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A Reflection on Sraffa’s Revolution in Economic Theory
An attempt is made to interpret Sraffa’s PCMC in terms of Constructive Type Theory of the Brouwer/MartinLöf variety. To facilitate this interpretation, PCMC is considered a precise mathematical text and Sraffa’s subtitle is considered to be meaningful in this sense.
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Not, perhaps, as good a
philosophical case for Sraffa, in general, and
PCMC in particular, as made by Sinha (and Sen; see also McGuiness
2008, pp. 15–16 & 229).
I use ‘precise’ instead of ‘rigorous’, mainly because I have
never read a definition of
rigour or
rigorous, in the standard mathematical literature.
Minimalist does not imply
minimum.
In the
intuitive sense customary, for example, in constructive and computable mathematics.
Economic theory, of whatever ideology, is—at most—syntactical, hardly semantical. I shall, however, not develop this line of thought further (in this paper). It will take me away from the main task of studying, from an algorithmic point of view,
PCMC, to consider Jevons, Walras, Fisher, Pareto, Barone, von Mises, von Hayek, Lange, and even Scarf (see, in particular, Sreenivasan
2005, p. 92).
I suspect that Knuth confines the notion of computers to the
digital kind; I am more eclectic, allowing the notion to refer to
analog and hybrid computers, as well (especially with
PCMC in mind).
I mention this aspect only because Alister Watson refers to Frank Ramsey as adhering to
extensions (Watson
1938, p. 444), who are—according to Sraffa, in
PCMC, pp. vivii—two of three mathematicians, the third being Besicovitch, to whom he is indebted for ‘invaluable mathematical help.’ I shall have more to say about these three mathematicians, in the context of my
PCMC thesis on intuitionistic constructive mathematics.
I suspect that Sraffa knew of
Specker sequences, recursive (un)decidability, undecidable disjunctions, the distinction between Cauchy’s and Dedekind’s definitions of real numbers—and, in general, Gödel’s results on the tradeoff between truth and provability, Turing’s on the
Halting Problem for Turing Machines, and the
Goodstein sequences. However, his adherence to intuitionistic constructive mathematics, in
PCMC, does
not depend on any familiarity with these things.
On the 20th of December, 1980, Sraffa wrote in my copy of
PCMC (italics added):
This book has the advantage of being compact.
Obviously, Sraffa does not mean compact in any mathematical sense!
The notion of
system in
PCMC is clarified in Sraffa’s interchange with Newman, reprinted in Bharadwaj (
1970).
The example of a mathematical economist of classical persuasions talking at cross purposes with a constructivist mathematical economists is illustrated in the above discussion between Newman and Sraffa; when the latter asks the former to
prove (p. 426, in Bharadwaj,
op.cit.), it is in the sense of
constructive existence proofs that is meant (I think). Consequently, I don’t think Newman understood Sraffa’s admonishment at all.
It is Brouwer’s
intuitionism and Russell’s
type theory, along with Bishop’s
constructive mathematics that was the motivation for MartinLöf’s inspiration. Ramsey’s revised type theory, developing the truthtable scheme of the early Wittgenstein, was decisive in the second edition of the
Principia Mathematica by Whitehead and Russell. Russell’s original motivation for introducing types in his logicism had been due to the logical paradox of too many or too large sets in Frege’s naïve conception of sets and the discovery of the BuraliForti paradox (see Russell
1937; van Heijenoort (edited),
2000, especially pp. 124–128). Brouwer’s introduction of intuitionism, in opposition to Hilbert’s formalism, was (partly, at least) due to what I think is Hilbert’s equally naïve introduction of formalism, on the basis of his axiomatization of geometry. Sraffa did not subscribe to axiomatic formalism or any kind of logicism, in
PCMC. Most—if not all—of the formalisers of mathematical economics, are followers of Hilbert.
However, Nordström et al. (
1990, chapter 2)), which I use for programming
PCMC in MartinLöf’s intuitionistic type theory interpret the rule about proofs of propositions as one of many possibilities.
If Sraffa, in
PCMC, had ‘adopted’ matrix notation, the obvious claim by all adherents of classical mathematics, in economics (irrespective, again, of ideology), would have been to question why the
PerronFrobenius theorem had not been invoked—as if such a claim was not made by all and sundry, anyway! Note, also, that Leontief (
1941) wrote out the equations—in ‘longhand’!—of interindustry analysis without any appeal to matrix algebra.
For these purposes, I add Wittgenstein and Turing to this list of three mathematicians, who were decisive in influencing the intuitionistic constructive mathematics of
PCMC.
Although I have the utmost respect for the Wittgenstein publications of McGuiness, I class the following observation about Alister Watson (McGuiness
2008, p. 8) in the same dubious class as the one about Sraffa (McGuiness
2002, p. x; italics added):
Later Alister Watson, perhaps the only one of the charmed circle of young Apostles to remain with him, contributed in somewhat the same way presentations of Wittgensteinian ideas on the foundations (or lack of foundations) of mathematics.Sraffa, whom Wittgenstein regarded as his severest critic, spent much of his time during the years of his closest association with the philosopher on the writing and rewriting of a devastating review of a paper by Hayek (who survived cheerfully enough).
First of all, it was a review of a
book by Hayek; secondly, till at least 1974, Hayek did
not ‘survive cheerfully’—in fact he was a
bitter man!
There is a curious error in Galavotti (
op.cit.), where the references to Majer in footnote 37, on p. 22 and footnote 1, on p. 197, are to two different journals! It is, obviously, a trivial error.
Gibson (
2010), p. 75, observes:
Note that one of the Examiners was A. S. Besicovitch—Fellow of Trinity, advisor to Sraffa— Wittgenstein sparring partner; …
‘Sparring partner’—of philosophy, mathematics, philosophy of mathematics, or?
The Mathematical Association of America, using a grant obtained from the National Science Foundation, established a Committee on Production of Films; the fourth of the films produced by this Committee, was the Besicovitch lecture on the geometric construction involved in his 1920s initial solution of the Kakeya problem (see Besicovitch,
op.cit., p. 697).
See Pál (
1921). I may mention that Pál developed Matsusaburo Fujiwara’s formulation of Soichi Kakeya’s problem.
Besicovitch (op. cit., p. 697) gives the area of the circle answering the Kakeya question as
exactly equal to.78 (=
πr
^{2}), whereas it can only be exactly approximated. Similarly, the area of the equilateral triangle, with side length 4/√3 is ≈ 0.58, not—as Besicovitch writes—exactly so.
I was greatly helped, in this interpretation of
PCMC, by a reading of Geuvers et al. (
2007).
Niels Henrik Abel’s famous late 1870s result, which I conjecture one or the other of the many mathematicians around Sraffa, would have appraised him, or—versatile as he was—he may have independent knowledge of this significant exact unsolvability.
In late 1979, when Sraffa walked with me in the ‘backs’, he stopped near King’s College, and said:
You know, I used to be a member of this College; but after the ‘Book” [he was referring to Keynes’ General Theory], it was impossible to live with Kahn! Robertson took me as a Fellow to Trinity.
If not for the difficulties of a personal relationship with Wittgenstein, I am sure Sraffa would have added the eminent philosopher’s name to the three mathematicians he listed as being indebted to, for the results in
PCMC. See Wittgenstein’s midtolate 1940s letters to Sraffa republished in McGuiness (
2008), in particular the sentence overheard by Smythies, of Sraffa saying to Wittgenstein (
op.cit., p. 339):
I won’t be bullied by you, Wittgenstein.
Incidentally, Sraffa and Wittgenstein were admitted to Trinity fellowships, on the same day, in 1938 (McGuiness, ibid., p. 288).
Remark 3 and Conjecture III are inspired by Kreisel (
1967).
See the first footnote in Bickford et al. (
2018).
One should not be led into thinking that Walrasian General Equilibrium is computable because it is at times claimed that Scarf’s algorithm can help compute. Scarf’s Computable General Equilibrium is
“neither
computable
nor constructive in the strict
mathematical
senses” (Velupillai
2006, p. 360). A list of noncomputable or nonconstructive results proved by Velupillai are to be found also in (Zambelli,
2010, pp. 37–38). For
“variations in the theme of conning in mathematical economics” see Velupillai (
2007) and for the
“unreasonable ineffectiveness of mathematics in economics” see Velupillai (
2005).
One should not be conned into thinking that arguments expressed in English prose, as it is the case for many neoRicardians, cannot be formalized in the form of mathematical propositions or sentences. The logical structure of these arguments remain the same. In Sraffa’s PCMC the propositions, whether presented in English or formulated in a mathematical language, have a definite computable algorithmic content. The same, however, is not true for the propositions made by many of Sraffa’s followers. A good example would be the claims made by many that Sraffa’s PCMC is dealing with a convergent (or with an already converged process towards) longrun equilibrium. Such claims are never supported by arguments that are constructive or algorithmic. Therefore, this interpretation of the Sraffa’s schemes of being relative to a longrun equilibrium is postulated or assumed, rather than demonstrated. Surely, it is not something assumed by Sraffa himself.
Even Pasinetti, who in his work has made propositions that can in fact be reformulated constructively, has not followed Sraffa’s method. As early as 1980, in his review of Pasinetti (
1977) Velupillai wrote:
There is a crucial distinction between the methodology followed by Pasinetti and that followed by Sraffa in proving the important propositions. [...] The distinction seems to be that Sraffa, whenever he gives an explicit proof, invariably gives us a constructive proof, whereas all the proofs Pasinetti (and almost everyone else who has attmpted to formalize and generalize Sraffa) gives, follows the method of formalist mathematicians (Velupillai 1980, pp. 64–65)
The computations of selfreplacing prices (Zambelli
2018b), of the standard commodity and of the subsystems (Velupillai, Vela with Zambelli
1993), VelupillaiFredholmZambelli algorithm (Zambelli et al.
2017, see Appendix) (which is an algorithmisation of chapter 12 of PCMC) and algorithms used in Zambelli (
2018a) and in Venkatachalam and Zambelli (
2021) can all be traced back to Sraffa’s PCMC or explicitly based on it.
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 Title
 Definitions, Assumptions, Propositions and Proofs in Sraffa’s PCMC
 DOI
 https://doi.org/10.1007/9783030472061_13
 Author:

K. Vela Velupillai
 Publisher
 Springer International Publishing
 Sequence number
 13
 Chapter number
 Chapter 13