1 The Birth of Science - MSRI

11.11 Recovery and Crisis of Scientific Methodology 349
the first modern edition of Menelaus saw the light of day in 1758, eight
years before Lambert wrote his work. 186
Hyperbolic geometry, too, is closely related to the spherical geometry of
Antiquity, and it is not surprising that Lobachevskii devotes to the latter a
good part of his New principles of geometry. Through his critical analysis of
the Euclidean set of postulates, Lobachevskii made a landmark contribution
to science as is generally agreed — but this precisely because he lived
in a culture that had never before created an axiomatic system comparable
to the Hellenistic ones. page 427
How slowly the scientific method was recovered is hidden from most.
For example, a student that takes a course in mathematical analysis and
encounters several theorems bearing Cauchy’s name never hears that the
now-standard statements of these theorems do not correspond to actual
theorems in that mathematician’s works. For Cauchy studied numerical
quantities, not the geometric magnitudes of Euclid, and for numbers there
was no rigorous theory analogous to the Euclidean one. Thus the “Cauchy
criterion” for the convergence of a sequence cannot be proved in the absence
of a theory of real numbers (which Cauchy lacked). As we have
remarked, 187 mathematical analysis became a scientific theory only after
the Euclidean notion of proportionality was reinstated by Weierstrass and
Dedekind, in 1872.
Up to that point, however, although mathematics had expanded widely,
especially in the direction of analysis, the maximum rigor that it had managed
to obtain in its base was that of Euclid, who remained unsurpassed
after twenty-two hundred years. To settle accounts with this cumbersome
character, it was necessary to finally face him on his own turf. This was
first attempted by David Hilbert with his Grundlagen der Geometrie, which
appeared in 1899, concluding a intense effort started by, among others,
Pasch and Peano. 188
Around the same time Peano formulated his axiomatization of arithmetic,
systems of axioms were created for various other branches of mathematics,
and several Hellenistic theories were revived, including propositional
logic and semantics. In areas more distant from mathematics, too:
besides dream theory, already discussed in Section 7.3, we may mention
186 The Greek text of Menelaus’ work has perished. The 1758 edition is a Latin translation from
Arabic and Hebrew manuscripts, edited by Edmund Halley. See [Menelaus/Krause] for a modern
critical edition.
187 See page 40.
188 Hilbert tried to improve on Euclid’s choice of postulates. Whether he succeeded is open to
discussion; the greater rigor obtained came at the cost of denying the postulates any meaning that
might relate them to experience. This created a problem of self-foundations in mathematics that
has not proved solvable.
Revision: 1.11 Date: 2003/01/06 07:48:20