attempt to clarify disreputable issues such as divergent series (Boyer, 1985: 562,
566), but as an exploration of particular issues rather than a general methodologi-
cal concern.
- This helps explain why innovations in basic axiomatic systems now came from
schoolteachers on the periphery of the mathematical world, such as Lobachevsky
in Russia, Bolyai in Hungary, and Grassmann while teaching at a German Gym-
nasium. I explore this further in the section on England, “The Advantages of
Provinciality.”
- This is not to say that physical interpretations of non-Euclidean geometry were
not offered; for example, elliptic geometry can be considered the geometry of a
surface of a sphere. Gauss around 1820 attempted to measure angles among
mountain peaks in order to test which geometry fit (Kline, 1972: 867–880).
Non-Euclidean geometry was more a flashpoint of public recognition than an
underlying shift in the way mathematicians worked. N-dimensional geometries,
which had been used occasionally in purely technical manipulations by d’Alembert
and Lagrange, broke the connection with the physical world even more decisively
by the 1840s. Older mathematicians such as Gauss had anticipated some of these
conceptions by following out pathways available in existing mathematical practice;
but they failed to see the potential significance because they were still operating in
a frame of reference, grounded in the lack of differentiation between mathematics
and scientific application, in which mathematical concepts must have intuitive
meaning. Gauss’s unwillingness to publish many of his ideas, in contrast with
Cauchy’s ambitious rush into print in the new attention space of pure math, shows
the key part the social context plays in selecting ideas from an unfocused back-
ground and putting them into the center of attention.
- All categorical propositions are to be analyzed into propositions asserting or
denying existence; “all men are mortal” becomes “an immortal man does not
exist,” and “all triangles have 180 degrees” becomes “there exists no triangle that
does not have 180 degrees” (Johnston, 1972: 292–293; Lindenfeld, 1980: 48–53).
- This parallels Brentano’s doctrine, developed around the same time, that the mind
intends objects; the object in general is, so to speak, a slot to be filled with particular
objects. In Brentano, the doctrine is mixed with empirical psychology; Husserl, a
network offshoot of both Brentano and Frege, extracts from their concepts a
general phenomenology.
- To put it another way: an entire functional statement may be taken as content for
a second-order statement; thus, existence may be treated as a second-order concept,
but it is invalid to collapse the levels and treat them as an attribute of a first-order
concept within the nested statements. This is the error Descartes made when he
argued that divinity implies existence just as the concept of a triangle implies truths
about its angles. In the same way, the universal quantifier (“all”) is a second-order
concept whose complexities are hidden in ordinary language (Kneale and Kneale,
1984: 504; Coffa, 1991: 73).
- This is the Leibnizian problem of identity of indiscernibles again. Frege points out
that it cannot be solved by distinguishing different positions in space or time, since
the issue arises once again in this context.
Notes to Pages 699–702^ •^1013
