nominalism bis

[Mikhail Katz]

The recent thread on nominalism seems to have petered out, possibly
because many mathematicians don't really see what the consequences
would be of deciding between nominalism, platonism, etc.  Ian Hacking
put it nicely in his book "Why is there philosophy of mathematics at
all?" as follows on page 75:

"None of the current favorites - platonism, nominalism, and
structuralism, say - has or implies any view about whether current
mathematics is the contingent result of a random walk, the Latin
Model, or, inevitably, the Butterfly Model."

Here Hacking contrasts a model of a deterministic (genetically
determined) biological development of animals like butterflies (the
egg-larva-cocoon-butterfly sequence), with a model of a contingent
historical evolution of languages like Latin.  In historical work,
emphasizing determinism over contingency can easily lead to
anachronism.

To give the discussion more flesh, I would mention a couple of
examples of what the implications would be of thinking of the
evolution of mathematics in terms of the Latin model:

(1) One could envision the possiblity that the events that led to the
establishment of set theory as a foundation could have occurred
differently, and category theory may have evolved as an early
foundation, rather than its competitor set theory.  Thinkers like
Frege and Peano at the end of the 19th century were already thinking
in terms that are strikingly modern in some respects.

(2) Sticking to set theory, one can envision the possibility that
Frege, Peano, and others may have developed a foundation that
incorporates more of the current (at the time) PRACTICE of 19th
analysis, whose trademark was the reliance on infinitesimals,
resulting in axiomatic foundations more along the lines of Nelson's
Internal Set Theory, where infinitesimals are found within the
ordinary real line.

My point is that if one thinks of evolution of analysis in
teleological terms, it is much harder to conceive of the possibilities
(1) and (2).

M. Katz