[FOM] FOM: Remedial mathematics?,?

Jeremy Shipley jeremyrshipley at gmail.com
Tue May 24 15:54:30 EDT 2011

> In any event, I suggest that what Mr. Shipley has in mind by
> "existentialism" is really what Hersh, and philosophers of mathematics, more
> typically mean by Platonism.

Not quite. A Kantian or neo-Kantian might be an "existentialist" (in
my sense) insofar as the objects of mathematics are given in intuition
then described in axiomatization but not a Platonist holding that they
are mind independent. Alternatively, some one like Stewart Shapiro
might be described as an "essentialist" but also a Platonist, though I
believe he distinguishes his ante rem structuralism from Plato's
Platonism. Indeed, in your quote Hersh lists Hilbert as a Platonist
(although I do not think this is clear), but he is my paradigm
"essentialist" because he held that we may freely postulate objects
corresponding to any consistent axioms.

Maybe those terms really are too cute in a way that is misleading. I
mean to be using them playfully, but not terrifically suggestively.
Insofar as the opposing views of Plato and Aristotle are suggested the
connection is very vague and loose.

On Tue, May 24, 2011 at 2:04 PM, Irving <ianellis at iupui.edu> wrote:
> Jeremy Shipley wrote:
>> I have a fondness for the following (maybe too cute) way of labeling
>> the positions taken by Frege and Hilbert in their old dispute over the
>> foundations of geometry, a dispute I have done some thinking and
>> writing about in the process of developing my philosophical views.
>> Existentialism (Frege): Existence (ie, intuition of logical and
>> geometric objects) precedes essence (ie, consistent axiomatic
>> systemization).
>> Essentialism (Hilbert): Essence precedes existence.
> I recall that, many years ago (probably some time in the early or
> mid-1980s), Reuben Hersh gave a colloquium talk in the mathematics
> department at the University of Iowa. I don't recall the specifics of that
> talk, but in its general tenor it went along the lines that, in their
> workaday world. most mathematicians are Platonists, working as though the
> mathematical structures with which they are working and which are the
> subject of theorems exist, whereas, on weekends, they deny the real
> existence of mathematical entities.
> In the description for Reuben Hersh's What Is Mathematics Really? (Oxford U.
> Press, 1997), Hersh's position is described (in part) as follows:
> 'Platonism is the most pervasive philosophy of mathematics. Indeed, it can
> be argued that an inarticulate, half-conscious Platonism is nearly universal
> among mathematicians. The basic idea is that mathematical entities exist
> outside space and time, outside thought and matter, in an abstract realm.
> ...In What is Mathematics, Really?, renowned mathematician Reuben Hersh
> takes these eloquent words and this pervasive philosophy to task, in a
> subversive attack on traditional philosophies of mathematics, most notably,
> Platonism and formalism. Virtually all philosophers of mathematics treat it
> as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that
> mathematics must be understood as a human activity, a social phenomenon,
> part of human culture, historically evolved, and intelligible only in a
> social context. Mathematical objects are created by humans, not arbitrarily,
> but from activity with existing mathematical objects, and from the needs of
> science and daily life. Hersh pulls the screen back to reveal mathematics as
> seen by professionals, debunking many mathematical myths, and demonstrating
> how the "humanist" idea of the nature of mathematics more closely resembles
> how mathematicians actually work. At the heart of the book is a fascinating
> historical account of the mainstream of philosophy--ranging from Pythagoras,
> Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert,
> Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw
> mathematics as a human artifact, including Aristotle, Locke, Hume, Mill,
> Peirce, Dewey, and Lakatos. ..."
> I don't know whether it makes a philosophical difference for his
> anti-Platonist attitude, but Hersh's early work was in applied mathematics,
> P.D.E.s, linear operator equations, and their concrete applications, e.g. to
> stochastic processes, rather than in abstract or "pure" mathematics.
> In any event, I suggest that what Mr. Shipley has in mind by
> "existentialism" is really what Hersh, and philosophers of mathematics, more
> typically mean by Platonism.
> Irving H. Anellis
> Visiting Research Associate
> Peirce Edition, Institute for American Thought
> 902 W. New York St.
> Indiana University-Purdue University at Indianapolis
> Indianapolis, IN 46202-5159
> URL: http://www.irvinganellis.info
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

Jeremy Shipley
Ballard and Seashore Doctoral Research Fellow
Department of Philosophy
The University of Iowa

jeremy-shipley at uiowa.edu
jeremyrshipley at gmail.com

More information about the FOM mailing list