13. Definitions, Assumptions, Propositions and Proofs in Sraffa’s PCMC

Sinha (2016, op. cit., italics added), substantiates his case by quoting Sen 2003, p. 1240:

Sraffa’s economic contributions … cannot, in general, be divorced from his philosophical understanding

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. vi-vii—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 trade-off 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 Martin-Löf’s inspiration. Ramsey’s revised type theory, developing the truth-table 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 Burali-Forti 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 Martin-Löf’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 Perron-Frobenius 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 ‘long-hand’!—of inter-industry 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!

See also Russell (1931, p. 477). Till Heyting and Kolmogorov in the early 1930s, eminent English mathematicians referred to intuitionists as finitists—as, for example, in Hardy (1929), p. 5: ‘the finitists or intuitionists’.

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 (=πr²), 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 mid-to-late 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).

Note that this is not to be confused with Computational Economics. Computable Economics is much wider in scope and its foundations are firmly set based on results from metamathematics and Computability Theory (Velupillai 2000, 2010a).

Clearly, this implicitly adhers to the Church-Turing Thesis. On computation and the Church-Turing thesis, see Velupillai (2010a) and/or Davis (1958).

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 non-computable or non-constructive 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 neo-Ricardians, 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) long-run 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 long-run 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 self-replacing prices (Zambelli 2018b), of the standard commodity and of the subsystems (Velupillai, Vela with Zambelli 1993), Velupillai-Fredholm-Zambelli 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.