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"The Culture of Science"

November 20

Kris Tapp (Keck Fellow in Math)

"The Philosophies of
Mathematics"

Prepared by Anne Dalke

Additions, revisions, extensions are encouraged in the Forum

Participants

The disjunction between common knowledge and specialization becomes apparent when one does not know the history of the literature that lies behind the claims being made. The phenomenon Kris describes is not particular to math; most professional literatures "spiral inward," leading not to general answers but rather to more and more specialized ones. With the revolution in molecular biology, however, shared tools and methodologies allow for common conversations, even when the systems being treated are distant. Can the work of high level theoretical physicists not be evaluated because it has no empirical grounding? Are there fewer questions in nature? Are scientists drawing their questions from a smaller set than mathematicians do? Do mathematicians think that there are physical grounds for their theorems? In what sense do mathematical objects really exist?

In reply to such questions, Kris presented an overview of formalist systems, which all stop at certain undefined terms (such as point, line, plane), at certain axioms (rules about what can be proved), and at certain theorems (which have a very particular meaning for mathematicians: that which is derivable from agreed-upon axioms). These are purely formal systems, which computers can check; this is geometry disjoint from intution. For thirty years, beginning in the 1920's, mathematicians tried to shore up the foundations of their discipline, to re-package it as a formalist system. "Beautiful math" was created, but it didn't address the difficulty that real mathematicians don't produce formal proofs. The system was neither descriptive or prescriptive, but rather philosophical: it made mathematicians more comfortable with the rigor of their work. But Gödel's Theorem claimed that there could never be a data set large enough to prove both consistent and complete. This theorem has broad relevance for our intellectual lives: it says that the program of trying to achieve understanding by formal systems cannot succeed; to proceed on the assumption that the enterprise is to enunciate a fixed set of starting points and conclusions is to set oneself up for failure. We are constrained, and cannot complete that program. (For example, language cannot be analyzed purely in terms of the formal system of syntax.)

Math involves "pencil and paper" experiments only. In comparison with
other forms of inquiry, it does not need sensory input; the models it
works with--point, line, etc.--are defined outside the system. There was
a debate about whether mathematical concepts come from interactions with
the environment, or are purely symbolic manipulations; George Lakoff's
*Where Mathematics Comes From* was mentioned. It was
claimed, at first,
that there was no distinction between the reality of mathematical and
other objects, that "the issues are the same." Later, however,
descriptions were offered both of the excitement of being able to "strip
away" messy reality to manipulate an arithmetic system "that worked,"
and of the notion that mathematical objects are "completely knowable,"
and in that way fundamentally different from "real objects" such as
chairs. Math is a metaphor or model for real life. ("What is reality?"
was also asked, but NOT explored.)

If math is a figment of the mathematician's imagination, why is it so unreasonably effective in describing the world?

Although most mathematicians don't understand a given axiom, all accept a shared definition of truth: they allow themselves to create the machinery to prove the truth of a given theorem. There is in mathematics a sociological agreement about truth; although often confused with the common usage of the term, it only works within a limited system. It is based on the unreasonable belief that the gulf has been bridged between logicians and research being done in the forefront of the field. Although such researchers can tell which statements within a given system are contradictory or incomplete, or what the consequences for such claims might be, they do not, and CAN not, get to "ground zero." The projects of George Weaver, a philosopher who works on model theory, was described as an example: George looks to see what systems demonstrate a given set of axioms, and what such systems might look like.

When the humanists were (finally) allowed to enter the conversation, they suggested that mathematics--as a metaphor for the world, as a manipulation of symbols, as independent of sensory data--was very much akin to their own work. On what axis is such an affiliation grounded? There is certainly a history of descent which links math to the sciences; outside that phylogeny, however, there is a space of infinite dimensionality. Can a continuum be constructed on the basis of our old, much valued distinction of metaphor and metonymy? Is a more useful distinction based on "inside/outside," that is, on the degree of interaction with the physical world? Does another possible parameter have to do with the question of truth? If the "truth" of math derives from a set of axioms, not from the world; if the truth it proves is a chosen, arbitrary one, not accountable to the world, is it more like the work of humanists than that of the natural scientists? Or is the relationship between our various projects best figured, not in terms of a continuum, but rather as a circle, spiral or tube?

The brown bags will "bag it" next week, while we all have
Thanksgiving Dinner, and will resume on FRIDAY (note change
in day-of-the-week) December 6, when Tomomi Kinukawa will
bring a guest to campus. Judith Houck (Assistant Professor @
University of Wisconsin-Madison, in Medical History, Women's Studies,
History of Science, Center of Women's Health and Women's Health
Research), will speak with us about her work on the history of
menopause, as a way of introducing a more general conversation about
"Why Scientists Should Care About History."

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