A Fine-Grained Overview

5 December 2016



Constructive and Non-Constructive Perspectives

Whenever I discuss methodology, I eventually come around to discussing the difference between constructive and non-constructive methods, as this is a fundamental distinction in reasoning, though often unappreciated, and especially neglected in informal thought (which is almost all human thought). After posting Ex Post Facto Eight Year Anniversary I realized that the distinction that I made in that post between detail (granularity) and overview (comprehensivity) can also be illuminated by the distinction between the constructive and the non-constructive.

Two two pairs of concepts can be juxtapositioned in order to show the four permutations yielded by them. I have done the same thing with the dual dichotomies of nomothetic/ideographic and synchonic/diachronic (in Axes of Historiography) and with weak panspermia/strong panspermia and theological/technological (in Is astrobiology discrediting the possibility of directed panspermia?). The table above gives the permutations for the juxtaposition of detail/overview and constructive/non-constructive.

In that previous post I identified my theoretical ideal as a fine-grained overview, combining digging deeply into details while also cultivating an awareness of the big picture in which the details occur. Can a fine-grained overview be attained more readily through constructive or non-constructive methods?

In P or Not-P I quoted this from Alain Connes:

“Constructivism may be compared to mountain climbers who proudly scale a peak with their bare hands, and formalists to climbers who permit themselves the luxury of hiring a helicopter to fly over the summit.”

Changeux and Connes, Conversations on Mind, Matter, and Mathematics, Princeton, 1995, p. 42

This image makes of constructivism the fine-grained, detail-oriented approach, while non-constructive methods are like the overview from on high, as though looking down from a helicopter. But it isn’t quite that simple. If we take this idea of constructivists as mountain climbers, we may extend the image with this thought from Wittgenstein:

“With my full philosophical rucksack I can climb only slowly up the mountain of mathematics.”

Ludwig Wittgenstein, Culture and Value, p. 4

And so it is with constructivism: the climbing is slow because they labor under their weight of a philosophical burden. They have an overarching vision of what logic and mathematics ought to be, and generally are not satisfied with these disciplines as they are. Thus constructivism has an overview as well — a prescriptive overview — though this overview is not always kept in mind. As Jean Largeault wrote, “The grand design has given way to technical work.” (in the original: “Les grands desseins ont cédé la place au travail technique.” L’intuitionisme, p. 118) By this Largeault meant that the formalization of intuitionistic logic had deprived intuitionism (one species of constructivism) of its overarching philosophical vision, its grand design:

“Even those who do not believe in the omnipotence of logic and who defend the rights of intuition have acceded to this movement in order to justify themselves in the eyes of their opponents. As a result we find them setting out, somewhat paradoxically, the ‘formal rules of intuitionist logic’ and establishing an ‘intuitionistic formalism’.”

…and in the original…

“Ceux-la memes qui ne croient pas a la toute-puissance de la logique et qui défendent les droits de l’intuition, ont du, eux aussi, céder au mouvement pour pouvoir se justifier aux yeux de leurs adversaires, et l’on a vu ainsi, chose passablement paradoxale, énoncer les ‘regles formelles de la logique intuitioniste’ et se constituer un ‘formalisme intuitioniste’.”

Robert Blanché, L’axioimatique, § 17

But intuitionists and constructivists return time and again to a grand design, so that the big picture is always there, though often it remains implicit. At very least, both the granular and the comprehensive conceptions of constructivism have at least a passing methodological familiarity, as we see in the table above, on the left side, granular constructivism with its typical concern for the “right” methods (which can be divorced from any overview), but also, below that, the philosophical ideas that inspired the constructivist deviation from classical eclecticism, from Kant through Hilbert and Brouwer to the constructivists of our time, such as Errett Bishop.

These two faces of methodology are not as familiar with non-constructivism. In so far as non-constructivism is classical eclecticism (a phrase I have taken from the late Torkel Frazén), a methodological “anything goes,” this is the granular conception of non-constructivism that consists of formal methods without any unifying philosophical conception. This much is familiar. Less familiar is the possibility of a non-constructive overview made systematic by some unifying conception. The idea of a non-constructive overview is familiar enough, and appears in the Connes quote above, but it this idea has had little philosophical content.

There is, however, the possibility of giving non-constructive formal methodology an overarching philosophical vision, and this follows readily enough from familiar forms of non-constructive thought. Cantor’s theory of transfinite numbers, and the proof techniques that Cantor formulated (and which remain notorious among constructivists) is a rare example of non-constructive thought pushed to its limits and beyond. Applied to a non-constructive overview, the transfinite perspective suggests that a systematically non-constructive methodology would insistently seek a total context for any idea, by always contextualizing any idea in a more comprehensive setting, and pursuing that contextualization to infinity. Thus any attempt to think a finite thought forces us to grapple with the infinite.

A fine-grained overview might be formulated by way of a systematically non-constructive methodology — not the classical eclecticism that is an accidental embrace of non-constructive methods alongside constructive methods — that digs deep and drills down into details by non-constructive methods that also furnish a sweeping, comprehensive philosophical vision of what formal methods can be, when that philosophical vision is not inspired to systematically limit formal methods (as is the case with constructivism).

Would the details that would be brought out by a systematically non-constructive method be the same fine-grained details that constructivism brings out when it insists upon finitistic proof procedures? Might there be different kinds of detail to be revealed by distinct methods of granularity in formal thought? These are elusive thoughts that I have not yet pinned down, so examples and answers will have to wait until I have achieved Cartesian clarity and distinctness about non-constructive methods. I beg the reader’s indulgence for my inadequate formulations here. Even as I write, ideas appear briefly and then disappear before I can record them, so this post is different from what I imagined as I sat down to write it.

Here again I can appeal to Wittgenstein:

“This book is written for such men as are in sympathy with its spirit. This spirit is different from the one which informs the vast stream of European and American civilization in which all of us stand. The spirit expresses itself in an onwards movement, building ever larger and more complicated structures; the other in striving after clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its periphery — in its variety; the second at its center — in its essence. And so the first adds one construction to another, moving on and up, as it were, from one stage to the next, while the other remains where it is and what it tries to grasp is always the same.”

Ludwig Wittgenstein, Philosophical Remarks, Foreword

These two movements of thought are not mutually exclusive; it is possible to build larger structures while always trying to grasp an elusive essence. It could be argued that anything built on uncertain foundations will come to naught, so that we must grasp the essence first, before we can proceed to construction. As important as it is to attempt to grasp an elusive essence, if we do this, we risk the intellectual equivalent of the waiting gambit.

. . . . .

Constructivism and Non-constructivism

P or Not-P

What is the Relationship between Constructive and Non-Constructive Mathematics?

A Pop Culture Exposition of Constructivism

Intuitively Clear Slippery Concepts

Kantian Non-Constructivism

Constructivism without Constructivism

The Vacuous Identity Principle

Permutations of Infinitistic Methods

Methodological Differences

Constructivist Watersheds

Constructive Moments within Non-Constructive Thought

Gödel between Constructivism and Non-Constructivism

The Natural History of Constructivism

Cosmology: Constructive and Non-Constructive

Saying, Showing, Constructing

Arthur C. Clarke’s tertium non datur

A Non-Constructive World

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Wittgenstein wrote, “With my full philosophical rucksack I can climb only slowly up the mountain of mathematics.”

Wittgenstein wrote, “With my full philosophical rucksack I can climb only slowly up the mountain of mathematics.”

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Grand Strategy Annex

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project astrolabe logo smaller

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