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Discord Invitation

26th November 2015

Post

Constructive Moments within Non-Constructive Thought

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Satan presents Christ with a top-down, non-constructive view of the kingdoms of the world – one of the temptations in the wilderness.

In a series of posts I have considered constructive and non-constructive thought, especially as the distinction touches upon formal thought, but also more generally – the important distinction between constructive and non-constructive thought should not be reserved exclusively for the use of logicians and mathematicians. These posts include, but are not limited to, the following:

I realized later, after thinking further about my post on Einstein and the centennial of Einstein’s field equations of general relativity (cf. A Century of General Relativity), that one might well have a “constructive” perspective on non-constructive thought.

What I mean by this is that, without having made any special commitment to a constructive methodology, with the consequent focus on how a proposition is proved rather than that a proposition is proved, one might, through repeated efforts to prove a difficult theorem, have tried every imaginable avenue in pursuit of the proof, and, after much effort and much experience of the formal conceptual space in which the theorem is located, have acquired a constructive, step-by-step, bottom-up view of the theorem and its proof. This is what I will call a constructive moment within non-constructive thought.  

Were it not for these constructive moments within non-constructive thought, the classical tradition might have wandered even farther from constructivism. Indeed, the argument could be made that constructivism has its origins in the constructive moments within non-constructive thought. 

Generally speaking, and admitting of many exceptions, formal rationalists like Einstein also tend to be realists. Einstein was unambiguously a realist, and, in fact, his realism is what was at the base of his rejection of quantum theory as a complete description of the world. Realism, in turn, is often non-constructive in its methodology. So while we cannot simply call Einstein’s formal method non-constructive, his rationalism tended to realism, and his realism tended toward non-constructive methods.

Hilbert’s thought (which I compared to Einstein in A Century of General Relativity) is a very different matter. While Hilbert superficially defended Cantor’s work, Hilbert wanted to keep what he called Cantor’s “paradise” of set theory, but he wanted to have it with finite proof procedures proved to be consistent. In other words, Hilbert’s thought involves an overlapping of emerging constructivist ideas and the tradition of classical eclecticism that Hilbert inherited as part of the mathematics of his time. Hilbert wasn’t ready to make the dramatic break with the tradition that his contemporary and critic Brouwer had made, but he was clearly part of a growing dissatisfaction with the eclectic attitude of the tradition to methodology.   

In Hilbert, the constructive moments within non-constructive thought become more than isolated moments, and are linked up into a recognizable method, though this method has not yet self-consciously separated itself from the traditional of classical eclecticism. The constructivist revolt against the tradition of classical eclecticism (when the method is self-consciously separated from tradition) might be characterized as a revolution in formal thought – certainly Weyl thought so, as he said, “…and Brouwer—that is the revolution!” – and so may be understood in terms of the structure of mathematical revolutions, when a birfurcation emerges and multiple mathematical research programs emerge from the previous continuity and unity of the mathematical tradition.

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Tagged: constructivismnon-constructivismrealismformal thought