FOM: Platonism v. social constructivism

[Charles Silver]

Charlie Silver
Present Affiliation: Lecturer in Logic
                     Smith College
(Formerly: associate professor of computer science at Southeastern
Louisiana University.  Professional interests: foundations of math,
foundations of computer science, and (documentary) film)

	I have some questions to raise about the two vying views.  First,
I am not clear what Reuben Hersh's claims are on behalf of social
constructivism.  So, most of my questions will be about that. I don't know
exactly what Neil has in mind in his "Astonishing Dictum," but I think I
have a somewhat better idea about the claims of Platonism.


On Thu, 25 Dec 1997, Reuben Hersh wrote:

> >>  
> >Dear Prof. Tennant:  First let me quote your astonishing dictum:
> 
> "...any social-institutional factors shaping the
> 
> > history of mathematics as thus far developed by human beings is
> 
> > IRRELEVANT to the truth of Platonism (or any other philosophy of
> 
> > mathematics opposed to social constructivism). The same goes for
> 
> > whatever social-institutional factors might have shaped the history of
> 
> > mathematics in any other extra-terrestrial civilization. "
> 


> 	I refrain from any extra-terrestrial arguments.  According to the 
> 
> first sentence, since history runs up to the present, all 
> 
> social-institutional factors in present day mathematics 
> 
> are IRRELEVANT to the truth of Platonism.
> 
> I agree with that, if I can rephrase it:  to a
> 
> devout Platonist, no social-institutional (or social non-institutional)
> 
> factors in present or past mathematics can shake his/her Platonist
> 
> convictions.  


	The fact that a devout Platonist may not be easily converted by
any arguments isn't relevant to what I think may be the point here, which
is that social factors surrounding a practice do not serve in and of
themselves to tell us that there is no underlying reality beyond the
practice itself. Remember King Kong?  Consider two scenarios.  Scenario 1: 
There *is* a huge gorilla in the jungle beyond the fortress that has been
set up in the jungle to protect the natives.  Scenario 2: There is *not* a
huge or even a rather large gorilla beyond the fortress.  The point is
that the evidence for Kong is not to be found in the social practices of
the natives, since (by hypothesis) the practices are the same in both
cases. I think something along these lines may have been suggested by Neil
Tennant, but I don't know for sure.  Also, I can't tell whether Reuben
Hersh really intends to *oppose* Platonism by calling attention to
society's role in the development of mathematics.  Rather, I *think* he
may wish to say that social factors are important to an understanding of
what mathematics *is*.  And, from that perspective, I think it also
doesn't matter to him which scenario unfolds, 1 or 2.  (In 1, when food
and human sacrifices have been left outside the fortress, it is Kong who
claims them; in 2, other animals eat the food and the sacrificial humans
escape.  In both cases, the food and the human disappear by the next
morning.) That is, perhaps Reuben Hersh wishes to say that in both
scenarios what matters most are the social practices of the natives, not
whether Kong actually exists.  At any rate, I would appreciate some
clarification by both Neil Tennant and Reuben Hersh on what they wish to
claim. 


***	From Reuben Hersh's earlier post (around December 20): ***

> Subject: Re: FOM: What is mathematics, really?, Gen Intellectual
>Interest, Political Agendas

> Since my name is occasionally being mentioned on this list, I
> take the privilege of explaining myself, or rather my book, *What
> Is Mathematics, Really?*

	The part that most interests me is the meaning of 'Really'.  In
what sense is mathematics *really* (just?) social construction?  Hersh
says:

>	But there's a major chunk of reality that's
> neither transcendental, mental or physical.  What are
> money, war, religion, art, literature, music, patriotism, race hatred,
> universities, department stores, language, politics, government?  Real, 
> for sure.  But neither: (1) transcendental, (2) mental, or
> (2) physical. They are (4) social-cultural-historic realities. 

[I put the numbers in. C.S.]

>	Given this fourth choice for an answer to the title question,
> I claim that this is where math fits best.   

[...stuff snipped]

> This proposed answer to What is Mathematics? has not been popular in the
> philosophy of math community.  But it isn't new.  Leslie White proposed
>it explicitly, and in doing so gave credit to Emile Durkheim.

[More has been snipped]

>	In the last chapter, almost as an after thought, I raised the 
> question whether there is a correlation betweeen the humanist-Mainstream 
> split, and a political split betweeen right and left.  And I found, to
> no one's surprise, that humanism tends to associate with left,
> Mainstream with right. 

>	This is a distinction between philosophies, not between 
> mathematics!  I know nothing of aristocratic or humanitarian
> mathematics.

	I don't fully understand the above.  Is the point that there *is*
or *should be* one and only one mathematics?  If so, why?  That is, if
mathematics is *only* a kind of social construction, which presumably
satisfies certain deep social-psychological needs, then why shouldn't
there be multiple mathematics?  Sort of like having multiple basketball
leagues to support our need to watch more basketball games. Going back to
the Kong image, if scenario 2 is what really takes place (there's no Kong,
just other animals out there in the dark, scary jungle), then why
shouldn't there be multiple myths about jungle creatures, in order to
satisfy more persons' needs? That is, applying scenario 2 to mathematics,
why shouldn't there be distinct kinds of mathematics?


>	One big mistake I made, to which I plead naivete.  I didn't 
> recognize the danger of being associated with the black plague 
> of postmodernism.   (Actually, after some effort I have 
> failed to understand what postmodernism is.)

	To my mind, Reuben Hersh could be hedging his bets here, but I
hope he's not.  I hope he will define what he takes "the black plague of
postmodernism" to be and to explicitly disavow any connection between his
views and that "disease"--if he really wants to, of course.  For example,
is everything mere text and the only "underlying reality" a given person's
or a given society's textual interpretation?  Is that all that math
is--just text?  I think he does not wish to go this far.  But, I'm hoping
he will clarify.


>	I understand that the classification of mathematical reality 
> as social-cultural-historical raises difficult questions.  But 
> I think the chance of answering them is better than with any 
> other classification of mathematical reality. 

	Again, I would appreciate some clarification.  There is an
ambiguity in "mathematical reality" in the last sentence.  Platonists tell
us that there is an underlying mathematical reality of "real" objects
(residing in Plato's Heaven) and that is what mathematics is about--that
realm of objects.  Formalism and Logicism are normally taken to be
opposing positions about what the underlying mathematical reality truly
is.  That is, all three views seem to be more or less on a par, opposed to
each other about the underlying reality of mathematics.  It seems to me
that Reuben Hersh's view is not on the same level as these views and thus
shouldn't be contrasted with them.  But, perhaps I'm wrong.  It seems to
me that the "reality" spoken of in the last sentence in the above
paragraph does not concern what mathematics is truly *about*, but concerns
only the historico-socio-culturo-... behavior of mathematicians.  Is the
position that mathematics is best understood in terms of mathematicians'
behavior over the centuries?  Their successes, their failures, their
rivalries, and so on?  And then, is mathematics *just* a record of these
historical developments.  Is it really *about*...nothing?  Please clarify. 


Charlie Silver

P.S. I want to take this opportunity to disagree mildly with one of
Hersh's claims.  He said, if I recall correctly, that Wiles worked on FLT
for seven years because of the prestige conferred by the mathematical
community on the solution to FLT.  Wiles says in the film that he first
became attracted to FLT as a small boy.  I'd like to suggest that for the
young boy there was something intrinsic to the problem itself, its purity,
its elegance, and so forth that attracted him to it.  It called out to
him, so to speak, perhaps even influencing him to take up mathematics as a
career. That is, initially, something *intrinsic* to the problem itself
attracted him, rather than the lure of later collegial approval.  I don't
deny that the (extrinsic) prestige of solving FLT certainly influenced
him--probably sustaining him through failure after failure--but I am also
saying that the intrinsic nature of the mathematical problem itself seems
to have played an early role.