If Aik is a component of a covariant tensor of rank 2, then its covariant
derivative with respect to xl is given by
Aik,l ∂Aik
∂x^1 (^) ∑
j = 1
n
{il,j}Ajk (^) ∑
j = 1
n
{kl,j}Aij.
The network of mathematicians now becomes more restricted; at some level it
is limited to the network of active mathematicians who are creating the
research front of mathematical truths.
For comparison, consider a statement made in the Chinese mathematics of
the Sung dynasty algebra in Figure E.1. The difficulty is not merely that we (if
we are Westerners) do not know the individual symbols in the same way that
Westerners usually cannot understand the equation 4 5 9 if it is written
but that we do not know the operations determining how these markers are
to be manipulated. Chinese mathematics was performed on a counting board
divided into squares (depicted in this text as a series of 3 3 or 3 4 matrices,
depending on how much of the counting board was used). The Sung algebra,
called the “celestial element method,” was a set of procedures for representing
expressions of constants and unknowns raised to various degrees by placing
number signs in particular places on the board surrounding the central element.
For example, in conventional modern European notation, the frame in the
middle of the first column on the right can be stated as xy^2 120 y 2 xy
2 x^2 2 x. The Chinese ideographs between the frames are an argument in
words (read from top to bottom and right to left), explaining how one algebraic
expression is to be transformed into another. This is a verbal rendition of
mathematical results; in actual practice, the mathematician uses a set of stand-
ard procedures for manipulating the counters on the board and thereby trans-
forming one expression into another. The physical operations and the symbolic
structure (not just the individual symbols) are different from the Cartesian rules
for moving expressions from one side an equal sign () to another; the
similarity is in the overall form of the practice, which allows derivation of
strings of mathematical expressions from one another.
A Platonist would say that the form of statement is irrelevant; that the
derivation from one mathematical expression to another is true, no matter if
it is written in verbal argument in Latin, in post-Cartesian symbolism, in Sung
algebra, or anything else. But Platonism is merely a theory; it assumes what
has to be proven, that mathematical truths exist somewhere in a realm that
has nothing to do with the human activities of making mathematical state-
ments. One may show this with a mathematical quasi-cogito: if I deny that a
mathematical statement must exist in the form of some particular kind of
Epilogue: Sociological Realism^ •^863