FOM: social construction and general intellectual interest

[michael Detlefsen]

I have a couple of things to say re. the recent discussion (by Sol Feferman
and Moshe Machover) on the nature of mathematics as 'social construction'
and the seemingly different matter of the 'general intellectual interest'
of fom by Harvey Friedman, Lou van den Dries, Steve Simpson, Michael Thayer
and others.

I. The first item is a question for Sol and Moshe. Why 'social'
construction rather than, say, invariant individual 'psychological'
construction? This is what I don't get. Or, rather, I should say that I get
the idea, but I don't see what particular feature(s) of our social aims and
lives you take to be essential forces which shape the distinctive
characteristics of mathematical knowledge and practice. In order to make a
serious case for mathematics as social construction, one must, it seems to
me, do this.

Objectivity or intersubjectivity seem unlikely to be able to bear such a
burden. Kant taught us this much. He said that what we really know about
our so-called 'objective' judgments is that they are 'involuntary' (nicht
selbst ausgedachten). He then distinguished between two different types of
cognitive or epistemic compulsions--one a subjective compulsion
(Ueberredung), the other an intersubjective or objective compulsion
(Ueberzeugung). The former he described as being 'privately valid' and as
having its ground only in the special or idiosyncratic character of the
individual subject. The latter is also an individual (as distinct from a
social) compulsion, but it compels every individual possessing reason in
the same way. Kant takes this type of intersubjectivity either to itself BE
objectivity or, if one accepts the 'presumption' that the explanation of
such agreement must reside in some special status of what is common to all
the individual experiences (viz. the object), to point to objectivity. Is
this a 'social' construction? If so, I wonder what it is that makes
something a 'social' construction.

II. Social/political ideals and mathematics. To illustrate more clearly
what is bothering me, let me develop a little history. The Pythagoreans
divided human beings into two classes, slaves and free. This division did
not represent something imposed upon humans by a political or cultural
regime. Rather, it was taken to reflect objective differences between
people. The free were 'self directed'. Slaves, on the other hand, had to be
directed (programed) by others--in the purest and oldest sense of the term,
they were 'automata'. For the free, THEORIA (self-contained reflection) was
the epistemic ideal. For slaves, cognition was limited to PRAXIS (a kind of
'knowing how' to do things, a knowledge instilled in slaves by training or
forced imitation, paradigmatically, through moving their limbs, etc. in the
way they were intended to move).

Such ideas made Plato's MENO a revolutionary tract. It called this
Pythagorean picture into question. In the Meno a SLAVE boy is
discovered/shown to have the roots of geometrical knowledge embedded within
him, independent of any possible training or programming. The dialogue
argues that all that is necessary to bring this knowledge to light, and to
create in the slave boy both an awareness of his ignorance and a thirst for
further knowledge (which thirst serves as a self-contained motive force
impelling him to search for understanding), is the right type of
environment ... namely, a Socratic environment which provides an artful
questioner capable of stimulating the boy's innate 'recollection'. The
route the boy follows in this ascent is a route of praxis which consists in
constructive (as opposed to axiomatic or theoretical proof or
demonstration) geometrical "activity". Both the division of slave and free
and the distinction between praxis and theoria were thus brought into
question.

The Eleatics pulled the other way. Their paradoxes lead them to distrust
proof via visualization (hence the kind of praxis-oriented constructive
geometric proof found in the Meno) and to demand conceptual or theoretical
(i.e. pertaining to 'theoria') instead. They believed that when there is
conflict between the visualizable and the intellectable (as in their
paradoxes), the former must yield to the latter. The reason is that genuine
judgement is not a visual but an essentially intellectual act. One must
assent or dissent with his mind (nous), not his senses. Hence, intellection
is more basic than and must be given precedence over visualization. If,
therefore, there are two basic types of 'knowers', those capable only of
pragmatic knowledge vs. those capable of theoretical knowledge, only the
latter will be in a position to be genuine mathematicians.

The influence of the Eleatics launched the axiomatic approach to
mathematics. Their basic view of the untrustworthiness of visualizability
can still be found in the attitude of many 19th century mathematicians
towards the paradoxes of analysis. Did those same 19th century thinkers
also side with them on the socio-political division of slave and free? I
don't think so.

This brings me to my question for Sol and Moshe (if indeed he counts
himself among the social construction crowd). What are the socio-political
ideals that they believe undergird mathematical ideals or practices today?
And which ideals and/or practices?

III. Now for the Harvey-Lou-Torkel-Michael brawl re. general intellectual
interest. Before making my substantive point, I'd like to make an ethical
point. Michael, I'm happy to hear what you have to say. Who you are and
what your position is doesn't matter to me. I believe that this discussion
group should be about ideas and arguments, not about employment status or
class membership. Harvey, I think it was wrong of you to address Michael
the way you did. Same with your remarks to Torkel (and Swedish
mathematicians generally). This group would profit most from your showing a
character as large as your intellect.

Now for the substantive point. In this I am in distinct sympathy with
Harvey ... and for theoretical reasons. Let me explain by continuining the
history sketched above, but skipping on to the next big development--the
Protestant reformation. The chief intellectual spokesmen for this were
Luther and Leibniz. Liebniz' dream of a calculus of thought was primarily
an ethical and socio-political ideal. He accepted the old Greek distinction
between higher and lower cognitive capacities in human beings. He viewed
this, however, as a moral tragedy. Love and love of justice, he said, bid
us try to overcome the differences in cognitive capacity resulting from the
'natural' distribution.

To do this we need to find some capacity or capacities that are (much) more
evenly distributed than native cognitive ability and find a way of
'substituting' the former for the latter. This is essentially what Leibniz
sought to do in his calculus of thought. He would substitute the evenly
distributed capacity to perform rudimentary mechanical acts
(praxis--executing a program) and the evenly distributed capacity of
recognizing rudimentary (and rudimentarily different) shapes and sequences
of such for the very uneven distribution of the full range of cognitive
capacities. In that way, he would make the full range of cognitive or
intellectual goods available to even those with the poorest natural gifts
... in his own phrase 'even the most stupid'.

I think Leibniz' dream is one that should still be pursued today. However,
I also think that we now know of essential limitations on it. I am
thinking, of course, of such classic fom-type results as Church's Theorem
and Goedel's Theorems. (Church's Theorem, for example, can be taken as
showing that we can't hope to overcome all natural differentials in ability
to detect invalidity by purely computational means.)

Surely, in posing limits to so magnificent a social vision as Leibniz' (who
thought it should even end the lust for warfare through pitting people in a
'computational' struggle over logical materiel rather than the more usual
types of blood-letting like the Thirty Years War) fom-type results like
Church's Theorem show themselves to be of the greatest "general
intellectual interest". I don't see anything in Lou's examples to compare
with this. (On the other hand, I don't know a lot about Lou's examples
either.) The modern-day Leibnizian, as she seeks to minimize the
'discouragement' to Leibniz' Program (the term I use for the above-sketched
logico-socio-political program of Leibniz) of Church's Theorem and the
like, will take a natural and well-motivated interest in the P=NP
completeness problem (and research in computational complexity and related
foundational matters in general). Thus the large potential for "general
intellectual interest" in the latter. If Lou or Michael knows of anything
to compare with this in non-fom higher mathematics, I'd like to hear what
it is. And I mean that sincerely. You'd be helping to educate me ... and
perhaps others on this list as well.



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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
U.S.A.
e-mail:  Detlefsen.1 at nd.edu
FAX:  219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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