requires a sociology of mathematics. The history of mathematics is the purest
case of the abstraction-reflexivity sequence. The objects which mathematicians
investigate are intellectual operations. It took many generations of continuous
intellectual competition before the community came to recognize this. At first
mathematics was regarded as a realm of objective objects—the Pythagoreans’
number pebbles and the Platonic Forms. It took successive layers of self-reflec-
tion for the network to discover that it could create new operations, new modes
of manipulating its symbolism, and investigate their consequences. Higher
mathematics consists in exploring this reflexivity of mathematical operations.
Since thinking is internalized from communication within a social network,
mathematics is the investigation of the pure, contentless properties of thought
as internalized operations of communication per se. This will become clearer
if we review the two great revolutions in modern mathematics.
mathematicians discover the pure
reflexiveness of intellectual operations
The takeoff of modern mathematics in the 1500s and early 1600s occurred by
turning mathematics into a problem-solving machine. The verbal methods by
which mathematical argument previously had been carried out were replaced
by a pervasive symbolism, and even more important, by an apparatus for
manipulating symbols according to rules. Mathematics became a technology
of symbols on paper, lined up in equations; the principles of how to move
symbols about so as to transform one into another mechanically became both
the means of solving particular problems and an arena in which reliable
generalizations were discovered. Methods were found for solving lower-order
problems (e.g., algebra gives general principles for classes of problems in
arithmetic). These methods in turn could be taken as lower-order patterns
about which still higher-order principles can be discovered. That is to say, the
lower-order operations can be collected under a symbol (numbers are symbol-
ized as algebraic unknowns x, y, and so on; adding, subtracting, and so forth
can be symbolized as a function; functions can be collected under functions of
functions), and the rules for operating with those higher-order symbols can in
turn be explored. As earlier problems were solved, extensions were suggested
(e.g., from rules for solving algebraic problems involving cubic equations, new
challenges were posed by competing mathematicians to solve quartic and
quintic equations, and so on). Suddenly there was an outpouring of mathe-
matical discoveries. This intellectual network began to pulsate with energy as
it explored a widening turf: the general features of algebraic equations and
trigonometry, which in turn cross-fertilized older areas of mathematics such as
geometry, giving increasing attention to the more difficult problems of solid
figures, and the conic sections which turned out to yield representations of
motion and open the way to the calculus.
848 •^ Meta-reflections