of previously accepted concepts of mathematical reality, strange mathematical
beasts which outraged traditionalists. A self-generating cycle was set in motion:
the conflicts over paradoxes energized stronger efforts to overcome them by
more careful re-digging of axiomatic foundations and by making clear the
operations of the mathematical machinery which produced them. Further
axiomatization and rigor made it possible to examine still other kinds of
axiomatic sets, and to discover still more comprehensive layers of abstract
space, within which could be found still more of the paradoxical breed. Not
only spatial intuition but any acceptance of self-evident procedures or concepts
came under suspicion.
Vast new realms of higher mathematics opened up. At the same time,
conflict grew acute between the most enthusiastic explorers of those realms of
free mathematical invention—such as Georg Cantor, with his multiple levels
of transfinite numbers—and opponents who declared that rigor must ground
itself upon firmly intuited reality. Mathematics, the time-honored standard of
absolute certainty and consensual agreement, now split into a foundational
conflict. At this point some mathematicians reached a meta-turf resembling a
branch of philosophy, and a new hybrid of mathematical philosophy arose.mathematician-philosopher hybrids
from descartes to russell
The key points of modern philosophy come when the mathematical network
spills over into philosophical terrain. This happens naturally because any topics
pursued at a sufficiently high level of abstraction and reflexivity leave the
specific problems of the original practitioners and encounter philosophical turf.
In his own interests and intentions, Descartes was a mathematician and (what
we would call) natural scientist. He was a key figure in proclaiming and
consolidating the takeoff of the mathematical revolution, synthesizing the
notational tools developed by his predecessors, and turning them into an
explicit machinery for the manipulation of equations. His famous analytical
geometry, published in conjunction with his Discourse on Method, displays
how even the most traditional parts of mathematics can be pushed forward
with this new machinery. In his larger philosophical statements, Descartes
extends to the entire realm of knowledge the confidence of the scientific and
mathematical revolutions, the feeling that rapid discovery and certainty are
linked through the use of the new methodological machinery. Mathematics was
his inspiration and standard for attempting the comprehensive deduction of
all certain knowledge; his goal was to deduce all scientific principles from the
properties of extended substance or matter, which now replaced the Neopla-
tonist and Aristotelean cosmologies.
The philosophical attention space was rejuvenated by the preliminary parts
850 •^ Meta-reflections