by Jacob Ritz
In the Critique of Pure Reason Immanuel Kant accounts for the possibility of synthetic a priori knowledge[i]by separating appearances from things-in-themselves, thereby limiting cognition to the bounds of experience. In response, it will be argued that while Kant successfully resolves the impasses of classical metaphysics, his solution unnecessarily limits thought to appearance. Alternatively, we will see that set theory is an example of synthetic a priori knowledge irreducible to appearance, thus revealing the possibility of post-critical ontology.[ii]
Kant’s aim to secure the validity of synthetic a priori knowledge originates in Hume’s problem of induction, which states that a necessary connection between a cause and an effect cannot be deduced a priori from the repeated observation of contingent instances. Instead, Hume argues that we create associations by habit due to a repeated succession of perceptions.[iii] The origin of causation is thus identified with the constant conjunction of cause and effect in experience rather than guaranteed by deductive reason. Given that causation is not the only pure concept which we employ, but in fact “metaphysics consists altogether of such concepts”, Kant is quick to recognise the significance of Hume’s argument.[iv] Although Kant concedes that Hume successfully problematises the pureconcepts of ‘dogmatic metaphysics’ since experience “never gives its judgements true or strict but only assumed and comparative universality,” he realises that we are now unable to distinguish the subjective and objective order of cognitions so that our knowledge of ourselves and our knowledge of the way the world appears to us become ultimately indistinguishable.[v] To adequately resolve these ambiguities, Kant finds it necessary to separate appearances and things-in-themselves.
The separation of appearances and things-in-themselves aims to disentangle the ambiguities of Hume’s empiricism to justify the possibility of synthetic a priori knowledge. While Kant concurs with Hume that, in the context of appearance, we can only observe contingent events, he claims that the objective reordering of the subjective succession of cognitions is a synthetic reorganization of the empirical order of perception.[vi] That is to say, if we are to get from the raw material of our immediate sense experiences to a structured experience of reality, we must coherently unite our observations through a synthesis. Thus, cognition and sensation are irreducible to each other.[vii] This is illustrated in the distinction between intuitions (immediate representations) in the sensibility, which comprises the content of our sensations, and concepts (mediated representations) in the understanding. Kant then separates pureintuitions from the content of sensation, arguing that “that in which alone sensations can be ordered…in a certain form cannot itself be sensation again.”[viii] In other words, when we are gathering our sensations together into an intelligible form, the form which we find most suitable to order these sensations cannot be the sensations themselves; the word ‘horse’ is not itself a literal horse. This legitimates the claim that the “pure form…in which everything manifold in the appearances is intuited…must be found a priori in the mind” which enacts synthesis or “the act of putting different representations together, and of comprehending their manifoldness in one item of knowledge.”[ix] Thus, the understanding and sensibility are only able to constitute appearances when employed in conjunction, while things-in-themselves are a bare regulative concept that we can know of but not about.[x] For instance, when I hold a book, I may detect many of its qualities such as its colour, its smell, its weight and so on. However, I never gain knowledge of the book itself, only knowledge about it via the qualities which it possesses. Given this synthetic principle that “every different empirical consciousness must be combined in a single self-consciousness,” Adorno states that in Kant’s “system the subject becomes…the guarantor of objectivity.”[xi] In nuce Kant shifts from traditional ontology, concerned with objects’ constitution, to a transcendental philosophy concerning “our manner of knowing objects insofar as this manner is to be possible a priori.”[xii] Causality is thus not a concept found in experience, as Hume thought, but is brought to experience by an act of a priori synthesis. However, synthesis relies upon sensible intuitions for content, so Kant opens himself up to the criticism of having reduced the source of knowledge to a merely empirical process. To avoid this, Kant constructs a sphere which unites both our intuition and the a priori.
The transcendental aesthetic unites intuitions and the a priori in space and time, explaining both the intuition of sensuous reality and the validity of synthetic a priori knowledge. For Kant, the pure concepts, which describe synthetic a priori knowledge, are coterminous with the pure intuitions of space and time given that the empirically given (which is spatio-temporal) is organised by the synthetic activity of our understanding. He argues these are pure because to intuit sensations as outside and alongside one another or as simultaneous or in succession, space and time must be posited a priori.[xiii] Yet it is only possible to speak of pure intuitions from a human perspective, since to subtract from subjectivity would nullify their meaning.[xiv] Kant therefore separates pure intuitions from things-in-themselves to guarantee the objectivity of synthetic a priori knowledge in appearance. Moreover, Kant refutes Hume’s claim that physical objects are “a cluster of qualities existing in some mind” because he holds that our finite intellect is unable to make objects exist ex nihilo and therefore he retains a concept, albeit problematic, of reality behind appearance which he calls the noumenon.[xv] Consequently, since the objectivity of cognition is bound by possible experience, “… the legitimate domain of metaphysics is sharply limited.”[xvi] However, is such a restriction ultimately necessary?
Kant separates things-in-themselves and phenomena in order to replace theocentric metaphysics with a theory of cognition. This replacement is possible because Kant restricts the noumenon to being a regulative concept which removes ideas of transcendence (for instance God or Freedom) from the realm of objective knowledge.[xvii] Thus, if reason attempts to determine suprasensible rather than merely sensible objects, it will exceed the bounds of possible experience finding only illusion.[xviii] This departs from a theocentric to an anthropocentric model of cognition which preserves the objectivity of scientific truth in appearance.[xix] For context, a key motivation for Kant carrying out his philosophical project was to provide a metaphysical basis for Newtonian physics, although without uncritically accepting Newton’s philosophical presuppositions.[xx] He thus wishes to renovate philosophical thinking so we may provide a rational justification for the novelties of modern physics, in contrast to pre-modern European philosophy which positioned itself principally as the handmaiden of theology. Adorno clarifies this paradigm shift, stating that Kant “commands reason to stop…to think the Absolute…[and] refuses to recognize any truth apart from scientific truth”.[xxi] Despite this project that he has embarked upon, Kant admits that he must “deny knowledge in order to make room for faith” since he is unable to assume “God, freedom, immortality” for use in practical reason unless he “deprive[s] speculative reason of its pretension to transcendent insights…”[xxii] Ironically, Kant maintains an important place for religious thinking by restricting theoretical reason from things-in-themselves, despite the enlightenment goals he set for himself.
Does transcendental philosophy then truly fulfill his Enlightenment ideal of maturity where we moderns leave behind the ‘childish’ beliefs of tradition and fully embrace reason?[xxiii] We have already established that Kant limits theoretical reason to the sensibility given that “all intuitions are extensive magnitudes” and further that the “principle for all intuition is…a principle for all (axioms) of mathematics.”[xxiv] That is to say, Kant holds that the theoretical knowledge which we construct in mathematics is ultimately dependent upon the empirical knowledge we have through observations of the world. However, if we look at set theory, the formal theory of multiples, a central axiom (or fundamental assumption) is the existence of the empty set, which is a set without any elements.[xxv] In other words, the empty set is a multiple without extension, and it is precisely the axiomatic positing of its existence which constitutes the rational construction of modern mathematics i.e., synthetic a priori knowledge. This is because to construct sets, we must begin a counting process, which begins with the set containing the empty set {Ø} (the counting of nothing), and then next counts the set containing both the empty set and the set containing the empty set {Ø, {Ø}}, and so on. The entire hierarchy of mathematical objects is thus constructed from a multiple without extension. Therefore, Kant’s claim that the empty intuition is a “mere fancy”, or ens imaginarium, appears to privilege presence over absence i.e., his system explains how we gain knowledge so that meaning becomes directly accessible through intuitions of objects such as our sense of red or blue. [xxvi] He thus forecloses the possibility that meaning is not directly given as in the case of the empty set, which ultimately has no meaning besides its status as a pure symbol. Instead, Kant privileges empirical knowledge as a fullness which grants us meaning in the world, thus reflecting the remnants of a religious thinking (as he seeks to ground human thought in some ultimate meaning) in his otherwise radically modern philosophy.
According to Kant’s understanding of knowledge, set theory is an example of synthetic a priori knowledge that presents an ontology irreducible to appearance because its only existential claim is that of the empty set axiom. One may thereby preserve Kant’s critical breakthrough without limiting reason to experience. Since human cognition requires the conjunction of concepts and sensible intuitions, Kant writes that “… space is nothing once we leave out of consideration the condition of the possibility of all experience…”[xxvii] Allison corroborates this in stating that, “Kant affirmed the strong thesis of the non-spatiotemporality of [things-in-themselves] …”[xxviii] In other words, if we leave the realm of experience, space and time will ultimately have no meaning as they describe relations among objects (which must appear in some space at some time in the first place). In terms of modern set theory, the transcendental ideality of space and time acquires a positive sense with regards to the empty set axiom: there exists a set such that no elements (extensive magnitudes) belong to this set.[xxix] In other words, I have a singular representation of a multiple without extension when I write the empty set, Ø. For example, say that I intuit a red ball and a blue ball, and if someone were to ask me the ways in which I may group these, I would have the case in which I have only the red ball, only the blue ball, both the red and blue ball, and finally the case in which I have neither ball. Therefore, the empty set is the intuition of a set without any elements or, in Kant’s terminology, a manifold without extension.
Set theory thus derives synthetic a priori knowledge from an object of non-sensible intuition i.e., it affirms the positive use of noumena in contrast to Kant who only admits a negative use, believing that “we cannot in any manner attain knowledge of [them].”[xxx] So, although Kant holds that pure categories provide analytic judgements about things-in-themselves, this can never be synthetic a priori knowledge.[xxxi] Correspondingly, he critiques the empty set before it was formalised in modern mathematics, stating that “nothing can enter into experience that would prove a vacuum,” for “…the void…lying outside the field of possible experience…can never come before the tribunal of the understanding, which has to decide only on such questions as concern the use to be made of given appearances…”[xxxii] Therefore, Kant holds that we are unable to develop concepts or synthetic a priori knowledge if our point of departure is the empty set because, as he acknowledges, we are unable to experience a vacuum. The understanding is thus limited to this intuited manifold.
In a significant passage of the Critique, Kant argues the intuited manifold is unable to be determined to be finite or infinite, so is ultimately indeterminant. [xxxiii] Alternatively, Georg Cantor, the founder of modern set theory, posits the existence of two kinds of definite multiples: consistent and inconsistent, where the former may be gathered into a unity and the latter may not.[xxxiv] Put simply, consistent multiplicities are those which may be formalised in the context of mathematics and are thus consistent with rationality, while inconsistent multiplicities are those absolutely infinite ‘objects’ which defy rational formalisation. Following further developments of Cantor’s original ideas, Bertrand Russell discovered the eponymous paradox where we cannot have a set (unity) of all sets, given that this would imply the set is included in itself which would lead us towards an infinite regress, a mathematical mise en abyme, which resists formalisation. Therefore, since the universe of sets is no longer able to be counted as a set or gathered into a unity, what is being counted is now understood to operate in the context of inconsistent multiplicity. The empty set therefore cannot be understood to be the beginning of a complete formal system where every element has its rightful place because the process of counting cannot presuppose or even anticipate a final unity. Instead, given that we can only presuppose inconsistent or unity-less multiplicity, for the counting process to even begin unifying sets, it must first count nothing as something, i.e., Ø is the mark of nothingness. Set theory thus constitutes synthetic a priori knowledge of ‘pure and merely intelligible objects’ by positing the unity-less empty set, in contrast to Kant’s reliance on the synthetic principle to guarantee the objective validity of pure concepts.[xxxv] In other words, set theory assumes that we may coherently name nothingness and then develop a rational understanding; the implicit ontology of set theory relies on a notion of nothingness which refers to the emptiness of actually existing things. Whereas Kant argues that there is an active principle (synthesis) which is able to unify our experiences and is thus the fundamental basis of our knowledge of the world; Kant’s ontology supposes the existence of a principal which is above actually existing things. This corroborates Adorno’s claim that Kant seeks to “salvage transcendence by concealing its existence at the heart of subjectivity,” indicating that, ironically, by ‘de-ontologising’ Kant’s epistemology (i.e., ridding ourselves of synthesis as our ultimate assumption) we may return to ontology, albeit not in the guise of classical metaphysics but, as we shall see, to a rational ontology of multiplicity.[xxxvi]
Admittedly, Kant’s mathematical context limits him to the content provided by Euclidean geometry, a mathematics whose assertions, such as “a straight line is the shortest distance between two points”, necessarily appeal to spatial intuition and are thus only valid for appearances.[xxxvii] However, since set theory can formalise all geometries and be a “foundation of everything,” Kant’s belief that a concept of things-in-themselves cannot determine objects is seriously problematised.[xxxviii] Badiou summarises this in stating that Kant “takes great pains to avoid…intellectual intuition,” but “skirts the truth…[because] the concept of object designates the point where phenomenon and noumenon are indistinguishable.”[xxxix] In other words, on the one hand, the general concept of an object is a pure concept of the understanding because a ‘general’ object is nowhere to be found in the empirical world. While, on the other hand, the concept of object can only pertain to phenomenality as such. Importantly, set theory respects Kant’s break with substance metaphysics because all sets are composed of the empty set and thus no primary substance is posited to re-substantialise things-in-themselves. Badiou thus asserts that set theory is ontology and that it uncovers “the structure of beings qua beings by taking the objects that constitute the content of thought and stripping them of every possible predicate…leaving nothing but multiplicities without unity.”[xl] In other words, Badiou takes the objects of set theory in order to abstract from the empirical qualities of our experience, leaving nothing behind but that which is describable by mathematics; the purely intelligible background upon which our experiences flourish. Consequently, we must admit that Kant’s synthetic unity (or counting) ontologically presupposes the empty set, indicating that:
…Nothing qua [the empty set] exists as the sole index of existence… [and is] a consolidation and radicalisation of Kant’s own subtractive gesture: it converts the thing-in-itself as external limit of empirical knowledge into the [empty set] as internal limit of mathematical knowledge[xli]
Thus, rather than reducing the noumenon to a limiting concept and denying its conceptual intelligibility “to keep the claims of sensibility within proper bounds”, noumena qua empty set become the minimal ground of a post-critical ontology of multiplicity.[xlii] That is to say, even though Kant revolutionised philosophy by putting it in permanent dialogue with the empirical sciences, an ontology which does not forsake rationality remains possible. This is because the multiples which compose the world may be accounted for merely by a process of counting that presupposes only a multiple without extension (or nothing). This thematic shift from critical epistemology to set-theoretic ontology reaffirms Kant’s radical break with classical metaphysics by maintaining beings’ insubstantiality while liberating reason from appearance.
To conclude, Kant provides an adequate justification for synthetic a priori knowledge within his philosophical context, as he rejects empty metaphysical concepts to affirm that the objectivity of appearance reflects rules of cognition. However, following Cantor’s set-theoretic revolution, to restrict cognition to appearance becomes unnecessarily limiting, since theoretical knowledge may be founded on the non-sense of the empty set rather than limited to sensible appearances. In this essay, I have demonstrated that we may maintain a philosophical approach which remains faithful to the ideals of scientific rationality without resorting to the religious metaphysics that Kant fought so hard to escape.
Jacob is an honours student in pure mathematics currently focusing on mathematical logic at the University of Queensland. His interest in early modern philosophy, 19th century German philosophy, 20th century French philosophy and the analytic tradition led him to study French and German language and culture, and further pursue his studies in mathematics.
ENDNOTES
[i] Non-trivial assertions which are valid independent of experience.
[ii] By post-critical ontology, we mean an ontology following Kant’s philosophical turn towards the mere consideration of the bounds of our knowledge of the empirical world.
[iii] David Hume, Treatise of Human Nature (NY: Prometheus Books, 1992), 265.
[iv] Immanuel Kant, Prolegomena to Any Future Metaphysics (Indianapolis: Hackett Publishing, 2001), 5.
[v] Immanuel Kant, Critique of Pure Reason (London: Penguin Publications, 2007), B4.
[vi] Kant, Critique of Pure Reason, A102/B138.
[vii] Dieter Henrich, Between Kant and Hegel: Lectures on German Idealism (Cambridge: Harvard University Press, 2004), 35.
[viii] Kant, Critique of Pure Reason, A20-21/B34-35.
[ix] Kant, Critique of Pure Reason, A20/B34-35, B102-103.
[x] Lucien Goldmann, Immanuel Kant (London: Verso Books, 2011), 134.
[xi] H.E. Allison, Kant’s Transcendental Idealism: An Interpretation and Defense (US: Yale University Press, 2004), 167; Theodor Adorno, Kant’s Critique of Pure Reason (US: Stanford University Press, 2001), 33.
[xii] Kant, Critique of Pure Reason, A11-12/B25.
[xiii] J.W. Ellington, “Introduction,” in Prolegomena to Any Future Metaphysics, by Immanuel Kant, ix-xv, (US: Hackett Publishing. 2001), xii; Kant, Critique of Pure Reason, A24-25/B39, A31-32/B46-47.
[xiv] Kant, Critique of Pure Reason, A27-28/B43-44.
[xv] The epistemological counterpart to things-in-themselves.
[xvi] R.L Anderson, “The Introduction to the Critique: Framing the question,” in The Cambridge Companion to Kant’s Critique of Pure Reason, ed. P. Guyer (UK: Cambridge University Press, 2010), 92.
[xvii] Samuel Gardner, Kant and the Critique of Pure Reason, (London: Routledge, 1999), 50.
[xviii] Ellington, Introduction. In Prolegomena to Any Future Metaphysics, xiv
[xix] Allison, Kant’s Transcendental Idealism: An Interpretation and Defense, xv-xvi.
[xx] Otfried Höffe, Immanuel Kant (NY: State University of New York Press, 1994), 63.
[xxi] Adorno, Kant’s Critique of Pure Reason, 76.
[xxii] Kant, Critique of Pure Reason, Bxxx.
[xxiii] Immanuel Kant, An Answer to the Question: ‘What is Enlightenment?’ (London: Penguin Publications, 2009), 1.
[xxiv] H.E. Allison, Kant’s Transcendental Idealism: An Interpretation and Defense (US: Yale University Press, 2004), 95-96; Kant, Critique of Pure Reason, B202; Höffe, Immanuel Kant, 95.
[xxv] I prefer that the reader proceed with an understanding of the term ‘multiple’ in the most naïve way possible, rather than explicitly giving a definition of a multiple. The reason for this request is because if we specify what a multiple is before discussing the mathematical formalism in set theory, we will unduly restrict the generality of our constructions. Ultimately, a multiple has no explicit definition but is only implicitly defined as that which can be formalised in set theory.
[xxvi] Kant, Critique of Pure Reason, A292/B348-349.
[xxvii] Kant, Critique of Pure Reason, A52/A76, A28/B44.
[xxviii] Allison, Kant’s Transcendental Idealism (US: Yale University Press, 2004), 7.
[xxix] Penelope Maddy, Naturalism in Mathematics (UK: Oxford University Press, 1997), 40.
[xxx] Kant, Critique of Pure Reason, B308; Gardner, Kant and the Critique of Pure Reason, 202.
[xxxi] Allison, Kant’s Transcendental Idealism (US: Yale University Press, 2004), 17.
[xxxii] Kant, Critique of Pure Reason, B281-282/A229-230.
[xxxiii] Kant, Critique of Pure Reason, B551-552.
[xxxiv] Georg Cantor, “Letter to Dedekind”, in Handbook in Logic: From Frege to Gödel, ed. Jean Van Heijenoort, (Cambridge: Harvard University Press, 1967), 114.
[xxxv] Crucially, since set theory is a discourse, it does not contradict the discursivity of human cognition in Kant, although rejects conceiving the mind as merely finite. See Allison, Kant’s Transcendental Idealism, 12/77; Paul Guyer, Kant and the Claims of Knowledge (UK: Cambridge University Press, 1987), 171-172.
[xxxvi] Adorno, Kant’s Critique of Pure Reason, 222.
[xxxvii] However, the historical supersession of Euclidean Geometry and Newtonian Physics does not refute Kant since these are merely examples of synthetic a priori knowledge rather than an integral part of his critical revolution. See Höffe, Immanuel Kant, 93; Ellington, ‘Introduction,’ in Prolegomena to Any Future Metaphysics, xi-xiv; Lisa Shabel, “The Transcendental Aesthetic,” in The Cambridge Companion to Kant’s Critique of Pure Reason, ed. Paul Guyer (UK: Cambridge University Press, 2010), 114.
[xxxviii] Kenneth Kunen, Foundations of Mathematics (London: College Publications, 2012), 7; Gardner, Kant and the Critique of Pure Reason, 281.
[xxxix] Alain Badiou, Logics of Worlds (London, Bloomsbury Publishing, 2019), 191.
[xl] Alain Badiou. Being and Event (London: Bloomsbury Publishing, 2007), 14; Peter Wolfendale, The Noumenon’s New Clothes (UK: Urbanomic, 2019), 262.
[xli] Wolfendale, The Noumenon’s New Clothes, 262.
[xlii] Kant, Critique of Pure Reason, A256/B311-312.
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