THE LIMITS OF SENSE AND REASON: A Line-By-Line Critical Commentary on Kant’s “Critique of Pure Reason,” #14–Prelude to Kant’s Philosophy of Mathematics.

By Robert Hanna

[I] was then making plans for a work that might perhaps have the title, “The Limits of Sense and Reason.” I planned to have it consist of two parts, a theoretical and a practical. The first part would have two sections, (1) general phenomenology and (2) metaphysics, but this only with regard to its method. (Letter to Marcus Herz, 21 February 1772 [C 10: 129])

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Previous Installments:

#1: Introduction to The Limits of Sense and Reason

#2: Bii/GW91 The Motto

#3: Aiii/Biii/GW93–97 The Dedication

#4: Avii-ix/GW99 Preface to the First (A) Edition.

#5: Axi note/GW100–101 Preface to the First (A) Edition

#6: Axi note/GW100–101 Preface to the First (A) Edition

#7: Axii-xiv/GW101–102 Preface to the First (A) Edition

#8: Axv-xvi/GW102–103 Preface to the First (A) Edition

#9: Axvi-xvii/GW103 Preface to the First (A) Edition

#10: Axvii-xx/GW103–104 Preface to the First (A) Edition

#11: Axxi-xxii/GW104–105 Preface to the First (A) Edition

#12: Bviii-ix/GW106–107 Preface to the Second (B) Edition

#13: Bix-x/GW107 Preface to the Second (B) Edition

A Note on References to Kant’s Works

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The first five installments in this series followed the 2019–2020 version of THE LIMITS OF SENSE AND REASON, aka LSR.

Starting with installment #6, subsequent installments follow the revised and updated 2021 version of LSR.

In any case, you can read or download a .pdf of the complete text of the 2021 version of LSR HERE.

Because LSR is an ongoing and indeed infinite task, revised and updated .pdfs of the complete text will be uploaded to that URL on a regular basis.

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CPR TEXT B x-xii/GW107–108 Preface to the Second (B) Edition

Mathematics and physics are the two theoretical cognitions of reason that are supposed to determine their objectse a priori, the former entirely purely, the latter at least in part purely but also following standards of sources of cognition other than reason.

Mathematics has, from the earliest times to which the history of human reason reaches, in that admirable people the Greeks, traveled the secure path of a science. Yet it must not be thought that it was as easy for it as for logic — in which reason has to do only itself — to find that royal path, or rather itself to open it up; rather, I believe that Bxi mathematics was left groping about for a long time (chiefly among Egyptians), and that its transformation is to be ascribed to a revolution, brought about by the happy inspiration of a single man in an attempt from which the road to be taken onward could no longer be missed, and the secure course a science was entered on and prescribed for all time and to an infinite extent. The history of this revolution in the way of thinking — which was far more important than the discovery of the way around famous Cape — of the lucky one who brought it about, has not been preserved for us. But the legend handed down to us by Diogenes Laertius — who names the reputed inventor of the smallest elements of geometrical demonstrations, even of those that, according to common judgment, stand in no need of proof — proves that the memory of the alteration wrought by the discovery of this new path in its earliest footsteps must have seemed exceedingly important to mathematicians, and was thereby rendered unforgettable. A new light broke upon the first person who demonstrated the isoscelesa triangle (whether he was called “Thales” or some other name). For he found that what he to do was Bxii not to trace what he saw in this figure, or even trace its mere concept, and read off, as it were, from the properties of the figure; but rather that he had to produce the latter from what he himself thought into the object and presented (through construction) according to a priori concepts, and that in order to know something securely a priori he had to ascribe to the thing nothing except what followed necessarily from what he himself had put into it in accordance with its concept.

e Objecte

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COMMENTARY

Kant says that mathematics, like pure general logic, has been on the secure path of science since the time of the Greeks.

But unlike the science of pure general logic, which, as the original or Ur-science in relation to all the other authentic sciences, found it comparatively easy to open up that royal road — because in pure general logic “reason has only to do with itself” (CPR Bx), in the sense that it captures the formal essence of all human rationality, whether theoretical or practical — the scientific road for mathematics was a somewhat bumpier and more difficult one.

Kant isn’t entirely explicit as to the reason for this, but seems very likely that he thinks the problem was that in its earliest manifestations — for example, amongst the Egyptians and Mesopotamians — mathematics was tied too closely to merely empirical facts and measurement practices.[i]

Nevertheless, during the Ionian and Pythagorean period, mathematics was transformed from a quasi-empirical science into an inherently a priori science by a “revolution in the way of thought” (Revolution der Denkart), carried out by a single thinker — whether

Thales, or someone else — who, according to legend, first demonstrated that the base angles of an isosceles triangle are equal.[ii]

This crucial notion of a “way of thought” (Denkart)is closely related to what, in his anthropological writings, Kant calls a “way of thinking” (Denkungsart), and explicitly identifies with a person’s moral character (AM 25: 1385).

In turn, a person’s moral character is later identified by Kant with the self-organizing, goal-directed, individual constitution of a person’s causally efficacious power of choice or will (CPRA539–541/B567–569, A548–557/B576–585) (MM 6: 407).

Correspondingly, Kant also says that it’s possible for a person to transform or revolutionize her own moral character by resolutely choosing and thereby willing to live according to a new set of principles.

Hence a thinker who carries out a “revolution in the way of thought” also carries out a revolution in her way of living, for better or worse.

There’s clearly a deeply affinity and continuity here between, on the one hand, Kant’s proto-existentialist thoughts about metaphysics, resolute choice, and the meaning of a person’s life, and on the other, Augustine’s, Pascal’s, Kierkegaard’s kindred thoughts.[iii]

Indeed, this becomes fully evident in Religion within the Bounds of Mere Reason, via what Kant calls the “revolution of the heart” and the “revolution of the will,” consisting in a radically free change or reversal in attitude, or Gesinnung, that moves someone from the way of our natural condition of radical evil, to the way of life according to and for the sake of the Categorical Imperative:

If by a single and unalterable decision a human being reverses the supreme ground of his actions by which he was an evil human being (and thereby puts on a “new man”), he is, to this extent, by principle and attitude of mind, a subject receptive to the good; but he is a human being only in incessant laboring and becoming, i.e., he can hope … to find himself upon the good (though narrow) path of constant progress from bad to better. For him who penetrates to the intelligible ground of the heart (the ground of all the maxims of the power of choice) … this is the same as actually being a good human being … and to this extent the change can be considered a revolution. (Rel 6: 48)

In Religion, Kant also says, rather misleadingly and mysteriously, that this revolution of the rational human heart-and-will happens independently of space and time.

But that’s only to say that it is necessarily underdetermined by and not strongly supervenient on, all the natural, sensory, contingent facts in space and time, not that it is a supernatural act, outside of space and time.

So too in the Tractatus, Wittgenstein says that all the natural facts in the world can remain exactly the same while the metaphysical and ethical subject freely undergoes a radical change in attitude and spontaneously initiates a Gestalt shift in her whole life that moves her from being unhappy (half-hearted, inauthentic) to being happy (wholehearted, authentic):

If good or bad willing changes the world, it can only change the limits of the world, not the facts; not the things that can be expressed in language.

In brief, the world must thereby become quite another. It must so to speak wax or wane as a whole. The world of the happy is quite another than that of the unhappy.[4]

Or otherwise put, Kant’s Critical reflections on the nature of metaphysics, just like the Tractarian Wittgenstein’s, are clearly every bit as existential and practical as they are scientific and theoretical.

They both think we possess a radically spontaneous capacity to change our lives for the better by revolutionizing our hearts and wills.

Now coming back to the cognitive side of things, according to Kant, what, more precisely, was this revolution in a thinker’s “way of thought,” which was also a revolution in that thinker’s way of living?

It was that instead of deriving the properties of a Euclidean geometric figure from actual shapes or diagrams by inductive generalization, this revolutionary thinker recognized that the properties of a Euclidean geometric figure could be necessary or essential properties of that figure if and only if

(i) these necessary properties flowed from “what he himself thought into the object and presented (through construction) according to a priori concepts,” and

(ii) “that in order to know something securely a priori he had to ascribe to the thing nothing except what followed necessarily from what he himself had put into it in accordance with its concept” (CPR Bxii).

Bounded in a nutshell, this is Kant’s philosophy of mathematics, and I’ll have ample occasion later in LSR to examine Kant’s various attempts to spell it out in more detail.

But for now, the crucial points are:

first, that the properties of a mathematical object are themselves necessary or essential properties of that object only if they necessarily conform to a rational human cognitive subject’s spontaneous act of “construction” (Konstruktion) that presents the object according to a priori concepts, and

second, that a rational human cognitive subject’s “securely” (sicher) knowing these necessary or essential mathematical properties a priori consists in her clearly and distinctly representing her own act of a priori construction.

In other words, objectively (and thereby also subjectively) certain a priori knowledge in mathematics is an act of self-knowledge, such that the intrinsic connection between a subject’s consciously-experienced evidence for asserting her belief on the one hand, and the truth-maker of that belief on the other hand, is guaranteed by the thesis of transcendental idealism.

Or in still other words, what this self-labelled revolutionary thinker discovered is the transcendental idealism-based solution to The Problem of Cognitive Semantic Luck with respect to a priori knowledge in mathematics.

Two critical points should be flagged here for closer examination later, however.

First, although Kant’s example is drawn from (proto-) Euclidean geometry, his account of the necessity and a priori knowability of mathematics is supposed to hold not only for Euclidean geometry, but for all of mathematics as Kant understood it, including arithmetic, algebra, trigonometry, and the differential calculus.

But no indication is given here of how the generalization from (proto-)Euclidean geometry to the rest of mathematics, either just as Kant understood mathematics, or more generally, is supposed to go.

Second, the crucial notion of a construction isn’t spelled out here at all.

This naturally leads to several questions.

Do constructions require, or are they restricted to, either physical diagrams or visual thinking?

If so, then how do they avoid Berkeley’s worries, formulated in his famous critique of Locke’s abstractionist theory of general ideas,[v] about the non-generality of mental imagery?

But if constructions don’t require or aren’t restricted to physical diagrams or visual thinking, then in what sense do they “construct” anything?

And what, more generally, does it mean to construct a concept or to construct according to concepts?

Quite apart from the difficulty of interpreting Kant’s own texts, answering these questions is also made extremely difficult by the many different senses of “construction” and “constructivism” at play in post-Kantian philosophy, for example, constructivist mathematics (including Intuitionism), constructivist epistemology, constructivist ethics, “strict constructionism” in legal theory, and so-on, all of which, although they trace their provenance to Kantian doctrines, are importantly different from them, so they can significantly distort our view of what Kant himself meant.

The one thing we are told in the B Preface, slightly later on, is that mathematical cognition and reasoning proceeds “through the application of concepts to intuition” (CPR Bxiv).

So, at the very least, mathematical construction involves the display, exhibition, or presentation of geometric, arithmetical, algebraic, trigonometric, etc., concepts in specifically intuitional terms.

This, in turn, basically conforms to what Kant has to say about the nature of mathematical knowledge and reasoning much, much later in the book, in the Transcendental Doctrine of Method (CPR A712–738/B740–766).

This later account also crucially involves the seminal notion of imaginational schematism, however, which is not even introduced until CPR A137–147/B176–187.

Suffice it to say, for the time being, that Kant’s theory of mathematical construction is best understood as a theory of construal or interpretation.

That is, mathematical construction takes us from discursive and essentially conceptual information about pure general logical form, to non-discursive and essentially non-conceptual perceptual information about the world, via the mediating domain of formal-intuitional, imaginational-schematic, and essentially spatiotemporal structural information.

This information, in turn, at once encodes pure general logical-conceptual information on the one hand, and also models it in formal-ontological terms that are also inherently pre-formatted for all and only worlds of possible human experience on the other hand, thus converting pure logic into pure (and yet also empirically applicable) mathematics.

But the more complete story about this special epistemic, semantic, and ontological process, which was an original discovery of Kant’s, and on which he places so much emphasis in his theory of synthetic a priori cognition in mathematics, can wait for subsequent phases of the commentary.

NOTES

[i] See, e.g., C. Boyer, A History of Mathematics (revised edn., New York: Wiley & Sons, 1994), chs. 2–3.

[ii] The original CPR text, in both the A and B editions, actually reads “equilateral triangle” (gleichseitigen Triangel); but in a letter to Schütz of 25 June 1787 (C 10: 466), Kant indicates that he intended “isosceles triangle” (gleichschenklichten Triangel). In any case, in addition to the proof of the equality of the base angles of the iscoceles triangle, four other basic demonstrations in (proto-) Euclidean geometry were attributed to Thales: (i) that an angle inscribed in a semi-circle is a right angle, (ii) that a circle is bisected by a diameter, (iii) that the pairs of vertical angles formed by two intersecting lines are equal, and (iv) that if two triangles are such that they share two angles and a side, then they are congruent.

[iii] See, e.g., R. Green, Kierkegaard and Kant: The Hidden Debt (New York: SUNY Press, 1992); M. Kosch, Freedom and Reason in Kant, Schelling, and Kierkegaard (Oxford: Oxford Univ.Press, 2010); and R. Stern, Understanding Moral Obligation: Kant, Hegel, and Kierkegaard (Cambridge: Cambridge Univ. Press, 2012).

[iv] L. Wittgenstein, Tractatus Logico-Philosophicus, trans. C.K. Ogden (London: Routledge & Kegan Paul, 1981), prop. 6.43, p. 185.

[v] See G. Berkeley, Treatise Concering the Principles of Human Knowledge (Indianapolis, IN: Hackett, 1982), Introduction, pp. 7–22.

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