[FOM] FOM: Remedial mathematics?,?

Irving ianellis at iupui.edu
Tue May 24 15:04:14 EDT 2011

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 

'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

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