Saturday, October 27, 2012

What Is Mathematics - And Sraffa

An Unsurveyable Rule For Generating A Real Number In Binary Format

Noah Smith offers a definition: "Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules." ("Unlearning Economics" has also recently written on mathematics in economics). To me, the manipulation of meaningless symbols is a powerful form of reasoning. Taking this definition as is, I think two questions can be raised here:

  • What is the interest that mathematicians find in these rules and these symbols in the historical circumstances current at the time?
  • What does it mean to follow a rule?

Ludwig Wittgenstein is the philosopher most known, I think, for raising the question of what it means to follow a rule. Any summary of his views will be controversial, but I suppose one can fairly say that he adopted an anthropological point of view, at least for some purposes. Describing how to follow a rule by another rule raises the prospect of an infinite regression. Rather, one might show how people do actually follow a rule, how these uses and practices work pragmatically in some form of life. I find it difficult to see how such description conveys the logical must, so to speak, of many rules. But Wittgenstein was alive to this difficulty. He notes that a judge does not seem to treat a statute book as a manual of anthropology.

Furthermore, Wittgenstein spent quite some time in elaborating how these ideas relate to the philosophy of mathematics. His views on the foundations of mathematics seems to have been constructivist and included questioning whether mathematics needs a foundation. Wittgenstein has frequently been labeled an anti-foundationalist. From this viewpoint, one might question whether existence proofs that do not specify how to construct the relevant object can be reformulated. And one even ends up doubting the meaningfulness of defining the real numbers as, say, any set isomorphic to a set of certain equivalence classes of Cauchy-convergent sequences of rational numbers. The use of the notion of infinity remains, I guess, as a standard topic in the philosophy of mathematics.

It seems one of my favorite economists, Piero Sraffa, was an important stimulus in Wittgenstein's development of these views. Sraffa has been said to have led Wittgenstein to see the importance of an anthropological point of view. Sraffa's masterpiece, The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory, is written in a unique style, not less in the presentation of the mathematics underlying the economics in the book. Sraffa frequently provides outlines of algorithms for constructive existence proofs, maybe most famously for the Standard Commodity. So Sraffa and Wittgenstein might be said to have shared a certain attitude to the philosophy of mathematics, although I do not expect to ever see oral discussions on this topic to be well documented. Sraffa's book can also be said to address only a limited range of topics in economics. An earlier statement of his seems to suggest that he thought room should exist in economics for non-formal treatment of some topics:

"The causes of the preference shown by any group of buyers for a particular firm are of the most diverse nature, and may range from long custom, personal acquaintance, confidence in the quality of the product, proximity, knowledge of particular requirements and the possibility of obtaining credit, to the reputation of a trademark, or sign, or a name with high traditions, or to such special features of modelling or design in the product as - without constituting it a distinct commodity intended for the satisfaction of particular needs - have for their principal purpose that of distinguishing it from the products of other firms. What these and the many other possible reasons for preference have in common is that they are expressed in a willingness (which may frequently be dictated by necessity) on the part of the group of buyers who constitute a firm's clientele to pay, if necessary, something extra in order to obtain the goods from a particular firm rather than from any other." -- Piero Sraffa (1926). "The Laws of Returns Under Competitive Conditions", Economic Journal (Dec.): pp. 544-545.

Whatever you think of the speculations in this post, I think some conclusions are nearly inarguable. Advocates and opponents of the use of mathematics in economics do not neatly divide between mainstream and non-mainstream economists. In particular, one important non-mainstream economist, Piero Sraffa, demonstrated one approach to mathematical economics, while still being aware of the limits to formalism in economics. Furthermore, any comprehensive scholarly study of the philosophy of mathematics will necessarily look at his work as long as Wittgenstein's later views are considered germane to such scholarship.


Charles Stewart said...

Thank you for this post - I enjoy explrations into parallelisms between Wittgenstein and Sraffa. I think highly of Wittgenstein as someone who asked very good questions, but he also took quite a few hard-to-defend positions.

With respect to the foundations, he famously annoyed Alan Turing with his arguments against Goedel's incompleteness theorems, and here his skepticism about the utility of formal argument I think went too far. But he made a good point along the way: meaning is not the flexible and mobile concept we tend to think it is, and different users of concepts use their concepts in different ways. One response to the Banach-Tarski paradox (the one about cutting up a ball in 3d point-set space into pieces and rearranging them into two balls each the same size of the original), is that non-measurable sets do not have much to do with our normal shape concepts. If "shape" is a family resemblance concept, mathematicians and non-specialists have it made up from different family members.

My view is that Wittgenstein did not reject classical mathematics. Instead he rejected the usefulness of concepts from mathematics outside mathematics - a point opposed to the fashion for applicability and cross-fertilisation of scholarly disciplines. His skeptical remarks went beyond questioning classical mathematics, but also he raised ultrafinist doubts about constructive mathematics. The gap between ultrafinitist and constructive mathematics is quite a bit bigger, in terms of the difficulty of proving things, than the distance between constructivist and classical mathematics.

But it is dangerous to assume that Wittgenstein reaches the skeptical conclusions his arguments seem to be headed towards - in the PI he sometimes sets up skeptical positions in order to demolish them, and Wittgenstein is quite often called an anti-skeptical philosopher.

Robert Vienneau said...

Thanks. I have not heard of ultrafinitism before. Do you know of Floyd and Putnam's 2000 paper, "A Note on Wittgenstein's 'Notorious Paragraph' About the Godel Theorem"? I have not read it in a while, but as I remember it, they defend Wittgenstein's mathematics.

Migeru said...

Wittgenstein in 1920s could be forgiven for being a formalist, but not us in the 21st century.

It's as if economists are stuck in a time warp before Goedel's undecidability theorem. Debreu should have known better.

Logic is not calculation, proof is not algorithmic, and mathematics is not mindless manipulation of symbols according to syntactic rules, pace Noah Smith.