FOM: social construction of mathematics?

Reuben Hersh rhersh at
Wed Mar 18 23:24:17 EST 1998

On Wed, 18 Mar 1998, Randall Holmes wrote:

> (this is from Randall Holmes)
> This is a comment on the views expressed by Hersh and others concerning
> the "social construction" of mathematics.
> To make any sense of such views, they need to be placed in a larger
> context.  

Mathematics is the study of formal structure.  
	That is one view of what mathematics is.  You erroneously state
	it as a simple fact, not a controversial opinion.

One may
> attempt to answer Hersh or Hersh-like interlocutors by pointing out
> all the structure which exists in the world and asking if it depends
> on social consensus.  

	It is important to keep straight the distinction between
	"the world" (I take it you mean the exterior, physical world)
	and "mathematics," which we invent and create from day to day,
	(of course motivated and inspired often by "the world.")

One is not going to get much satisfaction this
> way, because the viewpoint that spawns these kinds of views is one which
> holds that we impose structure on the world rather than find it there.

	Not my viewpoint.  In my view, the world has structure, which we try to 
	To describe and explain the world (and also for internal mathematical 
	motivations), we invent mathematics.

> If one does not acknowledge that there is real formal structure in the
> world, then one cannot understand the objective character of mathematics.

	This needs an argument to carry conviction.

> Formal structure (whether it exists Platonically or inheres in a more
> Aristotelean manner in objects, independently of us in either case) is
> what mathematics is about.

	Again, erroneously presenting a controversial opinion as a settled fact.

> If structure is supposed to be imposed by minds, then there is the
> further problem of why it is that structure in general (including
> scientific laws and other things, not just mathematics) is
> intersubjective.  

	Since we agree that structure in the physical world is not "imposed
	by minds", your "further problem" is a "non-problem."

We know as individuals that we didn't really create
> all of it.  Thus the retreat to "society" or "culture" or (cleverly)
> "language" as the place where structure is created (by mental
> activity, but collaboratively, bit by bit).  This view leads
> inevitably to the idea that mathematics is a social construction.

	The view that mathematics is a social construction comes,
	on the contrary, from (1) being a mathematician and
	watching what mathematicians say and do (2) rejecting
	the other possibilities, that it is a transcendental
	immaterial "reality" (Platonism) or a mental process
	(Brouwer) or meaningless formal calculations (formalism.)

> This conclusion is absurd, 

	This is what you're supposed to demonstrate, not
	pronounce "ex cathedra."

but the underlying assumption also has the
> power of confusing the faculties of its adherents enough that they
> have trouble seeing this.

	How do you know that?  You  say I
	am confused.  Argue that.  Don't dabble in phony psychology.
> It is impossible to argue with Hersh or any others like him (others have
> posted similarly) without addressing the basic assumption.  The structure
> of the world was not created by human individuals or human societies.
> Scientific laws are not sociological phenomena; neither are mathematical
> theorems.  Both science and mathematics are human activities, but they
> are activities devoted to the study of real aspects of the real world.

	By "real world" do you mean the physical world, as I suppose?  
	Surely you know that most research and publication
	in pure math has nothing to do with any real or otherwise aspects
	of the real world.  Science isn't math, math isn't science.
	They overlap, they help each other.  Science is based ultimately
	on empirics, on some sort of physical observation or measurement.
	Math is not.  It is based on the joint thinking, mutual criticism,
	historically developed ideas  of people.  Not any old ideas,
	but the kind of ideas that can be checked, verified, confirmed, with
	virtual unanimity.
> Unfortunately, the assumption in question is so deeply rooted that it makes
> conversation with its adherents very difficult; each side keeps missing the
> point of what the other side is saying.  I am deeply convinced that a
> necessary consequence of the assumption that I am challenging here is that
> the world is entirely unintelligible; it is an intellectual poison which
> has already caused an enormous amount of damage.
	I have already repudiated twice (I think) the "deeply rooted"
	"offending" "assumption"  you are "challenging".  You could have saved 
	some trouble by asking me what I think instead of figuring it out for 
	yourself, and getting it wrong.

> If the offending assumption is abandoned, the objective character of
> mathematics becomes obvious and unproblematic.  There are still problems
> with the philosophy of mathematics (resolving the Plato vs. Aristotle
> question mentioned in passing above, for example) but everything becomes
> much clearer.
	Everything is not "much clearer" if you confound, conflate
	and confuse math and science.  

> And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
> the gates of Cantor's paradise, that the       | Boise State U. 

(disavows all) 
> slow-witted and the deliberately obtuse might | holmes at
> not glimpse the wonders therein. |

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