A Formulation of Naturalism

16 November 2009


Hallett Cantorian Set Theory

One of my favorite books on set theory is Michael Hallett’s Cantorian Set Theory and Limitation of Size. While I have read some rather critical notices of the book, I am not the only one who appreciates it. I found a customer review at Amazon by William D. Fusfield that reads, “This is BY FAR the best and most INTERESTING book available on how Cantor developed his key ideas about transfinite sets, large cardinals, ordinals etc.”

Since Cantor is remembered for legitimizing the infinite as a mathematical concept, it might be a little surprising to hear that Hallett attributes “finitism” to Cantor, but by finitism Hallett does not mean any of the range of constructivist or strict finitist positions staked out by those who deny the legitimacy of the actual infinite and set theory, but rather it describes what we may call the methodological finitism of Cantor’s approach to the transfinite numbers he defined by way of set theory.

Finite mathematics is largely uncontroversial and commands the consensus of almost all who take an interest in the matter, however much they disagree on other parts of mathematics. Thus Hallett formulates what he calls Cantor’s principle of finitism thus:

“The transfinite is on a par with the finite and mathematically is to be treated as far as possible like the finite.” (p. 7)

This I would call methodological finitism. A little further on, on page 32, Hallett quotes Weyl thus:

“…for set theory, there is no difference in principle between the finite and the infinite.”

Hallett then comments:

“…the unity which Weyl points to is so much a fundamental part of Cantorianism (at least when we substitute ‘transfinite’ for ‘infinite’) that I have called it Cantor’s principle of finitism.”

A week ago I was musing about naturalism while making a longish drive and it occurred to me that something parallel to this approach could be used in a formulation of naturalism. “Parallel” is the key term here as were are talking about very different things with naturalism and the transfinite. What strikes me about Hallett’s formulation is the innate good sense of “as far as possible.” This stands in contrast to polarizing and absolutist definitions that employ formulations like “nothing but” or some equivalent of an extremal clause.

In contemporary science, scientific materialism is largely uncontroversial and commands the consensus of almost all interested parties. But from a philosophical standpoint materialism is as dissatisfying as finitism. If you can focus on the science and not think much about the materialism, you’ll be fine. But if the whole object of your interest in science is to illuminate the world and to come to a better understanding of it over all (as is my own interest), then one cannot only not avoid thinking about scientific materialism, one is obligated to think about it carefully.

At this point, then, I would suggest a methodological naturalism parallel to Hallett’s formulation of methodological finitism in Cantor: “Naturalism is on a par with materialism, and philosophically is to be treated as far as possible like materialism.” Or one could formulate it thus: “The natural is on a par with the material and scientifically is to be treated as far as possible like the material.”

Such a formulation would acknowledge both the success and the limitations of classical materialism that views the world entire as “nothing but” matter in motion — Democritean atoms whirling in the void — a classically reductionist formulation. Methodological naturalism as I have formulated it above, parallel to Hallett, would follow classical materialism as far as possible, and would only depart from materialism when that materialism was unsustainable in light of the evidence. And at this point I do not mean to suggest that one makes a transition from matter in motion to a non-naturalistic account of the world. On the contrary, it is at this point that naturalism shows itself to be as distinct from materialism as the infinite is distinct from the finite. Naturalism takes the spirit of materialistic explanation forward into areas that patently cannot be treated in terms of matter in motion.

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Naturalism: a Series

1. A Formulation of Naturalism

2. Two Thoughts on Naturalism

3. Naturalism: Yet Another Formulation

4. Joseph Campbell and Kenneth Clark: Bifurcating Naturalisms

5. Naturalism and Object Oriented Ontology

6. Naturalism and Suffering

7. Transcendental Non-Naturalism

8. Methodological Naturalism and the Eerie Silence

9. Some Formulations of Methodological Naturalism

10. Darwin’s Cosmology: A Naturalistic World

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Grand Strategy Annex

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