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.
@incollection{shabel2005,
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
ToC

Introduction (p.29)

Representational Methods in Mathematical Practice (p.32)

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

Kant’s Response (p.44)
References (p.49)
Overview
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 birdseyeview 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 mindindependent 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)
Important
**Important:**
“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
**Quote:**
“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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“in Descartes’s system the unit segment, to which any other
representative segment stands in relation, is problemspecific: 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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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 subjectmatter of pure
mathematics.’’ See passages at AT 7:65 and AT 7:80.”
(p.36)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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 nonsensuous 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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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)
**Interesting:**
“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)