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.

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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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Monday


Ludwig Wittgenstein and L. E. J. Brouwer

Brouwer and Wittgenstein were contemporaries, with the whole of Wittgenstein’s years contained within those of Brouwer’s (Wittgenstein lived 1889 to 1951 while Brouwer lived the longer life from 1881 to 1966). It is mildly ironic that even as Brouwer’s followers were playing down his mysticism and trying to extract only the mathematical content from his intuitionist philosophy (even the faithful Heyting distanced himself from Brouwer’s mysticism), Wittgenstein’s writings reached a much larger public which resulted in the mystical content of Wittgenstein’s works being played up and the early Wittgenstein himself, very much the logician following in the tradition of Frege and Russell, presented as a mystic.

Not only were Brouwer and Wittgenstein contemporaries, but we also know that Brouwer played a little-known role in Wittgenstein’s return to philosophy. After having written the Tractatus Logico-Philosophicus and then disappearing into the mountains of Austria to become a village schoolmaster in Trattenbach, some of those philosophers that continued to seek out Wittgenstein in his self-imposed exile convinced him to go to a lecture in Vienna in March 1928. The lecture was delivered by Brouwer (Brouwer gave two lectures; Wittgenstein is said to have attended one of them). Wittgenstein was said to have listened to the lecture with a surprised look on his face (sort of like G. E. Moore saying that Wittgenstein was the only student that looked puzzled at this lectures). So it may be the case that Brouwer played a pivotal role in the transition from the thought of the early Wittgenstein to the thought of the later Wittgenstein. (Matthieu Marion has argued this thesis.)

Wittgenstein’s distinction between saying and showing, a doctrine that dates from the Tractatus (cf. sections 4.113 and following), is often adduced in expositions of his alleged mysticism. According to Wittgenstein’s distinction, some things can be said but cannot be shown, while other things can be shown but cannot be said. While to my knowledge Wittgenstein never used the term “ineffable,” that which can be shown but cannot be said would seem to be a paradigm case of the ineffable. And since Wittgenstein identified a substantial portion of our experience as showable although unsayable, the ineffable seems then to play a central role in his thought. This puts Wittgenstein firmly in the company of figures like, say, St. Symeon the New Theologian (also, like Wittgenstein, an ascetic), which makes the case for his mysticism.

An extract from St. Symeon on the ineffable: “The grace of the all-holy spirit is given as earnest money of the souls pledged in marriage to Christ. Just as a woman without a pledge has no certainty that the union with the groom will occur within a certain length of time, so does the soul have no firm assurance that it will be re-united to its God and Master for all eternity. The soul cannot be certain that it will achieve this mystic, ineffable union nor that it will enjoy its inaccessible beauty if it does not have the pledge of His grace and does not consciously have that grace within.” (Krivocheine, Basil and Gythiel, Anthony P., In the Light of Christ: Saint Symeon, the New Theologian 949–1022, St. Vladimir’s Seminary Press, 1986, p. 367)

Brouwer was a bit more explicit in his doctrine of ineffability than was Wittgenstein, and he repeatedly asserted that the language of mathematics was a necessary evil that approximated but never fully captured the intuitive experience of mathematics, which he understood to be a free creation of the human mind. This comes across both in his early mystical treatise Life, Art, and Mysticism, which is pervaded by a sense of pessimism over the evils of the world (which include the evils of mathematical language), and his more technical papers offering an exposition of intuitionism as a philosophy of mathematics. But, like Wittgenstein, Brouwer does not (to my limited knowledge) actually use the term “ineffable.”

There is another ellipsis common to both Brouwer and Wittgenstein, and that is despite Brouwer’s openly professed intuitionism, which can be considered a species of constructivism (this latter is a point that needs to be separately argued, but I will only pass over it here with a single mention), and despite the strict finitism of the later Wittgenstein, which can also be considered a species of constructivism, neither Brouwer nor Wittgenstein employ Kantian language or Kantian formulations. No doubt there are implicit references to Kant in both, but I am not aware of any systematic references to Kant in the work of either philosopher. This is significant. Both Brouwer and Wittgenstein were philosophers of the European continent, where the influence of Kant remains strong even as his reputation waxes and wanes over the generations.

Kant was an early constructivist, or, rather, a constructivist before constructivism was explicitly formulated, and therefore sometimes called a proto-constructivist — although I have pointed out an obvious non-constructive dimension to Kant’s thought despite his proto-constructivism (which I do not deny, notwithstanding Kant’s non-constructive arguments in the first Critique). Kant’s classic proto-constructivist formulation is that the synthetic a priori truths of mathematics must be constructed, or “exhibited in intuition.” It is this latter idea, of a concept being exhibited in intuition, that has been particularly influential. But what does it mean? Obviously, a formulation like this has invited many interpretations.

The approaches of Brouwer and the later Wittgenstein could be considered different ways of exhibiting a concept in intuition. Brouwer, by casting out the law of the excluded middle from mathematics (at least in infinitistic contexts), assured that double negation was not equivalent to the truth simpliciter, so that even if you know that it is not the case that x is false, you still don’t know that x is true. (On the law of the excluded middle cf. P or not-P.) The later Wittgenstein’s insistence upon working out how a particular term is used and not merely settling for some schematic meaning (think of slogans like “don’t ask for the meaning, ask for the use” and “back to the rough ground”) similarly forces one to consider concrete instances rather than accepting (non-constructive) arguments for the way that things putatively must be, rather than how they are in actual fact. Both Wittgenstein’s finitism and Brouwer’s intuitionism would look with equal distaste upon, for example, proving that every set can be well-ordered without actually showing (i.e., exhibiting) such an order — also, the impossibility of exhaustively showing (i.e., exhibiting in intuition) that every set can be well-ordered if one acknowledges an infinity of sets.

I give this latter example because I think it was largely the perceived excesses of set theory and Cantor’s transfinite number theory that were essentially responsible for the reaction among some mathematicians that led to constructivism. Cantor was a great mathematical innovator, and his radical contributions to mathematics spurred foundationalists like Frege (who objected to Cantor’s methods but not his results) and Russell to attempt to construct philosophico-mathematical justifications, i.e., foundations, that would legitimize that which Cantor had wrought.

The reaction against infinitistic mathematics and foundationalism continues to the present day. Michael Dummett wrote in Elements of Intuitionism, a classic textbook on basic intuitionistic logic and mathematics, that:

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

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

In other words, mathematics would show its justification; in contrast, the foundationalist project to assure the legitimacy of the flights of non-constructive mathematics was wrong-headed in its very conception, because nothing that we say is going to change the fact that non-constructive thought that derives its force from proof, i.e., from what is said, does not show its justification on its face. Its justification must be established because it does not show itself. This is what “foundations” are for.

Note: There is also an element of intellectual ascesis in Dummett’s idea of a conservative extension of a theory, and this corresponds to the asceticism of Wittgenstein’s character, and, by extension, to the asceticism of Wittgenstein’s thought — asceticism being one of the clear continuities between the earlier and the later Wittgenstein — like the implicit development of constructivist themes.

But it was not only the later Wittgenstein who reacted with others against Cantor. It seems to me that the saying/showing distinction of the Tractatus is a distinction not only between that which can be said and that which can be shown, but also a distinction between that which is established by argument, possibly non-constructive argument, and that which is exhibited in intuition, i.e., constructed. If this is right, Wittgenstein showed an early sensitivity to the possibility of constructivist thought, and his later development might be understood as a development of the constructivist strand within his thinking, making Wittgenstein’s development more linear than is often recognized (though there are many scholars who argue for the unity of Wittgenstein’s development on different principles). The saying/showing distinction may be the acorn from which the oak tree of the Philosophical Investigations (and the subsequently published posthumous works) grew.

For the early Wittgenstein, the distinction between saying and showing was thoroughly integrated into his idea of logic, and while in the later sections of the Tractatus the mysticism of what which can only be shown but cannot be said becomes more evident, it is impossible to say whether it was the logical impulse that prevailed, and served as the inspiration for the mysticism, or whether it was the mystic impulse that prevailed, and served as the pretext for formulating the logical doctrines. But the logical doctrines are clearly present in the Tractatus, and serve as the exposition of Wittgenstein’s ideas, even up to the famous metaphor when Wittgenstein says that the propositions of the Tractatus are like a ladder than one must cast away after having climbed up and over it.

Just as there is a mathematical content to Brouwer’s mysticism, so too there is a logical content to Wittgenstein’s mysticism. It is, in fact, likely that Wittgenstein’s distinction between saying and showing was suggested to him by what is now called the “picture theory of meaning” given an exposition in the Tractatus. Few philosophers today defend Wittgenstein’s picture theory of meaning, but it is central to the metaphysics of the Tractatus. For Wittgenstein, the logical structure of a proposition can be shown but not said. Since for Wittgenstein in his Tractarian period, “The facts in logical space are the world” (1.13), and “In the proposition the thought is expressed perceptibly through the senses” (3.1) — i.e., the proposition literally exhibits its structure in sensory intuition — thus, “The proposition is a picture of reality.” (4.01) One might even say that a proposition exhibits the world in intuition.

Today these formulations strike us as a bit odd, because we think of anything that can be formulated in logical terms as a paradigm case of something that can be said, and very possibly also something that may not be showable. For us, logic is a language is among languages, and one way among many to express the world; for the early Wittgenstein, on the contrary, logic is the structure of the world. It shows itself because the world shows itself, and after showing itself there is nothing more to be said. The only appropriate response is silence.

As we all know from the final sentence of the Tractatus, whereof one cannot speak, thereof one must remain silent. According to the Wittgenstein of the Tractatus, all scientific questions can be asked and all scientific questions can be answered (shades of Hilbert’s “Wir müssen wissen. Wir werden wissen.” — which Per Martin-Löf has called Hilbert’s solvability axiom, and which is the very antithesis of Brouwer’s rejection of the law of the excluded middle), but even when we have answered all scientific questions, the problems of life remain untouched.

As implied by the early Wittgenstein’s insistence upon the solvability of all scientific questions, the metaphysics of Brouwer and Wittgenstein were very different. Their common constructivism does not prevent their having fundamental, I might even say foundational, differences. Also, while Wittgenstein comes across in a melancholic fashion (a lot like Plotinus, another philosophical mystic), he is not fixated on the evils of the world in the same way that Brouwer was. If both Brouwer and Wittgenstein can be called mystics, they are mystics belonging to different traditions. Brouwer was a choleric mystic while Wittgenstein was melancholic mystic.

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Nazca to Ica

23 January 2012

Monday


A short distance north of Nazca, along the Panamericana, and situated between the designs of the “hands” (“manos“) and the “tree” (“arbol“), there is a tower (the “Torre Mirador”) that can be climbed, probably about 40 or 50 feet in height, in order to view some part of the lines of Nazca without flying over them. This close-up view of the lines clearly reveals the construction methods that I quoted yesterday (in Lines in the Desert) from Mason’s The Ancient Civilizations of Peru — stones have been removed from within geometrically defined areas and the removed stones have been piled at the edges of the designs. The piled stones not only represent the space cleared, but the piles themselves serve to make the demarcation between cleared and non-cleared areas all the more obvious, making the distinction more visually striking.

This construction technique was also used at nearby Palpa, and continues to be effective in the present day, as driving along the Panamericana (once outside the archaeologically preserved area) one sees a variety of messages spelled out in the desert, from the initials and names of individuals to fairly elaborate advertisements for small roadside stores.

In my naïveté I though that any intrepid visitor of sufficient curiosity might walk out into the desert and and look at the construction of the lines for themselves, but the desert has been fenced off along the Panamericana to prevent further damage to the lines, and once made aware of the threat it becomes immediately obvious how damaged many of the lines and figures are, which accounts for some of the difficulty in seeing some of the patterns from the air. Some — but not all.

Much is revealed by a close inspection (as one can gain from the tower along the Panamericana) that is lost in a distant view from the air, just as much is revealed in a distant inspection from the air that is close in the close-up view from near the ground. This is a perfect concrete illustration of what I was recently writing about in relation to the distinction between constructive and non-constructive thought (in P or not-P). In this post (on my other blog) I employed an image taken from Alain Connes to illustrate the constructive/non-constructive distinction such that the constructive perspective is like that of a mountain climber while the non-constructive perspective is like that of a visitor who flies over the summit of a mountain laboriously climbed by the other.

Any thorough investigation will want to make use of both perspectives in order to obtain the most comprehensive perspective possible — even though each perspective has its blind spots and its shadows that compromise our perspective on the whole. Indeed, it is precisely because each perspective incorporates deficits specific to the perspective that one will want to supplement any one perspective without another perspective with a different set of specific deficits. Between two or more fundamentally different perspectives on any one state-of-affairs there is the possibility of constructing the comprehensive conception that is excluded by any one perspective in isolation.

The two perspectives offered on the Nazca lines by the tower and an airplane flyover also reminded me of a point that I imperfectly attempted to make in my post on Epistemic Orders of Magnitude, in which I employed aerial photographs of cities in order to demonstrate the similar structures of cities transformed in the imagine of industrial-technological civilization. This similarity in structure may be masked by one’s experience of an urban area from the perspective of passing through the built environment on a human scale — i.e., simply walking through a city, which is how most people experience an urban area.

Now, in light of what I have subsequently written about constructivism, I might say that our experience of a built environment is intrinsically constructive, except for that of the urban planner or urban designer, who must see (or attempt to see) things whole. However, the urban planner must also inform his or her work with the street-level “constructive” perspective or the planning made exclusively from a top-down perspective is likely to be a failure. Almost all of the most spectacular failures in urban design have come about from an attempt to impose, from the top down, a certain vision and a certain order which may be at odds with the organically emergent order that rises from the bottom up.

This reflection gives us yet another perspective on utopianism, which I have many times tried to characterize in my attempts to show the near (not absolute) historical inevitability of utopian schemes transforming themselves upon their attempted implementation into dystopian nightmares — the utopian planner attempts to design from a purely non-constructive perspective without the benefit of a constructive perspective. This dooms the utopian plans to inevitable blindspots, shadows, and deficits. The oversights of a single perspective then, in the fullness of time, create the conditions for cascading catastrophic failure.

Historically speaking, it is not difficult to see how this comes about. After the astonishing planned cities of early antiquity, many from prehistoric societies that have left us little record except for their admirably regular and disciplined town plans, Europeans turned to a piecemeal, organic approach to urbanism. Once this approach was rapidly outgrown when cities began their burgeoning growth with the advent of the Industrial Revolution, it was a natural response on the part of Haussman-esque planners to view organic urbanism as a “failure” that necessitated replacement by another model that envisioned the already-built environment as a tabula rasa to be re-built according to rational standards. Cities henceforth were to be wholly planned to address to inadequacies of the medieval pattern of non-planning, which could not cope with cities with populations that now numbered in the millions.

I have observed elsewhere (in my Political Economy of Globalization) that many ancient prehistoric societies were essentially utopian constructions over which a god-king presided as a living god, present in the flesh among his people, and indeed some of the most striking examples of ancient town planning date from societies that exhibited (or seem to have exhibited) this now-vanished form of order. For only where a god-king is openly acknowledged as such can a social order based upon living and present divinity within the said social order be possible.

Nazca, however, does not seem to have been based on this social plan of a divinely-sanctioned social order which can bring utopian (and therefore likely non-constructive, top-down) planning into actual practice because of the physical presence of the god in the midst of his people. The book that I cited yesterday, The Ancient Civilizations of Peru by J. Alden Mason, has this to say of Nazca society:

“…the general picture seems to be one of a sedentary democratic people without marked class distinctions or authoritarianism, possibly without an established religion. There is less difference in the ‘richness’ or poverty of the graves, and women seem to be on an equality with men in this respect. The apparent absence of great public works, of extensive engineering features, and of temple pyramids implies a lack of authoritarian leadership. Instead, the leisure time of the people seems to have been spent in individual production, especially in the making of quantities of perfect, exquisite textiles and pottery vessels. This seems to indicate a strong cult of ancestor-worship. Cloths on which an incredible amount of labor was spent were made especially for funerary offerings and interred with the dead. The orientation seems to have been towards individualized religion rather than towards community participation, dictation, coercion, and aggression.”

J. Alden Mason, The Ancient Civilizations of Peru, Penguin Books, 1968, p. 85

Such egalitarian societies focused on the satisfaction of consumer demands were rare in the ancient world, but we should not be surprised that it was an egalitarian society, organized constructively from the bottom up, that produced the astonishing lines in the desert of the Nazca. Without an aerial perspective, the making of these lines was a thoroughly constructivistic undertaking, not even counter-balanced by a non-constructive perspective, which has only been obtained long after the Nazca civilization has disappeared, leaving only traces of itself in the dessicated sands of the desert.

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While I am posting this a couple of days after the fact, this entire account was written in longhand on the day here described.

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Sunday


Dated futurism is one of my guilty pleasures, and I have written about this previously in A Hundred Years of Futurism. Recently I’ve been reading a number of mid-twentieth century futurist works for some research I am doing. These are not the wide-eyed adolescent takes on the future, but intended to be sober analyses of what one book calls The Most Probable World. This is a project in the spirit of George Friedman’s The Next 100 Years, which I have discussed several times (cf. Ecological Succession in Cultural Geography).

The wide-eyed enthusiasm for possible futures is pure fun, but the serious attempts to try to understand a likely future constitute futurism of another order, and it deserves to be treated separately, if only because of the intentions of the author. While the science fiction scenarios have sometimes come closer to the truth than some overly-serious attempts to futurism (the latter at times approaching self-parody), this kind of nearly-chance correspondence bears some resemblance to the Gettier paradox, which can be intuitively understood as the fact that a non-functioning clock is precisely correct twice a day, but when a stopped clock is correct in indicating the time, it is not correct for the right reason.

Some of these “serious” (for lack of a better term) works of futurism are more sociological than futurist in character, and can only be called futurist in virtue of their discussion of present trends with a strong implication that the trend under discussion will be a central thread in the developments of the immediate future. In this sense, the sort of sober “futurist” works to which I am here referring needn’t even mention the future or prediction. The future is understood to be embodied in the pregnant present, if only we can recognize the inchoate future in embryo.

I would like to suggest that these works of sober futurism are distinct from works of enthusiasm because they are based on a method, however imperfectly put into practice, and this is the method of the historical a priori imagination. In several previous posts I have had occasion to refer to R. G. Collingwood’s conception of the historical a priori imagination. This is given in the Epilogomena to his The Idea of History, as follows:

“I have already remarked that, in addition to selecting from among his authorities’ statements those which he regards as important, the historian must in two ways go beyond what his authorities tell him. One is the critical way, and this is what Bradley has attempted to analyse. The other is the constructive way. Of this he has said nothing, and to this I now propose to return. I described constructive history as interpolating, between the statements borrowed from our authorities, other statements implied by them. Thus our authorities tell us that on one day Caesar was in Rome and on a later day in Gaul ; they tell us nothing about his journey from one place to the other, but we interpolate this with a perfectly good conscience.”

“This act of interpolation has two significant characteristics. First, it is in no way arbitrary or merely fanciful: it is necessary or, in Kantian language, a priori. If we filled up the narrative of Caesar’s doings with fanciful details such as the names of the persons he met on the way, and what he said to them, the construction would be arbitrary: it would be in fact the kind of construction which is done by an historical novelist. But if our construction involves nothing that is not necessitated by the evidence, it is a legitimate historical construction of a kind without which there can be no history at all.”

“Secondly, what is in this way inferred is essentially something imagined. If we look out over the sea and perceive a ship, and five minutes later look again and perceive it in a different place, we find ourselves obliged to imagine it as having occupied intermediate positions when we were not looking. That is already an example of historical thinking ; and it is not otherwise that we find ourselves obliged to imagine Caesar as having travelled from Rome to Gaul when we are told that he was in these different places at these successive times.”

“This activity, with this double character, I shall call a priori imagination; and, though I shall have more to say of it hereafter, for the present I shall be content to remark that, however unconscious we may be of its operation, it is this activity which, bridging the gaps between what our authorities tell us, gives the historical narrative or description its continuity. That the historian must use his imagination is a commonplace; to quote Macaulay’s Essay on History, ‘a perfect historian must possess an imagination sufficiently powerful to make his narrative affecting and picturesque'; but this is to underestimate the part played by the historical imagination, which is properly not ornamental but structural. Without it the historian would have no narrative to adorn. The imagination, that ‘blind but indispensable faculty’ without which, as Kant has shown, we could never perceive the world around us, is indispensable in the same way to history: it is this which, operating not capriciously as fancy but in its a priori form, does the entire work of historical construction.”

The Idea of History, Epilegomena: 2: The Historical Imagination, R. G. Collingwood, Oxford: Oxford University Press (1946)

This is more than I have quoted from Collingwood previously, because I wanted to give a better sense of his exposition. Collingwood calls his method “constructive” (in contradistinction to being “analytic”), but from a formal point of view it is the antithesis of constructive, it is a non-constructive inference of what must be, made on the basis of what is known to be the case.

But I think that Collingwood wanted to call his method “constructive” because he wanted to bring attention to the essentially conservative and traditional aspect of historical thought that he felt himself to be describing. It is one of the remarkable aspects of Collingwood’s conception that it is both metaphysically bold and methodologically conservative. As Collingwood notes, we have no scruples in deducing that when Caesar traveled from Rome to Gaul that he covered the intervening geographical region. This is, in a sense, a necessary truth, and in so far as it is a necessary truth, it is an a priori truth — furnished by imagination.

In works of history, we can make logical deductions as to what must have happened on the basis of connecting two points in history separated by the discrete period of time. In works of futurism, we cannot do this. We have only one point at which the facts are know, and this is the present. And often the present is known far more imperfectly than we would like to admit. As time passes, and we learn more and more about the past, we realize how little we knew of the present when it was in fact present.

Thus futurism labors under a double burden of knowing only half of what is needed to logically extrapolate the historical a priori imaginative narrative, as well as knowing this half highly imperfectly. Despite these substantial handicaps, we can still stand on the firm ground of methodological naturalism in making necessary deductions about the future.

We know that the future must follow from the present as the present has followed from the past. We know furthermore that there will be some future, and that it will be filled with some content, even if we don’t know what that content is. This makes futurism profoundly non-constructive.

Beyond these logical deductions from the very structure of time itself, we know empirically and inductively that things never quite develop as we expect things to develop, meaning that trends that seem to be important in the present often come to nothing, while world-historical events often seem to emerge suddenly if not violently from subtle trends in the present that are often evident only in hindsight.

A better appreciation of non-constructivism as a method of formal reasoning, as well as of subtle trends in the present that are neglected in favor of more obvious trends, would give us a better picture of the content of history that will shape the future. Both of these are highly difficult intellectual undertakings. Despite the fact (which you will know if you are familiar with the literature of formal reasoning) that constructivism is considered a marginal if not ideological mode of thought, I find it remarkable that constructivism has been given several systematic expositions, for example, in the work of Brouwer, Heyting, Dummett, and Beeson, among many others, while non-constructivism, the default form of formal reasoning that makes no special stipulations, has been given no explicit formulation. This is an ellipsis that not only is felt in formal thought, but as we can see here is also felt in historical thought.

As for the empirical and inductive dimension of futurism, a thorough and dispassionate survey of the present, undertaken in a frame of mind informed by parallels with past neglected trends, might reveal a number of threads of historical trends in the present which might hold the key to unexpected developments in the future.

While futurism remains marginal, it is not beyond hope in being given a firmer intellectual basis than it has enjoyed to date. What I have suggested above may be taken as a research program for putting futurism on a more solid footing.

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Sunday


Postmetaphysical Thinking (Studies in Contemporary German Social Thought), Jürgen Habermas

The most successful critiques of a tradition are those critiques that not only represent a countervailing tradition of thought to that being critiqued, but that also permanently change the nature of the thought that has been critiqued. One of the strengths of philosophical thought, i.e., one of the things that gives philosophical thought its perennial nature, is that it is continually engaged in a process of self-critique. In other words, philosophers are always criticizing other philosophers, but through this process we learn things. Another term for this is reflexivity. (Of course, dear reader, I understand that many poor souls feel that philosophy is a useless enterprise, as I have recently written about in connection with science; I pity these poor souls, and I hope you will join me in this commiseration.)

Philosophers have not only criticized other philosophers but have, in particular, engaged in a reflexive critique of metaphysics for centuries, not withstanding the fact that these philosophers themselves formulated alternative metaphysics, and only philosophers practice metaphysics. Leibniz reacted against Scholastic metaphysics, which he called “vain philosophy,” Kant reacted against Leibnizian metaphysics (primarily in the form given it by Christian Wolff), positivism and empiricism reacted against idealism in the Kantian tradition. In more recent developments, structuralists reacted against the subjectivism of phenomenology, post-structuralists reacted against structuralism, and so forth.

During the twentieth century almost all philosophers, even those of deeply divergent traditions, analytical and continental alike, explicitly rejected metaphysics and set forth programs of philosophical thought that would proceed on the basis of philosophy without metaphysics at all. But as Mark Twain famously said that the rumors of his death had been greatly exaggerated, so too the rumors of the death of metaphysics have been greatly exaggerated. Indeed, I recently wrote about The Apotheosis of Metaphysics, noting how the recently emergent school of object-oriented ontology takes the metaphysical distinction between appearance and reality to a new and greater heights.

A critique that changes all subsequent iterations of a tradition constitutes an intellectual revolution. Cantor’s set theory and transfinite numbers implicitly constituted a critique of finitism among mathematicians. Since then, when finitism has been re-asserted after Cantor, it is a finitism very changed by the fact of Cantorism, which cannot be ignored. The intuitionism of Brouwer is a very different creature than the careless dismissal of the infinite as a mere façon de parler, as the great mathematician Gauss contended:

“…so protestiere ich gegen den Gebrauch einer unendlichen Größe als einer vollendeten, welche in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine ‘Façon de parler,’ indem man eigentlich von Grenzen spricht, denen gewisse Verhältnisse so nahe kommen als man will, während andern ohne Einschränkung zu wachsen verstattet ist.”

Metaphysics today is like post-Cantorian constructivism — every idea that is employed ultimately refers back, whether explicitly or implicitly, to the conceptual revolution, even if the response to that revolution is reactionary or counter-revolutionary. The anti-metaphysical animus of twentieth century philosophy was something of an intellectual revolution, and the post-metaphysics of today bears the marks of its influence.

Continental philosopher Jürgen Habermas wrote a book about Postmetaphysical Thinking, in which he lays out what he sees as the primary thematic motives underlying metaphysical thinking, and those countervailing thematic motives that have emerged as drivers of postmetaphysical thinking. Readers will not be surprised to hear that I view Habermas’ post-metaphysical thinking as simply a new iteration of metaphysics: a new species of metaphysics emergent from descent with modification. And, I think, Habermas seems to be aware of this, and his critique is tinged with a diagnostic air; he understands that, if you wait long enough, the bus of philosophy will always make another stop.

Ideas, like bodies, contain traces of their past. This is as true of metaphysical ideas as of any other more familiar ideas. Descent with modification makes of an evolved entity a palimpsest in which the history of that entity can be patiently teased out by the careful eye that looks beneath the surface. As Ortega y Gasset said of man, we can say of ideas: ideas have not an essence, but a history. Or, rather, ideas have both an essence and a history.

The history of an idea is marked by the continual reassertion of the essential character of idea, but the essential character is also marked by its historical evolution. The continuity of ideas in history displays a high degree of historical viability, which is to say that the rate of change of ideas is slower than that of, say, social institutions (which are frequently shaped by ideas), but more rapid than that of geological features (which, I have argued, shape ideas in the very long term). Since we routinely make use of a concept like geological time, we might also posit an ideational time as being the scale of time at which ideas evolve. Indeed, this might be taken as a definition of what I initially called integral history, but which I have since come to call (in my own, personal iteration of ideational descent with modification) metaphysical history, as a division within the more comprehensive context of ecological temporality.

This reflection gives me some food for thought in the development of my conception of ecological temporality, into which I can hopefully more fully integrate the idea of an ideational temporality.

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Saturday


The arch-atheist Jean-Paul Sartre

Despite having posted on this twice recently in A Note on Sartre’s Atheism and More on Sartre’s Atheism, I haven’t yet finished with this (as though one could ever be finished with an idea!).

I have, in a couple of posts, quoted a line from Sartre’s “Existentialism is a Humanism” lecture that ends with I must confine myself to what I can see:

I do not know where the Russian revolution will lead. I can admire it and take it as an example in so far as it is evident, today, that the proletariat plays a part in Russia which it has attained in no other nation. But I cannot affirm that this will necessarily lead to the triumph of the proletariat: I must confine myself to what I can see.

For corroboration from a fellow Frenchman and a fellow novelist consider this from Balzac’s Louis Lambert (not his most admired novel, but perhaps his most philosophical novel), delivered by the novel’s protagonist:

“To think is to see,” he said one day, roused by one of our discussions on the principle of human organization. “All science rests on deduction, — a chink of vision by which we descend from cause to effect returning upward from effect to cause; or, in a broader sense, poetry, like every work of art, springs from a swift perception of things.”

Honoré de Balzac, Louis Lambert, translated by Katharine Prescott Wormeley, Boston: Roberts Brothers, 1889, p. 39

Fellow Frenchman and philosopher Descartes offers more than corroboration: he stands at the foundation of the tradition from which both Balzac and Sartre come. In his most systematic work, the Principles of Philosophy (Book I, ix), Descartes presents an all-encompassing conception of thought, as is appropriate for the philosopher who is the locus classicus of the cogito:

By the word thought, I understand all that which so takes place in us that we of ourselves are immediately conscious of it; and, accordingly, not only to understand (INTELLIGERE, ENTENDRE), to will (VELLE), to imagine (IMAGINARI), but even to perceive (SENTIRE, SENTIR), are here the same as to think (COGITARE, PENSER). For if I say, I see, or, I walk, therefore I am; and if I understand by vision or walking the act of my eyes or of my limbs, which is the work of the body, the conclusion is not absolutely certain, because, as is often the case in dreams, I may think that I see or walk, although I do not open my eyes or move from my place, and even, perhaps, although I have no body: but, if I mean the sensation itself, or consciousness of seeing or walking, the knowledge is manifestly certain, because it is then referred to the mind, which alone perceives or is conscious that it sees or walks.

On the one hand, one can view these accounts as tributes to the visible and the tangible, except that Descartes, who stands at the origin of the tradition, can in no way be assimilated to materialism. On the other hand, and more interestingly, all of these accounts can be understood as expressions of various degrees of constructivism — mostly unconsciously formulated constructivism, but nevertheless an awareness that our thought must be disciplined by experience in a rigorous way if it is not to go terribly wrong. This is also a Kantian orientation, as we observed in Temporal Illusions, and Kant is counted as an ancestor of contemporary constructivism.

Skeptics have always demanded that truths be exhibited. We saw this in our previous posts about Sartre’s atheism, taking Doubting Thomas as the paradigm of the skeptic, who must needs touch the wounds of Christ with his own hands before he will believe that it is the same Christ who was crucified and subsequently risen.

It is a feature of constructivist thought, and most especially intuitionism, to reject the law of logic that is called (in Latin) tertium non datur or the Law of the Excluded Middle (LEM, or just EM). This simply states that, of two contradictory propositions, one of them most be true (“P or not-P“). Intuitively, it seems eminently reasonable, except that we all know of instances in ordinary experience that cannot be adequately described in a black-or-white, yes-or-no formulation. Non-constructive reasoning makes unlimited use of the law of the excluded middle, and as a consequence holds that all propositions have definite truth values even if we haven’t yet determined the truth value or even if we can’t determine the truth value. This can lead to strange consequences, like the famous Aristotelian example of the sea fight tomorrow: either there will be a sea battle tomorrow or there will not be a sea battle tomorrow. We don’t know at present which is true, but if we accept the logic of non-constructive reasoning, we will acknowledge that one of these propositions is true while the other is false.

The law of the excluded middle implies the principle of bivalence — the principle that there are two and only two logical values, namely true and false — and bivalence in turn implies realism. Realism as a philosophical doctrine stands in opposition to constructivism. Plato is the most famous realist philosopher, and believed that all kinds of things were real that common sense and ordinary experience don’t think of as being “real,” while at the same time disbelieving in the reality of the material world. Thus Plato is something of an antithesis to the kind insistence upon the tangibility and visibility upon which the skeptic and the materialist rely.

It is interesting, then, in the context of Sartre’s atheism and his insistence upon relying upon the seen, which we have now come to recognize as a kind of constructivism, to contrast the very different viewpoint represented by William James. One of James’ most famous essays is “The Will to Believe” in which he lays down the criteria for legitimate belief even where sufficient evidence is lacking. William James offers, “a defence of our right to adopt a believing attitude in religious matters, in spite of the fact that our merely logical intellect may not have been coerced.” Among the criteria that James invokes is when a choice is forced, which he describes like this:

…if I say to you: “Choose between going out with your umbrella or without it,” I do not offer you a genuine option, for it is not forced. You can easily avoid it by not going out at all. Similarly, if I say, “Either love me or hate me,” “Either call my theory true or call it false,” your option is avoidable. You may remain indifferent to me, neither loving nor hating, and you may decline to offer any judgment as to my theory. But if I say, “Either accept this truth or go without it,” I put on you a forced option, for there is no standing place outside of the alternative. Every dilemma based on a complete logical disjunction, with no possibility of not choosing, is an option of this forced kind.

Logical disjunction is another name used for the law of the excluded middle. Here James reveals himself as a realist, if not a Platonist, in matters of the spirit, just as we saw that Sartre revealed himself as a constructivist, if not an intuitionist, in matters of the spirit. The point I am making here is that this is not merely a difference of belief, but a difference in logic, and a difference in logic and reaches up into the ontology of each and informs an entire view of the world. People tend to think of logic, if they think of logic at all, as something recondite and removed from ordinary human experience, but this is not the case. Logic determines the relationship that we construct with the world, and it organizes how we see the world.

Nietzsche wrote in a famous line (or, perhaps I should say, a line that ought to be more famous than it perhaps is) that the nature and degree of an individual’s sexuality reaches into the highest pinnacles of his spirit. I agree with this, but I would add that the nature and kind of an individual’s logic — be it constructivist or non-constructivist — also reaches into the highest pinnacles of his spirit and indeed informs the world in which his spirit finds a home… or fails to find a home.

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Tuesday


Luigi Nono

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

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

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

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

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

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

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

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

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

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

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

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

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

detlefsen

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

handbook of metaphysics and ontology

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

James Robert Brown

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

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

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

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A Non-Constructive World

19 April 2009


An imaginary illustration of Protagoras teaching.

An imaginary illustration of Protagoras teaching.

Further Ontological Ruminations

In yesterday’s Ontological Ruminations: Six Protagorean Propositions on the Nature of Man and the World I laid down several ontological principles of a Quasi-Protagorean bent. Protagoras (ca. 490– 420 BC; Greek: Πρωταγόρας), you will recall, was one of the greatest of the Presocratics, and was famous for having said, “Man is the measure of all things: of things which are, that they are, and of things which are not, that they are not.” This is one of those rare philosophical quotations that is sufficiently famous to have survived more than two thousand years and is recognized even by those with no interest in philosophy. So when I noted yesterday that the presence of man in the world as an agent constituting the world, it was, in essence, a Protagorean observation.

Beyond the immediate Protagorean interest, my six propositions of yesterday also suggested the non-constructive character of the world, but this requires explanation to make any sense whatsoever. When I speak of constructive and non-constructive in this forum, I mean the terms in their logical, and not their social, signification. What is the logical significance of constructive and non-constructive? That is difficult to easily sum up.

Constructivism in logic, mathematics, and formal thought generally is an embarras de richesse: there are a remarkable number of distinct formulations of constructivism as well as degrees of constructivism. I was just skimming several philosophical reference works for a simple and comprehensive definition of constructivity that would cover its various manifestations, and I couldn’t find anything satisfying.

There are logical approaches to constructivism, some of which involve logic without the Law of the Excluded Middle and others of which forbid the use of “existence” axioms that posit an entity without giving a method for constructing the entity, and there are finitist approaches to constructivism that deny or limit infinitistic propositions or methods, or which confine legitimate thought to finite assertions, and there are again predicative forms of constructivism that proscribe the use of impredicative definitions and methods.

Hopefully from the above (which is admittedly rather compressed and inexact) it should at least emerge that constructivists generally place limits on formal (or ontological) thought that would not otherwise be observed.

Constructive thought is pervasively influential today for a variety of reasons, ranging from essentially constructive nature of computer science, which makes itself felt in our lives in countless ways today because of the role of computers in our lives, to the increasingly constructivistic character of the sciences.

Physics has been turned into a constructivist undertaking without much notice of this profound change in perspective, yet it retains patently idealistic strains within the generally constructive drift—especially the presumption of the rationality of the world, i.e., its amenability to rational explanation, and mathematization of physics and its consequent idealizations and simplifications. Take, for example, the claim of the impossibility of travel at the velocity of light — if a philosopher deduced properties of the world from mathematical equations he would be a laughing stock, but physicists do so with impunity.

Physicists have taken the mantle of speculation from philosophers; science today is much more speculative than philosophy ever was, and the careful pedanticism of contemporary philosophers looks like a parody of scientific method intended to elicit laughter.

Moreover, the world itself seems constructive. Indeed, the constructivity of the world on a quantum scale is dramatically demonstrated by the failure of the law of the excluded middle and bivalence for quantum states: the logic of quantum theory is a logic without tertium non datur.

We see the extent to which the world is constructive when we contemplate the gradual, piecemeal way in which any actuality would need to approach any infinity. Just as we cannot reach aleph null by adding one repeatedly to any arbitrarily large number, so we cannot attain infinite mass by adding increments of mass to any arbitrarily large mass, nor can we shrink any arbitrarily small but finite quantity to nothing by gradually reducing it in size by a finite number of steps.

In mathematics, these limits have not the same function that they have in physics, because we can conduct thought experiments in which time and temporal processes have no place. But all that it subject to the laws of physics is also subject to the laws of time, and time will not allow us more than a constructive infinity of successively adding discrete quantities a finite number of times. This process can only yield an infinite result after the passage of an infinite quantity of time.

For all that, the world is still non-constructive, and even incorrigibly non-constructive.

The particular non-constructive aspect of the world that featured in yesterday’s exposition was the impredicativity of the world. An impredicative definition defines a given entity in terms of a whole of which it is a part. Impredicative reasoning makes use of impredicative definitions, and such are not terribly unusual. Any definition of an individual man that refers to all men is an impredicative definition, since the class of all men includes the individual so defined. And, more to the point in the present context, the world constructed by an agent who is part of that world is a non-constructive conception.

Not only is the world non-constructive and impredicative, but it is also indefinable in the traditional Aristotelian sense. In Aristotle, a term is defined by citing its genus and differentia. But the world has neither genus or differentia, and therefore cannot be defined. The world is a totality the eludes capture in formal thought. Or, as I put it in my Variations on the Theme of Life: “The world” is a metaphor for a concept that cannot be made literal.

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