The Overview Effect and Intuitive Tractability

26 March 2017

Sunday

Looking down on Earth from above may not only make us reevaluate out relationship to the planet, but may also help us to understand the planet better.

Science is a way to better understand the world, but science itself is not always easy to understand, and we often find that, after clarifying some problem through science, we must then clarify the science so that the science makes sense to us. Some call this science communication; I call it the pursuit of intuitive tractability.

While it is not part of science proper to seek intuitively tractable formulations, it is part of human nature to seek intuitively tractable formulations, as we are more satisfied with science formulated in intuitively tractable forms than with science that is not intuitively tractable. For example, there is, as yet, no intuitively tractable formulation of quantum theory, and this may be why Einstein famously wrote in a letter to Max Born that, “Quantum Mechanics is very impressive. But an inner voice tells me that it is not yet the real thing.”

When the concept of zero was introduced into mathematics, it was thought to be an advanced and difficult idea, but we now teach a number system starting with zero to children in primary school. In a similar way, the Hindu-Arabic system of numbers has displaced almost every other system of numbers because it is what I would paradoxically call an intuitive formalism, i.e., it is a formalization of the number concept that is both adequate to mathematics and closely follows our intuitive conception of number. Mathematics is easier with Hindu-Arabic numerals than other numbering systems because this numbering system is intuitively tractable. There are other formalisms for number that are equally valid and equally correct, but not as intuitively tractable.

The pursuit of intuitive tractability has also been evident in geometry, and especially the axiomatic exposition of geometry that begins with postulates accepted ab initio as self-evident, and which has been the model of rigorous mathematics ever since Euclid. Euclid’s fifth postulate, the famous parallel postulate, is difficult to understand and was a theoretical problem for geometry until its independence was proved, but whether or not the fifth postulate was demonstrably independent of the other postulates, Euclid’s opaque exposition did not help. Here is Euclid’s parallel axiom from the Elements:

“If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.”

Almost two thousand years later, in 1846, John Playfair formulated what we now call “Playfair’s axiom,” which tells us everything that Euclid’s postulate sought to communicate, but in a far more intuitively tractable form: “In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.” Once this more intuitively tractable formulation of the parallel postulate was available, Euclid’s formulation was largely abandoned. There is, then, a process of cognitive selection, whereby the most intuitively tractable formulations are preserved and the less intuitively tractable formulations are abandoned.

Those concepts that are the most intuitively tractable are those concepts that are familiar to us all and which are seamlessly integrated into ordinary thought and language. I have called such concepts “folk concepts.” Folk concepts that have persisted from their origins in our earliest evolutionary psychology up into the present have been subjected to the cognitive equivalent of natural selection, so that we can reasonably speak of folk concepts as having been refined and elaborated by the experience of many generations.

In a series of posts — Folk Astrobiology, Folk Concepts of Scientific Civilization, and Folk Concepts and Scientific Progress — I have considered the nature of “folk” concepts as they have been frequently invoked, and it is natural to ask, in the light of such an inquiry, whether there is a “folk Weltanschauung” that is constituted by a cluster of folk concepts that naturally hang together, and which inform the pre-scientific (or non-scientific) way of thinking about the world.

Arguably, the idea of a folk Weltanschauung is already familiar by a number of different terms that philosophers have employed to identify the concept (or something like the concept) — naïve realism or common sense realism, for example. What Husserl called “natürliche Einstellung” and which Boyce Gibson translated as “natural standpoint” and Fred Kersten translated as “natural attitude” could be said to approximate a folk Weltanschauung. Here is how Husserl describes the natürliche Einstellung:

“I am conscious of a world endlessly spread out in space, endlessly becoming and having endlessly become in time. I am conscious of it: that signifies, above all, that intuitively I find it immediately, that I experience it. By my seeing, touching, hearing, and so forth, and in the different modes of sensuous perception, corporeal physical things with some spatial distribution or other are simply there for me, ‘on hand’ in the literal or the figurative sense, whether or not I am particularly heedful of them and busied with them in my considering, thinking, feeling, or willing.”

Edmund Husserl, Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy: First Book: General Introduction to a Pure Phenomenology, translated by Fred Kersten, section 27

Husserl characterizes the natural attitude as a “thesis” — a thesis consisting of a series of posits of the unproblematic existence of ordinary objects — that can be suspended, set aside, as it were, by the phenomenological procedure of “bracketing.” These posits could be identified with folk concepts, making the thesis of the natural standpoint into a folk Weltanschauung, but I think this interpretation is a bit forced and not exactly what Husserl had in mind.

Perhaps closer to what I am getting at than the Husserlian natural attitude is what Wilfrid Sellars has called the manifest image of man-in-the-world, or simply the manifest image. Sellars’ thought is no easier to get a handle on than Husserl’s thought, so that one never quite knows if one has gotten it right, and one can easily imagine being lectured by a specialist in the inadequacies of one’s interpretation. Nevertheless, I think that Sellers’ manifest image is closer to what I am trying to get at than Husserl’s natürliche Einstellung. Closer, but still not the same.

Sellars develops the idea of the manifest image in contrast to the scientific image, and this distinction is especially given exposition in his essay Philosophy and the Scientific Image of Man. After initially characterizing the philosophical quest such that, “[i]t is… the ‘eye on the whole’ which distinguishes the philosophical enterprise,” and distinguishing several different senses in which philosophy could be said to be a synoptic effort at understanding the world as a whole, Sellars introduces terms for contrasting two distinct ways of seeing the world whole:

“…the philosopher is confronted not by one complex many dimensional picture, the unity of which, such as it is, he must come to appreciate; but by two pictures of essentially the same order of complexity, each of which purports to be a complete picture of man-in-the-world, and which, after separate scrutiny, he must fuse into one vision. Let me refer to these two perspectives, respectively, as the manifest and the scientific images of man-in-the-world.”

Wilfrid Sellars, Philosophy and the Scientific Image of Man, section 1

Sellars’ distinction between the manifest image and the scientific image has been quite influential. A special issue of the journal Humana Mente, Between Two Images: The Manifest and Scientific Conceptions of the Human Being, 50 Years On, focused on the two images. Bas C. van Fraassen in particular has written a lot about Sellars, devoting an entire book to one of the two images, The Scientific Image, and has also written several relevant papers, such as “On the Radical Incompleteness of the Manifest Image” (Proceedings of the Biennial Meeting of the Philosophy of Science Association,Vol. 1976, Volume Two: Symposia and Invited Papers 1976, pp. 335-343). All of this material is well worth reading.

Sellars is at pains to point out that his distinction between manifest image and scientific image is not intended to be a distinction between pre-scientific and scientific worldviews (“…what I mean by the manifest image is a refinement or sophistication of what might be called the ‘original’ image…”), though it is clear from this exposition that the manifest image, however refined and up-to-date, has its origins in a pre-scientific conception of the world. (“It is, first, the framework in terms of which man came to be aware of himself as man-in-the-world.”) The essence of this distinction between the manifest image and the scientific image is that the manifest image is correlational while the scientific image is postulational. What this means is that the manifest image “explains” the world (in so far as it could be said to explain the world at all) by correlations among observables, while the scientific image explains the world by positing unobservables that connect observables “under the surface” of things, as it were (involving, “…the postulation of imperceptible entities”). Sellars also maintains that the manifest image cannot postulate in this way, and therefore cannot be improved or refined by science, although it can improve on itself by its own correlational methods.

I do not yet understand Sellars well enough to say why he insists that the manifest image cannot incorporate insights from the scientific image, and this is a key point of divergence between Sellars’ manifest image and what I above called a folk Weltanschauung. If a folk Weltanschauung consists of a cluster of tightly-coupled folk concepts (and perhaps a wide penumbra of associated but loosely-coupled folk concepts), then the generation of refined scientific concepts can slowly, one-by-one, replace folk concepts, so that the folk Weltanschauung gradually evolves into a more scientific Weltanschauung, even if it is not entirely transformed under the influence of scientific concepts. Science, too, consists of a cluster of tightly-coupled concepts, and these two distinct clusters of concepts — the folk and the scientific — might well resist mixing for a time, but the human mind cannot keep such matters rigorously separate, and it is inevitable that each will bleed over into the other. Sometimes this “bleeding over” is intentional, as when science reaches for metaphors or non-scientific language as a way to make its findings understood to a wider audience. This is part of the pursuit of intuitively tractable formulations, but it can also go very wrong, as when scientists adopt theological language in an attempt at a popular exposition that will not be rejected out-of-hand by the Great Unwashed.

Despite my differences with Sellars, I am going to here adopt his terminology of the manifest image and the scientific image, and I will hope that I don’t make too much of a mess of it. I will have more to say on this use of Sellars’ concepts below (especially in relation to the postulational character of the scientific image). In the meantime, I want to use Sellars’ concepts in a exposition of intuitive tractability. Sellars’ uses the metaphor of “stereoscopic vision” as the proper way to understand how we must bring together the manifest image and the scientific image as a single way of understanding the world (“…the most appropriate analogy is stereoscopic vision, where two differing perspectives on a landscape are fused into one coherent experience”). I think, on the contrary, that intuitively tractable formulations of scientific concepts can make the manifest image and the scientific image coincide, so that they are one and the same, and not two distinct images fused together. A slightly weaker formulation of this is to assert that intuitively tractable formulations allow us to integrate the manifest image and the scientific image.

Now I want to illustrate this by reference to the overview effect, that is to say, the cognitive effect of seeing our planet whole — preferably from orbit, but, if not from orbit, in photographs and film that make the point as unmistakably as though one were there, in orbit, seeing it with one’s own eyes.

Before the overview effect, we saw our planet with the same eyes, but even after it is proved to us that the planet is (roughly) a sphere, hanging suspended in space, it is difficult to believe this. All manner of scientific proofs of the world as a spherical planet can be adduced, but the science lacks intuitive tractability and we have a difficult time bringing together our scientific concepts and our folk concepts of the world — or, if you will, we have difficulty reconciling the manifest image and the scientific image. The two are distinct. Until we achieve the overview effect, there is an apparent contradiction between what we experience of the world and our scientific knowledge of the world. Our senses tell us that the world is flat and solid and unmoving; scientific knowledge tells us that the world is round and moving and hanging in space.

Once we attain the overview effect, this changes, and the apparent contradiction is revealed as apparent. The overview effect shows how the manifest image and the scientific image coincide. The things we know about ordinary objects, which shapes the manifest image, now applies to Earth, which is seen as an object rather than as surrounding us as an environment with an horizon that we can never reach, and which therefore feels endless to us. Seen from orbit, this explains itself intuitively, and an explicit explanation now appears superfluous (as is ideally the case with an axiom — it is seen to be true as soon as it is understood). The overview effect makes the scientific knowledge of our planet as a planet intuitively tractable, transforming scientific truths into visceral truths. One might say that the overview effect is the lived experience of the scientific truth of our homeworld. In this particular case, we have replaced a folk concept with a scientific concept, and the scientific concept is correct even as intuition is satisfied.

The use of the overview effect to illustrate the manifest and scientific images, and their possible coincidence in a single experience, is especially interesting in light of Sellars’ insistence that the scientific image is distinctive because it is postulational, and more particularly that it postulates unobservables as a way to explain observables. When, in a scientific context, someone speaks of unobservables or “imperceptible entities” the assumption is that we are talking about entities that are too small to see with the naked eye. The germ theory of disease and the atomic theory of matter both exemplify this idea of unobservables being observable because they are smaller than the resolution of unaided human vision. We can only observe these unobservables with instruments, and then this experience is mediated by complex instruments and an even more complex conceptual framework so that no one ever speaks of the “lived experience” of particle physics or microbiology.

In contrast to this, the Earth is unobservable to the human eye not because it is too small, but because it is too large. When shown scientific demonstrations that the world is round, we must posit an unobservable planet, and then identify this unobservable entity with the actual ground under our feet. This is difficult to do, intuitively speaking. We see the world at all times, but we do not see it as a planet. We do not see enough of the world at any one moment to see it as a planet. Enter the overview effect. Seeing the Earth whole from space reveals the entity that is planet Earth, and if one has the good fortune to lift off from Earth and experience the process of departing from its surface to then see the same from space, this makes a previously unobservable postulate into a concretely experienced entity.

We are in the same position now vis-à-vis our place within the Milky Way galaxy, and our place within the larger universe, as we were once in relation to the spherical Earth. Our accumulated scientific knowledge tells us where we are at in the universe, and where we are at in the Milky Way. We can even see a portion of the Milky Way when we look up into the night sky, but we cannot stand back and see the whole from a distance, taking in the Milky Way and pointing of the position of our solar system within one of the spiral arms of our galaxy. We know it, but we haven’t yet experienced it viscerally. We have to posit the Milky Way galaxy as a whole, the Virgo supercluster, and the filaments of galaxies that stretch through the cosmos, because they are too large for us to observe at present. They are partially observed, in the way we might say that an atom is partially observed when we look at a piece of ordinary material composed of atoms.

Our postulational scientific image of the universe in which we live is redeemed for intuition by experiences that put us in a position to view these entities with our own eyes, and so to see them in an intuitively tractable manner. Perhaps one of the reasons that quantum theory remains intuitively intractable is that the unobservables that it posits are so small that we have no hope of ever seeing them, even with an electron microscope.

Ultimately, intuitively tractable formulations of formerly difficult if not opaque scientific ideas is a function of the conceptual framework that we employ, and this is ultimately a philosophical concern. Sellars suggests that the manifest and scientific conceptual framework might be harmonized in stereoscopic vision, but he doesn’t hold out any hope that the manifest image can be integrated with the scientific image. I think that the example of the overview effect demonstrates that there are at least some cases when manifest image and scientific image can be shown to coincide, and therefore these two ways of grasping the world are not entirely alien from each other. Cosmology may be the point of contact at which the two images coincide and through which the two images can communicate.

The pursuit of intuitive tractability is, I submit, a central concern of scientific civilization. If there ever is to be a fully scientific civilization, in which scientific ways of knowing and scientific approaches to problems and their solutions are the pervasively held view, this scientific civilization will come about because we have been successful in our pursuit of intuitive tractability, and we are able to make advanced scientific concepts as familiar as the idea of zero is now familiar to us. Since the question of a conceptual framework in which rigorous science and intuitively tractable concepts can be brought together is not a scientific question, but a philosophical question, the contemporary contempt for philosophy in the special sciences is invidious to the effective pursuit of intuitive tractability. The fate of scientific civilization lies with philosophy.

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Overview Effects

The Overview Effect and Intuitive Tractability

Homeworld Effects

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The Church-Turing Thesis and the Asymmetry of Intuition

23 November 2012

Friday

Alonzo Church and Alan Turing

What is the Church-Turing Thesis? The Church-Turing Thesis is an idea from theoretical computer science that emerged from research in the foundations of logic and mathematics, also called Church’s Thesis, Church’s Conjecture, the Church-Turing Conjecture as well as other names, that ultimately bears upon what can be computed, and thus, by extension, what a computer can do (and what a computer cannot do).

Note: For clarity’s sake, I ought to point out the Church’s Thesis and Church’s Theorem are distinct. Church’s Theorem is an established theorem of mathematical logic, proved by Alonzo Church in 1936, that there is no decision procedure for logic (i.e., there is no method for determining whether an arbitrary formula in first order logic is a theorem). But the two – Church’s theorem and Church’s thesis – are related: both follow from the exploration of the possibilities and limitations of formal systems and the attempt to define these in a rigorous way.

Even to state Church’s Thesis is controversial. There are many formulations, and many of these alternative formulations come straight from Church and Turing themselves, who framed the idea differently in different contexts. Also, when you hear computer science types discuss the Church-Turing thesis you might think that it is something like an engineering problem, but it is essentially a philosophical idea. What the Church-Turing thesis is not is as important as what it is: it is not a theorem of mathematical logic, it is not a law of nature, and it not a limit of engineering. We could say that it is a principle, because the word “principle” is ambiguous and thus covers the various formulations of the thesis.

There is an article on the Church-Turing Thesis at the Stanford Encyclopedia of Philosophy, one at Wikipedia (of course), and even a website dedicated to a critique of the Stanford article, Alan Turing in the Stanford Encyclopedia of Philosophy. All of these are valuable resources on the Church-Turing Thesis, and well worth reading to gain some orientation.

One way to formulate Church’s Thesis is that all effectively computable functions are general recursive. Both “effectively computable functions” and “general recursive” are technical terms, but there is an important different between these technical terms: “effectively computable” is an intuitive conception, whereas “general recursive” is a formal conception. Thus one way to understand Church’s Thesis is that it asserts the identity of a formal idea and an informal idea.

One of the reasons that there are many alternative formulations of the Church-Turing thesis is that there any several formally equivalent formulations of recursiveness: recursive functions, Turing computable functions, Post computable functions, representable functions, lambda-definable functions, and Markov normal algorithms among them. All of these are formal conceptions that can be rigorously defined. For the other term that constitutes the identity that is Church’s thesis, there are also several alternative formulations of effectively computable functions, and these include other intuitive notions like that of an algorithm or a procedure that can be implemented mechanically.

These may seem like recondite matters with little or no relationship to ordinary human experience, but I am surprised how often I find the same theoretical conflict played out in the most ordinary and familiar contexts. The dialectic of the formal and the informal (i.e., the intuitive) is much more central to human experience than is generally recognized. For example, the conflict between intuitively apprehended free will and apparently scientifically unimpeachable determinism is a conflict between an intuitive and a formal conception that both seem to characterize human life. Compatibilist accounts of determinism and free will may be considered the “Church’s thesis” of human action, asserting the identity of the two.

It should be understood here that when I discuss intuition in this context I am talking about the kind of mathematical intuition I discussed in Adventures in Geometrical Intuition, although the idea of mathematical intuition can be understood as perhaps the narrowest formulation of that intuition that is the polar concept standing in opposition to formalism. Kant made a useful distinction between sensory intuition and intellectual intuition that helps to clarify what is intended here, since the very idea of intuition in the Kantian sense has become lost in recent thought. Once we think of intuition as something given to us in the same way that sensory intuition is given to us, only without the mediation of the senses, we come closer to the operative idea of intuition as it is employed in mathematics.

Mathematical thought, and formal accounts of experience generally speaking, of course, seek to capture our intuitions, but this formal capture of the intuitive is itself an intuitive and essentially creative process even when it culminates in the formulation of a formal system that is essentially inaccessible to intuition (at least in parts of that formal system). What this means is that intuition can know itself, and know itself to be an intuitive grasp of some truth, but formality can only know itself as formality and cannot cross over the intuitive-formal divide in order to grasp the intuitive even when it captures intuition in an intuitively satisfying way. We cannot even understand the idea of an intuitively satisfying formalization without an intuitive grasp of all the relevant elements. As Spinoza said that the true is the criterion both of itself and of the false, we can say that the intuitive is the criterion both of itself and the formal. (And given that, today, truth is primarily understood formally, this is a significant claim to make.)

The above observation can be formulated as a general principle such that the intuitive can grasp all of the intuitive and a portion of the formal, whereas the formal can grasp only itself. I will refer to this as the principle of the asymmetry of intuition. We can see this principle operative both in the Church-Turing Thesis and in popular accounts of Gödel’s theorem. We are all familiar with popular and intuitive accounts of Gödel’s theorem (since the formal accounts are so difficult), and it is not usual to make claims for the limitative theorems that go far beyond what they formally demonstrate.

All of this holds also for the attempt to translate traditional philosophical concepts into scientific terms — the most obvious example being free will, supposedly accounted for by physics, biochemistry, and neurobiology. But if one makes the claim that consciousness is nothing but such-and-such physical phenomenon, it is impossible to cash out this claim in any robust way. The science is quantifiable and formalizable, but our concepts of mind, consciousness, and free will remain stubbornly intuitive and have not been satisfyingly captured in any formalism — the determination of any such satisfying formalization could only be determined by intuition and therefore eludes any formal capture. To “prove” determinism, then, would be as incoherent as “proving” Church’s Thesis in any robust sense.

There certainly are interesting philosophical arguments on both sides of Church’s Thesis — that is to say, both its denial and its affirmation — but these are arguments that appeal to our intuitions and, most crucially, our idea of ourselves is intuitive and informal. I should like to go further and to assert that the idea of the self must be intuitive and cannot be otherwise, but I am not fully confident that this is the case. Human nature can change, albeit slowly, along with the human condition, and we could, over time — and especially under the selective pressures of industrial-technological civilization — shape ourselves after the model of a formal conception of the self. (In fact, I think it very likely that this is happening.)

I cannot even say — I would not know where to begin — what would constitute a formal self-understanding of the self, much less any kind of understanding of a formal self. Well, maybe not. I have written elsewhere that the doctrine of the punctiform present (not very popular among philosophers these days, I might add) is a formal doctrine of time, and in so far as we identify internal time consciousness with the punctiform present we have a formal doctrine of the self.

While the above account is one to which I am sympathetic, this kind of formal concept — I mean the punctiform present as a formal conception of time — is very different from the kind of formality we find in physics, biochemistry, and neuroscience. We might assimilate it to some mathematical formalism, but this is an abstraction made concrete in subjective human experience, not in physical science. Perhaps this partly explains the fashionable anti-philosophy that I have written about.

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Saying, Showing, Constructing

6 August 2012

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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Doing Justice to Our Intuitions: A 10 Step Method

15 May 2011

Sunday

Philosophical thought is often believed to be remote from the concerns of quotidian life. One of the reasons that I created this particular forum was to attempt to show the deep and systematic way that philosophical ideas penetrate even the most mundane and ordinary concerns of our daily lives.

Personally I don’t believe that a person can get out of bed in the morning without implicitly having formulated a philosophical judgment that life is worth living and therefore there is a reason to get out of bed, and not merely to lie there and do nothing. When people do lie in bed all day and do nothing they are diagnosed with a mental illness, because science is today the paradigm for dealing with such matters. However, we are under no obligation to participate in this paradigm, and we can recognize the possibility of an existential malaise that is the visceral corollary of the philosophical position that it is not worth the effort to get out of bed. This is only one of many ways in which a theoretical attitude can have practical consequences.

If philosophical ideas often seem distant from ordinary concerns, philosophical argument must seem an order of magnitude further removed from life, with its remarkable subtleties and its complex details that demand our careful attention, but I want to try to show how philosophical reasoning and argumentation have a basis in matters familiar to almost everyone, and are even at times closer to our intuition than arguments of science.

There is a passage from Carl Sagan’s book The Demon-Haunted World in which he gently makes fun of those who presume to offer up, as authoritative arguments, their gut feelings:

Often, I’m asked next, “What do you really think?”
I say, “I just told you what I really think.”
“Yes, but what’s your gut feeling?”
But I try not to think with my gut. If I’m serious about understanding the world, thinking with anything besides my brain, as tempting as that might be, is likely to get me into trouble. Really, it’s okay to reserve judgment until the evidence is in.

Carl Sagan and Ann Druyan, The Demon-Haunted World: Science as a Candle in the Dark, 1997, p. 180

While I am not without sympathy for Sagan’s point here, it strikes me as inadequate from a philosophical point of view. Sagan, whatever his reputation as a sage, was ultimately and spiritually a scientist. His thoughts are formulated like a scientist, and science-like observations (which presumably exclude gut feelings) are as crucial to science as science-like reasoning, science-like theories, and science-like predictions.

However, as philosophers we are not limited to science-like observations, any more than we are obligated to participate in the scientific paradigm of existential malaise as mental illness. In fact, as philosophers we not only have the intellectual right to pursue matters on the cusp of the ineffable, but in fact we have an intellectual duty and obligation to do so. We must go farther and test every possibility of evidence or we will fall short of full possibilities of theoretical thought.

Obviously, Sagan did not think that gut instincts constituted “evidence.” Certainly untutored instincts do not constitute scientific evidence, but they are nevertheless evidence of something, and this evidence is of the greatest philosophical interest. The point here is not whether or not our intuitions are evidence, but what the value of what evidence is, what that evidence means, and what place it ought to hold in a given body of knowledge.

There would probably be a way to formulate this in terms of Bayesianism (and hopefully some day I will take the time to work out this formulation), but I won’t pursue that at present. I will, however, pursue an alternative method to doing justice to our intuitions, instincts, and feelings.

Therefore, and without further ado, my sure-fire, quick-and-easy, step-by-step method for formulating a cogent philosophical argument merely on the basis of one’s gut instincts is as follows:

Step 1: Review the current positions and arguments in any area of philosophy that strikes your interest.

Step 2: Search your feelings for your visceral reactions to these ideas and arguments. (If you have no visceral reaction whatsoever to ideas, you probably aren’t cut out to be a philosopher.) You will notice that some of your visceral reactions to ideas will be sympathetic, and some will be antipathetic. That is to say, you will like some ideas, and other ideas you will dislike.

Step 3: Turn your attention to your viscerally negative reactions to some ideas. Examine these reactions carefully. Ask yourself, “Why do I react strongly against this idea?” Inquire carefully into your intellectual likes and dislikes.

Step 4: If you can bring your feelings to a level of explicit consciousness, you will notice that your antipathetic responses to some ideas usually follow from the fact that the ideas in question have ignored or contradicted something that you intuitively know to be the case, and perhaps also to be important. Ask yourself, “What is the intuition to which this idea has not done justice?”

Step 5: Bring your neglected or contradicted intuition to full and explicit consciousness. Develop a theoretical exposition of this intuition (or these intuitions, if there are several) on its own terms.

Step 6: Compare this exposition of your neglected intuition with ideas and arguments to which you felt an immediate sympathy. Does it tally with them? If yes, you can develop your exposition of your intuition in the context of known theories.

Step 7: If your neglected idea does not tally with existing ideas with which you are sympathetic, you will need to go up to a higher level of generality to find a systematic theoretical context in which you can formulate an exposition of your intuitions.

Step 8: If you can’t find any systematic theoretical context within which you can fit the exposition of your neglected intuition, then you will have to construct an entire metaphysics from scratch, and you’re in for a long, hard slog. Enjoy it.

Step 9: Once you have an exposition in a fully developed metaphysical context of some gut instinct to which current philosophical ideas and arguments do not do justice, confront those ideas and arguments with your now powerfully formulated exposition of their ellipses. Wait for the sparks to fly.

Step 10: If no sparks fly, and your powerful formulation of an ellipsis in contemporary philosophical thought falls dead-born from the press (or, rather, falls too low in Google rankings to ever be seen or read by anyone), prepare to die gracefully and await posthumous discovery and fame. For a philosopher, patience is a virtue and death is the least of considerations when it comes to the value of an idea.

So, there you have it — ten easy steps to philosophical wisdom, and a method for doing justice to matters of the intellect that the intellect sometimes neglects, to do justice to that which we know in our bones. Of course, if you know something in your bones that doesn’t mean that it’s true, only that it has a place in our thought. The next step is to determine what the proper place is in our thought for our instincts, intuitions, and feelings. That will require a further method.

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Studies in Mathematical Intuition

1. Epistemic Space

2. The Ethos of Formal Thought

3. Fractal Intuitions: Benoît Mandelbrot, R.I.P.

4. A Question for Philosophically Inclined Mathematicians

5. Fractal Intuitions: Fractals and the Banach-Tarski Paradox

6. Fractal Intuitions: A visceral feeling for epsilon zero

7. Adventures in Geometrical Intuition

8. Fractal Intuitions: A Note on Fractals and Banach-Tarski Extraction

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

10. Exaptations of Intuition

11. Geometrical Intuition and Epistemic Space

12. Saying, Showing, Constructing

13. One Hundred Years of Intuitionism and Formalism

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A Note on Fractals and Banach-Tarski Extraction

28 January 2011

Friday

Stephan Banach and Alfred Tarski

Further to my recent posts on fractals and the Banach-Tarski Paradox (A Question for Philosophically Inclined Mathematicians, Fractals and the Banach-Tarski Paradox, A visceral feeling for epsilon zero, and Adventures in Geometrical Intuition), I realized how the permutations of formal methodology can be schematically delineated in regard to the finitude or infinitude of the number of iterations and the methods of iteration.

Many three dimensional fractals have been investigated, but I don't know of any attempts to show an infinite fractal such that each step of the interation involves an infinite process. One reason for this as no such fractal could be generated by a computer even in its first iteration. Such a fractal can only be seen in the mind's eye. Among the factors that led to the popularity of fractals were the beautifully detailed and colored illustrations generated by computers. Mechanized assistance to intuition has its limits.

The Banach-Tarski Paradox involves a finite number of steps, but for the Banach-Tarski paradox to work the sphere in question must be infinitely divisible, and in fact we must treat the sphere like a set of points with the cardinal of the continuum. Each step in Banach-Tarski extraction is infinitely complex because it must account for an infinite set of points, but the number of steps required to complete the extraction are finite. This is schematically the antithesis of a fractal, which latter involves an infinite number of steps, but each step of the construction of the fractal is finite. Thus we can see for ourselves the first few iterations of a fractal, and we can use computers to run fractals through very large (though still finite) numbers of iterations. A fractal only becomes infinitely complex and infinitely precise when it is infinitely iterated; before it reaches its limit, it is finite in every respect. This is one reason fractals have such a strong hold on mathematical intuition.

A sphere decomposed according to the Banach-Tarski method is assumed to be mathematically decomposable into an infinitude of points, and therefore it is infinitely precise at the beginning of the extraction. The Banach-Tarski Paradox begins with the presumption of classical continuity and infinite mathematical precision, as instantiated in the real number system, since the sphere decomposed and reassembled is essentially equivalent to the real number system. There is a sense, then, in which the Banach-Tarski extraction is platonistic and non-constructive, while fractals are constructivistic. This is interesting, but we will not pursue this any further except to note once again that computing is essentially constructivistic, and no computer can function non-constructively, which implies that fractals are exactly what Benoît Mandelbrot said that they were not: an artifact of computing. However, the mathematical purity of fractals can be restored to its honor by an extrapolation of fractals into non-constructive territory, and this is exactly what an infinite fractal is, i.e., a fractal each step of the iteration of which is infinite.

Are fractals a mere artifact of computing technology? Certainly we can say that computers have been crucial to the development of fractals, but fractals need not be limited by the finite parameters of computing.

Once we see the schematic distinction between the finite operation and infinite iteration of fractals in contradistinction to the infinite operation and finite iteration of the Banach-Tarski extraction, two other possibilities defined by the same schematism appear: finite operation with finite iteration, and infinite operation with infinite iteration. The former — finite operation with finite iteration — is all of finite mathematics: finite operations that never proceed beyond finite iterations. All of the mathematics you learned in primary school is like this. Contemporary mathematicians sometimes call this primitive recursive arithmetic (PRA). The latter — infinite operation with infinite iteration — is what I recently suggested in A visceral feeling for epsilon zero: if we extract an infinite number of spheres by the Banach-Tarski method an infinite number of times, we essentially have an infinite fractal in which each step is infinite and the iteration is infinite.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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A visceral feeling for epsilon zero

30 October 2010

Saturday

In many posts to this forum, and most recently in a couple of posts about fractals — A Question for Philosophically Inclined Mathematicians and Fractals and the Banach-Tarski Paradox — I have discussed the cultivations of novel forms of intellectual intuition that allow us to transcend our native intuitions which make many demonstrable truths counter-intuitive. The cultivation of intuition is a long and arduous process; there is no royal road to it, just as Euclid once informed a king that there was no royal road to geometry.

The good news is that the more people work on difficult ideas, the easier they can make them for others. That is why it is often said that we see farther because we stand on the shoulders of giants. I have pointed out before that the idea of zero was once very advanced mathematics mastered by only a select few; now it is taught in elementary schools. People who are fascinated by ideas are always looking for new and better ways to explain them. This is a social and cultural process that makes difficult and abstract ideas widely accessible. Today, for example, with the emphasis on visual modes of communication, people spend a lot of time trying to come up with striking graphics and diagrams to illustrate an idea, knowing that if they can show what they are saying in an intuitively clear way, that they will make their point all the better.

What is required for this intuitivization of the counter-intuitive is a conceptual effort to see things in a new way, and moreover a new way that appeals to latent forms of intuition that can then be developed into robust forms of intuition. Every once in a while, someone hits upon a truly inspired intuitivization of that which was once counter-intuitive, and the whole of civilization is advanced by this individual effort of a single mind to understand better, more clearly, more transparently. By the painfully slow methods of autodidacticism I eventually came to an intuitive understanding of ε0, though I’m not sure that my particular way of coming to this understanding will be of any help to others, though it was a real revelation to me. Someplace, buried in my notebooks of a few years ago, I made a note on the day that I had my transfinite epiphany.

My recent discussion of the Banach-Tarski Paradox provides another way to think about ε0. I don’t know the details of the derivation, but if the geometrical case is anything like the arithmetical case, it would be just as easy to extract two spheres from a given sphere as to extract one. I’ve drawn an illustration of this as a branching iteration, where each sphere leads to two others (above but one). Iterated to infinity, we come to an infinite number of mathematical spheres, just as we would with the one-by-one iteration illustrated above. But, if for technical reasons, this doesn’t work, we can always derive one sphere from every previous sphere (I have also attempted to illustrate this (immediately above), which gives us a similar result as the branching iteration.

Notice that the Banach-Tarski Paradox is called a paradox and not a contradiction. It is strange, but it in no way contradicts itself; the paradox is paradoxical but logically unimpeachable. One of the things are drives home how paradoxical it is, is that a mathematical sphere (which must be infinitely divisible for the division to work) can be decomposed into a finite number of parts and finitely reassembled into two spheres. This makes the paradox feel tantalizingly close to something we might do without own hands, and not only in our minds. Notice also that fractals, while iterated to infinity, involve only a finite process at each step of iteration. That is to say, the creation of a fractal is an infinite iteration of finite operations. This makes it possible to at least begin the illustration of fractal, even if we can’t finish it. But we need not stop at this point, mathematically speaking. I have paradoxically attempted to illustrate the unillustratable (above) by showing an iteration of Banach-Tarski sphere extraction that involves extracting an infinite number of spheres at each step.

An illustration can suggest, but it cannot show, an infinite operation. Instead, we employ the ellipsis — “…” — to illustrate that which has been left out (which is the infinite part that can’t be illustrated). With transfinite arithmetic, it is just as each to extract an infinite number of arithmetical series from a given arithmetical series, as it is to extract one. If the same is true of Banach-Tarski sphere extraction (which I do not know to be the case), then, starting with a single sphere, at the first iteration we extract an infinite number of spheres from the first sphere. At the second iteration, we extract an infinite number of spheres from the previously extracted infinite number of spheres. We continue this process until we have an infinite iteration of infinite extractions. At that point, we will have ε0 spheres.

In my illustration I have adopted the convention of using “ITR” as an abbreviation of “iteration,” each level of iteration is indicated by a lower-case letter a, b, c, …, n, followed by a subscript to indicate the number of spheres extracted at this level of iteration, 1, 2, 3, …, n. Thus ITRanbn refers to the nth sphere from iteration b which in turn is derived from the nth sphere of iteration a. I think this schemata is sufficiently general and sufficiently obvious for infinite iteration, though it would lead to expressions of infinite length.

If you can not only get your mind accustomed to this, but if you can actually feel it in your bones, then you will have an intuitive grasp of ε0, a visceral feeling of epsilon zero. As I said above, it took me many years to achieve this. When I did finally “get it” I felt like Odin on the Day of the Discovery of the Runes, except that my mind hung suspended for more then nine days — more like nine years.

Odin was suspended for nine days upon the world tree Yggdrasil in his quest to know the secret of the Runes.

I will also note that, if you can see the big picture of this geometrical realization of epsilon zero, you will immediately notice that it possesses self-similarly, and therefore constitutes an infinite fractal. We could call it an infinite explosion pattern. All fractals are infinite in so far as they involve infinite iteration, but we can posit another class of fractals beyond that which involve the infinite iteration of infinite operations. We can only generate such fractals in our mind, because no computer could even illustrate the first step of an infinite fractal of this kind. This interesting idea also serves as a demonstration that fractals are not merely artifacts of computing machines, but are as platonically ideal as any mathematical object sanctioned by tradition.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Fractals and the Banach-Tarski Paradox

29 October 2010

Friday

In true Cartesian fashion I woke up slowly this morning, and while I tossed and turned in bed I thought more about the Banach-Tarkski paradox, having just written about it last night. In yesterday’s A Question for Philosophically Inclined Mathematicians, I asked, “Can we pursue this extraction of volume in something like a process of transfinite recursion, arriving at some geometrical equivalent of ε0?” The extraction in question is that of taking one mathematical sphere out of another mathematical sphere, and both being equal to the original — the paradox that was proved by Banach and Tarski. I see no reason why this process cannot be iterated, and if it can be iterated it can be iterated to infinity, and if iterated to infinity we should finish with an infinite number of mathematical spheres that would fill an infinite quantity of mathematical space.

All of this is as odd and as counter-intuitive as many of the theorems of set theory when we first learn them, but one gets accustomed to the strangeness after a time, and if one spends enough time engaged with these ideas one probably develops new intuitions, set theoretical intuitions, that stand one in better stead in regard to the strange world of the transfinite than the intuitions that one had to abandon.

In any case, it occurred to be this morning that, since decompositions of a sphere in order to reassemble two spheres from one original does not consist of discrete “parts” as we usually understand them, but of sets of points, and these sets of points would constitute something that did not fully fill the space that they inhabit, and for this reason we could speak of them as possessing fractal dimension. On fractal dimension, the Wikipedia entry says this of the Koch curve:

“…the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. It could be said that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the speculation that its dimension might best be described in a sense by a number between one and two. This is just one simple way of imagining the idea of fractal dimension.”

The first space filling-curve discovered by Giuseppe Peano (the same Peano that formulated influential axioms of arithmetic, though the axioms seem to ultimately derive from Dedekind) already demonstrated a way in which a line, ordinarily considered one dimensional, can be two dimensional — or, if you prefer to take the opposite perspective, that a plane, ordinarily considered to be two dimensional, can be decomposed into a one dimensional line. A fractal like the Koch curve fills two dimensional space to a certain extent, but not completely like Peano’s space-filling curve, and its fractal dimension is calculated as 1.26.

Hilbert's version of a space filling curve.

The Koch curve is a line that is more than a line, and it can only be constructed in two dimensions. It is easy to dream up similar fractals based on two dimensional surfaces. For example, we could take a cube and construct a cube on each side, and construct a cube on each side of these cubes, and so on. We could do the same thing with bumps raised on the surface of a sphere. Right now, we are only thinking of in terms of surfaces. The six planes of a cube enclose a volume, so we can think of it either as a two dimensional surface or as a three dimensional body. In so far as we think of the cube only as a surface, it is a two dimensional surface that can only be constructed in three dimensions. (And the cube or sphere constructions can go terribly wrong also, as if we make the iterations too large they will run into each other. Still, the appropriate construction will yield a fractal.)

This process suggests that we might construct a fractal from three dimensional bodies, but to do so we would have to do this in four dimensions. In this case, the fractal dimension of a three dimensional fractal constructed in four dimensional space would be 3.n, depending upon how much four dimensional space was filled by this fractal “body.” (And I hope you will understand why I put “body” in scare quotes.)

I certainly can’t visualize a four dimensional fractal. In fact, “visualize” is probably the wrong term, because our visualization capacity locates objects in three dimensional space. It would be better to say that I cannot conceive of a four dimensional fractal, except that I can entertain the idea, and this is a form of conception. What I mean, of course, is a form of concrete conception not tied to three dimensional visualization. I suspect that those who have spent a lifetime working with such things may approach an adequate conception of four dimensional objects, but this is the rare exception among human minds.

Just as we must overcome the counter-intuitive feeling of the ideas of set theory in order to get to the point where we are conceptually comfortable with it, so too we would need to transcend our geometrical intuitions in order to adequately conceptualize four dimensional objects (which mathematicians call 4-manifolds). I do not say that it is impossible, but it is probably very unusual. This represents an order of thinking against the grain that will stand as a permanent aspiration for those of us who will never fully attain it. Intellectual intuition, like dimensionality, consists of levels, and even if we do not fully attain to a given level of intuition, if we glimpse it after a fashion we might express our grasp as a decimal fraction of the whole.

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A seasonally-appropriate illustration of the Banach-Tarski paradox.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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A Question for Philosophically Inclined Mathematicians

28 October 2010

Friday

Given the astonishing yet demonstrable consequence of the Banach-Tarski paradox, it is the sort of thing that one’s mind returns to on a regular basis in order to savor the intellectual satisfaction of it. The unnamed author of the Layman’s Guide to the Banach-Tarski Paradox explains the paradox thus:

The paradox states that it is possible to take a solid sphere (a “ball”), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere. In other words, you’ve doubled the volume of the original sphere.

The whole of the entry at Wolfram Mathworld runs as follows:

First stated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated. (Five pieces are minimal, although four pieces are sufficient as long as the single point at the center is neglected.) A generalization of this theorem is that any two bodies in R3 that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (i.e., they are equidecomposable).

The above-mentioned Layman’s Guide to the Banach-Tarski Paradox attempts to provide an intuitive gloss on this surprising result of set theory (making use of the axiom of choice, or some equivalent assumption), and concludes with this revealing comment:

In fact, if you think about it, this is not any stranger than how we managed to duplicate the set of all integers, by splitting it up into two halves, and renaming the members in each half so they each become identical to the original set again. It is only logical that we can continually extract more volume out of an infinitely dense, mathematical sphere S.

Before I read this today, I’d never come across such a clear and concise exposition of the Banach-Tarski paradox, and in provides food for thought. Can we pursue this extraction of volume in something like a process of transfinite recursion, arriving at some geometrical equivalent of ε0? This is an interesting question, but it isn’t the question that I started out thinking about as suitable for the philosophically inclined mathematician.

When I was thinking about the Banach-Tarski paradox today, I began wondering if a sufficiently generalized formulation of the paradox could be applied to ontology on the whole, so that we might demonstrate (perhaps not with the rigor of mathematics, but as best as anything can be demonstrated in ontology) that the world entire might be decomposed into a finite number of pieces and then reassembled into two or more identical worlds.

With the intuitive gloss quoted above, we can say that this is a possibility in so far as the world is ontologically infinitely dense. What might this mean? What would it be for the world to be ontologically dense in the way that infinite sets are infinitely dense? Well, this kind of question goes far beyond intuition, and therefore lands us in the open-texture of language that can accommodate novel uses but which has no “natural” meaning one way or the other. The open-texture of even our formal languages makes it like a quicksand: if you don’t have some kind of solid connection to solid ground, you are likely to flail away until you go under. It is precisely for this reason that Kant sought a critique of reason, so that reason would not go beyond its proper bounds, which are (as Strawson put it) the bounds of sense.

But as wary as we should be of unprecedented usages, we should also welcome them as opportunities to transcend intuitions ultimately rooted in the very soil from which we sprang. I have on many occasions in this forum argued that our ideas are ultimately derived from the landscape in which we live, by way of the way of life that is imposed upon us by the landscape. But we are not limited to that which our origins bequeathed to us. We have the power to transcend our mundane origins, and if it comes at the cost of occasional confusion and disorientation, so be it.

So I suggest that while there is no “right” answer to whether the world can be considered ontologically infinitely dense, we can give an answer to the question, and we can in fact make a rational and coherent case for our answer if only we will force ourselves to make the effort of thinking unfamiliar thoughts — always a salutary intellectual exercise.

Is the world, then, ontologically infinitely dense? Is the world everywhere continuous, so that it is truly describable by a classical theory like general relativity? Or is the world ultimately grainy, so that it must be described by a non-classical theory like quantum mechanics? At an even more abstract level, can the beings of the world be said to have any density if we do not restrict beings to spatio-temporal beings, so that our ontology is sufficiently general to embrace both the spatio-temporal and the non-spatio-temporal? This is again, as discussed above, a matter of establishing a rationally defensible convention.

I have no answer to this question at present. One ought not to expect ontological mysteries to yield themselves to a few minutes of casual thought. I will return to this, and think about it again. Someday — not likely someday soon, but someday nonetheless — I may hit upon a way of thinking about the problem that does justice to the question of the infinite density of beings in the world.

I do not think that this is quite as outlandish as it sounds. Two of the most common idioms one finds in contemporary analytical philosophy, when such philosophers choose not to speak in a technical idiom, are those of, “the furniture of the universe,” and of, “carving nature at its joints.” These are both wonderfully expressive phrases, and moreover they seem to point to a conception of the world as essentially discrete. In other words, they suggest an ultimate ontological discontinuity. If this could be followed up rigorously, we could answer the above question in the negative, but the very fact that we might possibly answer the question in the negative says two important things:

1) that the question can, at least in some ways, be meaningful, and therefore as being philosophically significant and worthy of our attention, and…

2) if a question can possibly be answered the negative, it is likely that a reasonably coherent case could also be made for answering the question in the affirmative.

The Banach-Tarski paradox is paradoxical at least in part because it does not seem to, “carve nature at the joints.” This violation of our geometrical intuition comes about as a result of the development of other intuitions, and it is ultimately the clash of intuitions that is paradoxical. Kant famously maintained that there can be no conflict among moral duties; parallel to this, it might be taken as a postulate of natural reason that there can be no conflict among intellectual intuitions. While this principle has not be explicitly formulated to my knowledge, it is an assumption pervasively present in our reasoning (that is to say, it is an intuition about our intiutions). Paradoxes as telling as the Banach-Tarski paradox (or, for that matter, most of the results of set theory) remind us of the limitations of our intuitions in addition to reminding us of the limitations of our geometrical intuition.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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A Talent for Objectivity

28 January 2010

Thursday

Our contemporary usage of the term "talent" is derived from the New Testament parable of the talents.

While this may sound a bit odd, I have come to view objectivity as a talent. Some individuals have a talent for objectivity, and some decidedly do not. This is true for most talents. I also believe that the talent for objectivity is somewhat rare, perhaps not as rare as artistic vision and genius, but perhaps close to it.

Talent is closely related to intuitive or instinctive mastery of some skill, but it is not reducible to this. Dedicated individuals can cultivate and improve their talents, even while intuitive and instinctive mastery remains the standard by which all efforts are measured.

What do I mean by intuitive and instinctive mastery? Why is this the standard by which all efforts are measured? Think of it like this. Artistic vision and genius constitute a paradigm of talent. Take, for example, a person with no particular talent for drawing or painting or the visual arts (which describes myself as well as the majority of people in the world). Give such an individual intensive and extensive training in painting. Certainly their talent will improve. They might even attain a level at which they can paint competent if not beautiful pictures. But even if such a person devoted a lifetime of effort to improving their talent at painting, they would never paint like Botticelli or Jan van Eyck or Rembrandt.

On the other hand, take someone with the obvious natural talent of a Botticelli or a Rembrandt, and while their talent may show a marginal technical improvement from training, the essential gift of talent is not changed, and probably not improved. Indeed, some natural talents are spoiled by the attempt to give them the kind of formal training that turns mediocre talents into competent painters.

As I see it, objectivity is a talent analogous to a talent for painting. Anyone can attempt to exercise objective judgment, but to some people it comes naturally. If objectivity comes naturally to an individual, their objectivity will not be substantially improved by formal training intended to improve their judgment, and the training may in fact do harm to the natural intuitions that are the basis of untrained but instinctive judgment. If objectivity does not come naturally to an individual, or if an individual’s objectivity is merely of a mediocre caliber, training will improve that mediocre talent, but it will never bring the trained mediocrity to the point of intuitive objectivity (which latter may sound like something of an oxymoron).

I realize that all this may sound a little odd to the reader. Classical ideals such as objectivity, rationality, harmony, order, and proportion have not fared well in the modern world. Our aesthetic vision has been so transformed by modernity that we may well feel more attracted to the Dionysian riot of the irrational and the subjective than to the Apollonian appeal of the rational and the objective. The effects of such cultural changes are very real, but the lingering claim that classical ideals possess are no less real. Even a mind distracted by the grotesque, the horrific, and the tormented will still recognize the purity of the classical vision, regardless of that mind’s individual preferences.

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Seeing the World for What It Is

19 November 2009

Thursday

Roosting birds at Pass-a-Grille Beach, Florida; the natural world is a fitting point of departure not only for understanding nature through science, but also of understanding science through the philosophy of science.

Yesterday’s meditation upon The Fungibility of the Biome led me to think in very general terms about scientific knowledge. It is one of the remarkable things about contemporary natural science — following rigorously, as it does, the methodological naturalism toward which it has struggled over the past several hundred years since the advent of the Scientific Revolution — that the more complex and sophisticated it becomes, the more closely science is in touch with the details of ordinary experience. This is almost precisely the opposite of what one finds with most intellectual traditions. As an intellectual tradition develops it often becomes involuted and self-involved, veering off in oddball directions and taking unpredictable tangents that take us away from the world and our immediate experience of it, not closer to it. The history of human reason is mostly a history of wild goose chases.

Detail of a pelican from the above photograph.

In fact, Western science began exactly in this way, and in so doing gave us the most obvious example of an involuted, self-referential intellectual tradition that was more interested in building on a particular cluster of ideas than of learning about the world. This we now know as scholasticism, when the clerics and monks of medieval Europe read and re-read, studied and commented upon, the works of Aristotle. For a thousand years, Aristotle was synonymous with natural science.

The scholastics constructed a science upon the basis of Aristotle, rather than upon the world with Aristotle as a point of departure.

Aristotle is not to be held responsible for the non-science that was done in his name and, to add insult to injury, was called science. If Aristotle had been treated as a point of departure rather than as dogma to be defended and upheld as doctrine, medieval history would have been very different. But at that time Western history was not yet prepared for the wrenching change that science, when properly pursued, forces upon us, both in terms of our understanding of the world and the technology it makes possible (and the industry made possible in turn by technology).

Science forces wrenching change upon us because it plays havoc with some of the more absurd notions that we have inherited from our earlier, pre-scientific history. Pre-scientific beliefs suffer catastrophic failure when confronted with their scientific alternatives, however gently the science is presented in the attempt to spare the feelings of those still wedded to the beliefs of the past.

Once we get past our inherited absurdities, as I implied above, we can see the world for what it is, and science puts us always more closely in touch with what the world it is. Allow me to mention two examples of things that I have recently learned:

Example 1) We know now that not only does the earth circle the sun, and the sun spins with the Milky Way, but we know that this circling and spinning is irregular and imperfect. The earth wobbles in its orbit, and in fact the sun bobs up and down in the plane of the Milky Way as the galaxy spins. This wobbling and bobbing has consequences for life on earth because it changes the climate, sometimes predictably and sometimes unpredictably. But regularity is at least partly a function of the length of time we consider. The impact of extraterrestrial objects on the earth seems like a paradigmatic instance of catastrophism, and the asteroid impact that likely contributed to the demise of the dinosaurs is thought of as a catastrophic punctuation in the history of life, but we now also know that the earth is subject to periods of greater bombardment by extraterrestrial bodies when it is passing through the galactic plane. Viewed from a perspective of cosmological time, asteroid impacts and regular and statistically predictable. And it happens that about 65 million years ago we were passing through the galactic plane and we caught a collision as a result. All of this makes eminently good sense. Matter is present at greater density in the galactic plane, so we are far more likely to experience collisions at this time. All of this accords with ordinary experience.

Example 2) We have had several decades to get used to the idea that the continents and oceans of the earth are not static and unchanging, but dynamic and dramatically different over time. A great many things that remain consistent during the course of one human lifetime have been mistakenly thought to be eternal and unchanging. Now we know that the earth changes and in fact the whole cosmos changes. Even Einstein had to correct himself on this account. His first formulation of general relativity included the cosmological constant in order to maintain the cosmos according to its presently visible structure. Now cosmological evolution is recognized and we detail the lives of stars as carefully as we detail the natural history of a species. Now that we know something of the natural history of our planet, and we know that it changes, we find that it changes according to our ordinary experience. In the midst of an ice age, when much of the world’s water is frozen as ice and is burdening the continental plates as ice, it turns out that the weight of the ice forces the continents lower as they float in the magma beneath them. During the interglacial periods, when much or most of the ice melts, unburdened of the weight the continents bob up again and rise relative to the oceanic plates that have not been been weighted down with ice. And, in fact, this is how things behave in our ordinary experience. It is perhaps also possible (though I don’t know if this is the case) that the weight of ice, melted and now run into the oceans, becomes additional water weight pressing down on the oceanic plates, which could sink a little as a result.

Last night I was reading A Historical Introduction to the Philosophy of Science by John Losee (an excellent book, by the way, that I heartily recommend) and happened across this quote from Larry Laudan (p. 213):

…the degree of adequacy of any theory of scientific appraisal is proportional to how many of the [preferred intuitions] it can do justice to. The more of our deep intuitions a model of rationality can reconstruct, the more confident we will be that it is a sound explication of what we mean by ‘rationality’.

Contemporary Anglo-American analytical philosophers seem to love to employ the locution “deep intuitions” and similar formulations in the way that a few years ago (or a few decades ago) phenomenologists never tired of writing about the “richness of experience.” Certainly experience is rich, and certainly there are deep intuitions, but to have to call attention to either by way of awkward locutions like these points to a weakness in formulating exactly what it is that is rich about experience, and exactly what it is that is deep about a deep intuition.

And this, of course, is the whole problem in a nutshell: what exactly is a deep intuition? What intuitions ought to be considered to be preferred intuitions? I suggest that our preferred intuitions ought to be those most common and ordinary intuitions that we derive from our common and ordinary experience, things like the fact that floating bodies, when weighted down, float a little lower in the water, or whatever medium in which they happen to float. It is in this spirit that we recall the words that Robert Green Ingersoll attributed to Ferdinand Magellan:

“The church says the earth is flat, but I know that it is round, for I have seen the shadow on the moon, and I have more faith in a shadow than in the church”

The quote bears exposition. Almost certainly Magellan never said it, or even anything like it. Nevertheless, we ought to be skeptical for reasons other than those cited by the most familiar skeptics, who like to point out that the church never argued for the flatness of the earth. We ought to be skeptical because Magellan was a deeply pious man, who lost his life before the completion of his circumnavigation by his crew because Magellan was so intent upon the conversion to Catholicism of the many peoples he encountered. Eventually he encountered peoples who did not want to be converted, and they eventually took up arms and killed him in an entirely unnecessary engagement. But what remains interesting in the quote, and its implied reference to Galileo’s early observations of the moon, is not so much about flatness as about perfection. Aristotle in particular, and ancient Greek philosophy in general, held that the heavens were a realm of perfection in which all bodies were perfectly spherical and moved in perfectly circular motions through the sky. We now know this to be false, and Galileo was among the first to graphically demonstrate this with his sketches of superlunary mountains.

What does the word “superlunary” refer to? It is a term that derives from pre-Copernican (or, if you will, Ptolemaic) astronomy. When it was believed that the earth was the center of the universe, the closest extraterrestrial body was believed to be the moon (this happened to be correct, even if much in Ptolemaic astronomy was not correct). Everything below the moon, i.e., everything sublunary, was believed to be tainted and imperfect, contaminated with the dirt of lowly things and the stain of Original Sin, while everything above the moon, i.e., everything superlunary, including all other known extraterrestrial bodies, were believed to be free of this taint and therefore to be perfect, therefore unblemished. Thus it was deeply radical to observe an “imperfection” on the supposedly perfect spheres beyond the earth, as it was equally radical to discover “new” extraterrestrial bodies that had never been seen before, like the moons of Jupiter.

Both of these heresies point to our previous tendency to attribute an eternal and unchanging status to things beyond the earth. It was believed impossible to discover “new” extraterrestrial bodies because the heavens, after all, were complete, perfect, and unchanging. For the same reason, one should not be able to view anything as irregular as mountains or shadows on extraterrestrial bodies. Once we get beyond the absurd postulate of extraterrestrial perfection, we can see the world with our own eyes, and for what it is. And when we begin to do so, we do not negate the properties of perfection once attributed to the superlunary world as much as we find them to be simply irrelevant. The heavens, like the earth, are neither perfect nor imperfect. They simply are, and they are what they are. To attribute evaluative or normative content or significance to them, such as believing in their perfection, is only to send us off on one of those oddball directions or unpredictable tangents that I mentioned in the first paragraph.

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