Is Bourbaki the enemy?

Lars Syll, whose blog I always find well worth reading, has written an interesting short piece about the baleful influence of the Bourbaki school on economics. I agree with the main point, but I think that this raises a lot of complex issues which I want to partially wade through in this blog entry. I’m not sure that Bourbaki per se is the enemy. I’m hoping that a few of at least will make an attempt to bear with me as I make an attempt to wade through some difficult issues in the history of the philosophy of logic and mathematics as well as philosophy of science.  I’m  well aware that I am stepping into a lot of difficult and lengthy controversies, taking some stances some will quibble or quarrel with, and glossing over quite a bit. But I hope that the very complexity of these issues may at least stimulate some further thought and discussion.

Firstly, who was Bourbaki and why would some economists look at Bourbaki as the font of all that is unholy and evil in economics? Unless you are a mathemetician, or an economist who takes historical and methodological questions in economics seriously, the odds that you have heard of Bourbaki are probably slim and the odds that you repudiate it’s contributions as ultra vires are even lower. Until several years ago, few economists had heard of Bourbaki either, until Roy Weintraub wrote a book about the influence of the Bourbaki school on Gerard Debreu’s formal mathematical proof of the existence of a general equilibrium in a formal system of equations which ostensibly describe a competitive market economy. The book set off a lengthy exchange in the The Journal of Post Keynesian Economics about the relationship of mathematics to economics and especially to Post Keynesianism and to Heterodox Economics more generally. I should caution the reader that my own effort to publish on this debate resulted in what I thought was an egregious misreading by one referee and a series of nitpicking points that were IMO largely irrelevant and not even good nits to pick by another referee (which makes me the first author ever to complain about journal referees).

All that said, what is the Bourbaki controversy even about? It goes back to efforts by multiple luminaries such as Frege , Russell , Hilbert and others to derive the entire structure of logic from a few foundational axioms and then to derive mathematics from logic. Actually, this is probably a bit of a gloss on some fairly complex issues about the relationship of formalist to logicist programs. In the end, the program proved overly ambitious, but it did set forth the basis for expressing mathematical systems in purely formal languages. This gets us to some complex issues about the nature of purely analytical statements (if such can exist) and how, if at all, we can distinguish such statements from synthetic statements. I’m going to put another gloss on a number of difficult issues here and say that in my view, there is a common sense distinction to be made between a proposition about mathematics and a proposition about the external world. Argue if you wish, that Quine showed the irrelevance of this distinction, but IMO, that’s a misreading of Quine. What Quine showed was that when we are trying to create a sentence about external reality (or a semantic statement) we will ultimately hit a dead end if we try to really come up with a purely analytic sentence with semantic reference. If you take Quine and Russell seriously, which I do, then ultimately you leave open the possibility that you can judge propositions in mathematics as empirically true or false and thus you may wind up revising mathematics in light of further empirical evidence. But the math and the logic are the last things that get revised. Again, I think we can reject metaphysical distinctions between analytic and synthetic, but preserve our common sense understanding of sentences about mathematical systems and sentences about economic systems.

And this is where IMO, economics took a wrong turn.  Formalizing economics per se, even via the mechanisms of set theory and topography, though perhaps unnecessary and not terribly illuminating, does not mean that the axioms with economic meaning cannot adequately describe real world economies. Pierro Sraffa formalized, albeit using matrix algebra. Paul Samuelson formalized in a way in which his axioms with economic meaning were intended to be close enough approximations to economic reality to be workable in practice. Samuelson was not aping Bourbaki, but rather imitating Einstein’s physics, or at least claiming to do so. For right now, I will leave aside the Keynes is to Marshall as Einstein is to Newton controversy. Was the problem with Samuelson that he formalized? Or was the problem that his axioms with economic meaning are considered to be unrealistic? My argument is that the primary problem with Samuelson is the latter, not the former. And if you want to argue that someone’s sentences about the external world are true or false, then you had best be prepared for a long, hard slog because very few propositions can actually be reduced to the level of Tarski’s white snow. Most of our arguments in economics, especially between heterodox and orthodox economists are about general statements about general properties of the world and interpretations of history. In sum, I agree, Samuelson is wrong, but it takes a lot of work to show that: you can’t just pound the table and assert it. It’s also not clear to me at all that  we really gain anything by formalizing beyond the level of Alfred Marshall or Keynes.

And this brings me back to Bourbaki and Debreu. What Debreu provides us is a proof of the existence of an equilibrium in a formal system. Take that or leave it. As a realist, it’s hard for me to see the point. If I want to read about fictional economies, I’ll read China Mieville : he’s both more interesting and more realistic. China Mieville is indeed trying to make multiple points about the political economy in which we actually live, but he does not pretend to be doing anything other in the end then writing fiction. Debreu, intentionally or unintentionally, created a whole series of axioms with semantic meaning but then broke the rules for testing truth in a formal semantic system. In the end, Debreu’s analysis is an Abomination that impermissably mixes the analytic and the synthetic and purports to give us truth about the external world, when the most it can provide is consistency in a system of formal languages about a world that few of us can recognize.

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