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Class-theoretic paradoxes and
the neo-Kantian discarding of intuition
Christian Onof
Associate Research Fellow
Birkbeck College, London
e-mails: HYPERLINK "mailto:c.onof@philosophy.bbk.ac.uk" c.onof@philosophy.bbk.ac.uk; HYPERLINK "mailto:c.onof@imperial.ac.uk" c.onof@imperial.ac.uk
The influence of Kantian philosophy as stretching to the present day is a well-established fact. One aspect of Kants work which has had a mixed impact is his insistence upon a duality of two types of cognitive content, that provided by concepts, and that residing in intuitions. It was rejected by post-Kantian German idealism and the Neo-Kantians. In the following, I would like to return to the original rejection of Kants concept/intuition dualism. This will shift the focus to consequences, in the foundations of mathematics. I shall look at the impact of the neo-Kantian discarding of the Kantian requirement for an intuitive content as essential in the representation of an object. The present paper has the limited scope of examining the impact upon Freges foundational work in the philosophy of mathematics, and proposing a Kantian interpretation of a way out of the problems it encounters.
Discarding intuition: a brief historical sketch
Kant and the post-Kantians
That the discussion about intuition should have centred around perception is understandable insofar as the main interest is an account of our cognitive relation to a world of objects. But I would like to suggest that if one is to discuss Kants cognitive dualism concept/intuition, one should recall the key rle that intuition plays in the Transcendental Deduction of the categories. According to interpreters ranging from Allison to Heidegger, the transcendental synthesis of the imagination plays a pivotal rle in the deduction. The reason is that it is by providing a unity of the a priori forms of intuition that the categories apply to any manifold given in intuition. More specifically, determinations of time are the product of this synthesis, and this is made possible by the schematisation of the categories. For our purposes, the main point however, is that it is only because there is some a priori form which characterises all intuition that a priori concepts such as the categories can be seen to have objective reality, i.e. in particular apply to manifolds given in outer sense.
Given the central importance of the Transcendental Deduction for the whole first Critique, the a priori forms of intuition would therefore seem to have a key rle in Kants project. And the knowledge which these forms directly guarantee is, according to the Transcendental Aesthetic, mathematical, knowledge. The two consequences I draw from these observations are, first, that discarding the rle of intuition would seem to amount to a rejection of a key tenet, rather than just a dispensable systematic feature of Kants project; and second, that discussing the importance of intuition must involve looking at mathematical knowledge rather than focussing exclusively upon perceptual knowledge. These remarks will guide the investigation in this paper.
The starting point to an examination of the consequences of discarding the duality intuition/concept must be a historical survey of its sources. These lie in German Idealism. The search for an ultimate grounding of all knowledge which triggered Fichtes early work involved an attempt to provide an unconditional first principle. Such unconditionality was not compatible with the endorsement of any dualism lying beyond the scope of the foundation in the first principle. For Fichte, the I as infinite activity produces the not-I, which forms the objectivity against which the finite I becomes conscious of itself. The I must, as a result, understand itself as finite to become aware of its infinite activity, and to grasp itself as infinite to understand its finitude. For Fichte, it is the imagination which enables these seemingly irreconcilable conceptions to be held together. Transcendental imagination, as for Kant, mediates between the understanding and sensibility. For Fichte, it grasps the infinite activity through its inhibition characterised by an intuitive manifold presented to the senses. In turn, this manifold provides that upon which the spontaneity of the understanding can exert itself. So, in effect, the duality between concept and intuition is transformed into a duality of perspectives in the Is self-understanding, a duality which is mediated by the imagination. Kants unquestioning acceptance of a duality of cognitive contents is thus overcome within a system which absorbs it into a duality of perspectives upon the nature of the I, both unconscious and infinite, and conscious and finite. Such a conception of the imagination is adopted by Hegel who sees it as the source of the duality subject/object, and thus as identifiable with Reason itself. Within such a system, there is no place for a proper duality concept/intuition.
The Neo-Kantians
The neo-Kantians, both of the Marburg and the South-West schools, endorsed this post-Kantian rejection of a duality concept/intuition. Historically, this rejection was seen as essential to the rejection of psychologism which characterised both the neo-Kantian schools and Husserlian phenomenology, with its conception of pure logic.
It is interesting however to note the impact that the exclusion of intuition had upon neo-Kantian epistemology. In view of what we saw at the outset to be the rle of the pure forms of intuition in accounting for the application of the categories, both neo-Kantian schools had to contend with the problem of providing a story for how the manifold given in sensation could be structured by pure concepts of the understanding. Since mathematics was viewed by Kant as the science of a priori intuition, we should expect the discarding of intuition to be reflected in a different understanding of the nature of mathematics.
Friedman (2000) describes the different approaches adopted by each school and how these reflect upon the understanding of mathematics. The Marburg school replaced the totally non-conceptual manifold of sensation with an already partly conceptualised manifold. This gets further conceptualised in the methodological progression of mathematical science. As a result, objective reality is defined as the limit of this progression. At any particular stage of the history of science, we only have a partial grasp of the object of knowledge. This amounts to a constructivist notion of what it is to be an object, and to a privileging of the knowledge provided by mathematical science. Such an epistemology of course requires that mathematics itself now be viewed as entirely conceptual, and thereby as belonging to logic. Such a view was made possible by the considerably broader conception of logic developed by the Marburg school, in comparison with Kants own notion of a science which is, ultimately that of syllogisms. In particular, for Natorp, logic includes the theory of relations. This enables Natorp to view the concept of number, which is based upon an asymmetric relation (that of the ordered series of numbers), as belonging to logic.
The Southwest school did not resort to this substitution of the manifold of sensation with a methodological progression of science. Rickert also insisted that the concept of number was not purely logical, although one must remember that the South-West schools notion of logic was practically identical to Kants, i.e. much narrower than the Marburg schools understanding of it. Nevertheless, the problem of accounting for the possibility of mathematics results from the gap left between the manifold of sensation and the logical forms of the understanding. More generally, the South-West schools epistemology exhibits a duality between the realm of sensation and that of logic. The solution proposed by Emil Lask was to take the categories rather than the forms of judgement as primitive. Kants metaphysical deduction of the categories therefore becomes irrelevant: logic is rather now understood as derived through abstraction from the already categorised object. The categorised object is thus taken as defining the realm of value. But the problem of grounding value, i.e. of explaining the binding between the realms of being and of validity, a problem solved by Kant within his epistemological framework by appealing to the transcendental subject, replaces the problem of the duality sensation/logic. The eventual outcome of this problem is that it will lead Heidegger to question the understanding of the subject and the timeless validity which lies at the heart of the philosophical tradition, a consequence which Friedman (2000, p.40-50) carefully unravels.
Russells paradox
Having noted the broadening of Kants conception of logic by the Marburg neo-Kantians (Cohen, Natorp, Cassirer), we also note that the neo-Kantian inclusion of mathematics in logic is in tune with the logicist programme of grounding mathematics on the basis of logic alone. The need for a proper foundation for mathematics became an important preoccupation for mathematicians at the end of the 19th century. During that century, mathematics experienced many developments which spawned foundational problems: the construction of non-Euclidean geometries (Riemann, Lobachevsky), the theory of transfinite numbers (Cantor), Hilbert was thus led to work on the foundations of geometry in the 1890s and later, in the 1920s, develop a formalist approach to grounding mathematics.
The initial work on grounding mathematics, however, saw it as the task of deriving mathematics from another theory, namely that of logic. The inspiration for this task lies in the development of the 19th century work by Bolzano on the grounding of the sciences upon pure logic. At the end of the 19th century, Frege published his work on the foundations of arithmetic. His ambitious attempt to derive arithmetic from logic however encountered a problem of inconsistency which Russell pointed out. Looking at this problem, I shall exhibit how it can be seen as a consequence of the ousting of intuition from epistemology.
This paradox is generally 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 class which is a member of itself, then it is one of its members, i.e. a class which is not a member of itself.
The paradox exhibits a problem for Freges axiomatic insofar as concepts have been exhibited for which one cannot define their extension. In an appendix to the Grundgesetze, Frege asks what are 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? Can we always infer from the extension of one concept's coinciding with that of a second, that every object which falls under the first concept also falls under the second? (Frege, 1964, p.127).
Standard resolutions of the paradox
If one cannot assign either of two mutually complementary (i.e. mutually exclusive and such that either one of these properties has to be instantiated) properties to an object, the first response to the paradox which ensues must involve questioning the nature of the property in question. The property of not being a member of oneself is a property which 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. The most famous solution to the paradox has adopted this approach. This is Russells theory of types. Russell avoids such classes as those 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 am always able to answer the question, as long as I understand how membership of the class is characterised. Such an understanding and the concurrent ability to distinguish circumstances in which the property in question is instantiated or not, are 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 Russells 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 which 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 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 identified.
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 is distinct from, and of more limited applicability than Kants. Indeed, although in mathematics, one might want to accept the identification of the possibility of having an intuitive representation with the possibility of constructing this representation, the intuitionists criterion refers to actual, not only possible constructions. Unlike Kants view, this does not address the fundamental question of what is an object, but only identifies objects constructed out of others, thus relying upon certain objects being given. Whether this is appropriate for mathematics is a question that cannot be addressed here. But if taken as a general characterisation of what can qualify as an object, the view that it has to be constructible out of more basic elements which are unproblematically viewed as having objective status is a view that is closer to phenomenalism (e.g. Ayer, 1978) than to Kant. On such a view, an object is the result of a construction out of the basic objects which are the sense-data we have access to in perceptual experience. This contrasts with Kants view that the postulate for cognizing the actuality of things requires perception () not immediate perception of the object itself the existence of which is to cognized, but still in connection with some actual perception in accordance with the analogies of experience, which exhibit all real connection in an experience in general (A225/B272). So, in summary, while objectivity for Kant lies in the possibility of an intuitive representation of an object that can be conceptualised in conformity with the principles of the pure understanding, the intuitionist approach relies only upon the actual conservation of the property of being an object through processes of construction. With this more limited notion of objectivity, the intuitionist only provides a limited diagnosis of why the property of not being a member of oneself leads to a paradox.
A Kantian approach
The approaches considered above took as their starting point the view that, if a paradox arises from the impossibility of assigning one of two complementary properties to an object, then the problem must lie exclusively in the property. But the solutions led to questioning the nature of the object (classes of all classes that are not members of themselves) itself insofar as this object is defined in terms of this very property. The other approach would therefore consist in directly questioning the nature of this object. This is a Kantian approach insofar as it examines the conditions under which there is an object of knowledge. The resolution which this approach leads to will necessarily be closely related to that proposed in the above realist approaches, for the reason noted that questioning the appropriateness of the property of not being a member of oneself leads to questioning the nature of this object. But its motivation is fundamentally different.
I would like to suggest that it is possible in this way to formulate a solution to Russells paradox which draws directly upon Kants understanding of objectivity. 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, it is the faculty of understanding which is at work in bringing a manifold of representations together. Intuitive representations of red objects can, for instance, be brought under a concept of class-hood. These intuitive representations are conceptualised intuitions (e.g. red objects with certain shapes and sizes, etc), and by bringing them under the concept of class-hood, one is viewing their property of forming a group.
The error of metaphysics, according to Kant, involves making knowledge claims for such cognitive syntheses in the absence of any intuition. This leads to making judgements, the truth-value of which cannot be assessed as they lie beyond all possible knowledge, i.e. the kind of knowledge which is available to a being endowed with a sensible intuition. Importantly, this is not to say that such syntheses of conceptual contents are meaningless. Kant makes it clear that I can talk of the self as having certain properties, even though I have no intuitions of the self.
When we consider the properties being a member of oneself and not being a member of oneself, it is obvious that no intuition is available for such a property. This follows from the fact that our intuition is spatio-temporal, and that an intuition of a member of a class will have to be a spatial part of the intuition of the class, and therefore cannot also be that class itself. Note that this is the intuition (in both senses of the word) which underlies Russells theory of types, although Russell would not acknowledge the spatial nature of such an insight. As a result, to make the claim that I know objects which are defined as classes that are not members of themselves, is to make a claim which is beyond the bounds of possible knowledge. Even though the understanding can grasp the synthesis of the concepts of class-hood and the property of not being a member of itself, it can have no knowledge of objects defined by a synthesis of these two concepts, because there is no intuition that will enable the synthesis of these concepts. This does not, of course, prevent us from having a grasp of the property not being a member of oneself such that, when an object is given, i.e. an intuition is already available conceptualised (under another concept), one can evaluate the truth-value of the claim that it has that property. Hence, one can claim that it is true of the class of all tea-cups that it has such a property.
Before continuing, two issues need to be mentioned to defend the possibility of such a Kantian approach. First, I have appealed to the spatiality of the intuition of sets. The notion that any kind of spatial intuition is required in mathematics is very controversial. The intuition claim however has been defended by a number of authors, and I refer here to Parsonss work (????). The spatial nature of the intuition however needs to be considered carefully, for instance, because of the problem of non-Euclidean geometries. I claim that no geometric notion of space is required here, but merely a topological one. This cannot be defended here, but I refer to the way in which Kant argues for the a priority of space in the Transcendental Aesthetic (A???/B???). Space is a condition for the representation of objects as outside me and distinct from one another. This defines topological requirements which do not involve geometric properties distinguishing Euclidean from other geometries.
Second, and potentially more problematic, is Kants notion of infinity. The possibility of a Kantian approach to set theory requires that the axiom of infinity in the ZF(C?) system be given a Kantian interpretation. A first response to this, is that the paradox dealt with in this paper does not, unlike the Burali-Forti paradox for instance, draw upon the axiom of infinity. However, a Kantian approach to the Russell paradox will only be really fruitful if it is part of a Kantian approach to set theory as a whole. There is no space to address this issue here, but the role of signs in mathematics should be used to go beyond the impossibility of a completed intuition of an infinite set. That is, in the case of the construction of non-finite ordinals such as + 1, an intuition of as a sign for the completed infinity is involved. There is no need for an intuition of the actual infinity of to be part of the intuition of + 1, which bypasses the conflict between a complete intuition and an infinite one. Indeed, such a use of signs is also at work in the construction of finite ordinals, and is operative every time we move from a collection of objects to the set containing that collection. So it is, arguably, consistent with the spirit of the construction of ordinal numbers. This issue cannot be further discussed in the context of this paper.
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The problem of metaphysical thinking as Kant presents it, is that the faculty of reason ignores these restrictions upon possible knowledge. It does so in specific ways which are connected with transcendental illusions that lead the faculty of reason to carry out hypostatizations of the contents of concepts when no intuition is available. These transcendental illusions are themselves unavoidable e.g. the illusion that the I of apperception refers to an entity (Grier, 2001; Allison, 2004).
In the case in hand, the illusion which holds us in its grip is that if a predicate is defined, it is possible to use it to define an object. I shall call it the formalistic illusion. It is not necessarily a widespread illusion in the sense of those identified by Kant, but it can be seen to have emerged in mathematical logic as a result of the extension of logic from Kants narrow syllogistic conception of it to include quantification in particular, as we shall see in more detail below. In the case in hand, the faculty of reason considers the property of not being a member of oneself as definitive of an object. As a result, it can consider the class of all individual objects defined by this property, and this turns out to be a problematic entity.
The transcendental realist solution
If one wants to object that, in the light of the problems encountered with this particular property, one would want to start with given objects and segregate those which have the property, then a class of objects has to be given at the outset, and these objects have to be classes. I can then form the class of all these objects which, as classes, are not members of themselves. Such a class, however, cannot be a member of itself because it is the result of a construction which starts with objects (which are classes), and moves up to classes of such classes in a way which excludes the possibility of the resulting class actually being a member of itself. That is, it is the construction of a class of such objects as distinct from the objects used in the construction which excludes its having the property of containing itself. This way of getting rid of the paradox constitutes the transcendental realist response to Russells paradox, insofar as it assumes that objects are given at the outset.
Insofar as it relies upon constructions of classes of objects which are distinct from the objects belonging to the class, the key may lie either in its being a construction, or in its specifying classes as distinct from the objects they comprise. The first option is, as we saw, that of the intuitionist approach to the problem. It has the disadvantage of constituting a requirement upon all new objects that is not grounded in an examination of objectivity (as Kants is), and leaves it open why the objects used as building blocks are exempt from such requirements.
The second option shifts the focus from the constructive nature of the constitution of new objects to the different types of objects thus generated. If a class is of a different type from one of its elements, the problem is averted. Russells (and Quines) solution is to introduce a hierarchy of types, so that what is described as a construction above amounts to a move to a higher type. The problem with this is that it only works as a solution if one starts off with objects of the same type. That is, when considering the objects as classes of which we isolated those which are not members of themselves, if Russells theory of types is to achieve its result of dispelling the paradox, it requires that the classes that are referred to are themselves of the same type. So, although this approach is appealing as a natural transcendental realist response to the problem in hand, it is ad hoc in that the solution it proposes relies upon the objects which are given at the outset having the kind of property specified in the solution.
The transcendental idealist solution
To understand in Kantian terms how a paradox arises, one must first note that it draws directly upon logical tools which were not available to Kant. Although Kant may have had the notion of universality defined as a function of unity of judgements, he did not have access to the quantifier ( in the variety of its uses. 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, which follows upon Cantors work in the theory of sets, leads to the construction of entities of which transcendental philosophy asks under what conditions they are indeed objects.
I claim that a 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 all such entities 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 . And this class can be viewed as a condition of the conditioned object EMBED Equation.3 .
There is however a difference between this formalistic illusion and Kants hypostatisation of transcendental ideas. Unlike Kants ideas of God, the soul and the world, 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 trees. The availability of such a process is ensured by construction, from the objects themselves to the class of all such objects. In the case of self-referential predicates, such a constructive process is not available. In such cases, the formalistic illusion leads to defining something which is not an object. In line with what we noted above, although we find a resolution of the paradox related to the intuitionists, it is not construction per se which is here definitive of what counts as an object, but the possibility (in principle, that is, in line with the principles of the pure understanding) of some intuitive representation which is conceptually determined as an object (see A156/B195). In the case of classes of objects, the possibility of a construction of the class coincides with the possibility in principle of an intuitive representation of the class.
This resolution of the paradox has the advantage of accounting for the fact that one is able to understand the property EMBED Equation.3 : not a member of oneself, even though one is not able to represent such a property in intuition. The distinction concept/intuition enables us to see that this understanding is conceptual: as long as an object x is already given, I can know the truth-value of EMBED Equation.3 . For such an object, I can have an intuition of it under another conceptualisation, and must then only examine whether this intuition can be brought under concept EMBED Equation.3 . But there is no intuitive representation of an object defined solely through property EMBED Equation.3 , and this gives rise to the formalistic illusion.
This also shows why a transcendentally idealistic perspective is required to resolve the paradox satisfactorily. That is because the paradox results from ignoring the conditions required for something to be an object, i.e. the possibility of an intuitive representation of a manifold determined by a concept. With self-referential properties, the grasp we have of them relies upon an object already being given. By ignoring this fact, the construction of the extension of such a property does not yield an object.
Allison, H. (2004) Kants Transcendental Idealism: An Interpretation and Defense, Yale University Press, Yale
Ayer, A.J. (1978) Language, Truth and Logic, Penguin, Harmondsworth
Frege, G. (1964) The Basic Laws of Arithmetic, University of California Press, Berkeley
Friedman, M. (2000) A Parting of the Ways: Carnap, Cassirer, and Heidegger, Open Court, Chicago
Grier, M. (2001) Kants Doctrine of Transcendental Illusion, Cambridge University Press, Cambridge
Hanna, R. (2002) Mathematics for Humans: Kant's Philosophy of Arithmetic Revisited, HYPERLINK "http://www.ingentaconnect.com/content/bpl/ejop" \o "European Journal of Philosophy" European Journal of Philosophy, 10(3) 328-352
Parsons, C. (1982) Kants philosophy of arithmetic, in Kant on Pure Reason, ed. R.C.S. Walker, Oxford University Press, Oxford, 13-40
Quine, W. V. O. (1980) New Foundations for Mathematical Logic, in From a Logical Point of View. 2d rev. ed., Harvard University Press, Cambridge, Massachusetts
Russell, B. (1996) The Principles of Mathematics. 2d. ed. Reprint, W. W. Norton & Company, New York
Sutherland, D. (2006) Kant on arithmetic, algebra, and the theory of proportions, Journal of the History of Philosophy, 44(4), 533-558
van Stigt, W.P. (1990) Brouwers Intuitionism, North Holland, Amsterdam
It must be mentioned that there is a default suspicion among British and American commentators in particular, towards Kants understanding of mathematics as knowledge involving intuition. This is largely due to the assumption that non-Euclidean geometry and formalism in mathematics must invalidate Kants views on the matter.
More would have to be said about the broader notion of intuition which is implied by the intuitionist move.
Further, we will see that this is more generally the case for classes of objects.
This leads of course to the intuitionists claim that what has not been shown to be true is neither true nor false.
As a result, an object which could (even in principle) never be directly perceived, such as a neutrino, is an object which clearly has a place through laws of empirical connection in Kants theory, whereas its existence for a phenomenalist is less clear. Note that the neutrinos place in the objective world is part of a theory and arguably involves regulative principles.
Note that the transcendental realist has no tools at her disposal to question the nature of objects, and hence would seem to have to consider some more or less ad hoc ways of limiting the applicability of the self-referential property not being a member of oneself.
There is no space here to discuss the sense in which I take even formal mathematics to involve spatio-temporal intuition. I refer to Parsons (1982), Hanna (2002), and Sutherland (2006).
Thanks to . for drawing my attention to the importance of this issue.
Note that one could argue that the potential (syncategorematic) infinite is intuitable. Recalling that the limitations of our senses for Kant are not epistemologically relevant (A???/B????), one can argue that a human being could, with an eternal life, have an intuition of the potential infinite (through enumeration of the natural numbers for instance).
It is one of many developments arising from the formalisation of logic, but it has implications that go further than purely formal innovations such as predicates with several places, i.e. EMBED Equation.3 .
This construction is discussed in more detail below.
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