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,

-- 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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