A Non-Constructive World

19 April 2009


An imaginary illustration of Protagoras teaching.

An imaginary illustration of Protagoras teaching.

Further Ontological Ruminations

In yesterday’s Ontological Ruminations: Six Protagorean Propositions on the Nature of Man and the World I laid down several ontological principles of a Quasi-Protagorean bent. Protagoras (ca. 490– 420 BC; Greek: Πρωταγόρας), you will recall, was one of the greatest of the Presocratics, and was famous for having said, “Man is the measure of all things: of things which are, that they are, and of things which are not, that they are not.” This is one of those rare philosophical quotations that is sufficiently famous to have survived more than two thousand years and is recognized even by those with no interest in philosophy. So when I noted yesterday that the presence of man in the world as an agent constituting the world, it was, in essence, a Protagorean observation.

Beyond the immediate Protagorean interest, my six propositions of yesterday also suggested the non-constructive character of the world, but this requires explanation to make any sense whatsoever. When I speak of constructive and non-constructive in this forum, I mean the terms in their logical, and not their social, signification. What is the logical significance of constructive and non-constructive? That is difficult to easily sum up.

Constructivism in logic, mathematics, and formal thought generally is an embarras de richesse: there are a remarkable number of distinct formulations of constructivism as well as degrees of constructivism. I was just skimming several philosophical reference works for a simple and comprehensive definition of constructivity that would cover its various manifestations, and I couldn’t find anything satisfying.

There are logical approaches to constructivism, some of which involve logic without the Law of the Excluded Middle and others of which forbid the use of “existence” axioms that posit an entity without giving a method for constructing the entity, and there are finitist approaches to constructivism that deny or limit infinitistic propositions or methods, or which confine legitimate thought to finite assertions, and there are again predicative forms of constructivism that proscribe the use of impredicative definitions and methods.

Hopefully from the above (which is admittedly rather compressed and inexact) it should at least emerge that constructivists generally place limits on formal (or ontological) thought that would not otherwise be observed.

Constructive thought is pervasively influential today for a variety of reasons, ranging from essentially constructive nature of computer science, which makes itself felt in our lives in countless ways today because of the role of computers in our lives, to the increasingly constructivistic character of the sciences.

Physics has been turned into a constructivist undertaking without much notice of this profound change in perspective, yet it retains patently idealistic strains within the generally constructive drift—especially the presumption of the rationality of the world, i.e., its amenability to rational explanation, and mathematization of physics and its consequent idealizations and simplifications. Take, for example, the claim of the impossibility of travel at the velocity of light — if a philosopher deduced properties of the world from mathematical equations he would be a laughing stock, but physicists do so with impunity.

Physicists have taken the mantle of speculation from philosophers; science today is much more speculative than philosophy ever was, and the careful pedanticism of contemporary philosophers looks like a parody of scientific method intended to elicit laughter.

Moreover, the world itself seems constructive. Indeed, the constructivity of the world on a quantum scale is dramatically demonstrated by the failure of the law of the excluded middle and bivalence for quantum states: the logic of quantum theory is a logic without tertium non datur.

We see the extent to which the world is constructive when we contemplate the gradual, piecemeal way in which any actuality would need to approach any infinity. Just as we cannot reach aleph null by adding one repeatedly to any arbitrarily large number, so we cannot attain infinite mass by adding increments of mass to any arbitrarily large mass, nor can we shrink any arbitrarily small but finite quantity to nothing by gradually reducing it in size by a finite number of steps.

In mathematics, these limits have not the same function that they have in physics, because we can conduct thought experiments in which time and temporal processes have no place. But all that it subject to the laws of physics is also subject to the laws of time, and time will not allow us more than a constructive infinity of successively adding discrete quantities a finite number of times. This process can only yield an infinite result after the passage of an infinite quantity of time.

For all that, the world is still non-constructive, and even incorrigibly non-constructive.

The particular non-constructive aspect of the world that featured in yesterday’s exposition was the impredicativity of the world. An impredicative definition defines a given entity in terms of a whole of which it is a part. Impredicative reasoning makes use of impredicative definitions, and such are not terribly unusual. Any definition of an individual man that refers to all men is an impredicative definition, since the class of all men includes the individual so defined. And, more to the point in the present context, the world constructed by an agent who is part of that world is a non-constructive conception.

Not only is the world non-constructive and impredicative, but it is also indefinable in the traditional Aristotelian sense. In Aristotle, a term is defined by citing its genus and differentia. But the world has neither genus or differentia, and therefore cannot be defined. The world is a totality the eludes capture in formal thought. Or, as I put it in my Variations on the Theme of Life: “The world” is a metaphor for a concept that cannot be made literal.

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