Monday, March 7, 2011

Realism: What (concluded)

We conclude our lexicographic exercise on the usage of the term Realism, with its application to mathematics.

The philosophy of mathematics has its own vocabulary for Realism and its rivals:  mathematical realism is called “Platonism” (in a somewhat Pickwickian sense of the term);  that we nonetheless treat this as a special case of Realism sensu lato, is itself tendentious, being part of our contention that coffee-cups are more like Riemannian manifolds than you might surmise, and that Riemannian manifolds are themselves (once you get to know them) more like coffee-cups  than is generally appreciated.

Here is an uncontroversial characterization:

Mathematical realists, or ‘Platonists’, have emphasized the non-mental nature of mathematical entities.  The addition function is not in any particular mind, nor is it the common property of all minds.  It has an independent, ‘objective’, existence.
-- Saul Kripke, Wittgenstein on Rules and Private Language (1982), p. 53

By whatever name, the  plainest rival to Platonism is Constructivism.  Thus Michael Dummett, “Wittgenstein’s Philosophy of Mathematics” (1959), reprinted in Truth and other enigmas (1978), p. 166:

In the philosophy of mathematics, Platonism stands opposed to various degrees of constructivism.  According to platonism, mathematical objects are there...

And Michael Dummett, “Realism” (1963), in Truth and other enigmas (1978), p. 163:

In mathematics, an anti-realist (i.e. constructivist) position  involves holding that a mathematical statement can be true only in virtue of actual evidence, that is, of our actually possessing a proof.

Note how Dummet uses “realist” (in a mathematical context) and “Platonism”  interchangeably.  He solidifies the identification by his use of the general term “nominalism” in a mathematical context as well. Truth and other enigmas (1978), p. 166:

As Frege showed, the nominalist objection to platonism -- that talk about ‘abstract entities’ is unintelligible -- is ill-taken.

Michael Dummett, “Truth” (1959), repr. in Truth and other enigmas (1978), p. 18:

Intuitionists speak of mathematics in a highly anti-realist (anti-platonist) way:  for them it is we who construct mathematics; it is not already there ….

Note:  only the term “constructivism” was (so far as I know) specifically confected to contrast with “Platonism”, in the way that nominalism is the antonym of realism.  But intuitionism is a whole program, not just an anti-stance.  Likewise Hilbert’s formalist program.  According to Reuben Hersh (“Some Proposals”, repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 16), 
Hilbert’s writings and conversation display full conviction that mathematical problems are questions about real objects

-- i.e., full-bore Platonism.  But for foundational reasons he championed a program called formalism.  As such, they need not logically stand in contradiction;  but in practical, psychological terms, they do tend to.  Again Hersh:
We can see the reason for the working mathematician’s uneasy oscillation between formalism and Platonism.

The view of a Dutch ‘Intuitionist’ constructivist, who balks at taking that simplest of progressions, the Natural Numbers (which even the nominalist Kronecker could swallow):

Brouwer … characterized his view on mathematics  as opbouwende wiskunde (‘constructive mathematics’) … [He] presents the natural numbers as a sequence characterized by a law, not as an infinite set.  [Yet] the continuum is introduced by a special continuum intuition.
-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 36

Surely this is to strain at a gnat  and swallow a camel.

Quine, in “On What There Is” (reprinted in From a Logical Point of View), rather surprisingly uses the term logicism where we would expect Platonism:

The three main mediaeval points of view regarding universals  are designated by historians as realism, conceptualism, and nominalism.  Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics  under the new names logicism, intuitionism, and formalism.

Russelian logicism, as I understand it, and as Wiki outlines it, was something quite other, or so I thought, bearing really no relationship to Platonism at all.  But Quine knew a vast amount about all this, so perhaps there is some deep connection that I am missing.  In any case, this terminology is not widely used.

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