Intuitions, Concepts and Wholes

Transcription

1 45 Intuitions, Concepts and Wholes Guillermo E. Rosado Haddock Universidad de Puerto Rico - Río Piedras guillermo.rosadohaddock@uprrp.edu Abstract: Kant s conception of mathematics as constructible in pure intuition is based on his arguments in the Transcendental Aesthetic on behalf of the intuitive nature of space and time. It is here shown that Kant s conclusion is completely unfounded, since one can reproduce those arguments on the basis both of the concept of a continuous manifold in Riemann s sense and of that of an extensive whole in Husserl s sense. Key words: pure intuition; Kant; wholes; sets. 1 Introduction In the Kritik der reinen Vernunft and very especially in the first part of the Doctrine of Method 1 Immanuel Kant argued for the constructibility of mathematical objects in pure intuition. According to Kant 2, philosophical knowledge is supposed to be rational knowledge from concepts, whereas mathematical knowledge is rational knowledge from the construction of concepts in pure intuition. Since intuitions are 1 Kritik der reinen Vernunft, A713f., B741f. 2 Ibid., A713, B741.

2 46 always singular, mathematical knowledge considers the general in the particular. Kant s conception of intuition had already been elucidated in the first part of the Critique, namely, in the Transcendental Aesthetic, especially in the Metaphysical Exposition, where he argued on behalf both of the a priori nature of space and of its intuitive nature. 3 We are not concerned here with the very questionable a priori nature of space, against which so much has been said and can be said, but with its intuitive nature. On this presumably intuitive nature of space (and of time) is based Kant s contention already mentioned that we construct mathematics in our pure intuition of space and time. In what follows we will first expose Kant s argumentation in his terms and then later show that the argumentation does not establish what Kant thinks it does. 2 Kant s Arguments In the third and fourth of the four arguments offered by Kant in the Metaphysical Exposition on behalf of the nature of space in the second edition of the Kritik der reinen Vernunft which correspond to the fourth and fifth arguments of the first edition - Kant wants to establish that space is not a discursive concept but an intuition. According to the first of the two arguments, one can only have a representation of a single space, and when one considers different spaces, one is solely considering parts of one and the same space. Moreover, such partial spaces cannot precede the unique space of which they are partial components, since they can only be considered as parts of the unique space, as sorts of internal delimitations of the unique whole space. Any concept of space would then be derivative, based on the a priori unique intuition of space. The second of the two arguments with which we are concerned contrasts our representation of space as that of a unique magnitude containing infinitely 3 For our purposes, see especially arguments 3 and 4, A25, B39-40.

3 47 many parts with that of a concept, which is a representation that seems contained in a possibly infinite number of single representations like the concept of horse in any single horse -, not one which contains an infinite number of partial representations, as is the case of space. The distinction envisioned by Kant in the two related arguments is made clear when one considers the traditional view of concepts, which Kant certainly presupposes. An empirical concept, like that of being a horse, is a representation present in an indefinite number of representations of concrete horses, not a representation that contains representations of different horses as parts obtained by delimitation or restriction of the representation of the concept of a horse. The same happens with non-empirical traditional concepts, like that of a natural number. The concept of natural number is not such that it contains the representations of the different natural numbers as restrictions or delimitations of the general representation of a natural number, as parts of this latter representation, but is in some sense contained in each representation of a single natural number in the same way in which the concept of a horse is contained in the representation of any singular horse. 3 A Few Remarks on Concepts and Manifolds Notwithstanding his genius, Kant was limited by a philosophical and scientific tradition that was soon to collapse. The nineteenth century saw revolutions (i) in physics with the advent of electromagnetism, (ii) in mathematics with the rigourization and arithmetization of analysis, the origins of abstract algebraic systems and, more importantly for our discussion, the advent of non-euclidian geometries, a fact that severely questions the relevance of intuition in mathematics, and finally (iii) in logic, a revolution that began with the work of George Boole and

4 48 culminated in the epoch-making contributions to logic of Gottlob Frege. Most importantly for our purposes, our present conception of concepts has little to do with the very restricted traditional view presupposed by Kant. Thus, we now speak of the mathematical concepts of a group or of a topological space, though they have little to do with traditional concepts as that of horse. In fact, already in the mid-nineteenth century one of the most distinguished mathematicians ever propounded the eradication of intuition from mathematics and its replacement by a broader understanding of the notion of concept. In his epoch-making monograph Über die Hypothesen, welche der Geometrie zugrunde liegen 4 the great Bernhard Riemann, without mentioning Kant by name but certainly having him in mind, questioned the traditional views on space in a most radical fashion. Not only did he consider physical space as just a particular case of the much more general concept of an n-fold extended magnitude, but made it perfectly clear that as to the determination of the exact nature of physical space no a priori considerations are available, but only empirical ones. Moreover, he neglected sensible intuition any role in geometrical considerations, obtaining the notion of an n-fold extended magnitude from a more general notion of magnitude. In fact, he drew 5 a very general division between the concept of a discrete manifold and that of a continuous one. He called 6 quanta the particular portions of a manifold, elements the quanta of discrete manifolds, and points those of the continuous ones. Moreover, he stressed that in the case of discrete manifolds, the quantitative comparison of quanta is obtained by counting, whereas in the case of continuous manifolds, the quantitative comparison of its quanta is obtained by measuring, which involves the superposition of portions of the manifold and, thus, presupposes the possibility of transporting a quantity used as a standard for the remaining quantities. Under that general purely conceptual division, the set of natural numbers 4 Über die Hypothesen, welche der Geometrie zugrunde liegen. See already p Ibid., p Ibid.

5 49 would certainly be a discrete manifold as well as the set of all horses. Space (as well as time), be it physical, intuitive or whatever, would certainly seem to be a continuous manifold. 7 Thus, the difference made by Kant in his argumentation on behalf of the intuitive nature of space (and time) can now be very easily explained using Riemann s distinction by pointing out that space is a continuous manifold, not a discrete one, and such that portions of space are obtained by division of the continuous manifold, and all parts and parts of parts of a continuous manifold are of exactly the same nature as the manifold. In contrast, in discrete manifolds, in which the comparison of different quanta is obtained by counting, there is no subdivision in parts homogeneous to the whole discrete manifold. Contrary to Kant s views, there is nothing intuitive about space, just a sort of general concept different from the more traditional concepts dealing with discrete manifolds. More generally, there is no such construction of mathematical objects in the pure intuition of space (and of time), as alleged by Kant: there is no ground for mathematical intuitionisms and constructivisms. 4 Brief Remarks on Sets and Wholes With the development of set theory in the second half of the nineteenth century and the beginning of the twentieth century, and its decisive role as the sole foundation of classical mathematics, in some sense the distinction between discrete and continuous manifolds lost some of its importance. Thus, one frequently spoke both of the set of real numbers, to which the number belongs as a member, and of the continuum of real numbers, to which is also supposed to belong. The doubleduty is most clearly present in general topology. A topology T is a family (i.e., a set or 7 Nonetheless, the great Riemann admits the possibility that in the infinitesimally small things could be different, not excluding the possibility that at the microcosmic level physical space could be discrete. See ibid, p. 20.

6 50 collection) of sets, usually called open sets. For any point in a topological space one can consider multiple neighbourhoods, namely, open sets containing the point. But this containment is ambiguous. The neighbourhood can be rendered as a collection of points or as a sort of region of which the point is an extremely small part. Moreover, open sets usually have accumulation points, and the set of all accumulation points of an open set O constitute what is sometimes called the derived set D of that open set. Together an open set and its derived set build what is usually called a closed set O c, which is the closure of the open set O. But the derived set can be rendered either as the set of its accumulation points or as a whole composed of all its accumulation points as parts. In some sense, the distinction between member of a set and part of a whole is washed out. Of course, one could counter-argue that the notion of continuity, though a topological notion, is not needed when considering only one topological space. However, when it is introduced, the notion of continuity does not change anything in the ambiguity present in general topology between discrete and continuous. The distinction pointed out by Riemann between discrete and continuous manifolds seems to have its roots much higher than the topological concept of continuity. Maybe we should look at the distinction between set and whole for some help. We are not going to discuss this issue in any detail. But it should be pointed out that some of Cantor s contemporaries seem to have quarrelled a little with the distinction between those two notions. Both Husserl 8 and more forcefully Frege 9 in their respective critical reviews of Ernst Schröder s Vorlesungen über die Algebra der Logik I accused Schröder of confusing the notions of set and whole. While Frege did not see any mathematical future for the notion of a whole, his younger rival dedicated the whole Third Logical Investigation to a very general discussion of a 8 See Husserl s Besprechung von E. Schröder, Vorlesungen über die Algebra der Logik I, 1891, reprint in Aufsätze und Rezensionen ( ), Hua XXII, 1983, pp. 3-43, especially pp See Frege s Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra der Logik, 1895, reprint in Kleine Schriften, revised edition 1990, pp , especially pp. 196, 198 and 210.

7 51 theory of parts and wholes. Of course, in that Investigation there is a lot that does not fit under a mathematical treatment of wholes, being concerned mostly with what Husserl considered as belonging to the synthetic a priori, Nonetheless, besides such considerations and the definition of synthetic a priori, there is also a treatment of more formal analytic aspects of wholes, including in 12 - Husserl s definitions of analyticity and of analytic necessity. More importantly for our purposes, in his philosophy of mathematics from approximately 1894 or 1895 on Husserl not only clearly distinguished between set and whole, but considered that a theory of parts and wholes had to be developed as one of the, to use some Bourbakian nomenclature, mother structures of the whole of mathematics on the same level as set theory and number theory. One should note that Husserl, though a structural Platonist, was no reductionist at all, and that set theory did not have for him any privileged status among the most fundamental mathematical disciplines. It should be mentioned here that the great Polish logician Stanislaw Lesniweski, certainly under some Husserlian influence, developed a theory of wholes and parts, which he called mereology, though, probably due to his nominalist leanings, as a sort of logical theory that would replace the presumably more ontologically committed set theory, not as another part of formal ontology at the side of and with equal rights as set theory, as conceived by Husserl. As a sort of final note, let us remember that in the Third Logical Investigation Husserl briefly discussed a less general but still very general theory of wholes, namely, the theory of extensive wholes, 10 which are such that the subdivision of the whole in parts can follow indefinitely without there being a determined hierarchy of parts of parts being nearer or farther from the whole or, in other words, a part of a part of a part of a whole does not need to be the result of an iterated division of the whole but could very well be the result of a first division. Interestingly enough, 10 Logische Untersuchungen, U. III, 17

8 52 Husserl mentioned spatial and temporal lines as examples of extensive wholes. 11 It would be a routine exercise for someone interested to show that Kant s arguments on behalf of the intuitive nature of space (and time) can also be easily expressed in the language of extensive wholes without any trace of reference to anything intuitive, simply by pointing out that space and time are extensive wholes, not sets in the restricted but precise sense in which sets are sets of discrete objects. A more general and bold enterprise, which I envisioned during my student days but never undertook, and which would require much fresher knowledge of mathematics and more spiritual strength than I presently have, is to develop mathematical analysis as exclusively based on the notion of extensive whole. References Cantor, G. Abhandlungen mathematischen und philosophischen Inhalts, Georg Olms, Hildesheim 1966 Frege, G. Die Grundlagen der Arithmetik 1884, Centenarausgabe, with an Introduction by Christian Thiel, Felix Meiner, Hamburg 1986 Frege, G. Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra der Logik, 1895, reprint in Kleine Schriften, Georg Olms, Hildesheim 1967, revised edition 1990, pp Husserl, E. Besprechung von E. Schröder, Vorlesungen über die Algebra der Logik I, 1891, reprint in Hua XXII, M. Nijhoff, The Hague 1979, pp Husserl, E. Logische Untersuchungen , Hua XVIII & XIX, Martinus Nijhoff, The Hague 1975 & 1984 Kant, I. Kritik der reinen Vernunft 1781, revised edition 1787, reprint of both editions, Felix Meiner, Hamburg 1930, third edition 1990 Lesniewski, S. Collected Works I, Kluwer, Dordrecht et al Ibid.

9 53 Riemann, B. Über die Hypothesen, welche der Geometrie zugrunde liegen 1867, third edition, Berlin 1923, reprint in Das Kontinuum und andere Monographien, Chelsea, New York 1973