Tuesday


Indiana Jones is adventure science at its most exciting, though the films are more often about looting and destroying sites rather than preserving them.

Indiana Jones is adventure science at its most exciting, though the films are more often about looting and destroying sites rather than preserving them.

In my recent paper “A Manifesto for the Scientific Study of Civilization” I argued that the study of civilization should be scientific, and that a scientific theory of civilization would be a formal theory. Prior to this, I argued in Rational Reconstructions of Time that a formal historiography is possible. What is the connection between these two claims? In A Metaphysical Disconnect I suggested that it is a philosophical problem that philosophies of time have not been tightly-coupled with philosophies of history. This implies that a formal theory of time could be tightly-coupled with a formal theory of history, and a formal theory of history would presumably encompass (or, at least, overlap) a theory of civilization. A formal theory of civilization, then, might ultimately follow from formal historiography.

I fully understand that these are strange claims for me to be making. What in the world do I mean by a formal theory of time, of history, or of civilization? How could a science of civilization be a formal science? What is a formal science, anyway? Despite the burgeoning growth of computer science in our time, which is the latest addition to the formal sciences, the very idea of the formal as a distinct category of thought (distinct, especially, from the material) seems odd and alien to us, and the distinction between the formal sciences and the natural sciences seems archaic. What are the formal sciences? Here is one view:

“To put it in Kantian terms, the formal sciences dealt with the Reine Anschauung as opposed to empirical data. By that they have been connected to the methodology of mathematics and logic, thereby being part of both the philosophical tradition and the newly won applications of mathematical sciences to the natural sciences and engineering. Both the object and the methods of the Formal sciences were recognized as different from the Natural and the Social sciences.”

“The Formal Sciences: Their Scope, Their Foundations, and Their Unity” by Benedikt Löwe, Synthese, Vol. 133, No. 1/2, Foundations of the Formal Sciences I (Oct.-Nov., 2002),pp. 5-11

In the same paper there is an explicit attempt to answer the question, “What are the Formal Sciences?” Two answers are given:

● Answer 1: “There is a profound duality in the classification of sciences according to their scientific approaches: some sciences are empirical, some are formal. The former deal with predictions and their falsification, the latter with the understanding of systems without empirical component, be it man-made systems (literary systems, the arts or social systems) or formal systems”.

● Answer 2: “Formal sciences are those that deal with the deductive analysis of formal systems (i.e., systems independent of direct human influence)”.

At present I am not going to analyze these differing definitions of the formal sciences, but I will leave them to percolate in the back of the mind of the reader in order to return to the question at hand: the study of civilization as a formal science, i.e., one formal science among many other formal sciences, however we choose to define them.

We can get a hint of what a formal science of civilization would look like from structuralist historians and historians of the Annales school, the chief representatives of the latter being Marc Bloch, Lucien Febvre, and Fernand Braudel. Marc Bloch’s two volume history of feudalism, in particular, stands out as a great achievement in the genre, with chapters devoted to features of feudal society rather than to great events and historical turning points. Whereas John Florio had Montaigne say that I describe not the essence but the passage, Bloch sought to describe not the passage, but the essence. (I previously quoted from Bloch in Hegel and the Overview Effect.)

There is (or, there will be) no one, single way to approach formal historiography, in the same way that there is no one, single axiomatization of set theory. Even if one agrees with Gödel that set theory describes a “well-determined reality” (a realist conception that most people today would agree describes the past, even if they would hesitate to say the same of set theory), there are, as yet, many distinct approaches to that reality. So too with formal historiography; there will be many distinct formalisms for the organization, exhibition, and exposition of the well-determined reality of history.

I reveal myself as being more of a traditionalist than Bloch by my preference for approaching a theory of civilization by way of a theory of history, and a theory of history by way of a theory of time. This is “traditional” in the sense that, as I have remarked many times in other places, it has been traditional to study civilization by studying history, rather than studying civilization as an object of knowledge in its own right. I retain the historical perspective, and indeed even many of the prejudices of historians (these come naturally to me), but I can also see beyond history sensu stricto and to a science of time, a science of history, and a science of civilization that lies beyond history even as it draws from the tradition all that that tradition has to offer.

Both the essentialist approach of Bloch and the Annales school, and my own quasi-historical approach to a formal science of civilization, may each have something to contribute to a theory of civilization. Obviously, these are not the only ways to study civilization. Civilization also can be studied as an empirical science — this is probably how most would conceive a science of civilization — and even as an adventure science. What is adventure science?

Together with Dr. Jacob Shively, I wrote an article about adventure science, Adventure Science Enters the Space Age, noting that “big science” has become the paradigm of scientific activity at the present time, but when individual human beings are able to go exploring they will be able to pluck the low-hanging fruit of exploration and discovery. Adventure science characterizes the earliest stage of a science when discoveries can be made simply by traveling to an exotic locale and being the first to describe some phenomenon never before documented by science. Such discoveries are difficult for us now, because the low-hanging fruit of terrestrial discovery has all been plucked, but once off Earth, new worlds will beckon with new discoveries waiting to be made. This will be a new Golden Age of adventure science.

Paradoxically, the science of civilization will become an adventure science (if it ever becomes one) quite late in its history, so that adventure science will characterize a science of civilization not in its earliest stages, but in its latest stages. But civilization has had a kind of early adventure science phase as well. Archaeology was once the paradigm of adventure science — as attested to by the cinematic adventures of Indiana Jones and the television adventures of Relic Hunter — when real life explorers entered jungles and deserts and swamps to search for long lost cities. Archaeology is perhaps the closest existing discipline that we have to a true science of civilization — archaeologists have many theories of civilization — so that the adventure science that archaeology once was, was at the same time (at least in part) an adventure science of civilization. And it may be so again, when xenoarchaeologists lead the way, looking for the ruins of alien civilizations.

All of the resources of contemporary big science, with its thousands of researchers and multi-generational socially-organized research programs, will be necessary in order to develop the science that will make possible the production of interstellar vessels. In my Centauri Dreams post, The Interstellar Imperative, I wrote, “A starship would be the ultimate scientific instrument produced by technological civilization, constituting both a demanding engineering challenge to build and offering the possibility of greatly expanding the scope of scientific knowledge by studying up close the stars and worlds of our universe, as well as any life and civilization these worlds may comprise.” Once starships become a reality, they will make possible the empirical study of civilizations, which will begin as an adventure science, the primary qualification for which will be a willingness to tolerate discomfort and to travel to distant places with a determination to document every new sight that one sees.

Geology will become an adventure science like this once again as soon as human beings have the freedom to travel around our solar system; biology and ecology will become adventure sciences once again as soon as we can visit other living worlds. The study of civilization will not become an adventure science until human beings are free to travel about the cosmos, so that this is a very distant prospect, but still a hopeful one. If we do not find a number of interesting civilizations to study, we will build a number of interesting civilizations, and eventually these will be studied in their turn. In this latter instance, the science of civilization will only become an adventure science after civilization has expanded throughout the cosmos, has forgotten the saga of its expansion, and then rediscovers itself across a plurality of worlds. And once again we will be forced to reckon with Hegel’s prescience for having said that the owl of Minerva takes flight only with the setting of the sun.

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'Anywhen' by Chris Foss perfectly expresses the mystery and adventure of exploration. Perhaps some day in the far future, the study of civilization will be an adventure science in which such exploration takes a central role.

“Anywhen” by Chris Foss perfectly expresses the mystery and adventure of exploration. Perhaps some day in the far future, the study of civilization will be an adventure science in which such exploration takes a central role.

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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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Thursday


There is passage in Foucault, in the preface to the English language of The Order of Things, after the more famous passage about the “Chinese dictionary” in Borges, in which he discusses a pathological failure of taxonomy. The theme of Foucault’s book, restated compellingly in this preface, is taxonomy — taxonomy in its most general (and therefore its most philosophical) signification. Taxonomy is a problem.

It appears that certain aphasiacs, when shown various differently coloured skeins of wool on a table top, are consistently unable to arrange them into any coherent pattern; as though that simple rectangle were unable to serve in their case as a homogeneous and neutral space in which things could be placed so as to display at the same time the continuous order of their identities or differences as well as the semantic field of their denomination. Within this simple space in which things are normally arranged and given names, the aphasiac will create a multiplicity of tiny, fragmented regions in which nameless resemblances agglutinate things into unconnected islets; in one corner, they will place the lightest-coloured skeins, in another the red ones, somewhere else those that are softest in texture, in yet another place the longest, or those that have a tinge of purple or those that have been wound up into a ball. But no sooner have they been adumbrated than all these groupings dissolve again, for the field of identity that sustains them, however limited it may be, is still too wide not to be unstable; and so the sick mind continues to infinity, creating groups then dispersing them again, heaping up diverse similarities, destroying those that seem clearest, splitting up things that are identical, superimposing different criteria, frenziedly beginning all over again, becoming more and more disturbed, and teetering finally on the brink of anxiety.

THE ORDER OF THINGS: An Archaeology of the Human Sciences, MICHEL FOUCAULT, A translation of Les Mots et les choses, VINTAGE BOOKS, A Division of Random House, Inc., New York, Preface

Taxonomy is the intersection of words and things — and just this was the original title of Foucault’s book, i.e., words and things — and Foucault brilliantly illustrates both the possibilities and problems inherent in taxonomy. Foucault had an enduring concern for taxonomy, and, as is well known, named his chair at the Collège de France the “History of Systems of Thought” — as though he were seeking a master taxonomy of human knowledge.

Foucault found madness and mental illness in the inability of a test subject to systematically arrange skeins of wool, since each attempted scheme of classification breaks down when it overlaps within another system of classification pursued simultaneously. One suspects that if the task placed before Foucault’s aphasiac were limited in certain ways — perhaps in the number of colors of wool, or the number of categories that could be employed — the task might become practical once a sufficient number of constraints come into play. But the infinite universe investigated by contemporary science is the very antithesis of constraint. There is always more to investigate, and as the sciences themselves grow and fission, begetting new sciences, the task of bringing order to the sciences themselves (rather than to the empirical phenomena that the sciences seek to order) becomes progressively more difficult.

The taxonomy of the sciences is more problematic that usually recognized. Consider these possible categories of science, not all of which are current today:

● natural sciences It is still somewhat common to speak of the “natural sciences,” with our intuitive understanding of what is “natural” as sufficient to classify a given study as an investigation into “nature.” What, then, is not a natural science? At one time there was a strong distinction made between the natural sciences and the formal sciences (q.v.)

● formal sciences The phrase “formal sciences” is rarely used today, though it is still a useful idea, comprising at least mathematics and logic and (for those who know what it is) formal mereology. Today the formal sciences might also include computer science and information science, though I haven’t myself ever heard anyone refer to these sciences as formal sciences. Since the mathematization of the natural sciences beginning with the scientific revolution, the natural sciences have come more and more to approximate formal sciences, to the point that mathematical physics has, at times, only a tenuous relationship to experiments in physics, while it has a much more robust relationship with mathematics.

● moral sciences Philosopher J. R. Lucas has written of the moral sciences, “The University of Cambridge used to have a Faculty of Moral Sciences. It was originally set up in contrast to the Faculty of Natural Sciences, and was concerned with the mores of men rather than the phenomena of nature. But the humane disciplines were hived off to become separate subjects, and when the faculty was finally renamed the Faculty of Philosophy, philosophy was indeed the only subject studied.”

● earth sciences The earth sciences may be understood to be a subdivision of the natural sciences, and may be strongly distinguished from the space sciences, but the distinction between the earth sciences and the space sciences, as well as these two sciences themselves, is quite recent, dating to the advent of the Space Age in the middle of the twentieth century. While the idea behind the earth sciences is ancient, their explicit recognition as a special division within the sciences is recent. I suspect that the fact of seeing the earth from space, made possible by the technology of the space age, contributed greatly to understanding the earth as a unified object of investigation.

● space sciences The space sciences can be defined in contradistinction to the earth sciences, as though science had a need to recapitulate the distinction between the sublunary and the superlunary of Ptolemaic cosmology; however, I don’t think that this was the actual genesis of the idea of a category of space sciences. The emergence of the “Space Age” and its associated specialty technologies, and the sciences that produced these technologies, is the likely source, but the question becomes whether a haphazardly introduced concept roughly corresponding to a practical division of scientific labor constitutes a useful theoretical category.

● social sciences The social sciences would obviously include sociology and cultural anthropology, but would it include biological anthropology? History? Political science? Economics? The social sciences often come under assault for their methodology, which seems to be much less intrinsically quantitative than that of the natural sciences, but are not social communities as “natural” as biologically defined communities?

● human sciences In German there is a term — Geisteswissenschaften — that could be translated as the “spiritual sciences,” and which roughly corresponds to the traditional humanities, but it is not entirely clear whether the human sciences coincide perfectly either with Geisteswissenschaften or the humanities. Foucault’s The Order of Things, quoted above, is subtitled, “An Archaeology of the Human Sciences,” and the human sciences that Foucault examines in particular include philology and economics, inter alia.

● life sciences I assume that “life sciences” was formulated as a collective term for biological sciences, which would include studies like biogeography, which might also be called an instance of the earth sciences, or the natural sciences. But the life sciences would also include all of medicine, which gives us a taxonomy of the medical sciences, though it does not give us a clear demarcation between the life sciences and the natural sciences. Does medicine include all of psychiatry, or ought psychiatric inquiries to be thought of as belonging to the social sciences?

● historical sciences I have written about the historical sciences in several posts, since S. J. Gould often made the point that that historical sciences have a distinctive methodology. In Historical Sciences I argued that there is a sense in which all sciences can be considered historical sciences. Indeed, one of the distinctive aspects of the scientific revolution has been to force human beings to stop assuming the eternity and permanence of the world and to see the world and everything in it as having a natural history. If everything has a natural history, then all investigations are historical investigations and all sciences are historical sciences — but if this is true, then Gould’s claim that the historical sciences have a unique methodology collapses.

There are also, of course, informal distinctions such as that between the “hard” sciences and the “soft” sciences, which is sometimes taken to be the distinction between mathematicized sciences and non-mathematicized sciences, and so may correspond to the rough distinction between the natural sciences and the social sciences, except the that the social sciences are now dominated by statistical methods and can no longer be thought of a non-mathematicized. This leads to problems of classification such as whether economics, for example, is a natural science or a social science.

For each of the science categories above we could attempt either an extensional or an intensional definition, i.e., we could give a list of particular sciences that fall under the category in question, or we could attempt to define the meaning of the term, and the meaning would then govern what sciences are so identified. An extensional definition of the earth sciences might involve a list including geomorphology, biogeography, geology, oceanography, hydrology, climatology, and so forth. An intentional definition of the earth sciences might be something like, “those sciences that have as their object of study the planet earth, its subsystems, and its inhabitants.”

Today we employ the sciences to bring order to our world, but how do we bring order to the sciences? Ordering our scientific knowledge is problematic. It is complicated. It involves unanticipated difficulties that appear when we try to make any taxonomy for the sciences systematic. Each of the scientific categories above (as well as others that I did not include — my list makes no pretension of completeness) implies a principled distinction between the kind of sciences identified by the category and all other sciences, even if the principle by which the distinction is to be made is not entirely clear.

The implicit distinction between the earth sciences and the space sciences has a certain intuitive plausibility, and it is useful to a certain extent, though recently I have tried to point out in Eo-, Exo-, Astro- the importance of astrobiology as unifying terrestrial biology and exobiology in a truly Copernican framework. While the attempted task of a taxonomy of the sciences is important, the nature of the task itself suggests a certain compartmentalization, and too much thinking in terms of compartmentalization can distract us from seeing the larger synthesis. Concepts based on categorization that separates the sciences will be intrinsically different from extended conceptions that emerge from unification. An exclusive concern for the earth sciences, then, might have the subtle affect of reinforcing geocentric, Ptolemaic assumptions, though if we pause for a moment it will be obvious that the earth is a planet, and that the planetary sciences ought to include the earth, and the the planetary sciences might be construed as belonging to the space sciences.

The anxiety experience by Foucault’s aphasiac is likely to be experienced by anyone attempting a systematic taxonomy of the sciences, as here, any mind, whether sick or healthy, might continue to extrapolate distinctions to infinity and still not arrive at a satisfactory method for taking the measure of the sciences in way that contributes both to the clarity of the individual sciences and an understanding of how the various special sciences relate to each other.

On the one hand, perfect rigor of thought would seem to imply that all possible distinctions must be observed and respected, except that not all distinction can be made at the same time because some cut across each other, are mutually exclusive, order the world differently, and subdivide other categories and hierarchies in incompatible schemes. To use a Leibnizean term, not all distinctions are compossible.

To invoke Leibniz in this context is to suggest a Leibnizean approach to the resolution of the difficulty: a Leibnizean conception of conceptual rigor would appeal to the greatest number of distinctions that are compossible and yield a coherent body of knowledge.

A thorough-going taxonomic study of human bodies of knowledge would reveal a great many possible taxonomies, some with overlapping distinctions, but it is likely that there is an optimal arrangement of distinctions that would allow the greatest possible number of distinctions to be employed simultaneously while retaining the unity of knowledge. This would be a system of compossible taxonomy, which might have to reject a few distinctions but which makes use of the greater number of distinctions that are mutually possible within the framework of methodological naturalism as this defines the scientific enterprise.

There are not merely academic considerations. The place of science within industrial-technological civilization means that our conception of science is integral with our conception of civilization; thus to make a systematic taxonomy of the sciences is to make a systematic taxonomy of a civilization that is based upon science. Such conception categories extrapolated from science to civilization will have consequences for human self-understanding and human interaction, which latter does not always take the form of “cultural exchanges” (in the saccharine terminology if international relations). Industrial-technological civilization is in coevolution with industrial-technological warfare, so that a taxonomy of science is also a taxonomy of scientific warfare. Our conception of science will ultimately influence how we kill each other, and how we seek peace in order to stop killing each other.

One of the most distinctive forms of propaganda and social engineering of our time is the creation from whole cloth of artificial and fraudulent sciences. Since science is the condition of legitimacy in industrial-technological civilization, social movements seeking legitimacy seek scientific justification for their moral positions, but the more that science is seen as a means to an end, where the end is stipulated in advance, then science as a process must be compromised because any science that does not tend to the desired socio-political end will be subject to socio-political disapproval or dismissal. While there is a limit to this, the limits are more tolerant than we might suppose: large, complex societies with large and diverse economies can sustain non-survival behavior for a significant period of time — perhaps enough time to conceal the failure of the model employed until it is too late to save the society that has become a victim of its own illusion.

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Grand Strategy Annex

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Monday


Geopolitics and Geostrategy

as a formal sciences


In a couple of posts — Formal Strategy and Philosophical Logic: Work in Progress and Axioms and Postulates of Strategy — I have explicitly discussed the possibility of a formal approach to strategy. This has been a consistent theme of my writing over the past three years, even when it is not made explicit. The posts that I wrote on theoretical geopolitics can also be considered an effort in the direction of formal strategy.

There is a sense in which formal thought is antithetical to the tradition of geopolitics, which latter seeks to immerse itself in the empirical facts of how history gets made, in contradistinction to the formalist’s desire to define, categorize, and clarify the concepts employed in analysis. Yet in so far as geopolitics takes the actual topographical structure of the land as its point of analytical departure, this physical structure becomes the form upon which the geopolitician constructs the logic of his or her analysis. Geopolitical thought is formal in so far as the forms to which it conforms itself are physical, topographical forms.

Most geopoliticians, however, have no inkling of the formal dimension of their analyses, and so this formal dimension remains implicit. I have commented elsewhere that one of the most common fallacies is the conflation of the formal and the informal. In Cartesian Formalism I wrote:

One of the biggest and yet one of the least recognized blunders in philosophy (and certainly not only in philosophy) is to conflate the formal and the informal, whether we are concerned with formal and informal objects, formal and informal methods, or formal and informal ideas, etc. (I recently treated this topic on my other blog in relation to the conflation of formal and informal strategy.)

Geopolitics, geostrategy, and in fact many of the so-called “soft” sciences that do not involve extensive mathematization are among the worst offenders when it comes to the conflation of the formal and the informal, often because the practitioners of the “soft” sciences do not themselves understand the implicit principles of form to which they appeal in their theories. Instead of theoretical formalisms we get informal narratives, many of which are compelling in terms of their human interest, but are lacking when it comes to analytical clarity. These narratives are primarily derived from historical studies within the discipline, so that when this method is followed in geopolitics we get a more-or-less quantified account of topographical forms that shape action and agency, with an overlay of narrative history to string together the meaning of names, dates, and places.

There is a sense in which geography and history cannot be separated, but there is another sense in which the two are separated. Because the ecological temporality of human agency is primarily operational at the levels of micro-temporality and meso-temporality, this agency is often exercised without reference to the historical scales of the exo-temporality of larger social institutions (like societies and civilizations) and the macro-historical scales of geology and geomorphology. That is to say, human beings usually act without reference to plate tectonics, the uplift of mountains, or seafloor spreading, except when these events act over micro- and meso-time scales as in the case of earthquakes and tsunamis generated by geological events that otherwise act so slowly that we never notice them in the course of a lifetime — or even in the course of the life of a civilization.

The greatest temporal disconnect occurs between the smallest scales (micro-temporality) and the largest scales (macro-temporality), while there is less disconnect across immediately adjacent divisions of ecological temporality. I can employ a distinction that I recently made in a discussion of Descartes, that between strong distinctions and weak distinctions (cf. Of Distinctions Weak and Strong). Immediately adjacent divisions of ecological temporality are weakly distinct, while those not immediately adjacent are strongly distinct.

We have traditionally recognized the abstraction of macroscopic history that does not descend into details, but it has not been customary to recognize the abstractness of microscopic history, immersed in details, that does not also place these events in relation to a macroscopic context. In order to attain to a comprehensive perspective that can place these more limited perspectives into a coherent context, it is important to understand the limitations of our conventional conceptions of history (such as the failure to understand the abstract character of micro-history) — and, for that matter, the limitations of our conventional conceptions of geography. One of these limitations is the abstractness of either geography or history taken in isolation.

The degree of abstractness of an inquiry can be quantified by the ecological scope of that inquiry; any one division of ecological temporality (or any one division of metaphysical ecology) taken in isolation from other divisions is abstract. It is only the whole of ecology taken together that a truly concrete theory is possible. To take into account the whole of ecological temporality in a study of history is a highly concrete undertaking which is nevertheless informed by the abstract theories that constitute each individual level of ecological temporality.

Geopolitics, despite its focus on the empirical conditions of history, is a highly abstract inquiry precisely because of its nearly-exclusive focus on one kind of structure as determinative in history. As I have argued elsewhere, and repeatedly, abstract theories are valuable and have their place. Given the complexity of a concrete theory that seeks to comprehend the movements of human history around the globe, an abstract theory is a necessary condition of any understanding. Nevertheless, we need to rest in our efforts with an abstract theory based exclusively in the material conditions of history, which is the perspective of geopolitics (and, incidentally, the perspective of Marxism).

Geopolitics focuses on the seemingly obvious influences on history following from the material conditions of geography, but the “obvious” can be misleading, and it is often just as important to see what is not obvious as to explicitly take into account what is obvious. Bertrand Russell once observed, in a passage both witty and wise, that:

“It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. 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. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved. And the same holds in other cases where self-evident things are to be proved.”

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

Russell here expresses himself in terms of symbolism, but I think it would better to formulate this in terms of formalism. When Russell writes that, “we invent some new and difficult symbolism, in which nothing seems obvious,” the new and difficult symbolism he mentions is more than mere symbolism, it is a formal theory. Russell’s point, then, is that if we formalize a body of knowledge heretofore consisting of intuitively “obvious” truths, certain relationships between truths become obvious that were not obvious prior to formalization. Another way to formulate this is to say that formalization constitutes a shift in our intuition, so that truths once intuitively obvious become inobvious, while inobvious truths because intuitive. Thus formalization is the making intuitive of previously unintuitive (or even counter-intuitive) truths.

Russell devoted a substantial portion of his career to formalizing heretofore informal bodies of knowledge, and therefore had considerable experience with the process of formalization. Since Russell practiced formalization without often explaining exactly what he was doing (the passage quoted above is a rare exception), we must look to the example of his formal thought as a model, since Russell himself offered no systematic account of the formalization of any given body of knowledge. (Russell and Whitehead’s Principia Mathematica is a tour de force comprising the order of justification of its propositions, while remaining silent about the order of discovery.)

A formal theory of time would have the same advantages for time as the theoretical virtues that Russell identified in the formalization of mathematics. In fact, Russell himself formulated a formal theory of time, in his paper “On Order in Time,” which is, in Russell’s characteristic way, reductionist and over-simplified. Since I aim to formulate a theory of time that is explicitly and consciously non-reductionist, I will make no use of Russell’s formal theory of time, though it is interesting at least to note Russell’s effort. The theory of ecological temporality that I have been formulating here is a fragment of a full formal theory of time, and as such it can offer certain insights into time that are lost in a reductionist account (as in Russell) or hidden in an informal account (as in geography and history).

As noted above, a formalized theory brings about a shift in our intuition, so that the formerly intuitive becomes unintuitive while the formerly unintuitive becomes intuitive. A shift in our intuitions about time (and history) means that a formal theory of time makes intuitive temporal relationships less obvious, while making temporal relationships that are hidden by the “buzzing, blooming world” more obvious, and therefore more amenable to analysis — perhaps for the first time.

Ecological temporality gives us a framework in which we can demonstrate the interconnectedness of strongly distinct temporalities, since the panarchy the holds between levels of an ecological system is the presumption that each level of an ecosystem impacts every other level of an ecosystem. Given the distinction between strong distinctions and weak distinctions, it would seem that adjacent ecological levels are weakly distinct and therefore have a greater impact on each other, while non-adjacent ecological levels are strongly distinct and therefore have less of an impact on each other. In an ecological theory of time, all of these principles hold in parallel, so that, for example, micro-temporality is only weakly distinct from meso-temporality, while being strongly distinct from exo-temporality. As a consequence, a disturbance in micro-temporality has a greater impact upon meso-temporality than upon exo-temporality (and vice versa), but less of an impact does not mean no impact at all.

Another virtue of formal theories, in addition to the shift in intuition that Russell identified, is that it forces us to be explicit about our assumptions and presuppositions. The implicit theory of time held by a geostrategist matters, because that geostrategist will interpret history in terms of the categories of his or her theory of time. But most geostrategists never bother to make their theory of time explicit, so that we do not know what assumptions they are making about the structure of time, hence also the structure of history.

Sometimes, in some cases, these assumptions will become so obvious that they cannot be ignored. This is especially the case with supernaturalistic and soteriological conceptions of metaphysical history that ultimately touch on everything else that an individual believes. This very obviousness makes it possible to easily identify eschatological and theological bias; what is much more insidious is the subtle assumption that is difficult to discern and which only can be elucidated with great effort.

If one comes to one’s analytical work presupposing that every moment of time possesses absolute novelty, one will likely make very different judgments than if one comes to the same work presupposing that there is nothing new under the sun. Temporal novelty means historical novelty: anything can happen; whereas, on the contrary, the essential identity of temporality over historical scales — identity for all practical purposes — means historical repetition: very little can happen.

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Note: Anglo-American political science implicitly takes geopolitics as its point of departure, but, as I have attempted to demonstrate in several posts, this tradition of mainstream geopolitics can be contrasted to a nascent movement of biopolitics. However, biopolitics too could be formulated in the manner of a theoretical biopolitics, and a theoretical biopolitics would be at risk of being as abstract as geopolitics and in need of supplementation by a more comprehensive ecological perspective.

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