Sunday, May 11, 2008

Contrasting Views On Sraffa's Mathematics

"...Sraffa's prices produce questions, besides whatever else, about the mathematics of his arguments." --S. N. Afriat (2008) "Sraffa's Prices", Sraffa or an Alternative Economics (ed. by G. Chiodi and L. Ditta), Palgrave Macmillan.
Here are two perspectives:
"I think that a very important difference exists between: (i) the process through which a mathematical result is reached, and (ii) a rigorous proof of the result. ... Regarding (i) I mean a sequence of mental objects: examples that appear to contain all of what is essential, graphical tools providing proofs that are only valid for dimensions two or three, incomplete proofs that appear as 'almost' correct, auxiliary constructions that show what is not immediately visible in the problem..."

...We know that all the results contained in Production of Commodities, Part I, can be restated in the language of standard mathematics (matrix theory, eigenvalues, eigenvectors, Perron-Frobenius Theorem, etc.) and rigorously proved. My opinion ... is that Sraffa's presentation is closer to the process that I have indicated by (i) in the Introduction, than to formal proofs. In some cases Sraffa's arguments are defective or insufficient, in others they introduce useless complications." --Marco Lippi (2008) "Some Observations on Sraffa and Mathematical Proofs with an Appendix on Sraffa's Convergence Algorithm", Sraffa or an Alternative Economics (ed. by G. Chiodi and L. Ditta), Palgrave Macmillan.
Lippi's position that Sraffa's mathematics contains defects is strengthed by his demonstration of a bug in Sraffa's algorithm for the construction of the standard commodity.

Is Velupilla in disagreement:
"From a purely mathematical point of view, PCC lacks nothing. The concerns in PCC are the solvability of equations systems and, whenever existence or uniqueness proofs are considered, they are either spelled out in completeness, albeit from a non-formal, non-classical point of view or detailed hints are given, usually in the form of examples, to complete the necessary proofs in required generalities. Pure laziness, inertia and ignorance of alternative traditions in mathematical philosophy have caused untold mischief and created an industry of re-casting and distorting PCC, a work of aesthetic purity and mathematical elegance, into a trivial application, to a large extent, of linear algebra." --Kumaraswamy Velupillai (2008) "Sraffa's Mathematics in Non-Classical Mathematical Modes", Sraffa or an Alternative Economics (ed. by G. Chiodi and L. Ditta), Palgrave Macmillan.
Velupilla is severely critical of the use of Perron-Fobenius theorems in the recasting of Sraffa's theory, when Sraffa essentially gave a constructive proof in demonstrating the existence of the standard commodity.

1 comment:

John Ryskamp said...

I said long ago that the problem with Sraffa's approach is that he uses what today we call constructivist mathematics. It is also called "natural mathematics" and it is the current incarnation of what used to be known as logicism, formalism and intuitionism.

It's the idiotic notion--derived from "paradoxes" which are not paradoxes--that in order to avoid paradox in argumentation, we must arbitrarily insert into every argument, the idea that mathematics is an inherent human function.

The result is that every argument which uses this mathematics, lacks logical content.

So your author is on the right track, but very ignorant of recent historiography in the history of set theory. Your author needs to read the following article. So do you:

Ryskamp, John Henry, "Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas" (June 17, 2008). Available at SSRN: