A Fine-Grained Overview

5 December 2016

Monday


overview-and-detail

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.

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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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Monday


Edmund Husserl wanted philosophy to become a rigorous science.

Edmund Husserl wanted philosophy to become a rigorous science.

In several posts I have discussed the need for a science of civilization (cf., e.g., The Future Science of Civilizations), and this is a theme I intended to continue to pursue in future posts. It is no small matter to constitute a new science where none has existed, and to constitute a new science for an object of knowledge as complex as civilization is a daunting task.

The problem of constituting a science of civilization, de novo for all intents and purposes, may be seen in the light of Husserl’s attempt to constitute (or re-constitute) philosophy as a rigorous science, which was a touchstone of Husserl’s work. Here is a passage from Husserl’s programmatic essay, “Philosophy as Strict Science” (variously translated) in which Husserl distinguishes between profundity and intelligibility:

“Profundity is the symptom of a chaos which true science must strive to resolve into a cosmos, i.e., into a simple, unequivocal, pellucid order. True science, insofar as it has become definable doctrine, knows no profundity. Every science, or part of a science, which has attained finality, is a coherent system of reasoning operations each of which is immediately intelligible; thus, not profound at all. Profundity is the concern of wisdom; that of methodical theory is conceptual clarity and distinctness. To reshape and transform the dark gropings of profundity into unequivocal, rational propositions: that is the essential act in methodically constituting a new science.”

Edmund Husserl, “Philosophy as Rigorous Science” in Phenomenology and the Crisis of Philosophy, edited by Quentin Lauer, New York: Harper, 1965 (originally “Philosophie als strenge Wissenschaft,” Logos, vol. I, 1911)

Recently re-reading this passage from Husserl’s essay I realized that much of what I have attempted in the way of “methodically constituting a new science” of civilization has taken the form of attempting to follow Husserl’s pursuit of “unequivocal, rational propositions” that eschew “the dark gropings of profundity.” I think much of the study of civilization, immersed as it is in history and historiography, has been subject more often to profound meditations (in the sense that Husserl gives to “profound”) than conceptual clarity and distinctness.

The Cartesian demand for clarity and distinctness is especially interesting in the context of constituting a science of civilization given Descartes’ famous disavowal of history (on which cf. the quote from Descartes in Big History and Scientific Historiography); if an historical inquiry is the basis of the study of civilization, and history consists of little more than fables, then a science of civilization becomes rather dubious. The emergence of scientific historiography, however, is relevant in this context.

The structure of Husserl’s essay is strikingly similar to the first lecture in Russell’s Our Knowledge of the External World. Both Russell and Husserl take up major philosophical movements of their time (and although the two were contemporaries, each took different examples — Husserl, naturalism, historicism, and Weltanschauung philosophy; Russell, idealism, which he calls “the classical tradition,” and evolutionism), primarily, it seems, to show how philosophy had gotten off on the wrong track. The two works can profitably be read side-by-side, as Russell is close to being an exemplar of the naturalism Husserl criticized, while Husserl is close to being an exemplar of the idealism that Russell criticized.

Despite the fundamental difference between Husserl and Russell, each had an idea of rigor and each attempted to realize in their philosophical work, and each thought of that rigor as bringing the scientific spirit into philosophy. (In Kierkegaard and Russell on Rigor I discussed Russell’s conception of rigor and its surprising similarity to Kierkegaard’s thought.) Interestingly, however, the two did not criticize each other directly, though they were contemporaries and each knew of the other’s work.

The new science Russell was involved in constituting was mathematical logic, which Roman Ingarden explicitly tells us that Husserl found inadequate for the task of a scientific philosophy:

“It is maybe unexpected and surprising that Husserl who was trained as a mathematician did not seek salvation for philosophy in the mathematical method which had from time to time stood out like a beacon as an ideal worthy of imitation by philosophers. But mathematical logic could not satisfy him… above all he fought for responsibility in philosophical research and devoted many years to the elaboration of a method which, according to him, was to secure for philosophy the status of a science.”

Roman Ingarden, On the Motives which Led Husserl to Transcendental Idealism, Translated from the Polish by Arnor Hannibalsson, Den Haag: Martinus Nijhoff, 1975, p. 9.

Ingarden’s discussion of Husserl is instructive, in so far as he notes the influence of mathematical method upon Husserl’s thought, but also that Husserl did not try to employ a mathematical method directly in philosophy. Rather, Husserl invested his philosophical career in the formulation of a new methodology that would allow the values of rigorous scientific practice to be expressed in philosophy and through a philosophical method — a method that might be said to be parallel to or mirroring the mathematical method, or derived from the same thematic motives as those that inform mathematical methodology.

The same question is posed in considering the possibility of a rigorously scientific method in the study of civilization. If civilization is sui generis, is a sui generis methodology necessary to the formulation of a rigorous theory of civilization? Even if that methodology is not what we today know as the methodology of science, or even if that methodology does not precisely mirror the rigorous method of mathematics, there may be a way to reason rigorously about civilization, though it has yet to be given an explicit form.

The need to think rigorously about civilization I took up implicitly in Thinking about Civilization, Suboptimal Civilizations, and Addendum on Suboptimal Civilizations. (I considered the possibility of thinking rigorously about the human condition in The Human Condition Made Rigorous.) Ultimately I would like to make my implicit methodology explicit and so to provide a theoretical framework for the study of civilization.

Since theories of civilization have been, for the most part, either implicit or vague or both, there has been little theoretical framework to give shape or direction to the historical studies that have been central to the study of civilization to date. Thus the study of civilization has been a discipline adrift, without a proper research program, and without an explicit methodology.

There are at least two sides to the rigorous study of civilization: theoretical and empirical. The empirical study of civilization is familiar to us all in the form of history, but history studied as history, as opposed to history studied for what it can contribute to the theory of civilization, are two different things. One of the initial fundamental problems of the study of civilization is to disentangle civilization from history, which involves a formal rather than a material distinction, because both the study of civilization and the study of history draw from the same material resources.

How do we begin to formulate a science of civlization? It is often said that, while science begins with definitions, philosophy culminates in definitions. There is some truth to this, but when one is attempting to create a new discipline one must be both philosopher and scientist simultaneously, practicing a philosophical science or a scientific philosophy that approaches a definition even as it assumes a definition (admittedly vague) in order for the inquiry to begin. Husserl, clearly, and Russell also, could be counted among those striving for a scientific philosophy, while Einstein and Gödel could be counted as among those practicing a philosophical science. All were engaged in the task of formulating new and unprecedented disciplines.

This division of labor between philosophy and science points to what Kant would have called the architectonic of knowledge. Husserl conceived this architectonic categorically, while we would now formulate the architectonic in hypothetico-deductive terms, and it is Husserl’s categorical conception of knowledge that ties him to the past and at times gives his thought an antiquated cast, but this is merely an historical contingency. Many of Husserl’s formulations are dated and openly appeal to a conception of science that no longer accords with what we would likely today think of as science, but in some respects Husserl grasps the perennial nature of science and what distinguishes the scientific mode of thought from non-scientific modes of thought.

Husserl’s conception of science is rooted in the conception of science already emergent in the ancient world in the work of Aristotle, Euclid, and Ptolemy, and which I described in Addendum on the Agrarian-Ecclesiastical Thesis. Russell’s conception science is that of industrial-technological civilization, jointly emergent from the scientific revolution, the political revolutions of the eighteenth century, and the industrial revolution. With the overthrow of scholasticism as the basis of university curricula (which took hundreds of years following the scientific revolution before the process was complete), a new paradigm of science was to emerge and take shape. It was in this context that Husserl and Russell, Einstein and Gödel, pursued their research, employing a mixture of established traditional ideas and radically new ideas.

In a thorough re-reading of Husserl we could treat his conception of science as an exercise to be updated as we went along, substituting an hypothetico-deductive formulation for each and every one of Husserl’s categorical formulations, ultimately converging upon a scientific conception of knowledge more in accord with contemporary conceptions of scientific knowledge. At the end of this exercise, Husserl’s observation about the different between science and profundity would still be intact, and would still be a valuable guide to the transformation of a profound chaos into a pellucid cosmos.

This ideal, and ever more so the realization of this ideal, ultimately may not prove to be possible. Husserl himself in his later writings famously said, “Philosophy as science, as serious, rigorous, indeed apodictically rigorous, science — the dream is over.”(It is interesting to compare this metaphor of a dream to Kant’s claim that he was awoken from his dogmatic slumbers by Hume.) The impulse to science returns, eventually, even if the idea of an apodictically rigorous science has come to seem a mere dream. And once the impulse to science returns, the impulse to make that science rigorous will reassert itself in time. Our rational nature asserts itself in and through this impulse, which is complementary to, rather than contradictory of, our animal nature. To pursue a rigorous science of civilization is ultimately as human as the satisfaction of any other impulse characteristic of our species.

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Axes of Historiography

3 November 2012

Saturday


axes of historiography

How do we orient ourselves within historiography? This may sound like an odd question; I will try to make it sound like a sensible question, and a question with relevance extending far beyond the bounds of historiography narrowly construed.

One way to orient oneself within historiography is to accept and elaborate upon a familiar schema of historical periodization. There are many from which to choose. For example, if one divides Western history into ancient, medieval and modern periods, and then goes on to describe the character of medieval civilization, this constitutes a kind of orientation within historiography. Others working on the medieval period will recognize your approach based on a received conception of periodization and will critique the effort accordingly.

While I often write about problematic issues in historical periodization, I am going to consider a very different orientation within historiography today, and this might be considered to be a methodological orientation, based on how one assesses and organizes the objects of historical knowledge.

A familiar distinction within historiography is that between the synchonic and the diachronic. I have written about this distinction in Synchronic and Diachronic Approaches to Civilization and Synchronic and Diachronic Geopolitical Theories. “Synchrony” and “diachrony” sound like forbidding technical terms, but the concepts they attempt to capture are not at all difficult. Synchrony is the present construed broadly enough to admit of short term historical interaction, while diachrony typically takes a narrower view but a longer span of time. Sometimes this is expressed by saying that synchrony is across time while diachrony is through time.

Another distinction often made is that between the nomothetic and the ideographic. Again, these are intimidating technical terms, but the ideas are simple. Nomothetic (which comes from the Greek “nomos” for “law” or “norm”) approaches are concerned with law-like transitions in time: cause and effect. For example, you intentionally touch a stove not knowing that it is hot, you burn your finger, you withdraw your hand and give a shout of pain. Ideographic approaches do not quite constitute the negation of cause and effect, but they focus on all that is merely contigent, accidental, and unpredictable in life. For example, while looking at some distraction out of the corner of your eye, you trip, and in seeking to catch your fall you touch a hot stove and burn your finger.

When we put together these two historiographical distinctions — synchronic and diachronic, nomothetic and ideographic — we get four possible permutations of historiographical methodology, as follows:

● nomothetic synchrony

Law-like interaction of all elements within a broadly-defined present

● ideographic synchrony

Contingent interactions of all elements within a broadly-defined present

● nomothetic diachrony

Law-like succession of related events through historical time (especially “deep time”)

● ideographic diachrony

Contingent succession of related events through historical time

This schematic representation of historiographical methodologies is in no wise intended to be exhaustive; I’m sure if I continued to think about this, all kinds of conditions, qualifications, and additions would occur to me. For example, one obvious way to give this much more subtlety and sophistication would be to define each of the above methodological orientations for each division of what I have called ecological temporality, i.e., define each method for each level of time, from the micro-temporality of lived experience to the meta-temporality of the unfolding of ideas in history. I’m not going to attempt to do this at present, I just wanted to give a sense of the simplified schematism I am employing here, which I hope has some relevance despite its simplicity.

All of this sounds very abstract, but if just the right intuitive illustrations of each concept can be found, the concepts will gain in concreteness and depth, and their usefulness will be immediately understood. I can’t claim that I have yet assembled the perfect intuitive illustrations for all four of these methodologies, but I will give you what I have at present, and as I continue to think about this I will (hopefully) add some telling examples.

Nomothetic synchrony, as a method of highlighting the law-like interaction of all elements within a broadly-defined present, is perhaps the most difficult to intuitively illustrate. What “the present” includes is ambiguous, but I have said that the present is “broadly-defined,” so you will understand that the present is not here the punctiform present but something more like “current events.” Current events are continually feeding back on themselves by being repeated in the media and iterated throughout numerous cultural channels. Not all of this feedback, and not all of these iterations, are law-like, but some are. For example, procedural rationality — laws, rules, and regulations intended to bring order and system to the ordinary business of life — constitutes a highly complex set of law-like interactions in the present. In natural history, in contradistinction to human history, ecology is, in a sense, an instance of nomothetic synchrony, and that genre of writing/study once called “nature studies” which focuses on life cycles and predictable patterns within a defined and limited ecosystem, habitat, or niche. Anything, then, that we can describe in ecological terms can also be described in terms of nomothetic synchrony, and since I have taken the trouble to define metaphysical ecology, this category is potentially highly comprehensive. For example, if we call sociology the ecology of society, or we call cosmology galactic ecology, these disciplines could both be treated in terms of nomothetic synchrony.

Ideographic synchrony as constituted by all contingent interactions within a broadly-defined present might be summed up as William James famously summarized sensory perception for an infant: “The baby, assailed by eyes, ears, nose, skin, and entrails at once, feels it all as one great blooming, buzzing, confusion.” Ideographic synchrony is a blooming, buzzing confusion. Anarchic processes like financial markets and warfare might be good illustrations of ideographic synchrony. Of course, markets are supposed to behave according to procedural rationality, and wars are supposed to be fought according to a strategy — but we have all heard of the “fog of war” and of battlefield “friction” (both concepts due to Clausewitz), as we have all heard that no plan survives contact with the enemy. Similarly, no trading strategy survives exposure to the market.

Nomothetic diachrony, the law-like succession of related events through historical time, is the paradigmatic form of historical thought, but more often than not an elusive ideal. Many “laws of history” have been proposed, but none have been widely accepted. The only law of history that has survived is not from history, but from biology: natural selection. Evolution, while often apparently random and pervasively contingent, is a perfect illustration of law-like transitions through deep time. The “big history” movement is also a paradigm case of nomothetic diachrony, with the central theoretical narrative being that of increasing complexity.

Ideographic diachrony, the contingent succession of related events through historical time, can be illustrated in several imaginative ways. The biography of an individual primarily consists of a tight focus on a contingent sequence of events (events in the life of one individual) through a period of time not limited to the broadly-defined present. Many writers like to dwell on the role of the merely contingent and even the spectacularly accidental in history, as with Pascal’s several remarks about how if Cleopatra’s nose had had another shape, history would be different — a particular theme that has been since taken up by others (as in Daniel J. Boorstin’s book, Cleopatra’s Nose: Essays on the Unexpected). There is also the famous rhyme about how “for want of a nail a kingdom fell” which also focuses on the disproportionate historical influence of accidental contingencies. The “butterfly effect” is another illustration.

These four concepts — nomothetic synchrony, ideographic synchrony, nomothetic diachrony, and ideographic diachrony — provide a kind of methodological orientation in historiography. But it is more than merely methodological, since particular methods imply particular metaphysical orientations as well. Someone who holds the cataclysmic conception of history — based upon a denial of human agency — is likely to pursue an ideographic methodology rather than a nomothetic methodology. However, the four conceptions of history that I have defined don’t neatly map on the four methodologies defined above, so I can’t just connect these two quadripartite schemas straight across, showing that each conception of history has an associated methodology.

It’s more complicated than that. It usually is with history.

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For more on the axes of historiography see Ecological Temporality and the Axes of Historiography.

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Tuesday


Aristotle as portrayed by Raphael

Aristotle claimed that mathematics has no ethos (Metaphysics, Book III, Chap. 2, 996a). Aristotle, of course, was more interested in the empirical sciences than his master Plato, whose Academy presumed and demanded familiarity with geometry — and we must understand that for the ancients, long before the emergence of analytical geometry in the work of Descartes (allowing us to formulate geometry algebraically, hence arithmetically), that geometry was always axiomatic thought, rigorously conceived in terms of demonstration. For the Greeks, this was the model and exemplar of all rigorous thought, and for Aristotle this was a mode of thought that lacked an ethos.

Euclid provided the model of formal thought with his axiomatization of geometry. Legend has it that there was a sign over the door of Plato's Academy stating, 'Let no one enter here who has not studied geometry.'

In this, I think, Aristotle was wrong, and I think that Plato would have agree on this point. But the intuition behind Aristotle’s denial of a mathematical ethos is, I think, a common one. And indeed it has even become a rhetorical trope to appeal to rigorous mathematics as an objective standard free from axiological accretions.

In his famous story within a story about the Grand Inquisitor, Dostoyevsky has the Grand Inquisitor explain how “miracles, mystery, and authority” are used to addle the wits of others.

In his famous story within a story about the Grand Inquisitor, Dostoyevsky has the Grand Inquisitor explain how “miracles, mystery, and authority” are used to addle the wits of others.

Our human, all-too-human faculties conspire to confuse us, to addle our wits, when we begin talking about morality, so that the purity and rigor of mathematical and logical thought seem to be called into question if we acknowledge that there is an ethos of formal thought. We easily confuse ourselves with religious, mystical, and ethical ideas, and since the great monument of mathematical thought has been mostly free of this particular species of confusion, to deny an ethos of formal thought can be understood as a strategy to protect and defend of the honor of mathematics and logic by preserving it from the morass that envelops most human attempts to think clearly, however heroically undertaken.

Kant famously said that he had to limit knowledge to make room for faith.

Kant famously stated in the Critique of Pure Reason that, “I have found it necessary to deny knowledge in order to make room for faith.” I should rather limit faith to make room for rigorous reasoning. Indeed, I would squeeze out faith altogether, and find myself among the most rigorous of the intuitionists, one of whom has said: “The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof…”

Western asceticism can be portrayed as demonic torment or as divine illumination; the same diversity of interpretation can be given to ascetic forms of reason.

Though here again, with intuitionism (and various species of constructivism generally), we have rigor, denial, asceticism — intuitionistic logic is no joyful wisdom. (An ethos of formal thought need not be an inspiring and edifying ethos.) It is logic with a frown, disapproving, censorious — a bitter medicine justified only because it offers hope of curing the disease of contradiction, contracted when mathematics was shown to be reducible to set theory, and the latter shown to be infected with paradox (as if the infinite hubris of set theory were not alone enough for its condemnation). Is the intuitionist’s hope justified? In so far as it is hope — i.e., hope and not proof, the expectation that things will go better for the intuitionistic program than for logicism — it is not justified.

Dummett has said that intuitionistic logic and mathematics are to wear their justification on their face:

“From an intuitionistic standpoint, mathematics, when correctly carried on, would not need any justification from without, a buttress from the side or a foundation from below: it would wear its own justification on its face.”

Dummett, Michael, Elements of Intuitionism, Oxford University Press, 1977, p. 2

The hope that contradiction will not arise from intuitionistic methods clearly is no such evident justification. As a matter of fact, empirically and historically verifiable, we know that intuitionism has resulted in no contradictions, but this could change tomorrow. Intuitionism stands in need of a consistency proof even more than formalism. There is, in its approach, a faith invested in the assumption that infinite totalities caused the paradoxes, and once we have disallowed reference to them all will go well. This is a perfectly reasonable assumption, but one, in so far as it is an article of faith, which is at variance with the aims and methods of intuitionism.

And what is a feasible proof, which our ultra-intuitionist would allow? Have we not with “feasible proof” abandoned proof altogether in favor of probability? Again, we will allow them their inconsistencies and meet them on their own ground. But we shall note that the critics of the logicist paradigm fix their gaze only upon consistency, and in so doing reveal again their stingy, miserly conception of the whole enterprise.

“The Ultra-Intuitionistic Criticism and the Antitraditional program for the foundations of Mathematics” by A. S. Yessenin-Volpin (who was arguing for intellectual freedom in the Soviet Union at the same time that he was arguing for a censorious conception of reason), in Intuitionism and Proof Theory, quoted briefly above, is worth quoting more fully:

The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof, in spite of the traditional incompleteness theorem, which deals only with a very narrow kinds [sic] of proofs (which I call ‘formal proofs’). I define proof as any fair way of making a sentence incontestable. Of course this explication is related to ethics — the notion fair means ‘free from any coercion or fraud’ — and to the theory of disputes, indicating the cases in which a sentence is to be considered as incontestable. Of course the methods of traditional mathematical logic are not sufficient for this program: and I have to enlarge the domain of means explicitly studied in logic. I shall work in a domain wherein are to be found only special notions of proof satisfying the mentioned explication. In this domain I shall allow as a means of proof only the strict following of definitions and other rules or principles of using signs.

Intuitionism and proof theory: Proceedings of the summer conference at Buffalo, N.Y., 1968, p. 3

What is coercion or fraud in argumentation? We find something of an illustration of this in Gregory Vlastos’ portrait of Socrates: “Plato’s Socrates is not persuasive at all. He wins every argument, but never manages to win over an opponent. He has to fight every inch of the way for any assent he gets, and gets it, so to speak, at the point of a dagger.” (The Philosophy of Socrates, Ed. by Gregory Vlastos, page 2)

According to Gregory Vlastos, Socrates used the kind of 'coercive' argumentation that the intuitionists abhor.

What appeal to logic does not invoke logical compulsion? Is logical compulsion unique to non-constructive mathematical thought? Is there not an element of logical compulsion present also in constructivism? Might it not indeed be the more coercive form of compulsion that is recognized alike by constructivists and non-constructivists?

The breadth of the conception outlined by Yessenin-Volpin is impressive, but the essay goes on to stipulate the harshest measures of finitude and constructivism. One can imagine these Goldwaterite logicians proclaiming: “Extremism in the defense of intuition is no vice, and moderation in the pursuit of constructivist rigor is no virtue.” Brouwer, the spiritual father of intuitionism, even appeals to the Law-and-Order mentality, saying that a criminal who has not been caught is still a criminal. Logic and mathematics, it seems, must be brought into line. They verge on criminality, deviancy, perversion.

Quine was no intuitionist by a long shot, but as a logician he brought a quasi-disciplinary attitude to reason and adopted a tone of disapproval not unlike Brouwer.

The same righteous, narrow, anathamatizing attitude is at work among the defenders of what is sometimes called the “first-order thesis” in logic. Quine sees a similar deviancy in modal logic (which can be shown to be equivalent to intuitionistic logic), which he says was “conceived in sin” — the sin of confusing use and mention. These accusations do little to help us understand logic. We would do well to adopt Foucault’s attitude on these matters: “leave it to our bureaucrats and our police to see that our papers are in order. At least spare us their morality when we write.” (The Archaeology of Knowledge, p. 17)

Foucault had little patience for the kind of philosophical reason that seemed to be asking if our papers are in order, a function he thought best left to the police.

The philosophical legacy of intuitionism has been profound yet mixed; its influence has been deeply ambiguous. (Far from the intuitive certainty, immediacy, clarity, and evident justification that it would like to propagate.) There is in inuitionism much in harmony with contemporary philosophy of mathematics and its emphasis on practices, the demand for finite constructivity, its anti-philosophical tenor, its opposition to platonism. The Father of Intuitionism, Brouwer, was, like many philosophers, anti-philosophical even while propounding a philosophy. No doubt his quasi-Kantianism put his conscience at rest in the Kantian tradition of decrying metaphysics while practicing it, and his mysticism gave a comforting halo (which softens and obscures the hard edges of intuitionist rigor in proof theory) to mathematics which some have found in the excesses of platonism.

L. E. J. Brouwer: philosopher of mathematics, mystic, and pessimistic social theorist

In any case, few followers of Brouwer followed him in his Kantianism and mysticism. The constructivist tradition which grew from intuitionism has proved to be philosophically rich, begetting a variety of constructive techniques and as many justifications for them. Even if few mathematicians actually do intuitionistic mathematics, controversies over the significance of constructivism have a great deal of currency in philosophy. And Dummett is explicit about the place of philosophy in intuitionistic logic and mathematics.

The light of reason serves as an inspiration to us as it shines down from above, and it remains an inspiration even when we are not equal to all that it might ideally demand of us.

Intuitionism and constructivism command our respect in the same way that Euclidean geometry commanded the respect of the ancients: we might not demand that all reasoning conform to this model, but it is valuable to know that rigorous standards can be formulated, as an ideal to which we might aspire if nothing else. And and ideal of reason is itself an ethos of reason, a norm to which formal thought aspires, and which it hopes to approximate even if it cannot always live up the the most exacting standard that it can recognize for itself.

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Studies in Formalism

1. The Ethos of Formal Thought

2. Epistemic Hubris

3. Parsimonious Formulations

4. Foucault’s Formalism

5. Cartesian Formalism

6. Doing Justice to Our Intuitions: A 10 Step Method

7. The Church-Turing Thesis and the Asymmetry of Intuition

8. Unpacking an Einstein Aphorism

9. Methodological and Ontological Parsimony (in preparation)

10. The Spirit of Formalism (in preparation)

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Grand Strategy Annex

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Tuesday


Luigi Nono

For the six months or so that I have been posting to this forum I have been quite preoccupied with intensely practical questions in history, economics, politics, diplomacy and how these spheres of activity are related in a substantive way to human nature.

When Heilbronner wrote his famous book about economists he called them the “the worldly philosophers,” which invites an implicit comparison to unworldly or otherworldly philosophers who do not concern themselves with the ordinary business of life. Be that as it may, it is possible, I think, to be both — thinking both the worldly and the unworldly by turns. Thus is the thinker likely to experience a dialectic not only within the realms of thought, but also as part of the ordinary business of his life.

I find myself today thrown back onto the most abstruse and obscure points of technical philosophy in my attempt to clarify my understanding of very worldly concepts that attempt to elucidate what Marshall called “the ordinary business of life.” I find that I am once again taking down my reference works on ontology, epistemology, and philosophy of logic from my bookshelves, and this, I think, is a good thing. The cross-fertilization of thought, whether inter-disciplinary or intra-disciplinary, is usually a source of fruitful meditation. In particular, I find myself working on the idea of constructivism.

Constructivism means many different things to many different persons. It would almost seem that a sense of “constructivism” has been defined for every conceivable special field of inquiry or endeavor. There is constructivism in the visual arts, and a constructivism in music, and a constructivism in sociology, and, what most concerns me, a constructivism in the philosophy of logic and mathematics.

Dr. David C. F. Wright quoted his friend British composer Reginald Smith Brindle regarding a visit to Luigi Nono:

I went there mostly while he was composing Il Canto Sospeso, a politically orientated work of choral-orchestral character which involved the most abstruse constructivism I have ever come across. Mathematics governed every detail of the composition … the pitch of the notes, their duration, volume and sound character. In his study, there was a wall entirely covered with successions of numbers, notes and performance details and from this he extracted all the details of the composition. It seemed to me that all his intense constructivism was a certain formula for the creation of non-music, yet from recordings of his music, I got the impression of a highly sensitive artistry.

What Brindle describes is more commonly known as integral serialism or total serialism. The relation between constructivism and serialism is an interesting question in itself, but one that I will not address here. And while I don’t have a CD of Il Canto Sospeso, I did have a recording by the Arditti Quartet of Nono’s fragmente – stille, an idiotma, so I put this on as my theme music for constructivism.

I regard the philosophy of mathematics as the ultimate proving ground for all philosophical theories. One finds philosophical theories applied to the philosophy of mathematics in their purest form, and it is in their purest form that theories are seen in their nakedness, revealed to all the world for what they are. This is especially true for constructivism, but while constructivism is best tested by the austere ontology of logic and mathematics, it has universal implications.

Constructivism is a methodological concept, and the distinction between constructive methods and non-constructive methods recapitulates the ancient division between idealism and realism in ontology. One could say that constructivism is idealism put into practice as a method. What, then, is the method of idealism?

At present I am only trying to get clear about the concept of constructivism, its proper scope as a concept. I sent off an e-mail to the phil-logic discussion listserv and got some replies both on-list and off-list that provided some initial stimulation. It is, however, extraordinarily difficult to develop a sympathetic discussion on an e-mail listserv. Even when others are the list are interested in the idea, the tone of discussion can be brutal at times. There is a value in brutal honesty and openness of discussion, but there is also a value in having someone with whole one can share inchoate ideas and help to bring out what is valuable in them without destroying a fragile thought. However, I have no one to act as my intellectual second (i.e., kaishakunin, 介錯人) and thus I pour it out here instead.

It takes a true friend to perform the office of kaishakunin.

It takes a true friend to perform the office of kaishakunin.

I found an interesting discussion of constructivism in Detlefsen’s contribution to the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, in which Detlefsen does try to formulate the theoretical unity of constructivism, although he never touches on predicativism. Should predicativism be considered something utterly different? There is also a great discussion of constructivism by Michael Hallett in the Handbook of Metaphysics and Ontology edited by Hans Burkhardt and Barry Smith, published by Philosophia Verlag (Hallett’s article is “mathematical objects”). While these two discussions are a great starting point, they don’t get to the essence of the question that is troubling me at the moment.

detlefsen

The many varieties of constructivism are different not only in detail but also importantly different in conceptual scope. Intuitionism, finitism, predicativism, and other conceptions that might generally be called constructivistic in tendency all restrict classical formal reasoning, but there does not seem to be any prima facie unity in virtue of which all deserve to be called constructivist. One of my off-list responses from the phil-logic listserv suggested that there would be “push back” at any attempt to classify intuitionism as a form of constructivism.

handbook of metaphysics and ontology

James Robert Brown’s The Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures has a section on “Constructivist Approaches” that quotes from Errett Bishop’s Foundations of Constructive Analysis. I don’t have a copy of Bishop’s book, so this is helpful. Bishop, at least, explicitly identifies his approach as constructivist, unlike Brouwer or Heyting, Poincare or Weyl, Yesnin-Volpin and Gauthier, Kielkopf and Wittgenstein. This self-ascribed constructivist identity carries more weight than all the other uses of “constructivist” combined.

James Robert Brown

Perhaps constructivism in its pure form should be defined more narrowly, strictly in terms of the avoidance of pure existence proofs, for example. But if we define constructivism more narrowly, then it would seem that there is still a need for a concept under which would fall all those theories of formal reason that restrict what Torkel Franzen called “classical eclectism,” and which would include a narrowly defined constructivism as well as other doctrines previously called constructivist. What concept could we use to cover all instances of principled restrictions upon formal reasoning, and is there any unity of motive in formulating and propounding principled limitations on formal reasoning?

The obvious course of action would be to elucidate the principles embodied in all such doctrines, loosely called “constructivist” up until now, and seek to systematically interrelate them. In every police drama one sees on television, the detectives on a difficult case assemble a large bulletin board upon which they display symbols for clues, and then map the interrelations between clues in an attempt to find a pattern that will solve the case. We need the conceptual equivalent of this in order to understand constructivism.

Two other obvious courses of action present themselves: simultaneously driving down into the foundations of constructivist doctrines while also extrapolating their consequences to the utmost limit. A convergence or divergence of either development would point to fundamental commonality or fundamental incommensurability.

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