FOM: Platonism v. social constructivism

[Reuben Hersh]

Dear Professor Silver,

Thank you for an enjoyable, thought-provoking letter.  I will confine myself

(more or less) to direct answers to your questions, rather than take the

time and space to repeat what I said as well as I could in my book,

"What is Mathematics, Really?"

First of all, what about that "really"?  As I explain in the first

few pages of the book, there is another book, a famous classic, by

Courant and Robbins, called "What is Mathematics?"  Courant and

Robbins was a major influence on me, as was, later, Courant

himself.  Yet I found the title of their book a little misleading,

for it consists of exposition and demonstration of beautiful mathematics,

but no direct answer to the title question.  So I, somewhat jokingly,

somewhat in humble tribute, thought of my book as carrying out the

promise in their title.

	To someone who has not looked at my book, all

this may smack of an "in" joke; that was not my intention.

	Why do I renounce aristocratic vs. humanitarian math?  Because

Prof. Tragesser, in all good will I am sure, attributed such thoughts

to me in one of his postings.  I do not wish to be asked to explain

the meaning of those expressions.  Of course that doesn't mean denying

that there could be two versions of math.  In fact, there already are

at least two--"classical"  and constructivist.  The indeterminate

state of the Continuum Hypothesis  (CH to FOM'ers) opens another

possibility of 2 maths.  I don't know if the same can be said about

categorical foundations vs. sets.  I don't see this as a problem.

	Next, postmodernism.  I am in shock following

a column in the New York Times last Saturday by a person namned Edward 

Rothstein.  He shredded me up as a multiculturalist and postmodernist!!!!

Then there was a posting by Jon Barwise where, if I recall, he said he

would be careful about social constructivism for fear of "the black

plague" of postmodernism.  So I wrote that I had been naive not to foresee 

such problems.

	No, I certainly don't believe math (or poetry or fiction or

journalism) is no more than "text."  From what I hear of what's going

on in lit crit and related areas, that idea deserves to be

called a black plague.  One of the principal themes of my book is that

mathematics is a living practise of a human community, much more than

just piles or rows of journal articles and treatises.  In fact, as a

rule those articles are incomprehensible to any one not initiated into the

mathematical subculture of the authors.  This

is carefully explained in a section of the book called "Mathematics

Has a Front and a Back."

		In my book I avoided the term "social constructivism"

while acknowledging that it's used by some authors to whom I am

sympathetic.  But I did not want my thoughts to be tied to theirs.

So I adopted the name "humanism", wanting my point of view

to be judged on its own, not by association.  Nevertheless, the

commentators on my book on this list always say  "social constructivism."

	Do I intend to reduce the study of mathematics to history and

perhaps some kind of sociology, so that mathematics becomes about--nothing?

	No!  Not at all!  Just the opposite!

	 I seem to hear a notion that what isn't mental, physical,

or transcendental, is--nothing!  But that's the whole point of my list

of social-cultural-historical entities.  The death penalty, the baseball

pennant race, the stock market, anti-semitism, patriotism, Catholicism,

the Lubavitcher othodox community, the yen, the mark and the dollar,

and yes, multiculturalism and postmodernism--they are all real things!

None of them is--nothing.   And to say mathematics is the same sort of thing

is not to say that it's nothing.  Mathematical ideas, beliefs, facts,

theorems, are real objects.  Their reality resides in a social consensus,

of which the consensus of mathematicians is a crucial part.  When

humanity departs from the cosmos, sooner or later, all those ideas,

beliefs, theorems will disappear.  This is not to ignore the roots of

mathematics in physical reality.  When humanity disappears, there

will no longer be any integers as abstract objects.  There will

still be 9 planets, for example.   But the theorem that there are two

groups of order 9 will not "exist", because it is an idea, a conception,

that can exist only in minds as parts of a community or society.

	For us, here, today, the fact that there are two groups of 

order 9 is not about nothing, it is about our shared concepts of groups 

and "9".  It is objectively true in that sense--perhaps you would prefer 

"intersubjectively."

	My viewpoint is indeed on a level with and

challenges Platonism, formalism and logicism.  I affirm that

mathematics is real and meaningful, by proposing a social reality

and a social meaning.  If you have trouble with my question,

"What is mathematics, really?"  I can rephrase it:

In what sense is mathematics real, it what sense is it meaningful?

	Now, what about Platonism?

	Neal Tennant is right when he says that

nothing about the actual practise of math can refute Platonism.

	So far as I know, nothing can refute Platonism.  It's irrefutable.

With all respect, I am compelled to compare it with the existence

of G-d.  No one ever could or can prove there is no G-d.  Those who

don't believe in Him/Her don't make such a claim.  They merely hold

that the arguments or evidence for His/Her existence are unconvincing.

So it is with a transcendent mathematical reality.  To some, it seems

obvious, undeniable, that math always existed and always will exist

in some immaterial, inhuman sense or other.  To others, such a

belief seems groundless and extremely implausible.  If I may repeat,

I have never taken on the responsibility of converting Platonists

to humanism.  I merely undertook to explain another point of view,

that lets us see mathematics in terms of real mathematical life, and which

affirms the reality and meaningfulness of mathematics.