This is a really useful overview article on mathematics in the Early Modern period, with lots of good background info for understanding Kant’s conception of the field and its problems and challenges. Shabel sets up two demands, which she presents as being in tension with one another. The a priori demand is to explain how it is that mathematics is a form of knowledge that is both universal and necessary. The applicability demands is to explain how it is that mathematical knowledge is applicable to empirical objects. Traditional empiricist and rationalist views can provide one or the other but not both. Shabel argues that Kant’s conception of his own philosophy of mathematics was that transcendental idealism helps us to understand how both demands may satisfied together.

    Author = {Shabel, Lisa},
    Booktitle = {The Oxford Handbook of Philosophy of Mathematics and Logic},
    Crossref = {shapiro2005},
    Month = {Jun},
    Publisher = {Oxford University Press},
    Title = {A Priority and Application: Philosophy of Mathematics in the Modern Period},
    Year = 2005}

Annotations: shabel2005.pdf


  1. Introduction (p.29)

  2. Representational Methods in Mathematical Practice (p.32)

  3. ‘‘Mathematicians,’’ ‘‘Metaphysicians,’’ and the ‘‘Natural Light’’ (p.35)

  4. Kant’s Response (p.44)

References (p.49)


Overview: Excellent discussion of Kant vs. rationalist conceptions of mathematics. The discussion of Descartes and Newton is particularly good. I had trouble understanding parts of the discussion of Leibniz, particularly the discussion of symbolic representation. It is also unfortunate that more time was not spent discussing Wolff’s views (and perhaps those of Euler and Lambert) since Kant is the culminating figure in this history and these figures would have been more directly influential on him. But a good birds-eye-view of the continental European debate.

Shabel articulates two ‘explanatory demands’ on a philosophy of mathematics. First, that it satisfy a ‘demand for a priority’ — i.e. an explanation of how mathematical knowledge is a priori. Second, a ‘demand for applicability’ — i.e. an explanation of how mathematical reasoning and knowledge is applicable to the empirical world. This is especially pressing in light of the a priority of mathematics. Shabel argues that Kant took himself to be the only philosopher whose theory could satisfy both demands. She suggests that, so long as we accept his idealist premises, that he succeeds in this. (p.29)

Summary Text

“It appears that if Descartes’s arguments are accepted, his rationalist philosophy of mathematics satisfies both of the demands on a viable account of early modern mathematical practice, identified above. He accounts for the apriority of mathematics with his theory that the intellect, a faculty of mind independent of the bodily faculties of imagination and sensation, provides direct access to innate mathematical ideas of eternal and immutable natures. He accounts for the applicability of mathematical reasoning by identifying the essence of the natural material world with pure extension, the very object of our innate mathematical ideas: an explanation of how a priori mathematics applies to the natural world is easily forthcoming from a theory that directly identifies the essential features of the natural world with the subject matter of pure mathematics. Since, on Descartes’s view, our a priori mathematical knowledge systematizes mathematical truths about a mind-independent and really extended natural world, he thus appears to have satisfied both the apriority and the applicability demands.” (p.38)

“Newton, like Descartes, thus accounts for our applicable a priori mathematical cognition by claiming that we have a clear perception of the mathematical features of an extramental natural world. Both Descartes and Newton conceive this clear perception of the mathematical features of an extramental natural world to be illuminated by the natural light of reason, a metaphor for the sense in which our faculty of understanding is, on their views, an acute mental vision bestowed by a nondeceiving God.31 As before, it appears that Newton’s account of mathematical cognition satisfies both the apriority and the applicability demands: according to Newton, we have the necessary cognitive tools to acquire a priori knowledge of the mathematical features of the natural world.” (p.40)

“According to Kant, the metaphysicians’ account serves to detach the two primary features of mathematical cognition—apriority and applicability—in such a way that there is no explanatory or meaningful harmony between the universal and a priori mathematical laws known by the understanding and the substantive but a posteriori mathematical cognition of the natural world acquired via the ‘‘common sense.’’ Kant’s charge is that formal a priori mathematical cognition as the metaphysician understands it is altogether isolated from the domain of objects taken to be mathematically and scientifically describable; universal and a priori mathematical truths can only be about ‘‘useful fictions’’ and not ‘‘real things.’’ If our account of a priori mathematical cognition does not give us a priori knowledge of the objects of our possible experience, then our account has failed the apriority demand, at least as Kant conceives it.” (p.46)

“Given the mathematicians’ absolutist conception of space, our achieving a priori knowledge of the mathematical features of the spatiotemporal natural world (e.g., knowledge of the geometry of spatial objects) requires that we also achieve a priori knowledge of the features of the supranatural world (e.g., knowledge of space itself, conceived independently from both our understanding and the objects it is thought to contain). Kant considers this to be a kind of ‘‘confusion’': the ‘‘mathematician’’ achieves apriority only by extending the domain of applicability beyond the bounds of our possible experience. For Kant, the cost of apriority cannot (and need not) be that high. Thus, Kant takes the mathematicians’ account to fail the applicability demand in a special sense: on the mathematicians’ account, a priori mathematical cognition is applicable, but it is applicable beyond acceptable limits, that is, beyond the limits of our possible experience and knowledge of nature.” (p.47)

“Kant thus charges that both of his opponents ‘‘come into conflict with the principles of experience,’’ albeit in different ways. The metaphysician conflicts with the alleged apriority of those principles as applied to experience; the mathematician with the limits of their domain of applicability. While Kant’s predecessors each satisfied only one of the two demands on a successful account of mathematical cognition, Kant considers his own theory to satisfy both the apriority and the applicability demands without conflicting with the principles of experience.” (p.47)

“If we are willing to accept this last caveat—that the natural world comprises all and only those things that we have the cognitive capacity to represent—then we can secure the ‘‘certainty of experiential cognition’': our a priori mathematical claims have direct and complete purchase on (our experience of ) the natural world. Kant thus claims to have provided an account of mathematical cognition that satisfies both the apriority and the applicability demands with which we began—and that, moreover, does not conflict with, but rather helps to establish, the ‘‘principles of experience.''” (p.49)


“The first question comes to a demand for apriority : a viable philosophical account of early modern mathematics must explain the apriority of mathematical reasoning. The second question comes to a demand for applicability : a viable philosophical account of early modern mathematics must explain the applicability of mathematical reasoning. Ultimately, then, the early modern philosopher of mathematics sought to provide an explanation of the relation between the mathematical features of the objects of the natural world and our paradigmatically a priori cognition thereof, thereby satisfying both demands.” (p.31)

Highlighted Text

“The term ‘‘magnitude’’ was used to describe both the quantifiable entity and the quantity it was determined to have. That is, for the moderns, magnitudes have magnitude” (p.30)

“modern mathematical ontology included both abstract mathematical representations and their concrete referents. Accordingly, modern mathematical epistemology could not rest with an account of our cognitive ability to manipulate mathematical abstractions but had also to explain the way in which these abstractions made contact with the natural world.” (p.30)

“On Kant’s view, both the ‘‘mathematical investigators of nature’’ (who suppose that the spatial domain of geometric investigation is an eternal and infinite subsisting real entity) and the ‘‘metaphysicians of nature’’ (who suppose that the spatial domain of geometric investigation comprises relations among confused representations of real entities that are themselves ultimately nonspatial) fail to provide a viable account of early modern mathematics in the sense described above. In particular, the ‘‘mathematicians’’ fail to meet the applicability demand, and the ‘‘metaphysicians’’ fail to meet the apriority demand.6 Kant’s claim, of course, is that his own theory of synthetic a priori mathematical cognition meets with greater success.” (p.31)

“the modern mathematician’s task included systematizing the science of quantity. This required, first and foremost, a systematic method for rep- resenting real, quantifiable objects mathematically, as well as a systematic method for manipulating such representations. The real, quantifiable objects were conceived to include both discrete magnitudes, or those that could be represented numerically and manipulated arithmetically, and continuous magnitudes, or those that could be represented spatially and manipulated geometrically” (p.32)

“Descartes’s unit segment cannot be identified with the real unit interval [0, 1], nor with sections of the orthogonal number lines that we use to construct what we anachronistically call the ‘‘Cartesian’’ coordinate system. Descartes’s unit segment is a line segment of arbitrary length that stands for whatever particular magnitude functions as the unit in a particular problem. Even if the representational system were generalized and a fixed unit segment were chosen to represent the unit of magnitude functioning in any mathematical context, nevertheless the Cartesian unit segment would still serve as unit by virtue of the ratio in which it stands to the other magnitudes of the problem, as evidenced by their relative lengths, but not by virtue of its structure as a dense linear ordering.” (p.33)

“in Descartes’s system the unit segment, to which any other representative segment stands in relation, is problem-specific: for solution of a particular problem ‘‘we may adopt as unit either one of the magnitudes already given or any other magnitude, and this will be the common measure of all the others.‘‘9 Once the unit magnitude is chosen and represented as a particular finite line segment, representations of all other relevant magnitudes can be constructed in relation to that unit.” (p.33)

“the Cartesian representational system liberates formerly heterogeneous magnitudes: on the Cartesian system, the multiplication of two line segments yields another line segment, rather than the area of a rectangle. It follows that products, quotients, and roots of linear magnitudes can be construed to stand in proportion to the magnitudes themselves; likewise, the degree of algebraic variables need not be taken to indicate strictly geometric dimensionality.” (p.34)

“Once a single straight line is designated as unity, any positive integer can be straightforwardly represented as a simple concatenation of units. One advantage of the Cartesian representational system is that it allows the notion of number to expand beyond the positive integers so constructed: numbers can now be conceived as ratios between line segments of any arbitrary length and a chosen unit. Rational numbers are identified as those segments that are (geometrically) commensurable with unity, and irrational numbers those that are incommensurable” (p.34)

“I take it that whereas the Sixth Meditation demonstrates the existence of material things, the Fifth Meditation shows that if material things exist, then they have the properties that I clearly and distinctly perceive them to have, that is, all the properties which ‘‘viewed in general terms, are comprised within the subject-matter of pure mathematics.’’ See passages at AT 7:65 and AT 7:80.” (p.36)

“Descartes claims further that his perception of particular features of quantity is in harmony with his very own nature, ‘‘like noticing for the first time things which were long present within me although I had never turned my mental gaze on them before.‘‘20 This is a claim that is buttressed by the ‘‘wax argument’’ of the Second Meditation. There Descartes identifies extension, or extendedness, as that feature of the wax that makes it a material thing, its nature or essence. More important, perhaps, he identifies his own intellect as that cognitive tool that allows him to perceive the extendedness of all material things: Descartes’s perception of the essential feature of material substance is due neither to sensation nor to imagination, but to a ‘‘purely mental scrutiny’’ which enables his clear and distinct perception of pure extension. Note 20 – Descartes (1985), AT 7:64.” (p.37)

“for Newton, our geometrical cognition of space affords us knowledge of an entity that is infinitely extended, continuous, motionless, eternal, and immutable, but that is not itself corporeal and that can be conceived as empty of bodies. Moreover, space is a unified whole of strictly contiguous parts: the single infinite space encompasses every possible spatial figure and position that a bodily object might ‘‘materially delineate.‘‘27 Bodies, for Newton, are the movable and impenetrable entities that occupy space, and thus provide us with corporeal in- stances of spatial parts.” (p.39)

“Newton follows Descartes in positing a faculty of understanding as the real source of mathematical cognition, a tool with which we can comprehend the eternal and immutable nature of extension, which he conceives as infinite space.29 While sensation allows us to represent ‘‘materially delineated’’ bits of extension (i.e., bodies), and imagination allows us to represent indefinitely great extension, according to Newton only the faculty of understanding can clearly represent the true and general nature of space/extension.30 Note 30 – McGuire makes this point: ‘‘In this sense, then, the understanding possess [sic] a non-sensuous representation of infinite distance which has its ultimate ground in the real but uncreated nature of extension itself’’ (1983, p. 107). According to Newton, the understanding can likewise represent the finite but infinitesimally small quantities that are the basis of his calculus: ‘‘fluxions are finite quantities and real, and consequently ought to have their own symbols; and each time it can conveniently so be done, it is preferable to express them by finite lines visible to the eye rather than by infinitely small ones’’ (1982, p. 107). Of course, that these finite lines be literally ‘‘visible to the eye’’ is, on his own view, irrelevant. What matters is that the finite lines be mathematically manipulable symbols of infinitely small but nevertheless real quantities, the ultimate subject matter of the calculus.” (p.39)

“Leibniz resolves this apparent tension—between the sensible source of our notions of mathematical objects and the intelligible source of our justification of mathematical truths—with recourse to the ‘‘natural light’’ of reason. The ‘‘natural light’’ of reason is solely responsible for our recognition of the necessary truth of the axioms of mathematics and for the force of demonstrations based on such axioms. Mathematical demonstration thus depends solely on ‘‘intelligible notions and truths, which alone are capable of allowing us to judge what is necessary.’’ For Leibniz, then, our pure mathematical knowledge is formal knowledge of the logic of mathematical relations, which are not directly dependent on any sensible data. This pure mathematical knowledge might be described as verified by our sensible experience, but ultimately our ideas of the mathematical features of sensible things conform to the mathematical necessities that we understand by the natural light:” (p.42)

“According to Kant, mathematical cognition is cognition of our own intuitive capacities, of our own pure intuitions. Since Kant argues that our intuition of space is prior to and in- dependent of our experience of empirical spatial objects, geometric cognition is paradigmatically a priori; thus his account satisfies the apriority demand. But, on Kant’s view, mathematical cognition is also cognition of the empirical objects that we represent as having spatiotemporal form, that is, of the objects of our possible experience. Inasmuch as we have a priori geometric cognition of space, we have a priori geometric cognition of the spatial form of real spatial objects. The domain of our a priori mathematical cognition extends beyond the pure intuition of space to the formal conditions under which we represent empirical objects as being in space, thus allowing a priori mathematical cognition to find application in the realm of real empirical objects.” (p.44)

“Note that from Kant’s perspective, space and time are on this relationist view nevertheless ‘‘absolutely real’': the objects or appearances that stand in such spatiotemporal relations are conceived to be the source of our notions of space and time and, thus, to be ‘‘absolutely real,’’ that is, independent of what Kant takes to be transcendental conditions on experience. It follows from the metaphysicians’ view that the sub- ject matter of mathematics is derived from our experience of the natural world, and is not a factor in our own construction of that experience.” (p.45)

“Kant here takes both of his opponents to defend the absolute reality of space; on this basis, he judges that both opponents fail to solve the problem of identifying the a priori principles of our experience of the natural world.” (p.45)

Interesting “Kant admits that on such an account, mathematical cognition does not exceed what he takes to be the limits of possible experience: if mathematical cognition derives directly from our engagement with the realm of appearances, then our application of mathematical cognition to all and only appearances seems guar- anteed. This is a virtue of the metaphysicians’ account and explains the sense in which Kant accepts it as satisfying the applicability demand. But the metaphysi- cians’ account has, according to Kant, the fatal defect of not satisfying the apriority demand. As Kant puts it, the metaphysicians ‘‘must dispute the validity or at least the apodictic certainty of a priori mathematical doctrines in regard to real things (e.g., in space), since this certainty does not occur a posteriori….’’ Here Kant claims that because the original source of mathematical cognition is, on the metaphysicians’ view, experiential, our geometric cognition of the spatial features of empirical objects must be a posteriori."(p.47)

“Kant identifies the defect in the mathematicians’ account: on this view, our a priori mathematical knowledge is applicable to all—but not only—appearances. It attempts to extend our a priori mathematical knowledge beyond the domain of appearances without explanation or justification of that extension.” (p.47)

“Because space and time are forms of sensibility and cognitive sources of a priori mathematical principles, space and time ‘‘determine their own boundaries’’ and ‘‘apply to objects only so far as they are considered as appearances….Those alone are the field of their validity, beyond which no further objective use of them takes place.’’ Kant is making a claim about the connection between our way of intuiting, representing, and knowing the structure of space and our way of intuiting, representing, and knowing the features of the objects we experience to be in space: the former determines the latter. For this reason, Kant claims that our a priori representation of space determines its own domain of applicability, its ‘‘field of validity.''” (p.48)

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