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bbbx(((8`$%B,$$$$$$$$$$$$&hF)$]WWW$$$V%"""W@$$$"W$"""$ H(x"$l%0%")!)")""m$$!X%WWWWDDThe cost of discarding intuition: Russells paradox as Kantian antinomy
Christian Onof - Birkbeck College, London
Kants account of objective knowledge famously uses a dualism of types of representation: intuitions and concepts. This dualism came under renewed criticism from many of his followers. By eliminating intuition, the a priori judgements of mathematics were no longer viewed as requiring syntheses. This means that mathematical knowledge had to be analytic. As a result, its derivability from logic became a major focus of interest at the end of the 19th century. Freges proposal for this logicist programme however led to a paradox identified by Russell. This paper claims that this can be seen as a direct consequence of the discarding of intuition in epistemology. The paper proposes a Kantian solution to the paradox, and offers a diagnosis which identifies in what sense this paradox can be viewed as an antinomy. In this way, the paper seeks to illustrate the scope of the Kantian conception of antinomy against a background of post-Kantian developments in logic.
Paradoxes, Antinomies and Intuition
A number of authors have observed the analogy between the structure of certain paradoxes (set-theoretical in particular) and Kants antinomies. As Priest (1995) indicates, opinions are divided as to whether there is a real similarity here. Priest (1995) himself credits Kant with having identified the underlying pattern of paradoxes connected with the limits of our thought. Kants text favours viewing the antinomies as closely related to mathematical paradoxes: he refers favourably to Zenos resolution of Achilless paradox as providing the proper strategy for the resolution of the antinomies (A502-3/B530-1). It is true that, when it comes to mathematics, Kant states that nowhere () do false assertions disguise themselves and make themselves invisible; for mathematical proofs always have to proceed along the lines of pure intuition, and indeed always through a self-evident synthesis (A424-5/B452). But note the key role assigned to intuition in ensuring this privileged epistemic status of mathematics. This paper seeks to show the correctness of this remarks identification of a key role for pure intuition in avoiding paradoxes such as Russells.
To lend credence to the possibility of viewing certain set-theoretic paradoxes as fitting more or less broadly in the framework of Kantian antinomies, let us recall what the latter involve. In the antinomies, Kant considers opposing metaphysical theses about the world, and proposes arguments for such mutually exclusive conclusions. Kant views these arguments as valid, but their conclusions are incompatible. His solution is that Transcendental Idealism (TI) provides a way out of this paradoxical situation (e.g. A502-A506/B530-B534). The problem with the mathematical antinomies is the transcendental realist assumption that something like the world is given in itself as object of knowledge. On the contrary, TI questions the conditions under which such an object might be given. And its diagnosis is that the world is the posited unconditioned ground of all appearances of outer sense. This notion of world exempts itself from the conditions under which objects can be given according to TI. In particular, no intuition of something like the world as an object is possible since it does not conform to the principles of the understanding that govern all empirical objectivity.
It is the contention of this paper that, through overlooking the crucial role played by intuition in grasping any objective truth, paradoxes have emerged. The paper analyses Russells paradox, standard solutions, and proposes a Kantian approach. This explains why there is no paradox on a TI interpretation, and shows how the paradox arises, thus introducing the notion of formalistic illusion. Although such paradoxes originate in work on the foundation of mathematics, they have a wider role that can be understood as that of a Kantian antinomy.
The foundations of mathematics and Russells paradox
Freges Basic Law V states that the "course-of-values" (i.e. the values of the function for each argument) of the function u is the same as the course-of-values of the function v if and only if ("x) [u(x) = v(x)] (Frege, 1964). In particular, the course of values of a concept C is its extension, and an object o is a member of the extension of C if and only if C maps o onto the True. Russell pointed out a problem with this notion. There are some extensions which are members of themselves, and others not. For instance, the extension of the concept cup of tea is not an element of itself: in Freges terminology, the concept cup of tea would map this extension onto the False, because extensions are not cups of tea. In the case of the concept not a cup of tea, its extension is not a cup of tea: the concept not a cup of tea maps it onto the True and it therefore belongs to itself. Russell then considered the concept extension which is not an element of itself. For such a concept, what can we say of its extension? This extension is a member of itself if and only if the concept maps this extension onto the True. But that means this extension is an element of itself if and only if it is not an element of itself.
This paradox is often stated in terms of classes. There are classes which are not members of themselves. Thus, the class of all objects with the property of being a tree is not a tree. There are also classes which are members of themselves. Thus, the class of all objects which have the property of not being a tree is not a tree. Consider the class of all classes which are of the first type, i.e. not members of themselves. Is this class of the first or second type? If it is a class which is not a member of itself, then it is not one of the classes which are not members of themselves, i.e. it is a member of itself. But if we assume it is a member of itself, then it is, by definition, a class which is not a member of itself.
The paradox exhibits a problem with Freges axiomatic insofar as concepts have been identified for which one cannot define their extension. In an appendix to the Grundgesetze, Frege (1964, p.327) summarises the implications of Russell's paradox: is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases?
Standard resolutions of the paradox
The first response to the paradox involves questioning the nature of the property of not being a member of oneself. It is a property of an object that involves a reference to the object itself. It is a kind of property that is predicated of the object qua class, namely that this class contains the object itself. This is the self-referentiality which is at the core of the paradox.
One option to avoid the paradox therefore consists in forbidding such self-referentiality. Russells theory of types excludes classes that are not members of themselves by arranging propositional functions (i.e. sentences of the theory) into a hierarchy. At the bottom of the hierarchy are sentences about individuals, and as we go up the hierarchy, we find sentences about classes of such individuals, then sentences about classes of classes of individuals, and so on. The theory of types then stipulates that it is only possible to construct classes which contain elements of one type (Russell, 1996). Therefore the question of whether a class contains itself is meaningless in such a theory. This first of all blocks the possibility of classes that are members of themselves. Second, the class of all classes that are not members of themselves is therefore the class of all classes and it is, because of its type, not a member of itself, which dispels the paradox. This solution works, but at the cost of a somewhat ad hoc subdivision of entities into different types. Moreover, it does not reflect the important fact that we can make sense of a class that is a member of itself. That is, if asked whether a class of certain objects is a member of itself, I can answer the question, as long as I understand how membership of the class is characterised. The ability to distinguish circumstances in which the property is instantiated or not, is not accounted for by Russells theory.
Quine (1980) proposes a way out of the paradoxical situation, which does not thus divide entities, and therefore provides a response to the latter criticism. Quine imposes restrictions upon the classes that are allowed to exist. These are namely those whose definition does not involve an infringement of the rule introduced in Russells type theory. That is while not being a member of itself is meaningful, it leads to a defining condition for a class: class of all classes that are not members of themselves which is not acceptable in the context of type theory, and therefore such a class does not exist in Quines system. The criticism of ad hoc-ness does not however disappear as a result of this Quinean variation on the theory of types.
The intuitionist approach in mathematics (van Stigt, 1990) does provide an account of why the property of including all classes that are not members of themselves, is to be rejected as definitive of a proper class. This is simply because no constructive process can be exhibited that yields such an entity. So, in effect, the intuitionist makes it a condition of objectivity that the object of the theory be constructible and Russells paradox is not a problem for him. Such an approach is broadly Kantian in that it takes on board the Kantian idea that objectivity is a construct in which a concept is applied to a manifold. Although it does not refer to the Kantian notion of intuition as such, there is a loose sense in which the construction is a process which is in intuition insofar as it can be exhibited. It does not, however, provide any further diagnosis of why it is that a property such as is not a member of itself does not allow a class to be constructed out of it. The intuitionist solution locates the problem in the impossibility of using this property in a constructive process: this points to a problem in the property itself, but this problem is not thereby characterised any further.
This is not however a shortcoming as such if the intuitionist replies that this is his way of dealing with the notion of objectivity, i.e. an equating of objectivity with that which is constructible in some sense. But such a notion of objectivity comes at a cost. A good deal of traditional mathematics (some of which are useful for applications) are no longer available to the intuitionist (see e.g. Brown, 2008:131,134). There would have to be other grounds for accepting such consequences than the avoidance of Russells paradox. Indeed, today, the Zermelo-Fraenkel axiomatic for set theory, augmented by the axiom of choice (ZFC) is widely accepted. It includes the axiom of separation (or specification) which is designed to avoid Russells paradox.
A Kantian approach
The transcendental idealist solution: dispelling the paradox
The approaches considered above took as their starting point the view that, if a paradox arises from the impossibility of assigning one of two contrary properties to an object, then the problem must lie exclusively in the properties. But the solutions led to questioning the nature of the object (class of all classes that are not members of themselves) itself insofar as this object is defined in terms of this very property. An alternative approach would therefore consist in directly questioning the possibility of this object. This is a Kantian approach insofar as it examines the conditions under which there is an object of knowledge.
In attempting to define an object as something characterised by its being a class and by a further property specifying what kind of class it is, the faculty of understanding has the task of unifying the manifold of representations. Intuitive representations of red objects, whether perceptual or not, can, for instance, be brought under a concept of class-hood. These intuitive representations are already conceptualised intuitions (e.g. of red objects with certain shapes and sizes, from a finite set), because their referring to objects requires that the manifolds in intuition be brought under concepts. By bringing them under the further concept of class-hood, one is viewing their property of forming a class.
When dealing with an abstract object like a class, the question arises as to what the intuition could be. This is clearly related to the broader question of the nature of intuition in mathematics for Kant. Much has been written on this topic (e.g. Parsons, 1993) with respect to geometry and arithmetic in particular. Here, I want to address set theory only. This is sufficient for my purpose in this paper, but would also provide a grounding of mathematics in intuition insofar as axiomatic set theory is nowadays generally taken as foundational for mathematics. I propose to understand the requirement of intuitability as the constraint that one know how to represent any countable (denumerable) set in the form of a Venn diagram. This requirement is understood in principle, so that what is at stake is the existence of a rule which would enable the construction of an a priori intuition which is the Venn diagram representation of the set. This Venn diagram has place holders for the objects in line with the principle behind the representation of numbers in Segners point arithmetic (B15/16; B179). That is, what is at stake is the existence of a schema for the set (A141/B180). The only notion of space which is required here is that whose transcendental function is exhibited in the Transcendental Aesthetic (A23/B38): it enables the representation of objects outside me and as outside one another. I take the second part of the claim to apply to mathematical intuition (the first relates to a space which contains me, i.e. empirical space). This is indeed the pre-condition for the possibility of point arithmetic which involves the representation of a number through points (or strokes for Hilbert) placed next to one another. The synthesis of composition which is at work in generating magnitudes (B201) also governs the generation of Venn diagram representations of countable sets. The difference is that the Venn diagram, insofar as it is taken as a representation of a collection of empirical objects, is an a priori intuition that refers to empirical intuitions of the objects contained in the set. That is, while a number represents a rule for the counting of points of Segners arithmetic, the schema of a set stands for a rule generating a collection of empirical intuitions. A set is an object as long as its concept has such a schema governing the synthesis of composition that generates its representation. This is consistent with viewing numbers as classes of equivalence of sets with identical cardinality (Parsons, 1984:119).
When we consider the properties being a member of oneself and not being a member of oneself, no object can be constructed using such characterisations of a set. These are concepts without schemata. They cannot be constructed in intuition. Note that this argument based upon intuition corresponds to the intuition which underlies Russells theory of types, although Russell would not acknowledge the spatial nature of his insight. Unlike the Russellian and intuitionist approaches however, this resolution of the paradox has the advantage of accounting for ones understanding of property EMBED Equation.3 : not a member of oneself, even though one is not able to represent it in intuition. The distinction concept/intuition thus enables us to see that this understanding is conceptual, so that, as long as an object x is already given, I can know the truth-value of EMBED Equation.3 . This is because, if the object is already grasped under a concept, I can bring the same object under a different conceptualisation using analytical relations between concepts. It is then a simple matter to examine whether this intuition can be brought under concept EMBED Equation.3 .
The error of metaphysics, according to Kant, involves making knowledge claims for such cognitive syntheses in the absence of any intuition (A51/B75). This involves hypostatizations of the contents of concepts when no intuition is available. The lure of such transcendental illusions is a feature of rationality e.g. the illusion that the I of apperception refers to an entity (Grier, 2001; Allison, 2004). TI offers a way of avoiding them. In the present case, the illusion holding us in its grip is that if a predicate is defined, it can be used to define an object, the extension of that predicate. I shall call it the formalistic illusion.
Below, we shall show that the TI resolution of the paradox does not only amount to a way of dispelling it but, unlike the other approaches we reviewed, provides a diagnosis of how the illusion arises. But first, a brief historical note is in order. As Kitcher (1979, 257) points out, Frege diverges from Kant in believing that there are fundamental logical laws which assert the existence of unintuitable objects such as the courses-of-values that lead to Russells paradox. It is interesting that, in the Grundgesetze (Frege, 1964), Frege was aware of his parting from the Kantian tradition on this issue, but did not consider returning to the Kantian tradition when Russell pointed out how the notion of a course-of-values has paradoxical consequences. However, this does not imply that the proposal developed in this paper is fundamentally at odds with Freges enterprise. On the contrary, following Kitchers lead, it is interesting to note that Freges later work (Frege, 1969) suggests grounding arithmetic upon geometry, thus now denying that it can be derived from logic. Arithmetical knowledge is now viewed as having its source in pure intuition (Frege, 1969, 299-301). Importantly, Frege notes that (ibid., 288) the logical source of knowledge can lead us into error. It seems reasonable to speculate with Kitcher (1979, 261) that this is the moral of the Russell paradox. Resolving this paradox with a return to the Kantian notion of intuition can therefore be seen as drawing the consequences of Freges later views for his notion of course-of-values.
The transcendental idealist solution: a diagnosis of what generates the paradox
To understand in Kantian terms how the formalistic illusion arises, one must first note that it directly draws upon logical tools which were not available to Kant. Although Kant may have had the category of universality defined as a function of unity of judgements, he did not have access to the quantifier EMBED Equation.3 . In particular, quantification allows for the definition of classes through predication, i.e. as EMBED Equation.3 or EMBED Equation.3 . This Fregean development of logic leads to the construction of entities of which transcendental philosophy asks under what conditions they are objects.
I claim that, in line with Kants analysis in the Transcendental Dialectic, the formalistic illusion arises as a result of reasons demand for completeness of the conditions under which anything is given, and that this illusion amounts to treating the entities which arise from this procedure as objects. Let us recall that Kant refers to the function of reason in terms of its transcendental concept which is the totality of conditions to a given conditioned thing (A322/B379). If one considers an object EMBED Equation.3 to which the predicate EMBED Equation.3 applies, one can form the idea of the class EMBED Equation.3 of all objects with this property, i.e. the idea of the extension of EMBED Equation.3 . This class can be viewed as unconditioned condition for the conditioned object EMBED Equation.3 .
In so doing, I am using condition and conditioned in a broader sense than Kant, in line with the aforementioned extension of logic beyond Kants exclusive focus upon syllogisms. Indeed, Kant understands the condition of a given judgement, such as the predication EMBED Equation.3 , e.g. Caius is mortal (A322/B378), as the major of a syllogism whose conclusion is this judgement. That is,
I seek a concept containing the condition under which the predicate (the assertion in general) of this judgment is given (i.e. here, the concept human), and after I have subsumed [the predicate] under this condition, taken in its whole domain (all humans are mortal), I determine the cognition of my object according to it (Caius is mortal) (ibid.)
So, Kants condition is given by a predicate EMBED Equation.3 (human) which applies to object EMBED Equation.3 (Caius), through the syllogism: major: EMBED Equation.3 , minor: EMBED Equation.3 and conclusion: EMBED Equation.3
This sequence of conditions of will therefore form a regressive series EMBED Equation.3 in which each term is a judgement that is a condition of the previous term. The totality of all conditions can thus be seen to be the class EMBED Equation.3 .
One way of generalising this process is to construct a sequence of conditions by considering conjunctive properties. The conjunction EMBED Equation.3 is defined by: EMBED Equation.3 . So, consider properties EMBED Equation.3 which hold of x. We can construct the sequence EMBED Equation.3 . This sequence has the same entailment features as the syllogistic one above, namely, EMBED Equation.3 and is thus a regressive series of conditions in the sense of Kants Dialectic. Assume now that EMBED Equation.3 to be the class of all properties which hold of x, i.e. the class EMBED Equation.3 defined by the quantification: EMBED Equation.3 . The totality of all conditions of is therefore the class EMBED Equation.3 . But this is equivalent to the class EMBED Equation.3 , which shows that the identification of the totality of conditions for a given conditioned can be achieved through quantification rather than syllogistic regress.
Having clarified how we can generalise Kants notion of the totality of conditions for a given condition, we must note that, for a given judgement EMBED Equation.3 , there are in fact at least two ways of defining a totality of conditions: one can quantify, as above, over the predicates which apply to object EMBED Equation.3 , or over the objects of which EMBED Equation.3 can be predicated. So here, considering objects EMBED Equation.3 which have property EMBED Equation.3 , we construct composite objects EMBED Equation.3 We then define property F so that a class has property F if and only if all its members have property f. This enables one to define the totality of conditions through the sequence EMBED Equation.3 , which has the required entailment features: EMBED Equation.3 and is thus a regressive series of conditions. Through the stepwise procedure of grouping objects EMBED Equation.3 which share property f, we obtain the extension of f. Positing it is as an object is equivalent to positing that the class of all conditions EMBED Equation.3 , or equivalently, the quantified class EMBED Equation.3 is an object.
The claim that such a totality of conditions for a given conditioned actually identifies an object is not grounded, in either of these cases, for the reasons discussed by Kant at the start of the Dialectic. This is a generalisation of the hypostatisation of the content of the transcendental concept of reason that Kant defines as the totality of all conditions to a given conditioned thing (A322/B379). We note however a difference with Kants hypostatisation of transcendental ideas. This is that the ideas which Kant discusses, i.e. of God, the soul, and the world can indeed not be thought in any other way; as the Dialectic will show, they do not therefore represent objects. So the claim that the totality of conditions for a given conditioned identifies an object is, in these cases, false. It is on the contrary true that the extension of many concepts can form a perfectly unproblematic object of knowledge. Thus, the class of all trees is such an object. Why? Because rather than viewing it as a totality of conditions, I would know in principle how to go about identifying it, i.e. through a process by which all the manifold in intuition is brought under the concept the totality of all (currently existing) trees. The possibility of such a process is ensured by the specification of a procedure of exploration of our planet. In the case of countable classes of objects and other mathematical objects, the existence of a procedure defining intuitability coincides with the constructibility of an intuitive representation. In the case of self-referential predicates, such a construction is not possible however, which leads to a resolution of the paradox related to the intuitionists.
Let us finally note that the formalistic illusion arises from a problem that has a logical form closely related to that of a Kantian antinomy. Let us specify this. In the Kantian antinomy, an incompatibility between thesis and antithesis arises because, from the fact that I always have a self-contradictory concept of the unconditioned synthetic unity in the series on one side, I infer the correctness of the opposite unity, even though I have no concept of it (A340/B398). Thus the thesis and antithesis in turn consider the consequences of their denial and find them untenable, thus concluding to their validity (e.g. A427-9/B455-7). The logical form is therefore that of the modus tollens:
Thesis Antithesis (1)
~H !C1 H !C2
~ C1 ~ C2
__________ __________
H ~H
In the paradox of formalism no intermediary between EMBED Equation.3 and EMBED Equation.3 is required. Starting from the claim that the class of all classes that are not members of themselves is a member of itself, one concludes that it is not a member of itself, and vice-versa:
Thesis Antithesis (2)
~H !H H !~H
__________ __________
H ~H
Because of the analogous form, I shall refer to this as an antinomy, the formalistic antinomy. Because the paradox does not refer to any particular schematised categories, the notion of world that is at stake here is not the empirical world, but is defined by the very feature of quantification that is at stake in the extension of logic beyond Kants syllogistic understanding of it. This extension of logic beyond Kant enables us to consider any number of possible sets, and therefore to consider the set of all sets, which provides a notion of world.
With this notion, we see that the resolution of the antinomy is conform to Kants: the impossibility of a mathematical object such as the set of all sets that do not contain themselves equivalent to that of the set of all sets. Let us briefly show the equivalence between positing these objects: (a) the claim that the set of all sets not containing themselves is an object entails the former, because this requires that there be a set over which the quantification is allowed. The only set which can allow for such quantification is the set of all sets. (b) Conversely, if one considers the set of all sets and the property the property of not containing oneself, the set of all elements x of that set which satisfy this property exists.
Like Kants mathematical antinomies, the formalistic antinomy is resolved through transcendental idealisms denying the existence the world as an object, where the world is here defined as the set of all sets. The diagnosis which TI enables us to carry out with Russells paradox commends it as a metaphysical outlook. Indeed, for a transcendental realist the world of objects is given, so there ought not be any problem in carving it up as one wishes using well-defined predicates. It is difficult to see how transcendental realism can forbid such carving-up without introducing ad hoc restrictions such as Russells theory of types proposes.
Allison, H. (2004) Kants Transcendental Idealism: An Interpretation and Defense, Yale: Yale University Press
Baron, M.E. (1969) A Note on the Historical Development of Logic Diagrams: Leibniz, Euler and Venn, The Mathematical Gazette, Vol. 53, No. 384, 113-125
Brown, J.R. (2008) Philosophy of Mathematics. A Contemporary Introduction to the World of Proofs and Pictures, London: Routledge
Frege, G. (1964) The Basic Laws of Arithmetic, Berkeley: University of California Press
Frege, G. (1969) Nachgelassene Schriften, ed. H. Hermes, F. Kambartel and F. Kaulbach, Hamburg: Felix Meiner
Grier, M. (2001) Kants Doctrine of Transcendental Illusion, Cambridge: Cambridge University Press
Kitcher, P. (1979) Freges epistemology, The Philosophical Review, LXXXVIII, 235-262
Parsons, C. (1984) Arithmetic and the categories, Topoi, 3, 109-121
Parsons, C. (1993) On some difficulties concerning intuition and intuitive knowledge, Mind, 102, 233-246
Posy, C. (2008) Intuition and infinity: a Kantian theme with echoes in the foundations of mathematics, in Kant and Philosophy of Science today, ed. M. Massimi, pp.165-194
Priest, G. (1995) Beyond the Limits of Thought, Cambridge: Cambridge University Press
Quine, W. V. O. (1980) New Foundations for Mathematical Logic, in From a Logical Point of View, Cambridge, Massachusetts: Harvard University Press
Russell, B. (1996) The Principles of Mathematics. 2d. ed. Reprint, New York: W. W. Norton & Company
van Stigt, W.P. (1990) Brouwers Intuitionism, Amsterdam: North Holland
For an interesting survey of the historical significance of Venn diagrams, see Baron (1969). The requirement I propose is for countable sets only because the constructability of a corresponding intuition requires a rule, which can only be based upon iteration of an identical process. For non-countable sets, construction procedures based upon countable sets will be available (for instance Cauchy sequences or Dedekind cuts for the real numbers). But these sets will not be intuitable.
This is the case even if the synthesis cannot be completed (see next footnote).
The terminology mathematical object is used loosely here; it could in principle be any referent of a mathematical symbol. A typical complaint against a Kantian approach to set theory is that it cannot account for infinite sets. In fact, for countable sets, a rule of construction of the Venn representation of such a set is available. This rule is such that, although the whole set cannot be intuited, there is nothing unknown about any part of the intuition to be cons6FGHIstR s } 5
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