Section 5: Logic & Scientific Method

CARNAP, MENGER, POPPER: EXPLICATION, THEORIES OF DIMENSION, AND THE

GROWTH OF SCIENTIFIC AND MATHEMATICAL THEORIES

Dale M. Johnson

1949 Weybridge Lane

Reston, Virginia 20191-3621U.S.A.

e-mail: [e-mail]

Rudolf Carnap advocated a procedure of explication—a procedure of transforming an inexact con-
cept into a new exact concept—as a significant part of science and philosophy. Explication is a
form of concept or meaning analysis. Karl Popper argued strongly against the usefulness of expli-
cation per se, because it focuses on problems about concepts or meanings and conceptual exactness
and misses the central problems in a given problem-situation that drive the development of theories
in science or mathematics. Carnap thought that looking for exact concepts is often a useful activity
in science or mathematics. Popper thought that the underlying problems and resulting theories are
the important matters. More or less exact concepts may be a byproduct of solving problems and
constructing theories, but looking for them is never a primary activity.
A significant historical background for Carnap’s philosophical discussion of explication is Karl
Menger’s proposals for mathematical definitions of dimension and also of curve. Menger’s work in
this area began in 1921. He elaborated his mathematical theories of dimension and curve in his
books Dimensionstheorie (1928) and Kurventheorie (1932). According to Menger (Selected Papers
(1979), page 4) his ideas on scientific definition became popular in the early Vienna Circle. Rudolf
Carnap took Menger’s definitions of dimension and curve as paradigms for explication.
My study of the development of dimension theory strongly suggests that it should not be construed
as conceptual explication to any great extent; the really significant development of the theory is best
interpreted from the Popperian perspective. Menger’s definition of dimension needs to be consid-
ered along with others. His dimension theory needs to be viewed in a larger historical context as
part of the growth of alternative and competing mathematical theories. (Cf. my papers ‘The Prob-
lem of the Invariance of Dimension in the Growth of Modern Topology,’ 1979, 1981.)
Menger’s inductive definition of dimension is intuitive and beautiful. His own philosophical view
of making precise scientific definitions is appealing. However, Menger was not merely interested
in the definition; he sought a body of theorems, a theory. Similarly others proposing definitions of
dimension, e.g., Poincaré, Brouwer, Urysohn, Menger, Čech, and Hausdorff, were not merely clari-
fying or explicating concepts but were searching for theoretical consequences.
Founders of dimension theories wanted to solve some theoretical problems and not just provide
definitions. Strong divergences appeared among the theories. An especially strong one cropped up
between the Brouwer theory of dimension degree and the Urysohn-Menger theory of dimension,
such that there is a set that has Brouwer dimension degree 0 and that is infinite–dimensional accord-
ing to Urysohn and Menger. Further divergences among dimension theories were highlighted by
the spaces assumed, whether compact metric, separable metric, metric, or topological.
Popper’s expansive view of the growth of theories, unlike Carnap’s limiting notion of the explica-
tion of concepts, provides a better way to understand significant parts of the history of dimension
theory. Though Menger’s work may have motivated Carnap’s notion, it is better interpreted accord-
ing to Popper. Definitions and theories of functions, the integral, measure, and notions of probabil-
ity in their historical context are also likely to be understood better by taking Popper’s view. (Cf.
Popper, Realism and the Aim of Science, pages 261-278.)