Saturday


The periodic table, color-coded by the source of the element in the solar system. By Jennifer Johnson.

Many years ago, reading a source I cannot now recall (and for which I searched unsuccessfully when I started writing this post), I came upon a passage that has stayed with me. The author was making the argument that no sciences were consistent except those that had been reduced to mere catalogs of facts, like geography and anatomy. I can’t recall the larger context in which this argument appeared, but the observation that sciences might only become fully consistent when they have matured to the point of being exhaustive but static and uninteresting catalogs of facts, implying that the field of research itself had been utterly exhausted, was something I remembered. This idea presents in miniature a developmental conception of the sciences, but I think that it is a developmental conception that is incomplete.

Thinking of this idea of an exhausted field of research, I am reminded of a discussion in Conversations on Mind, Matter, and Mathematics by Jean-Pierre Changeux and Alain Connes, in which mathematician Alain Connes distinguished between fully explored and as yet unexplored parts of mathematics:

“…the list of finite fields is relatively easy to grasp, and it’s a simple matter to prove that the list is complete. It is part of an almost completely explored mathematical reality, where few problems remain. Cultural and social circumstances clearly serve to indicate which directions need to be pursued on the fringe of current research — the conquest of the North Pole, to return again to my comparison, surely obeyed the same type of cultural and social motivations, at least for a certain time. But once exploration is finished, these cultural and social phenomena fade away, and all that’s left is a perfectly stable corpus, perfectly fitted to mathematical reality…”

Jean-Pierre Changeux and Alain Connes, Conversations on Mind, Matter, and Mathematics, Princeton: Princeton University Press, 1995, pp. 33-34

To illustrate a developmental conception of mathematics and the formal sciences would introduce additional complexities that follow from the not-yet-fully-understood relationship between the formal sciences and the empirical sciences, so I am going to focus on developmental conceptions of the empirical sciences, but I hope to return to the formal sciences in this connection.

The idea of the development of science as a two-stage process, with discovery followed by a consistent and exhaustive catalog, implies both that most sciences (and, if we decompose the individual special sciences into subdivisions, parts of most or all sciences) remain in the discovery phase, and that once the discovery phase has passed and we are in possession of an exhaustive and complete catalog of the facts discovered by a science, there is nothing more to be done in a given science. However, I can think of several historical examples in which a science seemed to be converging on a complete catalog, but this development was disrupted (one might say) by conceptual change within the field that forced the reorganization of the materials in a new way. My examples will not be perfect, and some additional scientific discovery always seems to have been involved, but I think that these examples will be at least suggestive.

Prior to the great discoveries of cosmology in the early twentieth century, after which astronomy became indissolubly connected to astrophysics, astronomy seemed to be converging slowly upon an exhaustive catalog of all stars, with the limitation on the research being simply the resolving power of the telescopes employed to view the stars. One could imagine a counterfactual world in which technological innovations in instrumentation supplied nothing more than new telescopes able to resolve more stars, and that the task of astronomy was merely to supply an exhaustive catalog of stars, listing their position in the sky, intrinsic brightness, and a few other facts about the points of light in the sky. But the cataloging of stars itself contributed to the revolution that would follow, particularly when the period-luminosity relationship in Cepheid variable stars was discovered by Henrietta Swan Leavitt (discovered in 1908 and published in 1912). The period-luminosity relationship provided a “standard candle” for astronomy, and this standard candle began the process of constructing the cosmological distance ladder, which in turn made it possible to identify Cepheid variables in the Andromeda galaxy and thus to prove that the Andromeda galaxy was two million light years away and not contained within the Milky Way.

Once astronomy became scientifically coupled to astrophysics, and the resources of physics (both relativistic and quantum) could be brought to bear upon understanding stars, a whole new cosmos opened up. Stars, galaxies, and the universe entire were transformed from something static that might be exhaustively cataloged, to a dynamic and changing reality with a natural history as well as a future. Astronomy went from being something that we might call a Platonic science, or even a Linnaean science, to being an historical science, like geology (after Hutton and Lyell), biology (after Darwin and Wallace), and Paleontology. This coupling of the study of the stars with the study of the matter that makes up the stars has since moved in both directions, with physics driving cosmology and cosmology driving physics. One result of this interaction between astronomy and physics is the illustration above (by Jennifer Johnson) of the periodic table of elements, which prominently exhibits the origins of the elements in cosmological processes. The periodic table once seemed, like a catalog of stars, to be something static to be memorized, and divorced from natural history. This conceptualization of matter in terms of its origins puts the periodic table in a dramatically different light.

As the cosmos was once conceived in Platonic terms as fixed and eternal, to be delineated in a Linnaean science of taxonomical classification, so too the Earth was conceived in Platonic terms as fixed and eternal, to be similarly delineated in a Linnaean science of classification. The first major disruption of this conception came with geology since Hutton and Lyell, followed by plate tectonics and geomorphology in the twentieth century. Now this process has been pushed further by the idea of mineral evolution. I have been listening through for the second time to Robert Hazen’s lectures The Origin and Evolution of Earth: From the Big Bang to the Future of Human Existence, which exposition closely follow the content of his book, The Story of Earth: The First 4.5 Billion Years, from Stardust to Living Planet, in which Hazen wrote:

“The ancient discipline of mineralogy, though absolutely central to everything we know about Earth and its storied past, has been curiously static and detached from the conceptual vagaries of time. For more than two hundred years, measurements of chemical composition, density, hardness, optical properties, and crystal structure have been the meat and potatoes of the mineralogist’s livelihood. Visit any natural history museum, and you’ll see what I mean: gorgeous crustal specimens arrayed in case after glass-fronted case, with labels showing name, chemical formula, crystal system, and locality. These most treasured fragments of Earth are rich in historical context, but you will likely search in vain for any clue as to their birth ages or subsequent geological transformations. The old way all but divorces minerals from their compelling life stories.”

Robert M. Hazen, The Story of Earth: The First 4.5 Billion Years, from Stardust to Living Planet, Viking Penguin, 2012, Introduction

This illustrates, from the perspective of mineralogy, much of what I said above in relation to star charts and catalogs: mineralogy was once about cataloging minerals, and this may have been a finite undertaking once all minerals had been isolated, identified, and cataloged. Now, however, we can understand mineralogy in the context of cosmological history, and this is as revolutionary for our understanding of Earth as the periodic table understood in terms of cosmological history. It could be argued, in addition, that compiling the “particle zoo” of contemporary particle physics is also a task of cataloging the entities studied by physics, but the cataloging of particles has been attended throughout with a theory of how these particles are generated and how they fit into the larger cosmological story — what Aristotle would have called their coming to be and passing away.

The best contemporary example of a science still in its initial phases of discovery and cataloging is the relatively recent confirmation of exoplanets. On my Tumblr blog I recently posted On the Likely Existence of “Random” Planetary Systems, which tried to place our current Golden Age of Exoplanet Discovery in the context of a developing science. We find the planetary systems that we do in fact find partly as a consequence of observation selection effects, and it belongs to the later stages of the development of a science to attempt to correct for observation selection effects built into the original methods of discovery employed. The planetary science that is emerging from exoplanet discoveries, however, and like contemporary particle physics, is attended by theories of planet formation that take into account cosmological history. However, the discovery phase, in terms of exoplanets, is still underway and still very new, and we have a lot to learn. Moreover, once we learn more about the possibilities of planets in our universe, hopefully also we will learn about the varied possibilities of planetary biospheres, and given the continual interaction between biosphere, lithosphere, atmosphere, and hydrosphere, which is a central motif of Hazen’s mineral evolution, we will be able to place planets and their biospheres into a large cosmological context (perhaps even reconstructing biosphere evolution). But first we must discover them, and then we must catalog them.

These observations, I think, have consequences not only for our understanding of the universe in which we find ourselves, but also for our understanding of science. Perhaps, instead of a two-stage process of discovery and taxonomy, science involves a three-stage process of discovery, taxonomy, and natural history, in which latter the objects and facts cataloged by one of the special sciences (earlier in their development) can take their place within cosmological history. If this is the case, then big history is the master category not only of history, but also of science, as big history is the ultimate framework for all knowledge that bears the lowly stamp of its origins. This conception of the task of science, once beyond the initial stages of discovery and classification, to integrate that which was discovered and classified into the framework of big history, suggests a concrete method by which to “cash out” in a meaningful way Wilfrid Sellars’ contention that, “…the specialist must have a sense of how not only his subject matter, but also the methods and principles of his thinking about it, fit into the intellectual landscape.” (cf. Philosophy and the Scientific Image of Man) Big history is the intellectual landscape in which the sciences are located.

A developmental conception of science that recognized stages in the development of science beyond classification, taxonomy, and an exhaustive catalog (which is, in effect, the tombstone of what was a living and growing science), has consequences for the practice of science. Discovery may well be the paradigmatic form of scientific activity, but it is not the only form of scientific activity. The painstakingly detailed and disciplined work of cataloging stars or minerals is the kind of challenge that attracts a certain kind of mind with a particular interest, and the kind of individual who is attracted to this task of systematically cataloging entities and facts is distinct from the kind of individual who might be most attracted by scientific discovery, and also distinct from the kind of individual who might be attracted to fitting the discoveries of a special science into the overall story of the universe and its natural history. There may need to be a division of labor within the sciences, and this may entail an educational difference. Dividing sciences by discipline (and, now, by university departments), which involves inter-generational conflicts among sciences and the paradigm shifts that sometimes emerge as a result of these conflicts, may ultimately make less sense than dividing sciences according their stage of development. Perhaps universities, instead of having departments of chemistry, geology, and botany, should have departments of discovery, taxonomy, and epistemic integration.

Speaking from personal experience, I know that (long ago) when I was in school, I absolutely hated the cataloging approach to the sciences, and I was bored to tears by memorizing facts about minerals or stars. But the developmental science of evolution so intrigued me that I read extensively about evolution and anthropology outside and well beyond the school curriculum. If mineral evolution and the Earth sciences in their contemporary form had been known then, I might have had more of an interest in them.

What are the sciences developing into, or what are the sciences becoming? What is the end and aim of science? I previously touched on this question, a bit obliquely, in What is, or what ought to be, the relationship between science and society? though this line of inquiry is more like a thought experiment. It may be too early in the history of the sciences to say what they are becoming or what they will become. Perhaps an emergent complexity will arise out of knowledge itself, something that I first suggested in Scientific Historiography: Past, Present, and Future, in which I wrote in the final paragraph:

We cannot simply assume an unproblematic diachronic extrapolation of scientific knowledge — or, for that matter, historical knowledge — especially as big history places such great emphasis upon emergent complexity. The linear extrapolation of science eventually may trigger a qualitative change in knowledge. In other words, what will be the emergent form of scientific knowledge (the ninth threshold, perhaps?) and how will it shape our conception of scientific historiography as embodied in big history, not to mention the consequences for civilization itself? We may yet see a scientific historiography as different from big history as big history is different from Augustine’s City of God.

It is only a lack of imagination that would limit science to the three stages of development I have outlined above. There may be developments in science beyond those we can currently understand. Perhaps the qualitative emergent from the quantitative expansion of scientific knowledge will be a change in science itself — possibly a fourth stage in the development of science — that will open up to scientific knowledge aspects of experience and regions of nature currently inaccessible to science.

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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


Studies in Formalism:

The Synoptic Perspective in Formal Thought


In my previous two posts on the overview effect — The Epistemic Overview Effect and The Overview Effect as Perspective Taking — I discussed how we can take insights gained from the “overview effect” — what astronauts and cosmonauts have experienced as a result of seeing our planet whole — and apply them to over areas of human experience and knowledge. Here I would like to try to apply these insights to formal thought.

The overview effect is, above all, a visceral experience, something that the individual feels as much as they experience, and you may wonder how I could possibly find a connection between a visceral experience and formal thinking. Part of the problem here is simply the impression that formal thought is distant from human concerns, that it is cold, impersonal, unfeeling, and, in a sense, inhuman. Yet for logicians and mathematicians (and now, increasingly, also for computer scientists) formal thought is a passionate, living, and intimate engagement with the world. Truly enough, this is not an engagement with the concrete artifacts of the world, which are all essentially accidents due to historical contingency, but rather an engagement with the principles implicit in all things. Aristotle, ironically, formalized the idea of formal thought being bereft of human feeling when he asserted that mathematics has no ethos. I don’t agree, and I have discussed this Aristotelian perspective in The Ethos of Formal Thought.

And yet. Although Aristotle, as the father of logic, had more to do with the origins of formal thought than any other human being who has ever lived, the Aristotelian denial of an ethos to formal thought does not do justice to our intuitive and even visceral engagement with formal ideas. To get a sense of this visceral and intuitive engagement with the formal, let us consider G. H. Hardy.

Late in his career, the great mathematician G. H. Hardy struggled to characterize what he called mathematically significant ideas, which is to say, what makes an idea significant in formal thought. Hardy insisted that “real” mathematics, which he distinguished from “trivial” mathematics, and which presumably engages with mathematically significant ideas, involves:

“…a very high degree of unexpectedness, combined with inevitability and economy.”

G. H. Hardy, A Mathematician’s Apology, section 15

Hardy’s appeal to parsimony is unsurprising, yet the striking contrast of the unexpected and the inevitable is almost paradoxical. One is not surprised to hear an exposition of mathematics in deterministic terms, which is what inevitability is, but if mathematics is the working out of rigid formal rules of procedure (i.e., a mechanistic procedure), how could any part of it be unexpected? And yet it is. Moreover, as Hardy suggested, “deep” mathematical ideas (which we will explore below) are unexpected even when they appear inevitable and economical.

It would not be going too far to suggest that Hardy was trying his best to characterize mathematical beauty, or elegance, which is something that is uppermost in the mind of the pure mathematician. Well, uppermost at least in the minds of some pure mathematicians; Gödel, who was as pure a formal thinker as ever lived, said that “…after all, what interests the mathematician, in addition to drawing consequences from these assumptions, is what can be carried out” (Collected Works Volume III, Unpublished essays and lectures, Oxford, 1995, p. 377), which is an essentially pragmatic point of view, in which formal elegance would seem to play little part. Mathematical elegance has never been given a satisfactory formulation, and it is an irony of intellectual history that the most formal of disciplines relies crucially on an informal intuition of formal elegance. Beauty, it is often said, in the mind of the beholder. Is this true also for mathematical beauty? Yes and no.

If a mathematically significant idea is inevitable, we should be able to anticipate it; if unexpected, it ought to elude all inevitability, since the inevitable ought to be predictable. One way to try to capture the ineffable sense of mathematical elegance is through paradox — here, the paradox of the inevitable and the unexpected — in way not unlike the attempt to seek enlightenment through the contemplation of Zen koans. But Hardy was no mystic, so he persisted in his attempted explication of mathematically significant ideas in terms of discursive thought:

“There are two things at any rate which seem essential, a certain generality and a certain depth; but neither quality is easy to define at all precisely.

G. H. Hardy, A Mathematician’s Apology, section 15

Although Hardy repeatedly expressed his dissatisfaction with his formulations of generality and depth, he nevertheless persisted in his attempts to clarify them. Of generality Hardy wrote:

“The idea should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds. The theorem should be one which, even if stated originally (like Pythagoras’s theorem) in a quite special form, is capable of considerable extension and is typical of a whole class of theorems of its kind. The relations revealed by the proof should be such as to connect many different mathematical ideas.” (section 15)

And of mathematical depth Hardy hazarded:

“It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general more difficult) the idea.” (section 17)

This would account for the special difficulty of foundational ideas, of which the most renown example would be the idea of sets, though there are other candidates to be found in other foundational efforts, as in category theory or reverse mathematics.

Hardy’s metaphor of mathematical depth suggests foundations, or a foundational approach to mathematical ideas (an approach which reached its zenith in the early twentieth century in the tripartite struggle over the foundations of mathematics, but is a tradition which has since fallen into disfavor). Depth, however, suggests the antithesis of a synoptic overview, although both the foundational perspective and the overview perspective seek overarching unification, one from the bottom up, the other from the top down. These perspectives — bottom up and top down — are significant, as I have used these motifs elsewhere as an intuitive shorthand for constructive and non-constructive perspectives respectively.

Few mathematicians in Hardy’s time had a principled commitment to constructive methods, and most employed non-constructive methods will little hesitation. Intuitionism was only then getting its start, and the full flowering of constructivistic schools of thought would come later. It could be argued that there is a “constructive” sense to Zermelo’s axiomatization of set theory, but this is of the variety that Godel called “strictly nominalistic construtivism.” Here is Godel’s attempt to draw a distinction between nominalistic constructivism and the sense of constructivism that has since overtaken the nominalistic conception:

…the term “constructivistic” in this paper is used for a strictly nominalistic kind of constructivism, such that that embodied in Russell’s “no class theory.” Its meaning, therefore, if very different from that used in current discussions on the foundations of mathematics, i.e., from both “intuitionistically admissible” and “constructive” in the sense of the Hilbert School. Both these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principle aims of Russell’s constructivism… What, in Russell’s own opinion, can be obtained by his constructivism (which might better be called fictionalism) is the system of finite orders of the ramified hierarchy without the axiom of infinity for individuals…”

Kurt Gödel, Kurt Gödel: Collected Works: Volume II: Publications 1938-1974, Oxford et al.: Oxford University Press, 1990, “Russell’s Mathematical Logic (1944),” footnote, Author’s addition of 1964, expanded in 1972, p. 119

This profound ambiguity in the meaning of “constructivism” is a conceptual opportunity — there is more that lurks in this idea of formal construction than is apparent prima facie. That what Gödel calls a, “strictly nominalistic kind of constructivism” coincides with what we would today call non-constructive thought demonstrates the very different conceptions of what is has meant to mathematicians (and other formal thinkers) to “construct” an object.

Kant, who is often called a proto-constructivist (though I have identified non-constructive elements on Kant’s thought in Kantian Non-Constructivism), does not invoke construction when he discusses formal entities, but instead formulates his thoughts in terms of exhibition. I think that this is an important difference (indeed, I have a long unfinished manuscript devoted to this). What Kant called “exhibition” later philosophers of mathematics came to call “surveyability” (“Übersichtlichkeit“). This latter term is especially due to Wittgenstein; Wittgenstein also uses “perspicuous” (“Übersehbar“). Notice in both of the terms Wittgenstein employs for surveyability — Übersichtlichkeit and Übersehbar — we have “Über,” usually (or often, at least) translated as “over.” Sometimes “Über” is translated as “super” as when Nietzsche’s Übermensch is translated as “superman” (although the term has also been translated as “over-man,” inter alia).

There is a difference between Kantian exhibition and Wittgensteinian surveyability — I don’t mean to conflate the two, or to suggest that Wittgenstein was simply following Kant, which he was not — but for the moment I want to focus on what they have in common, and what they have in common is the attempt to see matters whole, i.e., to take in the object of one’s thought in a single glance. In the actual practice of seeing matters whole it is a bit more complicated, especially since in English we commonly use “see” to mean “understand,” and there are a whole range of visual metaphors for understanding.

The range of possible meanings of “seeing” accounts for a great many of the different formulations of constructivism, which may distinguish between what is actually constructable in fact, that which it is feasible to construct (this use of “feasible” reminds me a bit of “not too large” in set theories based on the “limitation of size” principle, which is a purely conventional limitation), and that which can be constructed in theory, even if not constructable in fact, or if not feasible to construct. What is “surveyable” depends on our conception of what we can see — what might be called the modalities of seeing, or the modalities of surveyability.

There is an interesting paper on surveyability by Edwin Coleman, “The surveyability of long proofs,” (available in Foundations of Science, 14, 1-2, 2009) which I recommend to the reader. I’m not going to discuss the central themes of Coleman’s paper (this would take me too far afield), but I will quote a passage:

“…the problem is with memory: ‘our undertaking’ will only be knowledge if all of it is present before the mind’s eye together, which any reliance on memory prevents. It is certainly true that many long proofs don’t satisfy Descartes-surveyability — nobody can sweep through the calculations in the four color theorem in the requisite way. Nor can anyone do it with either of the proofs of the Enormous Theorem or Fermat’s Last Theorem. In fact most proofs in real mathematics fail this test. If real proofs require this Cartesian gaze, then long proofs are not real proofs.”

Edwin Coleman, “The surveyability of long proofs,” in Foundations of Science, 14 (1-2), 2009

For Coleman, the received conception of surveyability is deceptive, but what I wanted to get across by quoting his paper was the connection to the Cartesian tradition, and to the role of memory in seeing matters whole.

The embodied facts of seeing, when seeing is understood as the biophysical process of perception, was a concern to Bertrand Russell in the construction of a mathematical logic adequate to the deduction of mathematics. In the Introduction to Principia Mathematica Russell wrote:

“The terseness of the symbolism enables a whole proposition to be represented to the eyesight as one whole, or at most in two or three parts divided where the natural breaks, represented in the symbolism, occur. This is a humble property, but is in fact very important in connection with the advantages enumerated under the heading.”

Bertrand Russell and Alfred North Whitehead, Principia Mathematica, Volume I, second edition, Cambridge: Cambridge University Press, 1963, p. 2

…and Russell elaborated…

“The adaptation of the rules of the symbolism to the processes of deduction aids the intuition in regions too abstract for the imagination readily to present to the mind the true relation between the ideas employed. For various collocations of symbols become familiar as representing important collocations of ideas; and in turn the possible relations — according to the rules of the symbolism — between these collocations of symbols become familiar, and these further collocations represent still more complicated relations between the abstract ideas. And thus the mind is finally led to construct trains of reasoning in regions of thought in which the imagination would be entirely unable to sustain itself without symbolic help.”

Loc. cit.

Thinking is difficult, and symbolization allows us to — mechanically — extend thinking into regions where thinking alone, without symbolic aid, would not be capable of penetrating. But that doesn’t mean symbolic thinking is easy. Elsewhere Russell develops another rationalization for symbolization:

“The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self- evident; and it is very hard to see whether one self- evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined.”

Bertrand Russell, Mysticism and Logic, “Mathematics and the Metaphysicians”

Russell formulated the difficulty of thinking even more strongly in a later passage:

“There is a good deal of importance to philosophy in the theory of symbolism, a good deal more than at one time I thought. I think the importance is almost entirely negative, i.e., the importance lies in the fact that unless you are fairly self conscious about symbols, unless you are fairly aware of the relation of the symbol to what it symbolizes, you will find yourself attributing to the thing properties which only belong to the symbol. That, of course, is especially likely in very abstract studies such as philosophical logic, because the subject-matter that you are supposed to be thinking of is so exceedingly difficult and elusive that any person who has ever tried to think about it knows you do not think about it except perhaps once in six months for half a minute. The rest of the time you think about the symbols, because they are tangible, but the thing you are supposed to be thinking about is fearfully difficult and one does not often manage to think about it. The really good philosopher is the one who does once in six months think about it for a minute. Bad philosophers never do.”

Bertrand Russell, Logic and Knowledge: Essays 1901-1950, 1956, “The Philosophy of Logical Atomism,” I. “Facts and Propositions,” p. 185

Alfred North Whitehead, coauthor of Principia Mathematica, made a similar point more colorfully than Russell, which I recently in The Algorithmization of the World:

“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle: they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”

Alfred North Whitehead, An Introduction to Mathematics, London: WILLIAMS & NORGATE, Chap. V, pp. 45-46

This quote from Whitehead follows a lesser known passage from the same work:

“…by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain.”

Alfred North Whitehead, An Introduction to Mathematics, London: WILLIAMS & NORGATE, Chap. V, pp. 45

In other words, the brain is saved effort by mechanizing as much reason as can be mechanized. Of course, not everyone is capable of these kinds of mechanical deductions made possible by mathematical logic, which is especially difficult.

Recent scholarship has only served to underscore the difficulty of thinking, and the steps we must take to facilitate our thinking. Daniel Kahneman in particular has focused on the physiology effort involved in thinking. In his book Thinking, Fast and Slow, Daniel Kahneman distinguishes between two cognitive systems, which he calls System 1 and System 2, which are, respectively, that faculty of the mind that responds immediately, on an intuitive or instinctual level, and that faculty of the mind that proceeds more methodically, according to rules:

Why call them System 1 and System 2 rather than the more descriptive “automatic system” and “effortful system”? The reason is simple: “Automatic system” takes longer to say than “System 1” and therefore takes more space in your working memory. This matters, because anything that occupies your working memory reduces your ability to think. You should treat “System 1” and “System 2” as nicknames, like Bob and Joe, identifying characters that you will get to know over the course of this book. The fictitious systems make it easier for me to think about judgment and choice, and will make it easier for you to understand what I say.

Daniel Kahneman, Thinking, Fast and Slow, New York: Farrar, Straus, and Giroux, Part I, Chap. 1

While such concerns do not appear to have explicitly concerned Russell, Russell’s concern for economy of thought implicitly embraced this idea. One’s ability to think must be facilitated in any way possible, including the shortening of names — in purely formal thought, symbolization dispenses with names altogether and contents itself with symbols only, usually introduced as letters.

Kahneman’s book, by the way, is a wonderful review of cognitive biases that cites many of the obvious but often unnoticed ways in which thought requires effort. For example, if you are walking along with someone and you ask them in mid-stride to solve a difficult mathematical problem — or, for that matter, any problem that taxes working memory — your companion is likely to come to a stop when focusing mental effort on the work of solving the problem. Probably everyone has had experiences like this, but Kahneman develops the consequences systematically, with very interesting results (creating what is now known as behavioral economics in the process).

Formal thought is among the most difficult forms of cognition ever pursued by human beings. How can we facilitate our ability to think within a framework of thought that taxes us so profoundly? It is the overview provided by the non-constuctive perspective that makes it possible to take a “big picture” view of formal knowledge and formal thought, which is usually understood to be a matter entirely immersed in theoretical details and the minutiae of deduction and derivation. We must take an “Über” perspective in order to see formal thought whole. We have become accustomed to thinking of “surveyability” in constructivist terms, but it is just as valid in non-constructivist terms.

In P or not-P (as well as in subsequent posts concerned with constructivism, What is the relationship between constructive and non-constructive mathematics? Intuitively Clear Slippery Concepts, and Kantian Non-constructivism) I surveyed constructivist and non-constructivist views of tertium non datur — the central logical principle at issue in the conflict between constructivism and non-constructiviem — and suggested that constructivists and non-constructivists need each other, as each represents a distinct point of view on formal thought.

In P or not-P, cited above, I quoted French mathematician 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 …the uncountable axiom of choice gives an aerial view of mathematical reality — inevitably, therefore, a simplified view.”

Conversations on Mind, Matter, and Mathematics, Changeux and Connes, Princeton, 1995, pp. 42-43

In several posts I have taken up this theme of Alain Connes and have spoken of the non-constructive perspective (which Connes calls “formalist”) as being top-down and the constructive perspective as being bottom-up. In particular, in The Epistemic Overview Effect I argued that in additional to the possibility of a spatial overview (the world entire seen from space) and a temporal overview (history seen entire, after the manner of Big History), there is an epistemic overview, that is to say, an overview of knowledge, perhaps even the totality of knowledge.

If we think of those mathematical equations that have become sufficiently famous that they have become known outside mathematics and physics — (as well as some that should be more widely known, but are not, like the generalized continuum hypothesis and the expression of epsilon zero) — they all have not only the succinct property that Russell noted in the quotes above in regard to symbolism, but also many of the qualities that G. H. Hardy ascribed to what he called mathematically significant ideas.

It is primarily non-constructive modes of thought that give us a formal overview and which make it possible for us to engage with mathematically significant ideas, and, more generally, with formally significant ideas.

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Note added Monday 26 October 2015: I have written more about the above in Brief Addendum on the Overview Effect in Formal Thought.

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Formal thought begins with Greek mathematics and Aristotle's logic.

Formal thought begins with Greek mathematics and Aristotle’s logic.

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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. The Overview Effect in Formal Thought

10. Methodological and Ontological Parsimony (in preparation)

11. Einstein’s Conception of Formalism (in preparation)

12. The Spirit of Formalism (in preparation)

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Wittgenstein's Tractatus Logico-Philosophicus was part of the efflourescence of formal thinking focused on logic and mathematics.

Wittgenstein’s Tractatus Logico-Philosophicus was part of an early twentieth century efflorescence of formal thinking focused on logic and mathematics.

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The Epistemic Overview Effect

14 September 2013

Saturday


earth-from-space-1

OVERVIEW from Planetary Collective on Vimeo.

The Overview Effect

The “overview effect” is so named for the view of the earth entire — an “overview” of the earth — enjoyed by astronauts and cosmonauts, as well as the change in perspective that a few of these privileged observers have had as a result of seeing the earth whole with their own eyes.

One of these astronauts, Edgar Mitchell, who was on the 1971 Apollo mission and was the sixth human being to walk on the moon, has been instrumental to bringing attention to the overview effect, and has written a book about his experiences as an astronaut and how it affected his perception and perspective, The Way of the Explorer: An Apollo Astronaut’s Journey Through the Material and Mystical Worlds. A short film has been made about the overview effect, and an institution has been established to study and to promote the overview effect, The Overview Institute.

Here is an extract from the declaration of The Overview Institute:

For more than four decades, astronauts from many cultures and backgrounds have been telling us that, from the perspective of Earth orbit and the Moon, they have gained such a vision. There is even a common term for this experience: “The Overview Effect”, a phrase coined in the book of the same name by space philosopher and writer Frank White. It refers to the experience of seeing firsthand the reality of the Earth in space, which is immediately understood to be a tiny, fragile ball of life, hanging in the void, shielded and nourished by a paper-thin atmosphere. From space, the astronauts tell us, national boundaries vanish, the conflicts that divide us become less important and the need to create a planetary society with the united will to protect this “pale blue dot” becomes both obvious and imperative. Even more so, many of them tell us that from the Overview perspective, all of this seems imminently achievable, if only more people could have the experience!

We have a hint of the overview effect when we see pictures of the Earth as a “blue marble” and as a “pale blue dot”; those who have had the opportunity to see the Earth as a blue marble with their own eyes have been affected by this vision to a greater extent than we can presumably understand from seeing the photographs. Here is another description of the overview effect:

When people leave the surface of the Earth and travel into Low Earth Orbit, to a space station, or the moon, they see the planet differently. My colleague at the Overview Institute, David Beaver, likes to emphasize that they not only see the Earth from space but also in space. He has also been a strong proponent that we describe what then happens as a change in world view.

Deep Space: The Philosophy of the Overview Effect, Frank White

In the same essay White then quotes himself from his book, The Overview Effect: Space Exploration and Human Evolution, on the same theme:

“Mental processes and views of life cannot be separated from physical location. Our “world view” as a conceptual framework depends quite literally on our view of the world from a physical place in the universe.”

Frank White has sought to give a systematic exposition of the overview effect in his book, The Overview Effect: Space Exploration and Human Evolution, which seeks to develop a philosophy of space travel derived from the personal experience of space by space travelers.

sunset

The Spatial Overview

There is no question in my mind that sometimes you have to see things for yourself. I have invoked this argument numerous times in writing about travel — no amount of eloquent description or stunning photographs can substitute for the experience of seeing a place for yourself with your own eyes. This is largely a matter of context: being in a place, experiencing a place as a presence, requires one’s own presence, and one’s own presence can be realized only as the result of a journey. A journey contextualizes an experience within the experiences required the reach the object of the journey. The very fact that one must travel in order to each a destination alters the experience of the destination itself.

To be present in a landscape means that all of one’s senses are engaged: one not only sees, but one sees with the whole of one’s peripheral vision, and when one turns one’s body in order to take in more of the landscape, one not only sees more of the landscape, but one feels one’s body turn; one smells the air; one hears the distinctive reverberations of the most casual sounds — all of the things that remind us that this is not an illusion but possesses all the chance qualities that mark a real, concrete experience.

I have remarked in other posts that one of the distinctive trends in contemporary philosophy of mind is that of emphasizing the embodiedness of the mind, and in this context the embodied mind is a mind that is inseparable from its sensory apparatus and its sensory apparatus is inseparable from the world with which it is engaged. When our eyes hurt as we look at the sun we are reminded by this visceral experience of sight — one might say overwhelming sight — that we experience the world in virtue of a sensory apparatus that is made of essentially the same materials as the world — that there is an ontological reciprocity of eye that sees and sun that shines, and it is only because the two share the same world and are made of the same materials that they stand in a relation of cause and effect to each other. We are part of the world, of the world, and in the world.

Presumably, then, to the present in space and feel oneself kineasthetically in space — most obviously, the feeling of a micro-gravity environment once off the surface of the earth — is part of the experience of the overview effect, as is the dramatic journey into orbit, which must remind the viewer of the difficulty of attaining the perspective of seeing the world whole. This is the overview effect in space.

temporal overview

The Temporal Overview

There is also the possibility of an overview effect in time. For the same reason that we might insist that some experiences must be had for oneself, and that one must be present spatially in a spatial landscape in order to appreciate that landscape for what it is, we might also insist that a person who has lived a long life and who has experienced many things has a certain kind of understanding of the temporal landscape of life, and it is only through a conscious knowledge of the experience of time and history that we can attain an overview of time.

The movement in contemporary historiography called Big History (which I have written about several times, e.g., in The Science of Time and Addendum on Big History as the Science of Time) is an attempt to achieve an overview experience of time and history.

I have observed elsewhere that we find ourselves swimming in the ocean of history, but this very immersion in history often prevents us from seeing history whole — which is an interesting contrast to the spatial overview experience, which which contextualization in a particular space is necessary to its appreciation and understanding. But contextualization in a particular time — which we would otherwise call parochialism — tends to limit our historical perspective, and we must actively make an effort to free ourselves from our temporal and historical contextualization in order to see time and history whole.

It is the effort to free ourselves from temporal parochialism, and the particularities and peculiarities of our own time, that give as a perspective on history that is not tied to any one history but embraces the whole of time as the context of many different histories. This is the overview effect in time.

Knowledge Tree

The Epistemic Overview

I would like to suggest that there is also an epistemic overview effect. It is not enough to be told about knowledge in the way that newspaper and magazine articles might tell a popular audience about a new scientific discovery, or in the way that textbooks tell students about the wider world. While in some cases this may be sufficient, and we must rely upon the reports of others because we cannot construct the whole of knowledge on our own, in many cases knowledge must be gained firsthand in order for its proper significance to be appreciated.

Elsewhere (in P or not-P) I have illustrated the distinction between a constructive and a non-constructive point of view being something like the difference between climbing up a mountain, clambering over every rock until one achieves the summit (constructive) versus taking a helicopter and being set down on the summit from above (non-constructive). (I have taken this example over from French mathematician Alain Connes.) With this image in mind, being blasted off into space and seeing the mountain from orbit is a paradigmatically non-constructive experience, and it is difficult to imagine how it could be made a constructive experience.

Well, there are ways. Once space technology becomes widely distributed and accessible, if a person were to build their own SSTO from off-the-shelf parts and then pilot themselves into orbit, that would be something like a constructive experience of the overview effect. And if we go on to create a vibrant and vigorous spacefaring civilization, making it into orbit will only be the first of many steps, so that a constructive experience of space travel will be to “climb” one’s way from the surface of the earth through the solar system and beyond, touching every transitional point in between. It has been said that the journey of the thousand miles begins with a single step — this is very much a constructivist perspective. And it holds true that a journey of a million miles or a billion miles begins with a single step, and that first step of a cosmic voyage is the step that takes us beyond the surface of the earth.

Despite the importance and value of the constructivist perspective, it has its limitations, just as the oft-derided non-constructive point of view has its particular virtues and its significance. Non-constructive methods can reveal to us knowledge that is disruptive because it is forced upon us suddenly, in one fell swoop. Such an experience is memorable; it leaves an impression, and quite possibly it leaves much more of an impression that a painstakingly gradual revelation of exactly the same perspective.

This is the antithesis of the often-cited example of a frog placed in a pot of water and which doesn’t jump out as the water is slowly brought to a boil. The frog in this scenario is a victim of constructivist gradualism; if the frog had had a non-constructive perspective on the hot water in which he was being boiled to death, he might have jumped out and saved himself. And perhaps this is exactly what we need as human beings: a non-constructive (and therefore disruptive) perspective on a the familiar life that has crept over us day-by-day, step-by-step, and bit-by-bit.

An epistemic overview of knowledge can give us a disruptive conception of the totality of knowledge that is not unlike the disruptive experience of the overview effect in space, which allows us to see the earth whole, and the disruptive experience of time that allows us to see history whole. Moreover, I would argue that the epistemic overview is the ultimate category — the summum genus — that must contextualize the overview effect in space and in time. However, it is important to point out that the immediate visceral experience of the overview effect may be the trigger that is required for an individual to begin to seek the epistemic overview that will give meaning to his experiences.

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