Historians of philosophy
differ in their strategies for seeking a context.
Some interpreters, such as Gaukroger or Buchdahl, read forward:
they take the period preceding and surrounding a given author
as the primary context. Others employ a strategy of reading backward. Friedman, in Kant and
the Exact Sciences, uses some
preceding material (especially in considering Kant’s
Newtonianism). But in addressing Kant’s philosophy of mathematics,
he reads backward from the perspective of
late-nineteenth-century developments in mathematics
and logic. He adopts attitudes that were not
available before the late nineteenth century about the relation
between logic and mathematics and about the subjectmatter of
mathematics itself, and he then interprets Kant by considering
how his work anticipated or fell short of the standards set by
these ways of thinking.
A primary aspect of
Friedman’s reconstruction concerns Kant’s
proposal that geometrical proofs require appeal to spatial intuition.
Kant makes the point most clearly in the Doctrine of Method
in the first Critique, where
he argues that, in geometry, synthetic
procedures relying on spatial intuition are needed; discursive logic and the analysis of concepts are
insufficient by themselves. Friedman
sees this appeal to spatial intuition as arising because
the logical resources available to Kant (monadic logic) were
inadequate for logically constructing continuous magnitude (either
the real number line, or a weaker subset of the reals, the rationals
together with square roots). For example, if Kant had been
asked to defend the proposition that a line-segment crossing the circumference of a circle (it starts
inside and ends outside the circle)
intersects that circumference, he could only have appealed to
constructive procedures that relied on spatial structure. After geometry had been interpreted on an
algebraic foundation in the nineteenth century, so
that line-segments and arcs of circles were constituted
as loci of point co-ordinates, a proof of this intersection could be provided algebraically. If one wished in this context to interpret the real number line
logically, one could construct a pointspace with
irrational co-ordinates (and thus betweenness relations appropriately
dense for the problem) by employing the dependence relations
for universal and existential quantifiers of modern polyadic logic. But Friedman has Kant realizing
that his own (monadic) logical resources could not
establish such a point-space, and turning to
iterative constructive procedures (in a spatial medium) to get it done. Accordingly, Kant would
demonstrate the appropriate infinity of
points, including the point of intersection, through infinitely (or indefinitely) iterated procedures of
construction (constructing one point,
then another, with compass-and-straight-edge procedures that
include square-root line-lengths).
This retrospective reading
ignores the facts that, in Kant’s time, geometry
was commonly considered to be more basic than algebra, and
geometrical structures were not thought to be composed of or constructed from points or point-sets.
The idea of deriving all geometrical
structures from algebraic relations was foreign to mathematics,
certainly at the basic level at which Kant taught and understood
mathematics. (Euler and others were laying the foundation for
algebraization, but Kant didn’t contend with that level of mathematics.)
In the Critique, Kant offered a good
philosophical reconstruction of the
actual procedures of proof used in Euclid’s geometry and its common eighteenth-century expressions.
Lisa Shabel has shown that these procedures did not
rely primarily on logical structure, but often
drew upon the spatial relations exhibited in diagrams constructible with only compass and straight-edge.
These constructive procedures were not used to
demonstrate the existence of an infinite structure;
infinite spatial structure (or continuous, in the sense of unbroken)
was assumed. For example, if a proof required placing a point on
a line-segment between its two end-points, the procedure relied
on the assumed spatial structure of the line-segment. That is, it was taken as given that all points of
the segment lie between the two end-points; a point located anywhere
on the segment was already known to be between
the end-points, and its existence need not be
proved. As Shabel argues, Kant’s discussions in the Critique captured the ineliminable role of such
appeals to spatial structure in the
proofs of the extant Euclidean geometry. In this context, questions
about the existence of the point where a line crosses a circle
do not arise; such problems first arise with the nineteenthcentury reconception of geometry in algebraic
terms.
A reconstruction of Kant’s
philosophy of mathematics should, at the
outset, pay close attention to the actual mathematical conceptions
and practices of Kant and his predecessors. By allowing
a later understanding of the problems and methods of geometry
to set the context, Friedman missed fundamental aspects of
Kant’s theory and achievement. Whereas Kant appealed to spatial
intuition because he recognized the role of spatial structure in Euclid’s proofs, Friedman instead
sees him as responding to questions
that arose only fifty or one hundred years later by employing
a counterpart to modern logical techniques. In writing the
history and philosophy of mathematics, it will be more fruitful to read forward, by asking how the
problems and methods of geometry were conceived at
one time and then came to be reconceived
later. Kant’s position will not be most fruitfully characterized
as ‘not yet using’ the later methods, or as ‘using this work-around’
to solve the later problems. Taking earlier mathematics and
philosophy on their own terms will help locate the specific
problems and opportunities that motivated or afforded later
developments.
I do not suggest that reading
backward is never useful. I do suggest
that reading forward is more often useful in setting context. Reading backward should come later, in
posing questions about shapes and themes in history.
Source : Sorell, Tom and
G. A. J. Rogers.(2005), Analytic Philosophy
and History of Philosophy (Oxford: Clarendon
Press).
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