidistant straight lines, but declines to use the definition in terms of distance because he does not yet have a definition of the minimal curve from a straight line to a straight line. But we can ignore that subtlety in this paper, as interesting as it may be.Leibniz almost surely got the idea of defining parallelism in terms of equidistance from Clavius (whose edition of Euclid Leibniz used) and Borelli, whose work he studied closely (see De Risi 2007, 80). On Clavius and Borelli, see Saccheri (1920, 87–91) and Heath's commentary in Euclid (1925, 194).

^17. My reading of these notes differs from that proposed by Adickes, who does not see Kant criticizing Euclid's definition (Ak 14:31), and that proposed by Webb (2006, 230–232), who reads Kant as endorsing a ‘proof’ of Euclid's axiom using Leibniz's definition. For a fuller defence of my reading, see Heis (forthcoming).

^18. On real definitions, see Leibniz (1996, III.iii.15, 19); ‘Meditations on Knowledge, Truth, and Ideas’ (Leibniz 1989, 25–26); ‘On Universal Synthesis’ (Leibniz 1970, 230–231); Letter to Tschirnhaus (Leibniz 1970, 194); Discourse on Metaphysics (Leibniz 1989, §24).In this paper, I feel free to cite passages from Leibnizian works that were unknown in the eighteenth century. Although this could be dangerous in contextual histories like this one, in this paper, it will do no damage. All of the doctrines I am ascribing to Leibniz are clearly expressed in New Essays and ‘Meditations’ – both works that Kant and his contemporaries knew very well.

^19.Discourse on Metaphysics (Leibniz 1989, §24). See also Leibniz (1996, III.iii.18) and Discourse (Leibniz 1989, §26, 1970, 230).

^20. On the following page, Leibniz provides an additional consideration, not found in Kant: that the same concept can have two real definitions, as – for example – an ellipse can be generated either by sectioning a cone or tracing a curve with a thread whose ends are fixed on the foci. He suggests, however, that there will still be one unique most perfect real definition.

^21. See Leibniz (1996, IV.xii.6), supplement to a Letter to Huygens (Leibniz 1970, 250), ‘On Contingency’ (Leibniz 1989, 28) and ‘On Freedom’ (Leibniz 1989, 96).

^22. I think this point helps explain another fundamental difference between Kant's and Leibniz's notion of real definition. Leibniz (1996, III.iii.18; 1989, 26) allows for a posteriori real definitions, since we can know, through experiencing an actual instance, that a concept is possible (Discourse on Metaphysics [Leibniz 1989, §24]). But Kant strongly denies that there can be real definitions of empirical concepts, partly because we can never be sure through experience that we have successfully identified marks that will pick out all instances of a concept (A727-8/B755-6).

^23. A similar complaint against Wolff is levelled by Lambert in 1771 (§11) and in 3 February 1766, Letter to Kant, Ak 10:64.

^24. My understanding of Lambert owes much to Laywine (2001, 2010) and Dunlop (2009).

^25. On necessity of simple concepts, see Lambert (1764, §653–§654, 1915, §36). Lambert alleges that Wolff did not fully appreciate the role of simple concepts (1771, §11–§18, 1915, §26). For Lambert, Leibniz was more cognizant of the importance of simples than Wolff. Still, though, he lacked a sure criterion for distinguishing simples from compounds, and lacked principles that would license (or preclude) simple combinations of simples (Lambert, 1771, §7–§8). From Lambert's point of view, it is striking that Leibniz (1989, 26) will claim that ‘the possibility of a thing is known a priori when we resolve it into its requisites, that is, into other notions known to be possible, and we know that there is nothing incompatible among them’ (see also Discourse [Leibniz 1989, §24]). But how do we know when the resolution has reached ‘simple, primitive notions understood in themselves’ (Leibniz 1970, 231)? And how would we know that there is nothing incompatible among these simples? Certainly not by their definitions.

^26. I do not think, though, that Kant would have agreed with Wolff that axioms are corollaries of definitions. Kant's position is therefore intermediate between Leibniz's and Lambert's. I cannot, however, defend this reading here. See Heis (forthcoming, §3).

^27. See Lambert (1764, §650): ‘the composition of individual marks is a means of attaining concepts and one can proceed arbitrarily insofar as the possibility of such a concept can be proven later (§65ff.). Now as long as the possibility has not yet been proved, the concept remains hypothetical’ (my emphasis). Lambert says that these hypothetical concepts (which presumably have only nominal definitions) can later become derived concepts [Lehrbegriffe] (which would then have real definitions).

^28. That Euclid's definition can be proved without the axiom of parallels (Lambert 1895b, §8); that Euclid's definition is real (Lambert 1915, §81, 1895a, §7–§8, 1895b, §3, §7, §10) and that Euclid's definition is therefore preferable (Lambert 1915, §79, 1895b, §4–§10, 1771, §12, §23–§24).

^29. These are Lambert's examples from (1764, part 1, §63).

^30. ‘The proof of a derivative concept a priori depends on its mode of genesis from axiomatic concepts. An a posteriori proof, however, depends on the mode of genesis of the thing’ (Lambert 1915, §92.9–§10). A derivative concept is a concept whose possibility has been proved from the axioms and postulates. This proof is a priori in the sense that it is a logical proof – done purely syllogistically – from what is conceptually prior: axioms and postulates that exhibit the immediately certain (im)possibilities of combining simples.Lambert's claim that genetic definitions are appropriate only for a posteriori concepts gives an interesting set of contrasting positions. For Wolff, both empirical and mathematical concepts can have genetic definitions; for Kant, only mathematical concepts do; for Lambert, only empirical concepts do.

^31. On < equilateral triangle>, see Lambert (1771, §20, 1915, §79, 1895a, #4); on < triangle>, see Lambert (1915, §79, 1895a, #6); on < parallel lines>, see Lambert (1895b, §3, 1895a, #7).

^32. According to Webb (2006, 219), Kant's view ‘fits Lambert like a glove’. A similar comment (including speculation about the role of Lambert's 3 February 1766 letter to Kant in shaping Kant's thinking) appears in Hintikka (1969, 43–44).

^33. For a penetrating discussion of Lambert's argument, see Dunlop (2009, §5).

^34. As an anonymous referee pointed out, it is noteworthy that in the passage quoted from Lambert (1771, §12), Lambert attributes the generality of the figure to the postulates, and does not mention the axioms. I have been claiming that for Kant, an instance of the concept < parallel lines> constructed according to Euclid I.31 would be general in Kant's sense, only if we are justified in believing Euclid's axiom of parallels. This difference is, I believe, revealing. For Lambert, postulates assert that two or more simple concepts < A>… < N> can be combined (i.e. that there are instances of < A and … and N>), while an axiom asserts that two or more simple concepts cannot be combined (i.e. there are no instances of < A and … and N>). If all that matters for making a definition real is that there provably are instances of the definiendum (as I claim is the case for Lambert), then in general the postulates will be sufficient to show that a definition is real. Axioms, on the other hand, would in general be necessary to show that there are no instances of a concept other than those that meet some condition. For example, we would need axioms to show that though there are instances of the concept < non-intersecting coplanar straights>, the concept < cut by a transversal making interior angles less than two rights> cannot be combined with < non-intersecting coplanar straights>. Postulates would be in general sufficient to show that all straights constructed according to the procedure described in I.31 are parallels. Axioms would be necessary to show that all and only straights constructed in that way are parallel. Lambert's real definitions need the first condition. Kant's real definitions require the second.

^35. See also Lambert (1895b, §11), where he compares giving a proof with solving an algebraic equation, and denies that the ‘representation of the thing’ can play any role in the proof.

^36. That the possibility of only derivative concepts requires proof, whereas axiomatic concepts do not (Lambert 1915, §45, 57, 66, 92.9, 1764, §652). That the possibility of concepts such as < straight line> does not need to be demonstrated (Lambert 1918, §21).

^37. Thanks to Alison Laywine for urging me to make sense of I.1 on my interpretation.