Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics

By David B. Malament | Go to book overview

8
Geometry as a Branch of Physics:
Background and Context for Einstein's
“Geometry and Experience”

MICHAEL FRIEDMAN

Albert Einstein's celebrated paper, “Geometrie und Erfahrung,” first presented as a lecture to the Prussian Academy of Sciences at Berlin in January of 1921 and then published in “expanded” form later in that same year, is a landmark in the philosophy of geometry.1 In particular, it provided a very clear and sharp version of the distinction between “pure” and “applied”— mathematical and physical—geometry that soon became canonical in twentieth-century scientific thought.2 According to this conception, mathematical geometry derives its certainty and purity from its “formallogical” character as a mere deductive system operating with “contentless conceptual schemata.” The primitive terms of mathematical geometry, such as “point,” “line,” “congruence,” and so on, do not refer to objects or concepts antecedently given (by some sort of direct intuition, for example), but rather have only that purely “formal-logical” meaning stipulated in the primitive axioms. These axioms serve therefore as “implicit definitions” of the primitive terms, and all the theorems of mathematical geometry then follow purely logically from the stipulated axioms:

Geometry treats of objects that arc designated with the words line, point, etc.
No kind of acquaintance or intuition of these objects is presupposed, but only
the validity of those axioms which are likewise to be conceived as purely for-
mal, i.e., as separated from every content of intuition and experience.… These
axioms are free creations of the human spirit. All other geometrical proposi-
tions are logical consequences of the (only nominalistically conceived) axioms.
The axioms first define the objects of which geometry treats. Schlick therefore
designated the axioms very appropriately as “implicit definitions” in his hook
on theory of knowledge.

The conception represented by modern axioimatics purifies mathematics
from all elements not belonging to it, and thus removes the mystical obscurity
that previously clung to the foundations of mathematics. But such a purified

-193-

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