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Discord Invitation

11th April 2015

Post with 2 notes

Constructivist Watersheds

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The emergence of set theory as a mathematical discipline was a watershed for constructivist thought, not merely because of the spectacularly non-constructive nature of some of its proofs, and not merely because the explicitly Platonist formulations of some of its originators gave constructivists a stalking horse for their slings and arrows, but above all because it was with set theory that all mathematicians were confronted with the comprehension principle had to accept some limitation on the comprehension principle (i.e., that any property defines a set) as a principle generative of sets.   

Unrestricted comprehension begat paradoxes and contradictions, so that a consistent set theory required some restrictions upon comprehension, and with this the fundamental underlying principle of constructivism was granted–that mathematical reasoning must acknowledge certain limitations, or, as it was already formulated in antiquity by Epicharmus of Syracuse, “Mortal man must think mortal and not immortal thoughts.”

In principle, then, the principle of constructivism was conceded, but this concession was not the victory of constructivism. Non-constructive thought has been as gradual in its concessions as constructive thought has been sweeping in its principles and gradual in its efforts to reformulate mathematics upon a constructive basis. In other words, there is a leap from granting a single concession to granting any systematic and methodological regime of limitations.

I previously discussed the role of limitation in constructivist thought in Constructivism Without Constructivism and The Vacuous Identity Principle, and I suggested the possibility that one might agree in principle that limitations are necessary to formal thought, even while not agreeing to any of the specific limitations set by various constructivist or finitist programs. In this way the concession of the principle of constructivism becomes a vacuous concept, entertained in principle but not agreed in practice. (I might point out that this accords quite well with the idea of classical eclecticism, on which cf. Epistemic Opportunism.)  However, there are areas of agreement in practice as well as in principle.

The vicious circle principle, and the explicit formulation of impredicativity following from the vicious circle principle, was, like set theory, a constructivist watershed. Even mathematical classicists like Russell agreed with proto-constructivists like Poincaré on the importance of avoiding impredicative formulations (i.e., not defining an individual in terms of a whole of which that individual is a part). However, the relatively recent formulation of the idea of impredicativity may be understood as simply the most (historically) recent formulation of a very ancient philosophical idea, viz. the need to avoid an infinite regress in reasoning or the meaninglessness of vacuous reflexivity (i.e., circular reference). 

As I have recently been writing about infinitistic thought (in Permutations of Infinitistic Methods, Inifintistic Historiography and Terrestrial Eocivilization, and Toward an Infinitistic Cosmology) I cannot but notice the implicit pathologicalization of infinitistic methods and infinitistic domains in constructivist thought, and the extent to which finitistic assumptions have penetrated into mainstream formal thought. I noted some time ago (in extraordinary sets) that the axiom of foundation in ZF set theory rules out so-called extraordinary sets, i.e., sets with an infinitely descending membership chain. While this does away with certain problematic sets, it also does away with a lot of interesting sets that are in no sense paradoxical, contradictory, or inconsistent.

As ZF stands at the center of set theoretical research today (though it is by no means the only area of research in set theory), the willingness to accept the ban on extraordinary sets as an acceptable cost is telling. Extraordinary sets involve an infinite regress (specifically, an infinite regress of membership), and so exemplify the ancient philosophical quandary of infinite regresses. Philosophers as well as mathematicians seem happy to be done with them.

Although the vicious circle principle has been accepted by constructivists and non-constructivists alike, the structure of our life in the world – we are beings in the world, part of a whole defined in relation to ourselves – means that we can only discuss the human condition in impredicative terms. Indeed, the human condition is impredicative (something I formulated quite some time ago in Ontological Ruminations), so that if we are to cultivate formal thought equal to the task of conceptualizing the human condition, this formal thought must be infinitistic even to the point of accepting impredicative reasoning. And, better than merely accepting and acknowledging impredicative reasoning, what we need is an active engagement with infinitistic and impredicative thought.

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Tagged: Epicharmus of Syracuseimpredicativityvicious circle principleformal thoughtconstructivismnon-constructivismconstructivisticextraordinary setshuman conditioninfinitisticaxiom of foundation

  1. geopolicraticus posted this