Re: Article in the Chronicle of Higher Education

["J. Barkley Rosser, Jr." <rosserjb@xxxxxxx>]

Ted,
      First of all I never identified math or
rigor with deduction.  You did.  One of
my arguments with Roy Weintraub,
which will probably prove to be not
really a disagreement after all once the
JPKE Symposium appears, apparently
this summer, is over Bourbakism and
its role in math econ, as I noted in another
message.   Bourbakist math is deductivism
in its most extreme form.  As described
by Weintraub, it was in this form that
econ became mathematized in the 1950s,
with the Arrow-Debreu model of general
equilibrium as the supreme example.
      Roy himself notes that such math has
now very much fallen out fashion among
mathematicians.  It is my observation (and
others' as well) that it is increasingly falling
out of fashion among economists as well.
     I fully agree that it is possible to be
rigorous without being precisely mathematical.
I would associate "rigor" with "logical" and
"consistent."   Properly done deductivism is
rigorous, but it is possible to be rigorous
without being deductivist.  And much deduction
is empty.  Rigor is no guarantee of a serious
argument.  It may be empty rigor, rigor mortis.
    There is also a problem regarding what is
"mathematical."  There is a tendency to equate
being mathematical with using equations, or
maybe with graphs or figures.  But, there are
people producing equations that are ridiculous
and empty and non-rigorous, covering up their
ignorance with apparent math.   Also, there are
economists who make rigorous mathematical
arguments in a strictly verbal form.  Joan Robinson
used to do this, even as she criticized the use of
"maths."  Likewise, when John Stuart Mill first
introduced nonlinear programming in his discussion
of international exchange value, he did so purely
verbally although completely rigorously.
     I fully agree that there is a problem of identity
versus diversity in whether a symbol represents
a "real reality" when what it represents can change
over time.  OTOH, econ certainly lends itself to a
mathematical approach if for no other reason than
that many things we are interested in in economics
are quantitative and easily measured by numbers,
prices, quantities, outputs, populations, etc.
      Also, Keynes did not get the Nobel because
he died in1946, and it did not begin being given
until 1969.  Relatively non-mathematical economists
have won the Nobel, including Hayek, Myrdal, and
Douglas North, to name a few.
Barkley Rosser
----- Original Message -----
From: "Ted Winslow" <egwinslow@xxxxxxxxxx>
To: <pkt@xxxxxxxxxxxxxxxx>
Sent: Monday, January 27, 2003 6:26 PM
Subject: Re: Article in the Chronicle of Higher Education


> Barkley wrote:
>
> > I agree with Paul Davidson regarding the
> > use of math and rigor.  Arguing against math
> > and rigor, per se, just leads to heterodox
> > economics being marginalized and disregarded.
>
> For the reasons spelled out by Whitehead and Keynes, "rigor" can't
> reasonably be identified with "math" or, more generally, with deductive
> reasoning from axioms treated as universally valid.  That it is so
> identified and that this identification persists in the face of reasonable
> argument showing it to be a mistake suggest (as Keynes himself pointed
out)
> that the identifications (like the ideas of money cranks) are anchored
> psychologically in a way that makes them immune to rational critique.
> Perhaps this is the reason Keynes is misunderstood, marginalized and
> disregarded while John Nash is given the "Nobel".
>
> These claims may, of course, be mistaken.  Here again is Whitehead's
> argument about the limits of "deduction" generally and of "math"
> specifically.  How is the point it makes about "algebra" taken account of
in
> the applications of algebra in orthodox and heterodox economics?
>
> "In this lecture we seek the evidence for that conception of the universe
> which is the justification for the ideals characterizing the civilized
> phases of human society.
>     "We have been assuming as self-evident the many actualities, their
forms
> of coordination in the historic process, their separate importance, and
> their joint importance for the universe in its unity.  It must be clearly
> understood, as stated in earlier lectures, that we are not arguing from
> well-defined premises.  Philosophy is the search for premises.  It is not
> deduction.  Such deductions as occur are for the purpose of testing the
> starting points by the evidence of the conclusions.
>     "A special science takes the philosophic assumptions and transforms
them
> into comparative clarity by narrowing them to the forms of the special
topic
> in question.  Also even in reasoning thus limited to special topics, there
> is no absolute conclusiveness in the deductive logic.  The premises have
> assumed their limited clarity by reason of presuming the irrelevance of
> considerations extraneous to the assigned topic.  The premises are
conceived
> in the simplicity of their individual isolation.  But there can be no
> logical test for the possibility that deductive procedure, leading to the
> elaboration of compositions, may introduce into relevance considerations
> from which the primitive notions of the topic have been abstracted.  The
> mutual conformity of the various perspectives can never be adequately
> determined.
>     "The history of science is full of such examples of sciences bursting
> through the bounds of their original assumptions.  Even in pure abstract
> logic as applied to arithmetic, it has within the last half century been
> found necessary to introduce the doctrine of types to correct the
omissions
> of the original premises.
>     "Thus deductive logic has not the coercive supremacy which is
> conventionally conceded to it.  When applied to concrete instances, it is
a
> tentative procedure, finally to be judged by the self-evidence of its
> issues.  This doctrine places philosophy on a pragmatic basis.  But the
> meaning of 'pragmatism' must be given its widest extension.  In much
modern
> thought, it has been limited by arbitrary specialist assumptions.  There
> should be no pragmatic exclusion of self-evidence by dogmatic denial.
> Pragmatism is simply an appeal to that self-evidence which sustains itself
> in civilized experience.  Thus pragmatism ultimately appeals to the wide
> self-evidence of civilization, and to the self-evidence of what we mean by
> 'civilization.'
>     "Before we finally dismiss deductive logic, it is well to note the
> function of the 'variable' in logical reason.  In this connection the term
> variable is applied to a symbol, occurring in a propositional form which
> merely indicates any entity to which the propositional form can be validly
> applied, so as to constitute a determinate proposition.  Also the
variable,
> though undetermined, sustains its identity throughout the arguments.  The
> notion originally assumed importance in algebra, in the familiar letters
> such as x, y, z indicating any numbers.  It also appears somewhat
> tentatively in the Aristotelian syllogisms, where names such as
'Socrates,'
> indicate 'any man, the same throughout the argument.'
>     "The use of the variable is to indicate the self-identity of some use
of
> 'any' throughout a train of reasoning.  For example in elementary algebra
> when x first appears it means 'any number.'  But in that train of
reasoning,
> the reappearance of x always means 'the same number' as in the original
> appearance.  Thus the variable is an ingenious combination of the
vagueness
> of any with the definiteness of a particular indication.
>     "In logical reasoning, which proceeds by the use of the variable,
there
> are always two tacit presuppositions - one is that the definite symbols of
> composition can retain the same meaning as the reasoning elaborates novel
> compositions.  The other presupposition is that this self-identity can be
> preserved when the variable is replaced by some definite instance.
Complete
> self-identity can never be preserved in any advance to novelty.  The only

> question is, as to whether the loss is relevant to the purposes of the
> argument.  The baby in the cradle, and the grown man in middle age, are in
> some senses identical and in other senses diverse.  Is the train of
argument
> in its conclusions substantiated by the identity or vitiated by the
> diversity?
>     "We thus dismiss deductive logic as a major instrument for
metaphysical
> discussion.  Such discussion is concerned with the eliciting of
> self-evidence.  Apart from such self-evidence, deduction fails.  Thus
logic
> presupposes metaphysics."  (Whitehead, Modes of Thought, pp. 105-7)
>
> Ted
>
>