From:
Journal of the History of Philosophy

Volume 41, Number 3, July 2003

pp. 426-427 | 10.1353/hph.2003.0043

*Journal of the History of Philosophy* 41.3 (2003) 426-427

Timothy Smiley, editor.Mathematics and Necessity: Essays in the History of Philosophy. New York: Oxford University Press, 2000. Pp. ix + 166. Cloth, $35.00.

*Mathematics and Necessity* contains essays by M. F.
Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given
to the British Academy in 1998. All concern the history of the
philosophical treatment of mathematics, necessity, or both,
although there is no single thread running through all three
essays.

Burnyeat focuses on Plato's understanding of mathematics in the
*Republic*, making use of the *Timeaus* in particular to
defend his interpretation. Plato claimed that in the ideal society
mathematics should be studied for a full ten years, which many have
thought excessive. To explain it, Burnyeat argues that the role of
mathematics is pervasive: ethics and politics are mathematical,
because in Plato's view mathematics is a constitutive part of
ethical understanding. On Burnyeat's interpretation, goodness is
part of the objective world, and mathematics is a science of
values. The link between mathematics and goodness in the objective
world is forged by the concord, harmony, and unity of proportions,
which express the goodness of the Divine Craftsman's beneficent
design. Proportions are both intrinsically good and the proper
study of mathematics.

Burnyeat explains why Plato thinks abstract kinematics (the pure
non-sensible counterpart to astronomy) and abstract harmonics lead
to knowledge of the good. Abstract kinematics concerns the "harmony
of the spheres" described in the *Republic*. Drawing heavily
on the *Timeaus*, Burnyeat argues that the circular motions of
the spheres are motions in the intelligence of the world soul and
that there are similar motions in human souls. The world soul and
human souls are spatial and have circular motions (invisibility and
intangibility make them incorporeal). Souls are the most important
subject matter of abstract harmonics—the study of good proportions.
A soul that understands the various mathematical disciplines and
their relations takes on the harmony and hence goodness of the
world soul. Such a soul serves as a model for organizing the social
world, a model which properly trained rulers can use to shape a
community. Burnyeat's well-argued essay is compelling and
exciting.

Ian Hacking's views provide a great contrast to Burnyeat's Plato. Hacking conducts a phenomenological inquiry into why some philosophers have become obsessed by mathematics as a source of philosophical inspiration. He holds that mathematics concerns a relatively minor part of human culture and that the influence of mathematics on philosophy is pernicious; he even describes it as an infection.

Hacking examines the views of six philosophers: Plato, Leibniz, Mill, Descartes, Lakatos, and Wittgenstein. Hacking's assessment of the six philosophers is at times disputable, but it is also interesting and stimulating, and serves his phenomenological inquiry.

Hacking traces philosophy's infection by mathematics to two factors. A mathematical proof can be perspicuous; that is, a proof can be "grasped," which requires seeing directly why something must be true and being able to recapitulate it. A mathematical proof can also anticipate facts about the world. He thinks that these two factors affect us in two ways: they give us the idea of a priori knowledge, and they dazzle us into unreasoned conviction in proofs. One might respond that, setting aside the feeling proofs give us, we are still justified in being impressed by them. Hacking argues that much of what impresses is based on a misunderstanding of their nature, which he attempts to correct.

According to Hacking, a mathematical demonstration involves shifts of meaning that analytically bind the proof and the theorem it establishes. They thereby mutually support each other and provide what he calls self-authentication. The correctness of the proof is supported by the "analytified" truth of the theorem and the theorem is considered true because proven.

This bootstrapping feature of mathematical proofs supports Hacking's deflationary account of necessity. Hacking suggests that the "must" of logical necessity rests upon our unwillingness to accept empirical counter-examples to the anticipated facts. The self-authentication of a proof and its theorem gives us new criteria for the application of terms in the theorem; those criteria will involve...

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