Page 85 - Contribution To Phenomenology

78 OSBORNE WIGGINS

The Origin of Mathematics

In order to advance his thesis concerning the construction of formal
systems out of more basic (lifeworldly) experiences, Lakoff adopts a view
of the foundations of mathematics put forward by the mathematician
Saunders Mac Lane (Lakoff, 1987, pp. 361-365). Mac Lane maintains that
if we conceive set theory as the foundation of mathematics, certain
questions in mathematics remain unanswered. Mac Lane mentions in
particular the question: Why does mathematics have the branches it has?
Mac Lane contends that "the grand set-theoretic foundation" of mathe-
matics,

does not adequately describe which are the relevant mathematical
structures to be built up from the starting point of set theory. A priori
from set theory there could be very many such structures, but in fact
there are a few which are dominant . . . .Some mathematical structures
(natural numbers, rational numbers, real numbers, Euclidean geometry)
are intended to be unique but other structures are built to have many
different models: group, ring, order and partial order, linear space and
module, topological space, measure space. The "Grand (Set-Theoretic)
Foundation" does not provide any way in which to explain the choice
of these concepts (Lakoff, 1987, p. 361)

Mac Lane proposes an answer to this question that resembles the
genetic thesis of phenomenology. Mac Lane writes,

The real nature of these structures does not lie in their often artificial
construction from set theory, but in their relation to simple mathematical
ideas or to basic human activities . . . mathematics is not the study of
intangible Platonic worlds, but of tangible formal systems which have
arisen from real human activities (Lakoff, 1987, p. 361).

Mac Lane thus seeks to correlate specific portions of mathematics with
specific kinds of human activities. He devises the following list of
correlated human activities and branches of mathematics:

counting: to arithmetic and number theory
measuring: to real numbers, calculus, analysis
shaping: to geometry, topology
forming (as in architecture): to symmetry, group theory
estimating: to probability, measure theory, statistics