Re: Article in the Chronicle of Higher Education

[Ted Winslow <egwinslow@xxxxxxxxxx>]

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