Monday, 8 August 2011

The History of Philosophy as Philosophy: (5). Reading Forward, Reading Backward

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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