FOM: Philosophy of Mathematics - why nothing works!

[Jeffrey Ketland]

I am certainly enjoying the recent discussions on FOM of philosophy of
mathematics -- ontological questions, questions about truth values,
determinability of these, completed infinities, etc. There is an excellent
recent short survey article (12 pages) of contemporary philosophy of
mathematics by Hilary Putnam. It's called:

    "Philosophy of Mathematics: Why Nothing Works", in Putnam 1994: Words
and Life (Harvard UP), pp. 499-512.

Putnam describes (and criticizes) the following 8 positions:

(1) Logicism: "mathematics is logic in disguise" (p. 499).
(2) Logical Positivism: "mathematical truths are true by virtue of rules of
language" (p. 500).
(3) Formalism: "set theory and non-constructive mathematics ... are just
"ideal" (and in themselves meaningless) extensions of "real" (finite
combinatory) mathematics" (p. 501)
(4) Platonism: "according to Godel, there really are mathematical objects,
and the human mind has a faculty different from but not totally disanalogous
to perception with the aid of which it acquires better and better intuitions
concerning the behaviour of these mathematical objects" (p. 503)
(5) Holism: "Quine has all along contended that mathematics has to be viewed
not all by itself but rather as part of the corpus of total science, and
that the necessity for quantification over mathematical objects if we are to
have a language rich enough for empirical science is the best possible
reason for taking the "posit" of sets exactly as seriously as we take any
other ontological "posit" .... sets and electrons are alike for Quine, in
being objects we need to postulate if we are to do science as we presently
do it" (p. 504).
(6) Quasi-Empirical Realism: "the idea [is] that there is something
analogous to empirical reasoning in pure mathematics" (p. 505).
(7) Modalism: "... we can reformulate classical mathematics so that instead
of speaking of sets, numbers, or other "objects", we simply assert the
*possibility* or *impossibility* of certain structures. "Sets are permanent
possibilities of selection" was the slogan." (p. 508).
(8) Intuitionism: "which accepts mathematical statements as meaningful while
rejecting the realist assumptions about truth (for example,
"bivalence"---every statement is true or false"). (p. 508). (Putnam mentions
Bishop-style constructive mathematics approvingly).

Putnam concludes with a section called "What Directions Should be Pursued?".
Putnam seems to go for a mixture of insights from the latter four, while
rejecting the first four.
I'd recommend any f.o.m.er interested in ph.o.m to have a look at this
article.

There are also four important contemporary positions that Putnam doesn't
discuss in any detail:

(9) Nominalism: most interesting and detailed version is developed in Hartry
Field's 1980 book "Science Without Numbers" (Princeton).
(10) Structuralism: several advocates these days (e.g., Stewart Shapiro
1997: "Philosophy of Mathematics: Structure and Ontology" (OUP) and Michael
Resnik 1997: "Mathematics as a Science of Patterns" (OUP)).
(11) Naturalism: c.f., Penelope Maddy's 1997 book "Naturalism in
Mathematics" (OUP).
(12) Predicative constructivism: c.f., Sol Feferman's 1998 book "In the
Light of Logic" (OUP).

Anyway, I think that positions (1)-(12) pretty much cover all bases in
contemporary philosophy of maths, although the mere names of these positions
only scratch the surface of what can be said either for or against them.
(Have I forgotten anything?).

Regards - Jeff

Jeffrey Ketland
Department of Philosophy
C15 Trent Building
University of Nottingham NG7 2RD
United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk