Parsimonious Formulations

30 December 2009

Wednesday


William of Ockham was one of the greatest philosophers of the Western tradition, who more-or-less single-handedly inaugurated the tradition of philosophical parsimony that still reigns today.

The principle of parsimony — also called Ockham’s Razor, after William of Ockham who gave the principle some of its most compelling formulations — is among the most venerable of principles in human thought. This must be one of the few medieval philosophical principles that remains a staple of thought even today. Few but Thomists would be able to make it through the bulk of the Summa Theologiae, and far fewer still would find much in it with which they could agree, but there are parts of Ockham that can be read like a contemporary. Ockham is among the very few medieval writers of whom we can say this, and he shares this status with the canonical texts of classical antiquity.

I can't think of much of anything in Saint Thomas Aquinas that we can read while nodding with approval.

Not long ago in A Formulation of Naturalism I cited Hallett’s book Cantorian Set Theory and Limitation of Size for its treatment of what Hallett called Cantor’s finitism, i.e., Cantor’s treatment of transfinite numbers as being like finite numbers as far as this methodological analogy could be made to hold. I suggested that a similar approach could be used to characterize naturalism in terms of materialism: we can treat naturalism like materialism by way of a methodological analogy that is employed as long as it can be made to work. Later, in Two Thoughts on Naturalism, I suggested that naturalism could be given a similar treatment vis-à-vis mechanism.

Such formulations — the transfinite in terms of the finite, and naturalism in terms of materialism or mechanism — are minimalist formulations. Conceptual minimalism makes the most it can from the fewest resources. This is an application of the principle of parsimony. It has always been felt most strongly in the formal sciences. Axiomatization is an expression of this spirit of minimalism. Łukasiewicz’s reduction of the propositional calculus to a single axiom is another expression of the spirit of parsimony, as is the Polish notation for symbolic logic that he first formulated. The later Russell’s formulations in terms of “minimum vocabularies” must be counted a part of the same tradition, though Russell’s parsimonious roots go much deeper and are perhaps expressed most profoundly in his theory of descriptions.

Russell's theory of descriptions, called a 'paradigm of philosophy' by Frank Ramsey, is a classically parsimonious theory.

The language of parsimony is pervasive throughout contemporary logic and mathematics, such as when one says that, for example, Von Neumann–Bernays–Gödel set theory is a conservative extension of Zermelo–Fraenkel set theory (ZF). There is even a conservativity theorem of mathematical logic that formalizes this approach to parsimony. Perhaps counter-intuitively, a conservative extension of a theory extends the language of a theory without extending the theorems that can be derived from the original (unextended) theory. Michael Dummett is sometimes credited with originating the idea of a conservative extension (by Neil Tennant, for example), and he wrote in his Frege: Philosophy of Mathematics that:

“The notion of a conservative extension makes sense only if the theory to be extended is formulated in a language more restricted than that of the extended theory.” (p. 297)

It sounds puzzling at first, but it shouldn’t surprise us. Quine noted that the more we conserve on the elements of our theory, the larger the apparatus of derivation must become, and vice versa: there is an inverse relationship between the two.

dummett michael

The short Wikipedia article on conservative extensions observes as follows:

“a conservative extension of a consistent theory is consistent. Hence, conservative extensions do not bear the risk of introducing new inconsistencies. This can also be seen as a methodology for writing and structuring large theories: start with a theory, T0, that is known (or assumed) to be consistent, and successively build conservative extensions T1, T2, … of it.”

Thus the methodologically parsimonious tool of conservative extensions has implications for theoretical work over all. One can imagine an entire theoretical discipline given over to gradual and incremental extensions of an originally modest theory, which implies a model of theoretical thought innocent of Kuhnian paradigm shifts and revolutions in knowledge.

Sufficiently gradual (and sufficiently conservative) extensions of an existing theory may obviate the need for a Kuhnian paradigm shift.

Sufficiently gradual (and sufficiently conservative) extensions of an existing theory may obviate the need for a Kuhnian paradigm shift.

Of course, all parsimonious theories must rely upon some original bold insight upon which later conservative extensions can build. Cantor’s informal insights into set theory and transfinite numbers begat such an embarrassment of riches that almost all subsequent mathematical thought has consisted of various restrictions and codifications of Cantor’s intuitive and informal ideas. There is scarcely anything in the history of science to compare with it, except for Darwin’s conceptual breakthrough to natural selection. But mathematical theory and biological theory are developed so differently that the resemblance of these two insights followed by decades (and, I would guess, coming centuries) and elaboration and qualification is easier to miss than to see.

There is an implicit recognition in the conceptualization of parsimonious formulations of the power of more sweeping formulations, the proactive character of conceptual innovation that goes beyond accepted formulations, even while there is at the same time an implicit recognition of the danger and perhaps also irresponsibility of such theorizing.

Some time ago I noted in Exaptation of the Law that the law has an intrinsic bias in favor of the past that makes it a conservative force in society. With the law, this influence is concrete and immediate, often deciding the fates of individuals. It strikes me now that the minimalism and parsimony of much (if not most) formal thought is intrinsically conservative in an intellectual sense, and constitutes the ontological equivalent of bias in favor of the past. This intrinsic bias of formal thought is likely to be less concrete and immediate than that of the law, but no less pervasive.

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