Measure for Measure
Þessi ritgerð var skrifuð fyrir áfangann "Kant’s Critique of Pure Reason” við Warwick-háskóla árið 2020. Hún fjallar um stærðfræðiheimspeki þýska heimspekingsins Immanuel Kant eins og hún birtist í Gagnrýni hreinnar skynsemi, verki sem kalla mætti höfuðrit hugsuðarins. Hugmyndir Kant um grundvöll og iðkun stærðfræðinnar eru hins vegar að mestu leyti taldar hafa elst ansi illa; stærðfræðin tók stakkaskiptum með og eftir Frege og Hilbert, og með tilkomu óevklíðskrar rúmfræði var leikvöllurinn svo gott sem endurskilgreindur. Kant hafði þá hugmynd, aftur á móti, að evklíðsk rúmfræði væri í reynd hvorki meira né minna en fullkomin og forskilvitleg vísindi rúms, vísindi sem ættu a priori grundvöll sinn í rúmi og tíma, formum skynhæfninnar. Samkvæmt Kant væri rúmfræði þar með fullkomin og röklega nauðsynlega sönn lýsing á rúminu fyrir alla mögulega reynslu. Eins og ég geri grein fyrir í ritgerðinni hafa nútímavísindamenn allt aðra skoðun en Kant og álíta stærðfræðina í reynd tungumálaleik eða rökfræðikonstrúkt sem á ekkert skylt við raunveruleikann; segja mætti að mottó þeirra sé: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality,” eins og haft er eftir Einstein í ritgerðinni. Í lok ritgerðarinnar færi ég rök fyrir því að þótt niðurstöður Kant hafi ekki staðist skoðun þá eigi spurningarnar sem hann lagði upp með í gagnrýni sinni enn rétt á sér.
Málverkið í haus heitir Jalais Hill, Pontoise, og er eftir Camille Pissarro frá árinu 1867.
Measure for Measure
Space, Time and the Mathematical Synthetic a priori in Kant
Introduction
Immanuel Kant’s (1724-1804) theory of transcendental idealism, expounded in his seminal Critique of Pure Reason, [1] has been immensely influential from the time of its original publication in 1781 until the present. In this complex work, Kant argues that philosophers have, for the most part, proceeded by way of justifying how the mind conforms to objects, instead of explaining how objects themselves must conform to the mind’s cognition. Furthermore, his “Transcendental Aesthetic” contains one of his more interesting positions, one on which our subject-matter in this essay is based. In the Aesthetic, Kant argues that space and time are forms of our faculty of sensibility, and that any appearance must conform to these intuitive a priori forms. Here Kant takes a bold stand against both Newton and Leibniz—which seemed at the time to be the only available coherent positions. Kant identifies both positions as “transcendental realism” and proposes in their stead his own critical doctrine of transcendental idealism. [2] The broad argument of the Aesthetic has mostly stood the test of philosophical time—it is still taken quite seriously and respectfully as at least a coherent if not viable metaphysical position.
Conversely, however, Kant’s understanding of mathematics—which happens to play a role of more or less importance in the Aesthetic—has fallen into comparatively grave disrepute. Since the publication of Kant’s magnum opus, mathematics and logic have undergone radical transformations. Geometry is no longer exclusively Euclidean, for instance, and the world witnessed the birth of mathematical logic—to name but two examples. We cannot do these revolutionary developments justice in an essay so short and narrow in focus—which means that any particular advance must go untreated here. However, there is a general sense in which Kant’s philosophy of mathematics simply has not “kept abreast,” so to speak—and it is precisely this angle which I wish to explore. As Michael Friedman has noted, Kant’s theory of mathematics seems to induce many to view the Aesthetic as an “unfortunate embarrassment” that must be rushed through and gotten over with. [3] This state of affairs immediately raises a few questions. First, how does Kant understand the relation between spatial and temporal intuition and the practice of mathematics—and second, to what extent does Kant’s argument in the Aesthetic rely upon this relation, if at all? Does the Aesthetic rely unduly upon an ancient and defunct conception of mathematics—or can we simply dismiss Kant’s appeal to geometry as irrelevant to the central argument?
I will explore these questions in the following essay, which will have a quadripartite structure. Firstly, I will account for Kant’s arguments in the Transcendental Aesthetic, explicating his stance on the nature of intuition, sensibility, space and time. Secondly, I will list some crucial aspects of Kant’s theory of mathematics, explaining why he argued that mathematics must produce synthetic a priori truths as well as delineating his arguments for how they do so. Thirdly, I will consider why modern mathematicians might be averse to this conception of mathematics, seeing as the modern conception makes of mathematics a purely axiomatic and ratiocinative logical construction completely divorced from any transcendental argument regarding the formal conditions of experience. In order to show how an encounter between Kant and modernity might play out, I will consider a particular case—that of Albert Einstein’s arguments in a lecture of his called “Geometry and Experience” from 1921. Fourthly and finally, I will conclude by offering some remarks on the differences separating Kant’s view of mathematics from those of modern and contemporary mathematicians, logicians and philosophers of mathematics.
I. Sensibility, Intuition, Space and Time in the first Critique
The Critique of Pure Reason is divided firstly into the Transcendental Doctrine of Elements and the Transcendental Doctrine of Method. The Doctrine of Elements, in turn, divides into the Transcendental Aesthetic and the Transcendental Logic. It is in the Transcendental Aesthetic that Kant argues that space and time are a priori conditions of sensible experience.
We will begin by defining a few of Kant’s technical terms, so as to be clear about their meanings. First and foremost is intuition. An intuition is that through which a cognition immediately relates to objects.[4] This can only occur, however, if these objects are given to the mind, which is to say that in order for the mind to have an intuition it must be affected in some way by the object that is then represented immediately in the intuition. Kant calls the mind’s receptive capability sensibility: “Objects are therefore given to us by means of sensibility, and it alone affords us intuitions [...].” [5] Intuitive representations have, however, not been thought until they are dealt with by the understanding. Before they are thought, they are merely intuited. [6] The understanding, then, is another faculty of the mind—its central processing unit, if you will. Raw affective data is captured by sensibility in sensation, and any sensory intuition Kant calls empirical. An otherwise undetermined object of sensory intuition—one which has not been given determinacy through concepts—is called an appearance. [7]
Any appearance for Kant has a hylomorphic character. That in the appearance which is sensed is the matter of the appearance, while the primary order of relations between the manifold of appearances is constituted by their form. [8] The forms of sensibility thus receive and order [9] the matters given by sensation, capturing the matter by informing it and in essence making it intuitive. Since the form of appearances orders the matter furnished by sensation, the form cannot itself be given by sensation and must reside a priori in the mind. Any empirical appearance consists, then, of an a posteriori or empirical component—the matter of sensation—and of an a priori component—the formal conditions under which this matter is subsumed or by which it is in-formed. If we then separate all categorical and sensory content from this empirical intuition, we are left with nothing but pure intuition. [10] As Kant will argue, there are then two pure forms of sensibility which we experience intuitively; space and time. Any given matter is situated in external and internal relations—spatial and temporal, respectively.
Let us move on, then, to the actual arguments for the ideality of space and time—beginning with space and then moving on to time. In the second edition of the Critique, Kant argues in two distinct modes of exposition—metaphysical and transcendental—that space and time are a priori forms of sensibility, necessary formal conditions for the possibility of any appearance whatsoever.
A. The Metaphysical and Transcendental Expositions of the Concept of Space
The Metaphysical Exposition of the concept of space is divided into four parts. Firstly, Kant argues, space cannot be an empirical concept which we derive from outer experience, since for there to be something which the mind can refer to as “outside” itself, the representation of space must first be presupposed. We cannot, then, begin from the empirical representation of a body which is “out there” and infer the reality of space therefrom, since the very qualifier “out there” is already a representation of space. [11]
Secondly, space is a necessary a priori representation, since we can never represent to ourselves the absence of space itself, although we may well imagine an empty space devoid of objects. Space, therefore, is to be thought of as a condition of the possibility of appearances. [12]
Thirdly, space is not a concept, but an intuition. A concept, for Kant, is a function of the understanding which brings a myriad representations under a singular representation, the concept. [13] Space, however, is not to be thought of as something which brings under itself various “spaces” since any single “space,” such as a given and defined room, is always retrospectively cordoned off from the infinite and general space of which it must already be a part. [14]
Fourthly, space cannot be thought of as a concept, since, as we have seen, concepts contain an infinite number of representations under themselves, while space represents an infinite and given magnitude in itself. The argument here is that while the concept of chair, for example, subsumes infinite possible instances of chairs under it, the concept itself is not infinitely varied in itself. It is rather the subsumed singular representations that are infinitely many, while the concept itself remains singular, and it must be so since it serves as the common point of referral for an indefinite number of instantiations. Space, on the other hand, is itself represented as an infinite magnitude, which distinguishes it from a concept, making it an intuition. [15]
The Transcendental Exposition, on the other hand, comprises an explanation of a concept (space, in this instance) as a principle from which a priori synthetic knowledge can be acquired. [16] In this exposition, Kant means to make clear how space as a form of sensibility makes geometrical knowledge possible, knowledge which he takes to be synthetic a priori. He argues that space cannot be understood as a concept, since geometry makes use of space in its a priori synthetic demonstrations. If space were a concept, he argues, no synthetic propositions could be obtained from its manipulation, since a concept, as we have seen, merely acts as a subsumptive function which unites a myriad representations under itself. From a concept, then, nothing synthetic can be gained, since whatever instances the concept subsumes are already contained in it—from the concept itself no propositions can be obtained which go beyond the concept. [17] Geometry, however, does provide us with synthetic propositions, Kant argues. Since geometry, on Kant’s understanding, is a science of space, it follows that space cannot be a concept, but is instead an intuition.
In part V of the Introduction to the first Critique, Kant argues that all mathematical judgments are synthetic. [18] His argument that 5+7 does not analytically entail the concept of 12 is infamous enough and has been explored extensively in the literature. The argument warrants a closer examination here, since Kant seems here to be arguing that space is intuitive (in part) because of the synthetic quality of mathematical judgments—or at the very least seems to take this supposed synthesis as evidence for his claims about space. This we will consider more closely in the second section of the essay.
B. The Metaphysical and Transcendental Expositions of the Concept of Time
Kant’s arguments for time being a form of sensibility are quite similar in form to his arguments for the formal ideality of space. In his Metaphysical Exposition, he argues firstly that time cannot be an empirically derived concept since for anything to appear to us empirically it must either be placed in a certain order before or after with reference to the succession of our inner states, or in simultaneity with something else, that is, represented along with something else in a singular inner state of temporal succession. The representation of time must thus be presupposed for any empirical appearance to have sense at all. [19]
Secondly, we are incapable of thinking of appearances outside of time, although we may imagine time to be void of appearances. Kant argues as he did with space that this makes time a necessary representation underlying or in-forming all sensible intuitions, a representation which is purely formal when considered in abstracto from any empirical content. [20]
Thirdly, we can postulate apodeictic principles through our intuition of time—principles such as one that states that events taking place in different moments of time are related by a mode of succession in a series of instants, while events taking place at the same moment are simultaneous. These principles have apodeictic certainty because of the a priori and formal nature of time, and are not something we discover by way of experience. [21]
Fourthly, time, in the same manner as space, is not a discursive or general concept, since parts of time are not independently subsumed under a concept of time, but are parts of a singular time. [22] Fifthly and finally, since time is given as infinite, it cannot be represented by a concept, which can only contain partializations or limitations of the infinite temporal sequence. Any concept of a determinate magnitude of time must therefore presuppose the immediate intuition of time. [23]
The third section of the Metaphysical Exposition, Kant goes on to say, should properly belong in a Transcendental Exposition, since it demonstrates how time as an intuitive a priori form of sensibility provides us self-certain principles by means of which we may adduce further synthetic knowledge. He goes on to explain how the temporal principle discussed in the third section is necessary for the explanation of the concept of motion or alteration of place, in which an object A both is and is not in the same place X—a proposition which would violate the law of non-contradiction were the states AX /~AX not differentiated by an ordering of temporal succession. Time as a form of sensibility thus forms the basis for any comprehensible possibility of alteration. This example serves to demonstrate how time is an a priori principle from which one can adduce synthetic propositions in the same way that space as an a priori principle provides a foundation for geometrical knowledge. [24]
II. Arithmetic and Geometry in Kant
Now that we have seen how Kant argues that space and time must be a priori formal conditions of appearances in general, let us examine in greater detail how he conceived of his philosophy of mathematics. We will begin with an attempt to appreciate how Kant conceived of the mental operations found in arithmetic, operations which on his view hinge on the intuitions of space and time. For starters, it would be illuminating to take a closer look at the argument regarding the union of 5 and 7 mentioned above.
Kant’s argument goes something like this: the union or sum of the numbers 7+5 does not contain anything further than that the two numbers be united. It does not already contain the number 12, Kant argues, since in order to perform addition per se, one must intuitively put together the constitutive numbers. One must “go outside” the concepts of 7 and 5 in order to produce the sum entailed by their union. Such a “going outside” would require us to “call in the aid of the intuition which corresponds to one of them.” [25] This means that in order to produce the sum, one must revert from a conceptual understanding of the separate numbers (which would seem analytic) back to their origins in a temporal-sequential synthesis of the spatial manifold—the number-concepts 5 and 7, then, would be reduced back to their original form as iteration of intuitive units or quanta. What this means is that any arithmetical operation (as well as any geometrical operation, as we shall see) relies upon an imagination of several distinct points in a formal spatial manifold which are counted together or synthesized over time, an operation which would then yield the product or sum prescribed for the mind performing the calculation—which also explains the apodeictic nature of the operation. [26]
To sum up, Kant contends that in order to perform addition, one must appeal to an a priori intuition of an infinite spatial manifold, within which one can then posit the discrete points—points which are then taken together or counted through temporal succession, each unit building upon what has come before—a synthesis which ultimately produces the sum. His more significant point here, it seems, is that a hypothetical mind which has had no a posteriori experience of language or elementary arithmetic would still be able to construct the quantity represented by our number-concept “12” by intuitively counting together the units which 5 and 7 are comprised of within the formal manifold of space over a period of time. Kant insists that any operation carried out in this manner is a priori in the sense that it does not rely on any empirical sensation or previous externally acquired experience to be consistently true and thus scientific. The same goes, Kant argues, for geometry, and in fact for all pure mathematics. [27]
Kant’s arguments regarding the a priori syntheticity of geometry are strikingly similar. The conceptual atoms of Euclidean geometry are points, atoms of which lines are composed. In order to produce a line, then, Kant argues, one must draw it through a series of discrete points set upon a spatial manifold. Kant again takes up the argument in his “Axioms of Intuition,” where he argues that: “I cannot represent to myself any line, no matter how small it may be, without drawing it in thought, i.e., successively generating all its parts from one point, and thereby first sketching this intuition.” [28] For Kant, the a priori synthetic truths of geometry are thus grounded in this iterative process of productive or generative imagination in the same way one must appeal to an intuition of space in order to count together two definite quantities such as 5 and 7.
Kant goes further, however, in his claims about this relation between mathematics and intuition. Later on in the Axioms of Intuition he makes the strong claim that the transcendental principles of space and time make pure mathematics applicable to objects of experience with the same apodeictic certainty found in pure mathematics. [29] This is to be expected, since the forms of intuition that Kant argues make up our scientific body of mathematical knowledge are the same forms that structure and inform any empirical appearance. For Kant, then, geometry provides us with unequivocal a priori scientific knowledge of objects of the senses: “what geometry says about [pure intuition] is therefore undeniably valid of [empirical intuition], and evasions, as if objects of the senses did not have to be in agreement with the rules of construction in space [...] must cease.” [30] This claim is directed against transcendental realists who would assert against Kant’s transcendental idealism that objects as they appear to the senses are things in themselves and that our mathematical knowledge is only an uncertain and approximate description of these things.
It seems evident that Kant is seeking here to preserve the apodeictic and certain nature of mathematics by grounding its self-evidence in the formal nature of space and time, and it is precisely this move which so aggravates his modern detractors. Let us then move on to consider the objections his critics make by considering what is arguably the most famous example: Albert Einstein’s contention that “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” [31]
III. Einstein contra Kant—Geometry Pure and Applied
A great place to start looking at the divide between the Kantian and the modern idea, a divide which seems to originate in a fundamental incoherence between modern exact sciences and Kant’s transcendental idealism, is with Albert Einstein’s famous contention in his lecture entitled “Geometry and Experience” that:
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. It seems to me that complete clearness as to this state of things first became common property through that new departure in mathematics which is known by the name of mathematical logic or “Axiomatics.” The progress achieved by axiomatics consists in its having neatly separated the logical-formal from its objective or intuitive content; according to axiomatics the logical-formal alone forms the subject-matter of mathematics, which is not concerned with the intuitive or other content associated with the logical-formal.” [32]
We might be tempted to read Einstein’s position as a blunt dismissal of Kant—insofar as it claims a complete separation between the “logical-formal” of mathematics from an “objective” or “intuitive” content—and to some extent, we might be correct to do so, even if Einstein does not have Kant in mind here. Einstein’s position is grounded in a revolutionary advance made in mathematics by David Hilbert, who succeeded in formalizing geometry and the theory of real numbers. We will not be able to cover this revolutionary development in great detail due to the scope of this essay, but we may suffice it to say that it consists of viewing mathematics as a logical discipline which operates first and foremost by way of axiomatization. [33]
To axiomatize means to “[make] explicit the logical structure of the corresponding domain of knowledge,” as well as to “fix the basic principles of a science.” [34] To axiomatize thus means, in a sense, to logicize a given discipline—that is, to make the discipline completely independent of any intuitions we may or may not have. Hilbert’s famous assertion that “One must be able to say at all times—instead of points, lines and planes—tables, chairs and beer mugs,” [35] makes this explicit: the axioms of mathematics describe mere logical entities, empty placeholders which do not refer to any idea of space and time or any entities based on spatio-temporal imagery. In contrast, the mathematical points Kant works with in the Transcendental Aesthetic and the Axioms of Intuition seem to be precisely such spatio-temporal imagery.
In his fascinating essay, “Kant’s Theory of Geometry,” prominent Kant scholar and commentator Michael Friedman argues quite convincingly that the difference between Kant and the modern sciences ultimately comes down to a difference in conceptions of logic. In Kant’s time, logic was strictly syllogistic or Aristotelian—but the whole discipline was revolutionized in the late 19th century with the introduction of Frege’s Begriffsschrift, among other things. [36] Syllogistic or Aristotelian logic operates by way of interrelated propositions in which predicates are affirmed or denied of a subject—“Every x is y, z is x, therefore z is y”—which makes it monadic or unary insofar as each proposition qua logical function takes only one argument, that is, the affirmation or denial of whether the predicate applies to the subject.
As Friedman explains, syllogistic logic is limited by its monadic relational structure and does not function very well when it comes to concepts such as infinite continuity—it simply does not allow for strict formal description of the sort. [37] Today, geometry has instead been formalized and axiomatized by way of a polyadic logic, one which can soundly generate an infinity of objects such as points logically—which is radically different from how Kant’s iterative approach operates—in which points are intuitively drawn upon the infinite manifold of space as the pure form of sensibility. [38] As Einstein rightly points out, this axiomatic or logical-formal revolution seems to have removed the mathematical object from the sphere of intuition altogether, simply by making of geometry a logical language-game of sorts.
Of course, this does not nullify the metaphysical questions concerning the status of the mathematical object or the role of any supposed a priori formal structure of the mind, be it a structure of sensibility or of the understanding, in the generation of these objects. This revolution simply suspends judgment with regard to these questions. Even so, some mathematicians, such as Hilbert, seem sensitive to them. Hilbert seems to have considered whether mathematics must not come to rest on some sort of intuition, although his conception of intuition is not the same as Kant’s: while Kant refers to sensible intuition, Hilbert’s intuition would properly rather belong to the understanding in Kantian terminology—a logical intuition, it seems. [39]
In any case, the larger point here is that Kant would need to argue that in order to, say, draw a line, one must procedurally generate it in space by way of sensible intuition—and his need stems from the limitations of the monadic, syllogistic logic that was available to him. Modern mathematicians and logicians criticize Kant on this point, since for them, geometry is a formal language-game which has nothing to do with spatial intuition and cannot be grounded in sensibility—although, as Friedman points out, it seems anachronistic and unfair to reproach Kant for failing to possess an understanding of logic which would not even come into being until after he died. [40]
IV. Concluding Remarks
Although Friedman’s argument that the abyss separating Kant and his modern critics comes down to a difference in logical theory is convincing enough, I am not certain that it can truly be said to describe the full extent of their disagreement. I would argue that we must also take account of the fact that metaphysical commitments necessarily precede these differing conceptions of logic. Kant is fully committed to the idea that the human mind has a formal structure which informs the world it perceives, and this, his transcendental idealism, is not simply rebuked by the introduction of a novel theory of logic. On the other hand, the axiomatic methodology which Einstein and so many others champion simply suspends judgment on thorny metaphysical matters such as how or whether geometry can describe the world at all—ultimately resorting to an empiricist or verificationist approach which attempts to see “how well” a given geometrical-logical narrative “fits” the world of experience by way of experimentation and refinement. It will not and perhaps can not commit itself to any critical discourse on how this is possible in the first place—it simply ignores Kant, when all is said and done. The modern rebuke against Kant not only rejects his answers, which may well be rejected as incorrect!—but his critical line of questioning—“How is pure mathematics possible?” [41]—seems to fall upon deaf ears as well.
This is strikingly evident in Einstein’s discourse within the aforementioned lecture. On the ancient method, Einstein argues, the atoms of geometry and mathematics are common knowledge: everyone intuitively knows what a point is and what a line is. The nature of these atoms are left to the philosopher, however, since they seem mysterious to some extent—where is this “point” and from whence does it originate? These are precisely the questions Kant is grappling with, and the philosopher produces his answers: a point is an abstract, discrete and atomic intuition of space founded upon the mind’s a priori form of sensibility which to some extent orders appearances, making them intelligible to the understanding. The modern view, however, simply acknowledges the atoms, insofar as they are axiomatically defined, to be “free creations of the human mind,” [42] leaving the speculative hanging in absolute apathy: the mathematician seems here to offer the philosopher an ultimatum of “take it or leave it—in any case, I don’t care.”
Whether this has a positive or negative effect is hard to say. The exact sciences do not explicitly rely upon metaphysics in their empirical inquiries: professedly they provide us only with answers which have caveats, uncertainties, likelihoods and possibilities. The metaphysician may yell themselves hoarse trying to account for the origins of the logical atoms at work in the geometer’s discipline, but the mathematician need not listen at all—they make themselves out to be busy producing workable results, proving theorems, solving problems. If anything, this attitude of Einstein’s strikes one as overly stand-offish. This is perhaps not all that surprising, given that Einstein seems to have had a similar stance in his debates with Henri Bergson, a speculative philosopher who above all else sought to bring metaphysics and science into a closer and more cooperative rapport—although however he may have succeeded or failed in his arguments against Einstein is another matter altogether. However this may be—whether Einstein is “cold” towards metaphysics or not—I believe that this dispute about the role of metaphysics in scientific inquiry is of serious importance for the divide at hand. [43]
In any case, the aforementioned modern complaints against Kant can serve only to criticize his arguments on the grounding of mathematics in spatial intuition—but as far as I can tell, they cannot refute his more general theory of space and time as forms of intuition. For all modern mathematicians know, appearances could very well be formally structured a priori by the space and time qua pure forms of intuition, even if Kant’s arguments in the Transcendental Exposition and Axioms of Intuition make too strong a claim in asserting the a priori certainty and applicability of pure geometrical propositions to empirical or physical space. The modern Kantian needs only make the weaker claim that the mind has certain a priori structures which necessarily inform any possible appearance, without the assertion that these structures must serve as principles for any particular geometry against any other, for example by asserting that these structures are explicitly Euclidean and not Riemannian.
For example, Henry Allison, a capable defender of Kant’s transcendental idealism, argues that even if Kant’s iterative conception of geometry is untenable, his “reliance” upon it in the Transcendental Exposition needs not invalidate his theory that space and time are ideal forms of intuition. He argues quite convincingly that while Kant does appeal to geometry as a synthetic a priori science in his Transcendental Exposition, this merely signifies that space as a form of sensiblity is necessary for the construction of such a science, not that it is sufficient. For Allison, the Transcendental Exposition is superfluous for Kant’s argument for the ideality of space and time and any consideration of geometry can simply be bypassed. [44]
Even then, several questions loom large—such as the aforementioned one on how pure geometry can be applied to appearances at all, as well as what status one should confer to logic if it can indeed bring us geometries which describe and thus serve as predictive models for empirical experience. Could logic itself have roots in our sensibility as an evolutionary byproduct? The ontological status of the mathematical object in Kant is as yet uncertain—or so it at least seems, if we are to take the aforementioned criticisms seriously. In any case, these questions—although interesting enough—quickly become all-too speculative and broad for our present essay, and we must leave them unexplored.
Kant’s understanding of time and space, as well as his conception of mathematics, should in any case not be taken too lightly—they still pose serious metaphysical questions for critical thinkers to this day. Even if revolutionary leaps have been made in formalization and axiomatization within mathematical logic—leaps which must be reckoned with by readers and interpreters of Kant—they do not entail that the Transcendental Aesthetic is in any manner an unfortunate embarrassment. The slew of metaphysical questions raised and answered by Kant still remain under serious consideration—and although mathematicians may choose to shun philosophers for tarrying with these perennial and unpragmatic problems, they may very well be blundering in doing so.
Bibliography
Cited works by Immanuel Kant:
Kant, Immanuel. Critique of Pure Reason. Translated and edited by Paul Guyer and Allen W. Wood. Cambridge, UK: Cambridge University Press, 2020.
Kant, Immanuel. Prolegomena to Any Future Metaphysics. Translated and Edited by Gary Hatfield. New York, NY: Cambridge University Press, 2004.
Other works cited or consulted:
Allison, Henry. Kant’s Transcendental Idealism: An Interpretation and Defense. New Haven and London: Yale University Press, 2004.
Einstein, Albert. Ideas and Opinions. Edited by Carl Seelig. Translated by Carl Seelig and Sonja Bargmann. New York, NY: Crown Publishers, 1960.
Ewald, William (ed.) From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume II. Oxford: Clarendon Press, 1999.
Friedman, Michael. “Kant’s Theory of Geometry” in The Philosophical Review 94, no. 4 (1985): 455-506. Accessed June 10, 2020. doi:10.2307/2185244.
Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries: Development and History. New York, NY: W.H. Freeman and Company, 1994.
Venturi, Giorgio. “The concept of axiom in Hilbert’s thought”. 2011. Unpublished. Retrieved from <https://pdfs.semanticscholar.org/d791/b8df5fa83a62ce4da828e72c09dae7ddb944.pdf> on June 12th, 2020.
Footnotes
[1] Immanuel Kant, Critique of Pure Reason. Translated and edited by Paul Guyer and Allen W. Wood. Cambridge, UK: Cambridge University Press, 2020. Henceforth referred to by standard A/B notation.
[2] A369: “[T]ranscendental realism […] regards space and time as given in themselves (independent of our sensibility).”
[3] Michael Friedman, “Kant’s Theory of Geometry” in The Philosophical Review 94, no. 4 (1985): 455-506. Accessed June 10, 2020. doi:10.2307/2185244. P. 455.
[4] A19/B33.
[5] Ibid.
[6] Ibid.
[7] A20/B34.
[8] Ibid.
[9] Henry Allison argues that “sensibility does not itself order the given data, that being the task of the understanding, or, more properly, the imagination,” but that sensibility plays the role of presenting them as “capable of being ordered.” I’m not certain I agree with Allison here in that sensibility does not itself order the data—as far as I can tell, the intuitions described by the concepts “before” and “after” or “behind” and “in front of” denote a certain intuitive ordering. There might thus be a distinction between primary and secondary ordering—or perhaps the disagreement comes down to a semantic uncertainty regarding the concept of “ordering”. See Henry Allison, Kant’s Transcendental Idealism: An Interpretation and Defense. New Haven and London: Yale University Press, 2004. P. 14.
[10] A20/B34.
[11] A23/B38. Cf. Allison’s account, p. 101-102 of Kant’s Transcendental Idealism.
[12] A24/B38-39.
[13] A68/B93.
[14] A25/B39-40.
[15] Ibid.
[16] B40-41.
[17] B40.
[18] B14-17.
[19] A30/B46.
[20] A31/B46.
[21] A31/B47.
[22] A31-32/B47.
[23] A32/B47-48.
[24] B48-49.
[25] B14-16. Cf. also Kant’s discussion in the Axioms of Intuition (A162-166/B201-207).
[26] Compare also Friedman, “Kant’s Theory of Geometry,” p. 498 fn. 62.
[27] A10/B14.
[28] A162/B203.
[29] A165/B206.
[30] Ibid.
[31] Albert Einstein, “Geometry and Experience” published in Ideas and Opinions. Translated by Sonja Bargmann. Edited by Carl Seelig. New York, NY: Crown Publishers, 1960. P. 233.
[32] Ibid.
[33] Giorgio Venturi, “The concept of axiom in Hilbert’s thought”. 2011. Unpublished. Retrieved from <https://pdfs.semanticscholar.org/d791/b8df5fa83a62ce4da828e72c09dae7ddb944.pdf> on June 12th, 2020. P. 2.
[34] Ibid.
[35] Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries: Development and History. New York, NY: W.H. Freeman and Company, 1994. P. 72.
[36] Friedman, “Kant’s Theory of Geometry,” p. 456.
[37] Ibid, p. 466.
[38] Ibid, p. 465.
[39] David Hilbert, “Logic and the Knowledge of Nature” in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume II. Translated and edited by William Ewald. Oxford: Clarendon Press, 1999. P. 1161. Cf. also Giorgio Venturi, “The concept of axiom in Hilbert’s thought”.
[40] Friedman, “Kant’s Theory of Geometry,” p. 506.
[41] Immanuel Kant. Prolegomena to Any Future Metaphysics. Translated and Edited by Gary Hatfield. New York, NY: Cambridge University Press, 2004. P. 32.
[42] Albert Einstein, “Geometry and Experience,” p. 234.
[43] An attitude similar to Einstein’s can be found in Hilbert’s “Logic and the Knowledge of Nature” in which Hilbert reproaches Kant and Hegel both for relying too heavily on what the author takes to be a priori conceptual deduction. See David Hilbert, “Logic and the Knowledge of Nature,” p. 1161-1162.
[44] See Henry Allison, Kant’s Transcendental Idealism: An Interpretation and Defense, p. 116-118.
