METAPHYSICAL  STATUS OF FACTORS	241
A second, and to my mind a more plausible, reply is to
admit that, in an intricate field like that of human measure-
ment, we must be content with approximations, and accept
the simplest approximations as the safest, though not per-
haps the surest, until we have concrete evidence to show
what form the further complications can be assumed to take.
Thus, the more cautious factorist can start by postulating
a priori an unknown analytic function in the most general
form — g = f (.#!, #2? ..., #„) ; he can then defend himself
for choosing linear or additive functions to begin with,
by claiming that the unknown function may be expanded
by Taylor's theorem, and that, for a preliminary estimate,
the first term in the expansion alone need be retained.^1
Such a way of bridging the gap seems satisfactory enough
so long as we assume that a quantitative formulation of some
kind or other is admissible ; it certainly seems sufficient for
all practical purposes. For theoretical purposes, however,
we ought, I think, frankly to recognize that the question—
which mathematical function is " most likely to be correct" ?
—is really meaningless in psychology, because psychological
phenomena, so far as we know, do not strictly obey any of
the familiar mathematical laws. But there is an alternative.
Our difficulty is this. Ideally, in our generalized arguments,
it is not sufficient to substitute symbols to represent the
unknown variables : we need further symbols to denote
unknown functional relations. This suggests that the best
tool for factor-analysis will not be quantitative at all:
it will be a development of symbolic logic * rather than o£
^1	Cf. Marks of Examiners, p. 251.    It is curious to discover that if we sub-
stitute multiplication for addition, many of the transformations regularly
made in factor-analysis still yield the same results.    This is most easily
demonstrated in the case of the simple hierarchy, where the * two-factor
hypothesis * can be shown not to depend on the assumption that the relations
are additive.    We may even calculate factor-saturations by talcing geometric
means instead of arithmetic, i.e. by a multiplication method rather than by
a summation (loc. «*., p. 283).
^2	Just as the addition and multiplication of simple quantities are special
cases of the addition and multiplication of matrices, so addition and multi-
plication as understood in arithmetic and algebra are really special cases of
more general operations carried out by logic.    It would not be difficult to
generalize most of the factorial arguments along these lines.    Thus, we have
already seen that the ^£ product theorem,^5 on which the hierarchy is based,
16