FOM: Platonism and social constructivism

[Moshe' Machover]

It seems to me that the debate about the social costructivist view of
mathematics has become somewhat confused by being wrongly focused on the
question as to the mode of existence of mathematical *objects*.

I wish to argue that the correct focus is on the status of mathematical
*propositions*.

Reuben Hersh keeps insisting that his view on the social nature of
mathematics is a--even the--reasonable alternative to platonism. I believe
this is a mistake.

Like Martin Davis and RH, I am a materialist regarding *physical* objects.
I believe that the three pebbles on the beach (now or in Jurassic times)
exist(ed) independently of anyone's consciousness of their existence.

Mathematics, at least *pure* mathematics, is not at all about material
objects but about abstract objects. The number 3, the property of threeness
that applies to the [set of] pebbles mentioned above, and the group of 6
symmetries of [the set of] these pebbles, are abstract objects.

Platonists believe that such abstract objects exist independently of
anyone's consciousness of their existence.

I don't believe this. Like Hersh, I hold not only that mathematical
activity is a social human activity, but that *inventing* abstract objects
is part of this activity.

It seems to me that Hersh thinks that this more or less settles the status
of mathematics. I believe, on the contrary, that this is where the really
crucial problems only start.

Humans make mathematics, but they do not make it as they please. Once
mathematical objects are invented, they obey ineluctable laws. It is this
*fact* that creates the illusion of platonism, and enables mathematicians
to think and reason *as though* matheamtical objects exist independently of
anyone's consciousness.

A central problem of the philosophy of mathematics is to account for the
ineluctability of mathematical laws.

For platonists, this is not a difficulty. (*Their* main difficulty is to
explain how we come to know about the matheamtical objects and the laws
governing them.) But for those who believe that mathematical abstract
objects are invented, it is a real difficulty.

Social constructivists hold that not only are mathematical *objects*
invented, but the truth values of *propositions* about these objects and
the laws governing them are negotiable, and settled by social convention.
According to this view, not only the objective existence of mathematical
*objects*, but also the certitude of matheamtical *theorems*, is illusory.
The main feature of social constructivism is not its denial of platonism,
but its absolutely relativistic [:-)] attitude to mathematical truth.

To me (and I think to the vast majority of mathematicians) this position
is not only deeply unsatisfactory, but self-evidently false. I think there
are absolutely true mathematical propositions. As an example I could quote
Lagrange's four-squares theorem; but I know that some would dispute this.
(`In what theory?') However, at the very least the theorem that *Lagrange's
theorem is provable in first-order Peano arithmetic* is surely an
indubitable example. It is simply true. Once you understand the (socially
agreed!) definitions of the terms used in such propositions, you have no
choice but to accept their truth. If you don't, you are *objectively
mistaken* (and not, for example, a holder of different but equally valid
cultural values).

Social constructivism not only fails to account for such facts, but
actually denies them.

Lest Hersh asks me for my own explanation, I freely admit that I don't have
one. If I did, you'd have heard about it long ago. :-)

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