THE RISE OF FUNCTIONS
The Greeks did, of course, have some familiarity with categories of
cognition such as “correspondence,” “dependence,” “mapping,” even
“binary relation,” which enter into our present-day notion of a function.
But the mere occurrence of such categories does not yet make for the
presence of functions. Even the occurrence of assertions or conclusions
which can be readily translated or transliterated into functional relations is
not yet enough. It is still necessary, and this is decisive, that something
also be “done” with those functions, or with the trains of reasoning corre-
sponding to them, that is, that some kind of “mathematical” operations
be performed with, or on, those functions, or with the trains of reasoning
that “stand in” for functions. Now, it is this kind of “operational” activity,
or only attitude, which it is difficult to discern in the realm of Greek thought,
mathematical or other.
Thus, although the Greeks — and, in fact, Aristotle single-handedly —
created the syllogistic aspects of our modes of formal deduction, they never
broke through to a satisfactory conception of “relation,” binary or n-nary,
reflexive, invertive, or correlative. Book 5 of Aristotle’s Metaphysics is a
dictionary of some basic philosophical terms, and among others it has a
lengthy entry on “relation” (pros ti). But the content of the entry is so
embarrassingly ordinary that philosophers and logicians in general are not
aware of it, and only “all-inclusive” commentaries of Aristotle take note
of it. And of an algebra of relations, as begun by Leibniz (1646-1716) and
insisted upon by the American mathematician-philosopher Charles Sanders
Peirce (1839-1914), there is hardly a trace among the Greeks. Furthermore,
historians of logic have been recently asserting, even heatedly, that the
Greeks remained creative in the field of logic even after Aristotle, and that,
specifically, some principles of our “modern” propositional modes of
implication were already discovered by some of the Stoics [2, pp. 6-8].
But, here again, of propositional functions, that is of propositional schemes
that involve “all” or “any,” there is no trace [2, p. 32].
Even the great Archimedes does not have functions in his thinking,
meaningfully that is. Isaac Newton’s treatise on mechanics [3] is ostensibly
composed in the style of Archimedes, that is, in terms of curves and geo-
metric paths, all without coordinates. Yet Newton’s treatise is, by its
internal direction, function-oriented, whereas the work of Archimedes is not.
For instance, Newton views the tangent to a planetary orbit at a point as
the limiting position of a secant through this fixed point and a variable
neighboring point of the orbit, meaning that he performs the operation of
differentiation on “hidden” coordinate functions. Archimedes however
adheres to the euclidean definition that a tangent to a curve is a straight line