Tuesday


Copernicus

Today we celebrate the 540th anniversary of the birth of Nicolaus Copernicus. The great astronomer was born 19 February 1473 in Toruń, now part of Poland. The name of Copernicus belongs with the short list of thinkers who not only changed the direction of civilization, but also the nature and character of Western civilization. Copernicus as the distinction of having a cosmology named in his honor.

We would do well to recall how radically our understanding of the world has changed in relatively recent years. Up until the advent of modern science, several ancient traditions of Western civilization had come together in a comfortingly stable picture of the world in which all of Western society was deeply invested. The Aristotelian systematization of Christian theology carried out by Thomas Aquinas was especially influential. Questioning this framework was not welcome. But science was an idea whose time had come, and, as we all know, nothing can stop the progress of an idea whose time had come.

Copernicus began questioning this cosmology by putting the sun in the center of the universe; Galileo pointed his telescope into the heavens and showed that the sun has spots, the moon has mountains, and that Jupiter had moons of its own, the center of its own miniature planetary system. Others took up the mantle and went even farther: Tycho Brahe, Johannes Kepler, and eventually Newton and then Einstein.

Copernicus was a polymath, but essentially a theoretician. One must wonder if Copernicus ever read William of Ockham, since it was Ockham along with Copernicus who initiated the unraveling of the scholastic synthesis, out of which the modern world would rise like a Phoenix from the ashes of the medieval world. Ockham provided the theoretical justification for the sweeping simplification of cosmology that Copernicus effected; it is not outside the realm of possibility that the later theoretician read the work of the earlier.

Today, when our knowledge of cosmology is expanding at breathtaking speed, Copernicus is more relevant than ever. We find ourselves forced to consider and to reconsider the central Copernican idea from every possible angle. The Fermi Paradox and the Great Filter force us to seek new insights into Copernicanism. I quite literally think about Copernicanism every day, making Copernicus a living influence on my thought.

As our civilization grows in sophistication, the question “Are we alone?” becomes more and more pressing. Arthur C. Clarke wrote, “Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.” This insight is profound in its simplicity. Thus we search for peer civilizations and peer life in the universe. That is to say, we look for other civilizations like ours, and for life that resembles us.

SETI must be considered a process of elimination, which I take to already have eliminated “near by” exocivilizations, although we cannot rule out the possibility that we currency find ourselves within the “halo” of a vanished cosmological civilization.

A peer civilization only slightly advanced over our own (say 100-500 years more industrial development), if it is in fact a peer and not incomprehensibly alien, would also be asking themselves “Are we alone?” They, too, would be equally terrified at being alone in the cosmos or at having another peer civilization present. Because we know that we exist as an industrial-technological civilization, and we know the extent to which we can eliminate peer civilizations in the immediate neighborhood of our own star, we can assume that a more advanced peer civilization would have an even more extensive sphere of SETI elimination. They would home in on us as incredibly interesting, as an exception to the rule of the eerie silence, in the same way that we seek out others like ourselves. That is to say, they would have found us, not least because they would be actively seeking us. So this may be considered an alternative formulation of the Fermi paradox.

Despite the growing tally of planets discovered in the habitable zones of stars, including nearby examples at Tau Ceti which lies within our SETI exclusion zone (which excludes only civilizations producing EM spectrum signals), there is no evidence that there are other peer civilizations, and advanced peer civilizations would already have found us — and they would be as excited by discovering us as we would be excited in discovering a peer civilization. There are none close, which we know from the SETI zone of exclusion; we must look further afield. Other peer civilizations would also likely have to look further afield. In looking further afield they would find us.

I don’t believe that any of this contradicts the Copernican principle in spirit. I think it is just a matter of random chance that our civilization happens to be the first industrial-technological civilization to emerge in the Milky Way, and possibly also the first in the local cluster of galaxies. We are, after all, an accidental world. However, it will take considerable refinement of this idea to show exactly how the uniqueness of human civilization (if it is in fact locally unique) is consistent with Copernicanism — and this keeps Copernicus in my thoughts.

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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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