Re: Occam's Razor

["Gunnar Tomasson" <gunnar.tomasson@xxxxxxxxxxx>]

Paul:

Here are David Hilbert's views - with editorial changes bracketed - on key
points under discussion:

A.  "When we are engaged in investigating the foundations of a science, we
must set up a system of axioms which contains an exact and complete
description of the relations subsisting between the elementary ideas of that
science.  The axioms so set up are at the same time the definitions of those
elementary ideas, and no statement within the realm of the science whose
foundations we are testing is held to be correct unless it can be derived
from those axioms by means of a finite number of logical steps."

B.  "The [economist], as his theories develop, often finds himself forced by
the results of his [research] to make new hypotheses, while he depends, with
respect to the compatibility of the new hypotheses with the old axioms,
solely upon these [research findings] or upon a certain [economic]
intuition, a practice which in the rigorously logical building up of a
theory is not admissible." ("Mathematical Problems," Lecture delivered
before the International Congress of Mathematicians at Paris in 1900.)

I submit that my following proposition accords with Hilbert's point A:

> >If, as I suggest, axiomatic structures become Theorems X, Y, and Z are
> >"non-overlapping" in the sense that the rules/games of (a) Chess, (b)
> >Contract Bridge, and (c) Gin Rummy may be said to be so, then the number
of
> >axioms is immaterial.

You comment on my proposition as follows:

> I am sorry to say but these comments below are irrelevant. If Theories X,
Y,
> and Z do not share any common axioms then your point is correct --BUT
> our discussion requires the recognition that all economic theories share
some
> common axiooms. It is not that on the first Monday economics is a game of
> bridge, on Tuesdays it is a game of gin rummy; on wednesday -- blackjack,
etc,
> Or ar you arguing that the rules of the economicgame change every day?

I disagree - Hilbert's point A applies to (X) Newton's Gravitational Model;
(Y) Einstein's Gravitational Model/General Relativity; and (Z) any would-be
Quantum Mechanical Model of Gravity all of which "share" or have in "common"
the "field" axiom.

I also submit that my following suggestion accords with Hilbert's point B:

 > >Indeed, as indicated by Einstein's comments on the General Theory's
"logical
> >completeness," remove one axiom and the whole axiomatic structure
("theory")
> >reduces to incoherence.

You comment as follows:

> Not necessarily as my example of he classical fixity of money wages
implies--
> IF one removes the fixity assumption, the ergodic axiom [so that
uncertainty
> is the motivating force behind a demand for liquidity] and the lack of
gross
> substituion (only) between liquid assets and illiquid products of
industry --
> while stil permitting gross sunstition among the products of industry --
and
> one has a comlete system (with fewer restrictive axioms) showing the truth
of
> the possibility of involuntary unemployent. The resulting complete logical
> system known as THE GENERAL THEORY OF EMPLOYMENT INTEREST AND MONEY

In response, I submit that Keynes' modus operandi represented "a practice
which in the rigorously logical building up of a theory is not admissible."

A case in point:

"I shall argue that the postulates of the classical theory are applicable to
a special case only and not to the general case, the situation which it
assumes being a limiting point of the possible positions of equilibrium."

For Hilbert's point B implies that it is inadmissible to represent "the
classical theory" as a limiting point on, as it were, an epistemological
continuum originating with the General Theory.

Gunnar


----- Original Message -----
From: "pdavidso" <pdavidso@xxxxxxx>
To: <pkt@xxxxxxxxxxxxxxxx>
Sent: Tuesday, January 28, 2003 8:43 PM
Subject: Occam's Razor


> >===== Original Message From Gunnar Tomasson <gunnar.tomasson@xxxxxxxxxxx>
> =====
> >Paul:
> >
> >Let me preface my comments on your points by noting that, when I proposed
to
> >do my Ph. D. thesis at Harvard on "the epistemological foundations of
> >science in general and economics in particular" back in the 1970s, then
> >Department Chairman Dwight Perkins wrote back to advise that no Harvard
> >faculty member was qualified to supervise a thesis on the subject matter.
A
> >few days ago, a fellow Gang8 member based in Europe advised that he had
> >wanted to do a thesis on a similar topic in the early 1990s but that,
after
> >searching far and wide (he contacted about 100 economics professors) for
> >someone with whom he could do such a thesis, the result was the same as
in
> >my case.
>
> I am not sure why you think this preface is relevant.
>
> >
> >Through the years, my attempts to engage top-rank economic scholars in
> >dialogue on the subject matter have yielded, inter alia, the following
> >responses:
> >
> >1. "There are few expert in both theoretical physics and economics, but I
> >am confident that any such will not agree that you have isolated a
> >contradiction in my Foundations." (P. A. Samuelson.)
> >
> >2. "Epistemology is a very hard subject." (F. H. Hahn.)
> >
> >3. And Milton Friedman advised that he had done no work on the subject
> >matter since his essay on 'The Methodology of Positive Economics' in the
> >early 1950s.
> >
>
>
> So what?
>
> >As for my own views on the subject matter, they reflect extensive
research
> >over a quarter century into the epistemological aspects of theoretical
> >physics, key aspects of which came into clear focus through my own
original
> >(intensive) research on atomic structure and related electromagnetic
> >phenomena in the 1970s.
>
>
> Also irrelevant as to how you reached your views.
> >
> >All that being said, let me address your points one by one.
> >
> >> If X. Y , and Z share common axioms, then what differentiates X from Y
> >from Z
> >> is any additional axioms added to the common axiomatic foundation. In
> >choosing
> >> which theory is the more general theory among X, Y and Z, the criteria
is
> >the
> >> one theory with the smallest axiomatic base is the more general theory.
> >> Unfortunately, Debreu's general [equilibrium]theory [as well as
> >Samuelson's
> >> FOUNDATIONS] does not search for the maximum generality[i.e., the
theory
> >with
> >> the fewest required axioms] but rather what Debreu and Samuelson
believe
> >is
> >> the "right generality".
> >
> >Comment:
> >
> >If Theorems X, Y, and Z are axiomatic structures in the sense that
> >Einstein's General Theory of Relativity is such, then they must share the
> >latter's distinguishing feature which Einstein summarized as follows in
> >1919:
> >
> >"The chief attraction of the theory lies in its logical completeness."
>
> Again I fail to see the rlevance of this Einstein statement relative to
our
> discussion.
>
> My argument relies on the Hilbert discussion of the 1930s showing that if
one
> has a consistent system a proposition will be true or false in that system
> else that system will not be complete. In the classical system with the
gross
> substitution axiom and the ergodic axiom, involuntary unemployment is
false.
> So the classical scholars must add another restrictive assumption, namely
the
> fixity of money wages and/or prices to get a complete system where
involuntary
> unemployment is true. The completeness of any system is tied to the
> decidability ofits axiomatic propositions.
>
> in Keynes general theory-- where gross substituion is not assumed (see
chapter
> 17 of the GT) involuntary unemployment is true (possible) even if the
system
> has flexible wages and prices.-- therefore it does not require the
additional
> restrictive assumption of nominal fixities to demonstrate the truth of
> involuntary unemployment -- see Chapter 18 of the GT.
>
> >That is to say, the "theory" is neither more nor less than a structure
> >constructed through deductive reasoning from a given set of axiomatic
> >premises.
>
> of course, so what?
> >
> >With its 'logical completeness' assured, there remains the question of
the
> >General Theory's applicability to real-world phenomena - on that point,
> >Einstein continued as follows:
> >
> >"If a single one of the conclusions drawn from it proves wrong, it must
be
> >given up; to modify it without destroying the whole structure seems to be
> >impossible."
> >
> Not true see my discussion of the fixity of nomiknal values above.
>
> >That is to say, the question of the "theory's" real-world applicability
> >remains an open question.
>
> What I am saying id that the GT is more general than the classical theory
and
> what Keynes showed in Chapter 18 is that flexibility in money wages
> exacerbated the problem of getting to full employment-- while the
classical
> system holds that such flexibility provides an automatic mechanism to
> restoring equilibrium if the system is shocked off its full employmeny
> equilibrium.
>
>
> >
> >Also, since the "theory's" logical completeness is defined with respect
to a
> >given set of axiomatic premises, Einstein acknowledged a few months
before
> >his death that "his castle in the air" - and "also most of current
> >physics" - would be reduced to "nothing" if the "field" axiom were judged
to
> >be inadmissible in principle
>
> What does Einstein's field axiom have to do with Keynes GT?
>
> >
> >> If you have two theories BOTH purporting to explain real world
involuntary
> >> unemployment equilibrium, the more general theory (i.e., the one with
the
> >> least number of restrictive axioms) MUST BE PREFERABLE OVER THE THE
THOERY
> >> THAT REQUIRES ADDITIONAL RESTRICTIVE AXIOMS-- see OCCAM's RAZOR,
Gunnar!!
> >
> >Comment:
> >
> >The applicability of Occam's Razor remains to be established.
>
> Here we have an obvious disagreement!
>
> >
> >If, as I suggest, axiomatic structures become Theorems X, Y, and Z are
> >"non-overlapping" in the sense that the rules/games of (a) Chess, (b)
> >Contract Bridge, and (c) Gin Rummy may be said to be so, then the number
of
> >axioms is immaterial.
> >
>
>
>
> I am sorry to say but these comments below are irrelevant. If Theories X,
Y,
> and Z do not share any common axioms then your point is correct --BUT
> our discussion requires the recognition that all economic theories share
some
> common axiooms. It is not that on the first Monday economics is a game of
> bridge, on Tuesdays it is a game of gin rummy; on wednesday -- blackjack,
etc,
> Or ar you arguing that the rules of the economicgame change every day?
>
>
> >Indeed, as indicated by Einstein's comments on the General Theory's
"logical
> >completeness," remove one axiom and the whole axiomatic structure
("theory")
> >reduces to incoherence.
>
> Not necessarily as my example of he classical fixity of money wages
implies--
> IF one removes the fixity assumption, the ergodic axiom [so that
uncertainty
> is the motivating force behind a demand for liquidity] and the lack of
gross
> substituion (only) between liquid assets and illiquid products of
industry --
> while stil permitting gross sunstition among the products of industry --
and
> one has a comlete system (with fewer restrictive axioms) showing the truth
of
> the possibility of involuntary unemployent. The resulting complete logical
> system known as THE GENERAL THEORY OF EMPLOYMENT INTEREST AND MONEY
>
>
> Paul
>
>
>
>
>
> Paul Davidson
> Editor, Journal of Post Keynesian Economics
> University of Tennessee
> SMC 503
> Knoxville, Tennessee 37996-0550
> phone # (561)369-1951; fax #(561)369-1951;
> email pdavidson@xxxxxxx
> http://econ.bus.utk.edu/davidsonextra/Davidson.html
>
>