Doug writes:
> I'm still
> wondering, though - what is gained by Stiglitz's use of mathematical
> reasoning. Does it express something that can't be expressed in words? Does
> it deepen the mystery surrounding the priesthood? Does it aim to persuade
> an audience that would find mere word non-rigorous? Does it lend an aura of
> precision to something that is by its very nature fundamentally imprecise?
> Maybe I'm overestimating the rigor of real math, but my impression is that
> a mathematical proof is pretty damn persuasive to the cohort of
> mathematicians, whereas stuff in econ remains controversial despite the
> appearance of mathematical proof.
I've been thinking about these and related questions too, partly as a
result of being asked to write the entry on "Mathematical
formulations of Marxian Economics" for an upcoming encyclopedia of
classical political economy. Following are some not-necessarily-
coherent comments emerging from this thought process.
I should mention that I'm more of a verbal type by aptitude and
background. It wasn't until 4 years into graduate school that I
began to think that mathematical argument had an important role to
play in studying political economic issues. Some of the blanks about
the nature of mathematical argument have been filled in by reading
George Spencer-Brown's LAWS OF FORM and Morris Kline's MATHEMATICS:
THE LOSS OF CERTAINTY. Both are amazing books which I recommend,
even though I don't fully understand them.
1) Does mathematics express something which can't be expressed in
words? Strictly speaking, no, since all of the primitives of a
mathematical system must necessarily be defined in words. In fact,
Spencer-Brown remarks, "One of the most beautiful facts emerging from
mathematical studies is this very potent relationship between the
mathematical process and ordinary language. There seems to be no
mathematical idea of any importance or profundity that is not
mirrored, with an almost uncanny accuracy, in the common use of
words, and this appears especially true when we consider words in
their original, and sometimes long forgotten, senses."
One could imagine, then, undertaking an essentially mathematical
argument in which none of the words have been replaced by symbols.
Two comments on such an exercise: first, it would be incredibly
tedious. The simplest equation system would become a royal chore
just to specify, and such things as the conditions for existence of a
(unique) solution to the system, or the actual derivation of a
solution (think of applying Cramer's rule in literary terms, e.g.)
would become a living nightmare. To put it the other way around, you
could think of mathematical symbols as a certain special type of
words. But this leads to the second comment: mathematical argument
involves a particular, highly restrictive use of concepts, in which
"that which is not allowed is forbidden", to quote again from Spencer-
Brown. That is, none of the ambiguity which makes literary prose
potentially so rich and multilayered (take FINNEGANS WAKE as an
extreme example) is allowed. Why not? This leads to the 2nd point.
2) Mathematical argument, understood in the sense of the latter
comment, permits certain types of conclusions which are beyond the
scope of prose argument, having to do with the necessary content, and
the necessary limits, of one's understanding. Thus, impossibility
theorems ("can't have both A and B"), characterization ("A if and only
if B") results and the like are the special province of mathematical
argument.
A related point: the process of mathematical argument, since it
requires the arguer to specify what s/he is talking about before s/he
talks about it, forces one to be conscious of lurking preconceptions
and ambiguities. Spencer-Brown again: "The discipline of mathematics
is seen to be a way, powerful in comparison with others, of revealing
our internal knowledge of the structure of the world, and only by the
way associated with our common ability to reason and compute."
A corollary: mathematics is necessarily different from, but a
companion of, dialectical argument. Broadly speaking, math is a
method of specifying the contents of a given entity. Elucidation of
these contents, however that process is understood, gives way
to the dialectic.
My favorite analogy here is jazz, since that has clearly been subject
to dialectical change in its history. Once a jazz musician discovers
new ground, s/he and others work to elucidate the content implicit in
it. A striking illustration of this comes from Ornette Coleman, one
of the originators of "Free jazz", a seemingly lawless permutation of
modern jazz: "I knew I was onto something when I found that I could
make mistakes." Alternatively, a jazz musician may, in discovering
the limits of a particular structure, go beyond those limits.
Thus Charlie Parker discovering bebop in that chili house in New
York, by taking the givens of swing and extrapolating beyond them.
The immediate critical reaction? That's not jazz! Subsequently, of
course, bebop gets folded into an enlarged conception of jazz which
establishes new limits....etc.
3) There are a large number of results in political economy whose
proof is inconceivable in the absence of mathematics. Relatedly,
there are a large number of conjectures which will remain such until
a mathematical proof is given. Since most people on PEN won't find
results in mainstream economics to be of much significance, I'll
limit myself to findings in Marxian economics:
A) Due essentially to mathematical argument, we know have a better
understanding of the necessary limitations in Marx's theory of the
tendentially falling rate of profit. For example, even if one
doesn't agree with the premises of Okishio's theorem, who would have
known that Marx's claim was inconsistent with those premises before
Okishio's proof? More generally, we now know that there is at best a
tenuous connection between a rising organic composition of capital
and a falling rate of profit, even under conditions in which the
latter might possibly hold.
B) Similarly, we have a better sense of the connection, and the lack
of connection, between labor values and prices of production. Two
results of mathematical argument in particular strike me as critical:
1) the connection between labor values and commodity prices depends
idiosyncratically on underlying conditions of production.
2) while there may be a correlation between labor values and
commodity prices, there is nothing unique about this; for example a
similar correlation would arise between commodity prices
and prices in the standard system using the "standard commodity" to
construct a numeraire.
3) Prices of production and labor values will *generically* diverge
in such a way as to permit surplus value given wealth inequalities,
positive profit or interest rates, and variations in the labor
intensity of production across sectors.
This result is interesting in that Marx takes the first two
conditions as a given in Volume I, and certainly doesn't rule out the
latter condition in his value-theoretic analysis. Similarly, one must
under certain circumstances qualify the result given strategic
difficulties in extracting labor from labor power, but this strategic
reasoning is not a part of Marx's value analysis.
As Doug suggests, there is lingering controversy about the ultimate
impact of these findings. But this is not to say that they haven't
deepened our understanding of the targeted issues.
4) Conversely, claims of necessity have frequently been made on this
net: e.g., subsumption or dispossession of labor power is
*necessary* for the existence of exploitation in Marx's strong or
primary sense (Jim Devine); labor values are *necessary* to
understanding the manifestation of such phenomena as unequal exchange
(Alan Freeman). However, the necessity (as opposed to the
plausibility) of neither claim has yet been established, and I can't
imagine how it would be absent mathematical argument or an exhaustive
investigation of *every* *single* historical case.
For what it's worth,
Gil Skillman
