Einstein on Geometrical Intuition
23 November 2017
Thursday — Thanksgiving Day
Studies in Formal Thought:
Albert Einstein (14 March 1879 – 18 April 1955)
Einstein’s Philosophy of Mathematics
For some time I have had it on my mind to return to a post I wrote about a line from Einstein’s writing, Unpacking an Einstein Aphorism. The “aphorism” in question is this sentence:
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
…which, in the original German, was…
“Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nict sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.”
Although this sentence has been widely quoted out of context until it has achieved the de facto status of an aphorism, I was wrong to call it an aphorism. This sentence, and the idea it expresses, is entirely integral with the essay in which it appears, and should not be treated in isolation from that context. I can offer in mitigation that a full philosophical commentary on Einstein’s essay would run to the length of a volume, or several volumes, but this post will be something of a mea culpa and an effort toward mitigation of the incorrect impression I previously gave that Einstein formulated this idea as an aphorism.
The first few paragraphs of Einstein’s lecture, which includes the passage quoted above, constitute a preamble on the philosophy of mathematics. Einstein wrote this sententious survey of his philosophy of mathematics in order to give the listener (or reader) enough of a methodological background that they would be able to follow Einstein’s reasoning as he approaches the central idea he wanted to get across: Einstein’s lecture was an exercise in the cultivation of geometrical intuition. Unless one has some familiarity with formal thought — usually mathematics or logic — one is not likely to have an appreciation of the tension between intuition and formalization in formal thought, nor of how mathematicians use the term “intuition.” In ordinary language, “intuition” usually means arriving at a conclusion on some matter too subtle to be made fully explicit. For mathematicians, in contrast, intuition is a faculty of the mind that is analogous to perception. Indeed, Kant made this distinction, implying its underlying parallelism, by using the terms “sensible intuition” and “intellectual intuition” (which can also be called “outer” and “inner” intuition).
Intuition as employed in this formal sense has been, through most of the history of formal thought, understood sub specie aeternitatis, i.e., possessing many of the properties once reserved for divinity: eternity, immutability, impassibility, and so on. In the twentieth century this began to change, and the formal conception of intuition came to be more understood in naturalistic terms as a faculty of the human mind, and, as such, subject to change. Here is a passage from Gödel that I have quoted many times (e.g., in Transcendental Humors), in which Gödel delineates a dynamic and changing conception of intuition:
“Turing… gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but is constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure.”
“Some remarks on the undecidability results” (Italics in original) in Gödel, Kurt, Collected Works, Volume II, Publications 1938-1974, New York and Oxford: Oxford University Press, 1990, p. 306.
If geometrical intuition (or mathematical intuition more generally) is subject to change, it is also subject to improvement (or degradation). A logician like Gödel, acutely aware of the cognitive mechanisms by which he has come to grasp logic and mathematics, might devote himself to consciously developing intuitions, always refining and improving his conceptual framework, and straining toward developing new intuitions that would allow for the extension of mathematical rigor to regions of knowledge previously given over to Chaos and Old Night. Einstein did not make this as explicit as did Gödel, but he clearly had the same idea, and Einstein’s lecture was an attempt to demonstrate to his audience the cultivation of geometrical intuitions consistent with the cosmology of general relativity.
Einstein’s revolutionary work in physics represented at the time a new level of sophistication of the mathematical representation of physical phenomena. Mathematicized physics began with Galileo, and might be said to coincide with the advent of the scientific revolution, and the mathematization of physics reached a level of mature sophistication with Newton, who invented the calculus in order to be able to express his thought in mathematical form. The Newtonian paradigm in physics was elaborated as the “classical physics” of which Einstein and Infeld, like mathematical parallels of Edward Gibbon, recorded the decline and fall.
Between Einstein and Newton a philosophical revolution in mathematics occurred. The philosophy of mathematics formulated by Kant is taken by many philosophers to express the conception of mathematics to be found in Newton; I do not agree with this judgment, as much for historiographical reasons as for philosophical reasons. But perhaps if we scrape away the Kantian idealism and subjectivism there might well be a core of Newtonian philosophy of mathematics in Kant, or, if you prefer, a core of Kantian philosophy of mathematics intimated in Newton. For present purposes, this is neither here nor there.
The revolution that occurred between Newton and Einstein was the change to hypothetico-deductivism from that which preceded it. So what was it that preceded the hypothetico-deductive conception of formal systems in mathematics? I call this earlier form of mathematics, i.e., I call pre-hypothetico-deductive mathematics, categorico-deductive mathematics, because the principles or axioms now asserted hypothetically were once asserted categorically, in the belief that the truths of formal thought, i.e., of logic and mathematics, were eternal, immutable, unchanging truths, recognized by the mind’s eye as incontrovertible, indubitable, necessary truths as soon as they were glimpsed. It was often said (and is sometimes still today said), that to understand an axiom is ipso facto to see that it must be true; this is the categorico-deductive perspective.
In mathematics as it is pursued today, as an exercise in hypothetico-deductive reasoning, axioms are posited not because they are held to be necessarily true, or self-evidently true, or undeniably true; axioms need not be true at all. Axioms are posited because they are an economical point of origin for the fruitful derivation of consequences. This revolution in mathematical rigor transformed the landscape of mathematical thought so completely that Bertrand Russell, writing in the early twentieth century could write, “…mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Here formal, logical truth is entirely insulated from empirical, intuitive truth. It is at least arguable that the new formalisms made possible by the hypothetico-deductive method are at least partially responsible for Einstein’s innovations in physics. (I have earlier touched on Einstein’s conception of formalism in A Century of General Relativity and Constructive Moments within Non-Constructive Thought.)
If you are familiar with Einstein’s lecture, and especially with the opening summary of Einstein’s philosophy of mathematics, you will immediately recognize that Einstein formulates his position around the distinction between the categorico-deductive (which Einstein calls the “older interpretation” of axiomatics) and the hypothetico-deductive (which Einstein calls the “modern interpretation” of axiomatics). Drawing upon this distinction, Einstein gives us a somewhat subtler and more flexible formulation of the fundamental disconnect between the formal and the material than that which Russell paradoxically thrusts in our face. By formulating his distinction in terms of “as far as,” Einstein implies that there is a continuum of the dissociation between what Einstein called the “logical-formal” and “objective or intuitive content.”
Einstein then goes on to assert that a purely logical-formal account of mathematics joined together with the totality of physical laws allows us to say something, “about the behavior of real things.” The logical-formal alone can can say nothing about the real world; in its isolated formal purity it is perfectly rigorous and certain, but also impotent. This marvelous structure must be supplemented with empirical laws of nature, empirically discovered, empirically defined, empirically applied, and empirically tested, in order to further our knowledge of the world. Here we see Einstein making use of the hypothetico-deductive method, and supplementing it with contemporary physical theory; significantly, in order to establish a relationship between the formalisms of general relativity and the actual world he didn’t try to turn back the clock by returning to categorico-deductivism, but took up hypothetic-deductivism and ran with it.
But all of this is mere prologue. The question that Einstein wants to discuss in his lecture is the spatial extension of the universe, which Einstein distills to two alternatives:
1. The universe is spatially infinite. This is possible only if in the universe the average spatial density of matter, concentrated in the stars, vanishes, i.e., if the ratio of the total mass of the stars to the volume of the space through which they are scattered indefinitely approaches zero as greater and greater volumes are considered.
2. The universe is spatially finite. This must be so, if there exists an average density of the ponderable matter in the universe that is different from zero. The smaller that average density, the greater is the volume of the universe.
Albert Einstein, Geometry and Experience, Lecture before the Prussian Academy of Sciences, January 27, 1921. The last part appeared first in a reprint by Springer, Berlin, 1921
It is interesting to note, especially in light of the Kantian distinction noted above between sensible and intellectual intuition, that one of Kant’s four antinomies of pure reason was whether or not the universe was finite or infinite in extent. Einstein has taken this Kantian antimony of pure reason and has cast it in a light in which it is no longer exclusively the province of pure reason, and so may be answered by the methods of science. To this end, Einstein presents the distinction between a finite universe and an infinite universe in the context of the density of matter — “the ratio of the total mass of the stars to the volume of the space through which they are scattered” — which is a question that may be determined by science, whereas the purely abstract terms of the Kantian antimony allowed for no scientific contribution to the solution of the question. For Kant, pure reason could gain no traction on this paralogism of pure reason; Einstein gains traction by making the question less pure, and moves toward more engagement with reality and therefore less certainty.
It was the shift from categorico-deductivism to hypothetico-deductivism, followed by Einstein’s “completion” of geometry by the superaddition of empirical laws, that allows Einstein to adopt a methodology that is both rigorous and scientifically fruitful. (“We will call this completed geometry ‘practical geometry,’ and shall distinguish it in what follows from ‘purely axiomatic geometry’.”) Where the simplicity of Euclidean geometry allows for the straight-forward application of empirical laws to “practically-rigid bodies” then the simplest solution of Euclidean geometry is preferred, but where this fails, other geometries may be employed to resolve the apparent contradiction between mathematics and empirical laws. Ultimately, the latter is found to be the case — “the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction” — and so it is Riemannian geometry rather than Euclidean geometry that is the mathematical setting of general relativity.
Einstein’s use of Riemannian geometry is significant. The philosophical shift from categorico-deductivism to hypothetico-deductivism could be reasonably attributed to (or, at least, to follow from) the nineteenth century discovery of non-Euclidean geometries, and this discovery is an interesting and complex story in itself. Gauss (sometimes called the “Prince of Mathematicians”) discovered non-Euclidean geometry, but published none of it in his lifetime. It was independently discovered by the Hungarian János Bolyai (the son of a colleague of Gauss) and the Russian Nikolai Ivanovich Lobachevsky. Both Bolyai and Lobachevsky arrived at non-Euclidean geometry by adopting the axioms of Euclid but altering the axiom of parallels. The axioms of parallels had long been a sore spot in mathematics; generation after generation of mathematicians had sought to prove the axiom of parallels from the other axioms, to no avail. Bolyai and Lobachevsky found that they could replace the axiom of parallels with another axiom and derive perfectly consistent but strange and unfamiliar geometrical theorems. This was the beginning of the disconnect between the logical-formal and objective or intuitive content.
Riemann also independently arrived at non-Euclidean geometry, but by a different route than that taken by Bolyai and Lobachevsky. Whereas the latter employed the axiomatic method — hence its immediate relevance to the shift from the categorico-deductive to the hypothetico-deductive — Riemann employed a metrical method. That is to say, Riemann’s method involved measurements of line segments in space defined by the distance between two points. In Euclidean space, the distance between two points is given by a formula derived from the Pythagorean theorem — d = √(x2 – x1)2 + (y2 – y1)2 — so that in non-Euclidean space the distance between two points could be given by some different equation.
Whereas the approach of Bolyai and Lobachevsky could be characterized as variations on a theme of axiomatics, Riemann’s approach could be characterized as variations on a theme of analytical geometry. The applicability to general relativity becomes clear when we reflect how, already in antiquity, Eratosthenes was able to determine that the Earth is a sphere by taking measurements on the surface of the Earth. By the same token, although we are embedded in the spacetime continuum, if we take careful measurements we can determine the curvature of space, and perhaps also the overall geometry of the universe.
From a philosophical standpoint, it is interesting to ask if there is an essential relationship between the method of a non-Euclidean geometry and the geometrical intuitions engaged by these methods. Both Bolyai and Lobachevsky arrived at hyperbolic non-Euclidean geometry (an infinitude of parallel lines) whereas Riemann arrived at elliptic non-Euclidean geometry (no parallel lines). I will not attempt to analyze this question here, though I find it interesting and potentially relevant. The non-Euclidean structure of Einstein’s general relativity is more-or-less a three dimensional extrapolation of the elliptic two dimensional surface of a sphere. Our minds cannot conceive this (at least, my mind can’t conceive of it, but there may be mathematicians who, having spent their lives thinking in these terms, are able to visualize three dimensional spatial curvature), but we can formally work with the mathematics, and if the measurements we take of the universe match the mathematics of Riemannian elliptical space, then space is curved in a way such that most human beings cannot form any geometrical intuition of it.
Einstein’s lecture culminates in an attempt to gently herd his listeners toward achieving such an impossible geometrical intuition. After a short discussion of the apparent distribution of mass in the universe (in accord with Einstein’s formulation of the dichotomy between an infinite or a finite universe), Einstein suggests that these considerations point to a finite universe, and then explicitly asks in his lecture, “Can we visualize a three-dimensional universe which is finite, yet unbounded?” Einstein offers a visualization of this by showing how an infinite Euclidean plane can be mapped onto a finite surface of a sphere, and then suggesting an extrapolation from this mapping of an infinite two dimensional Euclidean space to a finite but unbounded two dimensional elliptic space as a mapping from an infinite three dimensional Euclidean space to a finite but unbounded three dimensional elliptic space. Einstein explicitly acknowledges that, “…this is the place where the reader’s imagination boggles.”
Given my own limitations when it comes to geometrical intuition, it is no surprise that I cannot achieve any facility in the use of Einstein’s intuitive method, though I have tried to perform it as a thought experiment many times. I have no doubt that Einstein was able to do this, and much more besides, and that it was, at least in part, his mastery of sophisticated forms of geometrical intuition that had much to do with his seminal discoveries in physics and cosmology. Einstein concluded his lecture by saying, “My only aim today has been to show that the human faculty of visualization is by no means bound to capitulate to non-Euclidean geometry.”
Above I said it would be an interesting question to pursue whether there is an essential relationship between formalisms and the intuitions engaged by them. This problem returns to us in a particularly compelling way when we think of Einstein’s effort in this lecture to guide his readers toward conceiving of the universe as finite and unbounded. When Einstein gave this lecture in 1922 he maintained a steady-state conception of the universe. About the same time the Russian mathematician Alexander Friedmann was formulating solutions to Einstein’s field equations that employed expanding and contracting universes, of which Einstein himself did not approve. It wasn’t until Einstein met with Georges Lemaître in 1927 that we know something about Einstein’s engagement with Lemaître’s cosmology, which would become the big bang hypothesis. (Cf. the interesting sketch of their relationship, Einstein and Lemaître: two friends, two cosmologies… by Dominique Lambert.)
Ten years after Einstein delivered his “Geometry and Experience” lecture he was hesitantly beginning to accept the expansion of the universe, though he still had reservations about the initial singularity in Lemaître’s cosmology. Nevertheless, Einstein’s long-standing defense of the counter-intuitive idea (which he attempted to make intuitively palatable) of a finite and unbounded universe would seem to have prepared his mind for Lemaître’s cosmology, as Einstein’s finite and unbounded universe is a natural fit with the big bang hypothesis: if the universe began from an initial singularity at a finite point of time in the past, then the universe derived from the initial singularity would still be finite any finite period of time after the initial singularity. Just as we find ourselves on the surface of the Earth (i.e., our planetary endemism), which is a finite and unbounded surface, so we seem to find ourselves within a finite and unbounded universe. Simply connected surfaces of these kinds possess a topological parsimony and hence would presumably be favored as an explanation for the structure of the world in which we find ourselves. Whether our formalisms (i.e., those formalisms accessible to the human mind, i.e., intuitively tractable formalisms) are conductive to this conception, however, is another question for another time.
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An illustration from Einstein’s lecture Geometry and Experience
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Studies in Formalism
1. The Ethos of Formal Thought
6. Doing Justice to Our Intuitions: A 10 Step Method
7. The Church-Turing Thesis and the Asymmetry of Intuition
8. Unpacking an Einstein Aphorism
9. The Overview Effect in Formal Thought
10. Einstein on Geometrical intuition
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Wittgenstein’s Tractatus Logico-Philosophicus was part of an early twentieth century efflorescence of formal thinking focused on logic and mathematics.
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Transcendental Humors
4 December 2015
Friday
Portrait of Michel Eyquem de Montaigne (1533–1592) by Daniel Dumonstier, around 1578.
In the last of his essays, essay XIII of Book III, “Of Experience,” Montaigne wrote:
“The mind has not willingly other hours enough wherein to do its business, without disassociating itself from the body, in that little space it must have for its necessity. They would put themselves out of themselves, and escape from being men. It is folly; instead of transforming themselves into angels, they transform themselves into beasts; instead of elevating, they lay themselves lower. These transcendental humours affright me, like high and inaccessible places; and nothing is hard for me to digest in the life of Socrates but his ecstasies and communication with demons; nothing so human in Plato as that for which they say he was called divine; and of our sciences, those seem to be the most terrestrial and low that are highest mounted; and I find nothing so humble and mortal in the life of Alexander as his fancies about his immortalisation.”
Michel Eyquem de Montaigne, Essays, Book III, “Of Experience”
In writing of “transcendental humors” Montaigne has brilliantly co-opted the medieval physiology of “humors” and gone beyond it even while employing a language that his readers would have immediately recognized. In this passage Montaigne has managed to transcend his era even while employing the language and the concepts of his time.
In medieval western medicine it was believed that the body possessed four “vital humors” including blood, phlegm, yellow bile, and black bile. This was not only a medical idea, but also a psychological idea, as differences in temperament were ascribed to an excess or deficiency of a given humor. We retain traces of these ideas in our language, as when we describe an individual as “sanguine” or “phlegmatic.” These humors were human, all-too-human. This may sound a bit strange, but if the medieval imagination had comprised the possibility other beings on other worlds, it seems likely that such an imagination would have posited other, alien humors that would have determined both the physical constitution and mental temperament of these other beings, and speculation on the character of ETI would have taken the form of suggesting what other kinds of humors there might possibly be.
It is possible that we, too, may be able to transcend the limits of our time even while continuing to employ the familiar linguistic and conceptual infrastructure that is as deeply embedded in contemporary history as Montaigne’s linguistic and conceptual infrastructure was deeply embedded in the thought of his time. It is an uncommon insight, but not an impossible insight, that throws away the ladder after having climbed up the same.
Perhaps this passage from Montaigne so appeals to me because it is so similar to my own way of thought. In my Variations on the Theme of Life I wrote (in section 572):
Biology of religion.–The more human, all-too-human a given phenomenon, the more certain it is to be called sacred or holy.
It almost sounds as though I am purposefully paraphrasing Montaigne, but when I wrote this I was not familiar with the passage from Montaigne quoted above.
One might well be accused of a “category error” to study religion in terms of biology, though in recent years this has become much more common, as quite a number of books on the evolutionary psychology of religion have appeared, though we can see above the idea is already present in Montaigne, and it occurs throughout Nietzsche, even if it is not as explicit there as we would hope, and Nietzsche lacked the detailed scientific background that would have made it possible for him to fully appreciate, and to fully develop, the idea.
We are now getting to the point at which such ideas can be formulated explicitly and given clear and unambiguous scientific content. But our linguistic and conceptual infrastructure, while it provides the basis of the possibility of our intellectual development and progress, remains limited, and moments of great insight are necessary to transcend the prejudices of our age and to begin to comprehend the ideas that, some hundreds of years from now, our descendants will be able to formulate in an explicit way.
Perhaps it is better for us at the present time that we cannot yet formulate our most elusive ideas explicitly. I am reminded of a passage from H. P. Lovecraft that I recently quoted in The Cosmos Primeval:
“The most merciful thing in the world, I think, is the inability of the human mind to correlate all its contents. We live on a placid island of ignorance in the midst of black seas of infinity, and it was not meant that we should voyage far. The sciences, each straining in its own direction, have hitherto harmed us little; but some day the piecing together of dissociated knowledge will open up such terrifying vistas of reality, and of our frightful position therein, that we shall either go mad from the revelation or flee from the light into the peace and safety of a new dark age.”
H. P. Lovecraft, “The Call of Cthulhu,” first paragraph
Lovecraft was half right, but he (like many others) failed to see, or refused to acknowledge (perhaps in Lovecraft it follows from a matter of principle), the possibility of progress in knowledge. While it is true that some go mad and some flee, while others exist on the cusp and madness and sanity, still others are able to look squarely at terrifying vistas of reality and to stare into the face of Medusa without turning to stone.
We are always engaged in the business of slowly and painstakingly assembling dissociated bits of knowledge into a larger and more comprehensive scheme, even if we are not aware that our thirst for comprehension and clarity (which thirst must certainly be accounted among the transcendental humors) is pushing us toward a revelation for which we are not prepared. Most this occurs on an historical scale of time, so that the frightening outlines of the world to come is only discerned dimly by us, and, as Lovecraft implied, this may be a mercy. But every once in a while, under the influence of especially strong transcendental humors, we may find ourselves suddenly face-to-face with the Medusa, quite unexpectedly. Such moments are definitive.
I have often quoted a passage from Kurt Gödel (most recently in Folk Concepts and Scientific Progress) about the possibility of progress in knowledge:
“Turing… gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but is constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure.”
“Some remarks on the undecidability results” (Italics in original) in Gödel, Kurt, Collected Works, Volume II, Publications 1938-1974, New York and Oxford: Oxford University Press, 1990, p. 306.
Without any intention of belittling Gödel, it is perhaps worthwhile to note in this context that Gödel himself lived on the verge of madness, and that his mental health deteriorated to the point that he essentially starved himself to death, like some western equivalent of an Indian Yogi (or, if you prefer, a starving Buddha, representations of which always have the same haunted eyes that one sees in the photographs of the logician). One can imagine Montaigne transported into another place or time, writing essays on Gödel or a starving Buddha, neither of which he ever encountered, but each of which I think would have piqued his interest, as they represent those transcendental humors that have both plagued humanity with self-imposed ascetic rigors and which have equally advanced civilization in the most unexpected ways.
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Folk Concepts and Scientific Progress
2 August 2015
Sunday
For some philosophers, naturalism is simply an extension of physicalism, which was in turn an extension of materialism. Narrow conceptions of materialism had to be extended to account for physical phenomena not reducible to material objects (like theoretical terms in science), and we can similarly view naturalism as a broadening of physicalism in order to more adequately account for the world. (I have quoted definitions of materialism and physicalism in Materialism, Physicalism, and… What?.) But, coming from this perspective, naturalism is approached from a primarily reductivist or eliminativist point of view that places an emphasis upon economy rather than adequacy in the description of nature (on reductivism and eliminativism cf. my post Reduction, Emergence, Supervenience). Here the principle of parsimony is paramount.
One target of eliminativism and reductionism is a class of concepts sometimes called “folk” concepts. The identification of folk concepts in the exposition of philosophy of science can be traced to philosopher Daniel Dennett. Dennett introduced the term “folk psychology” in The Intentional Stance and thereafter employed the term throughout his books. Here is part of his original introduction of the idea:
“We learn to use folk psychology — as a vernacular social technology, a craft — but we don’t learn it self-consciously as a theory — we learn no meta-theory with the theory — and in this regard our knowledge of folk psychology is like our knowledge of the grammar of our native tongue. This fact does not make our knowledge of folk psychology entirely unlike human knowledge of explicit academic theories, however; one could probably be a good practising chemist and yet find it embarrassingly difficult to produce a satisfactory textbook definition of a metal or an ion.”
Daniel Dennett, The Intentional Stance, Chap. 3, “Three Kinds of Intentional Psychology”
Earlier (in the same chapter of the same book) Dennett had posited “folk physics”:
“In one sense people knew what magnets were — they were things that attracted iron — long before science told them what magnets were. A child learns what the word ‘magnet’ means not, typically, by learning an explicit definition, but by learning the ‘folk physics’ of magnets, in which the ordinary term ‘magnet’ is embedded or implicitly defined as a theoretical term.”
Daniel Dennett, The Intentional Stance, Chap. 3, “Three Kinds of Intentional Psychology”
Here is another characterization of folk psychology:
“Philosophers with a yen for conceptual reform are nowadays prone to describe our ordinary, common sense, Rylean description of the mind as ‘folk psychology,’ the implication being that when we ascribe intentions, beliefs, motives, and emotions to others we are offering explanations of those persons’ behaviour, explanations which belong to a sort of pre-scientific theory.”
Scott M. Christensen and Dale R. Turner, editors, Folk Psychology and the Philosophy of Mind, Chap. 10, “The Very Idea of a Folk Psychology” by Robert A. Sharpe, University of Wales, United Kingdom
There is now quite a considerable literature on folk psychology, and many positions in the philosophy of mind are defined by their relationship to folk psychology — eliminativism is largely the elimination of folk psychology; reductionism is largely the reduction of folk psychology to cognitive science or scientific psychology, and so on. Others have gone on to identify other folk concepts, as, for example, folk biology:
Folk biology is the cognitive study of how people classify and reason about the organic world. Humans everywhere classify animals and plants into species-like groups as obvious to a modern scientist as to a Maya Indian. Such groups are primary loci for thinking about biological causes and relations (Mayr 1969). Historically, they provided a transtheoretical base for scientific biology in that different theories — including evolutionary theory — have sought to account for the apparent constancy of “common species” and the organic processes centering on them. In addition, these preferred groups have “from the most remote period… been classed in groups under groups” (Darwin 1859: 431). This taxonomic array provides a natural framework for inference, and an inductive compendium of information, about organic categories and properties. It is not as conventional or arbitrary in structure and content, nor as variable across cultures, as the assembly of entities into cosmologies, materials, or social groups. From the vantage of EVOLUTIONARY PSYCHOLOGY, such natural systems are arguably routine “habits of mind,” in part a natural selection for grasping relevant and recurrent “habits of the world.”
Robert Andrew Wilson and Frank C. Keil, The MIT Encyclopedia of the Cognitive Sciences
We can easily see that the idea of folk concepts as pre-scientific concepts is applicable throughout all branches of knowledge. This has already been made explicit:
“…there is good evidence that we have or had folk physics, folk chemistry, folk biology, folk botany, and so on. What has happened to these folk endeavors? They seem to have given way to scientific accounts.”
William Andrew Rottschaefer, The Biology and Psychology of Moral Agency, 1998, p. 179.
The simplest reading of the above is that in a pre-scientific state we use pre-scientific concepts, and as the scientific revolution unfolds and begins to transform traditional bodies of knowledge, these pre-scientific folk concepts are replaced with scientific concepts and knowledge becomes scientific knowledge. Thereafter, folk concepts are abandoned (eliminated) or formalized so that they can be systematically located in a scientific body of knowledge. All of this is quite close to the 19th century positivist August Comte’s theory of the three stages of knowledge, according to which theological explanations gave way to metaphysical explanations, which in turn gave way to positive scientific explanations, which demonstrates the continuity of positivist thought — even that philosophical thought that does not recognize itself as being positivist. In each case, an earlier non-scientific mode of thought is gradually replaced by a mature scientific mode of thought.
While this simple replacement model of scientific knowledge has certain advantages, it has a crucial weakness, and this is a weakness shared by all theories that, implicitly or explicitly, assume that the mind and its concepts are static and stagnant. Allow me to once again quote one of my favorite passage from Kurt Gödel, the importance of which I cannot stress enough:
“Turing… gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but is constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure.”
“Some remarks on the undecidability results” (Italics in original) in Gödel, Kurt, Collected Works, Volume II, Publications 1938-1974, New York and Oxford: Oxford University Press, 1990, p. 306.
Not only does the mind refine its concepts and arrive at more abstract formulations; the mind also introduces wholly new concepts in order to attempt to understand new or hitherto unknown phenomena. In this context, what this means is that we are always introducing new “folk” concepts as our experience expands and diversifies, so that there is not a one-time transition from unscientific folk concepts to scientific concepts, but a continual and ongoing evolution of scientific thought in which folk concepts are introduced, their want of rigor is felt, and more refined and scientific concepts are eventually introduced to address the problem of the folk concepts. But this process can result in the formulation of entirely new sciences, and we must then in turn hazard new “folk” concepts in the attempt to get a handle on this new discipline, however inadequate our first attempts may be to understand some unfamiliar body of knowledge.
For example, before the work of Georg Cantor and Richard Dedekind there was no science of set theory. In formulating set theory, 19th century mathematicians had to introduce a great many novel concepts (set, element, mapping) and mathematical procedures (one-to-one correspondence, diagonalization). These early concepts of set theory are now called “naïve set theory,” which have largely been replaced by (several distinct) axiomatizations of set theory, which have either formalized or eliminated the concepts of naïve set theory, which we might also call “folk” set theory. Nevertheless, many “folk” concepts of set theory persist, and Gödel spent much of his later career attempting to produce better formalizations of the concepts of set theory than those employed in now accepted axiomatizations of set theory.
As civilization has changed, and indeed as civilization emerged, we have had occasion to introduce new terms and concepts in order to describe and explain newly emergent forms of life. The domestication of plants and animals necessitated the introduction of concepts of plant and animal husbandry. The industrial revolution and the macroeconomic forces it loosed upon the world necessitated the introduction of terms and concepts of industry and economics. In each case, non-scientific folk concepts preceded the introduction of scientific concepts explained within a comprehensive theoretical framework. In many cases, our theoretical framework is not yet fully formulated and we are still in a stage of conceptual development that involves the overlapping of folk and scientific concepts.
Given the idea of folk concepts and their replacement by scientific concepts, a mature science could be defined as a science in which all folk concepts have been either formalized, transcended, or eliminated. The infinitistic nature of science mystery (which is discussed in Scientific Curiosity and Existential Need), however, suggests that there will always be sciences in an early and therefore immature stage of development. Our knowledge of the scientific method and the development of science means that we can anticipate scientific developments and understand when our intuitions are inadequate and therefore, in a sense, folk concepts. We have an advantage over the unscientific past that knew nothing of the coming scientific revolution and how it would transform knowledge. But we cannot entirely eliminate folk concepts from the early stages of scientific development, and in so far as our scientific civilization results in continuous scientific development, we will always have sciences in the early stages of development.
Scientific progress, then, does not eliminate folk concepts, but generates new and ever more folk concepts even as it eliminates old and outdated folk concepts.
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Evolutionary Transcendence
7 December 2013
Saturday
Gödel, McGinn, and the Evolution of Mind
In so far as our human, all-too-human minds supervene upon human, all-too-human bodies (i.e., brains) — subject to all the ills that flesh is heir to — the biological evolution of the brain entails the evolution of the human mind. Indeed, one of the most interesting features of hominid evolution is brain evolution, which has been called The Runaway Brain.
Brain evolution has been a distinctive constituent of human evolution.
Why did the brain take off in a runaway escalating spiral among hominids but not in other species? The likely answer is that, at some point in our history — a history that extends well back in time before homo sapiens — human beings and our predecessor species engaged in direct competition engaging intellectual capacity — the survival of the smartest. The human mind, and its distinctive form of consciousness, is the result of an evolutionary “arms race” in intelligence (perhaps even constituting a Fisherian runaway, though not necessarily sexually dimorphic). The impressive brains of several mollusc species, which possess sophisticated camera eyes and a structural complexity prerequisite to controlling the coloration of the entire skin surface, never seems to have been drawn into a cognitive evolutionary spiral as happened with hominids.
It has been argued that the human brain has reached the practical limitations of biologically-based intelligence. A much larger brain would slow down signaling between the regions of the brain, and a much smaller brain would fall below the threshold necessary for consciousness, sentience, and intelligence. While this may be true for strictly biological entities, it leaves aside the possibility of the technological enhancement of an organic brain, or the possibility of pharmaceutical-based cognitive enhancement, in the form of drugs that would improve focus and concentration, or otherwise enhance cognitive function without organic changes to the basic structure and size of the human brain. So there may yet be room for improvement, even if our brains remain more-or-less the same, biologically and organically speaking.
The kind of cognitive improvement I would like to discuss here, however, is not technological or pharmaceutical, but the kind of cognitive improvement that we have seen throughout the development of hominids and the social institutions that hominids have created in order to facilitate cognitive function (like the cumulative effects of social learning, which David Christian of Big History fame has emphasized as a crucial stage in human development). This is a technology, too, if we count ideas as social technologies. The idea of zero is a mathematical technology that allows us to think much more efficiently and effectively in mathematical terms, even if our brains are not organically improved, and even if we are subject to an entire battery of logical fallacies and cognitive biases. (Brain plasticity means that learning new ideas means actual organic changes in the brain, in terms of restructuring neural pathways, but the overall structure and function of the brain remains intact; here it is useful to distinguish between ontogenic development and phylogenic development.)
The evolution of our ideas has been almost as slow as the evolution of our brains. If we look back on the long history of hominids and see the use of tools by early hominids, and eventually the use of fire, were developments of the first importance, but which remained static for literally millions of years. Later with homo sapiens came language, and later still came written language. Only in the past two hundred years have we added electronic telecommunications (the first telegraph for regular communication became operable in 1833), which have so greatly accelerated the utility of language, and allowed language to grow and branch out in new and unexpected ways, that we tend to forget that human beings spoke to each other for tens of thousands of years before anyone was able to put into practice the idea of written symbolic communication.
This at times painfully slow pace of development contrasts with our habit of speaking of “revolutions” in our intellectual development (like the Copernican revolution). Ideas build one upon another, with the earlier often being the condition of the possibility of the later, which makes these earlier ideas, in Kantian terms, the transcendental condition of later ideas. In the same way, our bodies in evolution — and our bodies as specific to our minds, i.e., our brains — are similarly constructed incrementally through history, with the earlier developments being the condition of the possibility of later developments.
The evolutionary and incremental development of our brain, our mind that supervenes on our brain, and our ideas that supervene on our minds, is slow and gradual and only reveals its radical character over the very long term. Without the discoveries of scientific historiography, which has restored to us the once-lost deep history of our species, we might assume that everything remains unchanged in an eternal and unchanging universe — which was, in fact, the dominant conception of human beings in the cosmos in the past.
The reader may find my title — evolutionary transcendence — a bit odd, perhaps not quite right, since we understand by evolution an immanent process, thoroughly integral with the mundane world, and not at all as something “above” or “independent of” the world. Indeed, it would make no sense at all to speak of evolution that is “above” the world. Nevertheless, evolution has, over the long term, repeatedly resulted in radical transcendence that supervenes upon the incremental Kantian transcendental conditions that hold for each stage of a developmental history.
In the Oxford English Dictionary, definition 1.b. of “transcendence” reads as follows:
The attribute of being above and independent of the universe; distinguished from immanence.
The relevant sense of “immanence” (adj. 1) reads as follows:
Indwelling, inherent; actually present or abiding in; remaining within.
Both of the terms — transcendence and immanence — are heavily laden with theological connotations, as the further definitions and examples from the OED make clear. This theological baggage makes the terms problematic, but, if we set aside the connotations of otherworldliness, there is no other word for the consequences of evolution over la longue durée than than later forms — of life and of ideas — transcend earlier forms. Thus is it that existential viability in an evolving world is predicated upon the ability to change, even to the point of essential change, or what Aristotle would have called metábasis eis állo génos (μετάβασις εἰς ἄλλο γένος) — a change into another kind of genus (or category — for Aristotle this was an illegitimate leap, a non sequitur). We usually think of evolution as, quite literally, a change into another kind of species, so the that idea of a change into another kind of genus presents itself as something more radical that evolution. This is merely a façon de parler, since all the biodiversity of the world — from species through genus, family, order, class, and so forth — is the result of repeated evolutionary branching that shapes geographical varieties in the shorter term and genuses and other biological classifications in the longer term.
As suggested above, the biological evolution of the brain bears upon the cognitive history of the mind, and the cognitive history of the mind bears upon the intellectual history of ideas. Driven by the imperative of existential viability, the organism must change, if it changes at all, in such a way as to remain viable — competitive — at every stage of its development. Thus each individual change is small, while the cumulative effect increases over time. And so with the mind and its ideas: at each stage of development the mind and its ideas must be viable in and of themselves, or result in catastrophic failure that marks the extinction of this particular line of development.
We have seen this incremental improvement in mind before in the work of Kurt Gödel. In my post Gödel’s Lesson for Geopolitics (as well as in Addendum on Technological Unemployment) I quoted Gödel as follows:
“Turing . . . gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but is constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure.”
“Some remarks on the undecidability results” (Italics in original) in Gödel, Kurt, Collected Works, Volume II, Publications 1938-1974, New York and Oxford: Oxford University Press, 1990, p. 306
Gödel here reveals an infinitistic conception of mind that is at home in an infinitistic and evolving universe. In the earlier post cited above I Further commented (referencing Fukuyama’s approach to the “end of history”):
“To suppose that human moral evolution had come to an end with the advent of the idea and implementation of liberal democracy, however admirable this condition is (or would be), is to suppose that we had tried all possible ideas for human society and that there will be no new ideas (at least, there will be no new moral ideas unless we change human nature through biotechnological intervention). I do not accept either that all ideas for society have been tried and rejected or that there will be no fundamentally new ideas.”
…and…
“Gödel is right. The human mind is always developing and changing. Because the mind is not static, it formulates novel ideas on a regular basis. It is a fallacy to conflate the failure of new ideas of achieve widespread socio-political currency with the absence of novel ideas. Among the novel ideas constantly pioneered by the dynamism of human cognition are moral and political ideas. In so far as there are new moral and political ideas, there are new possibilities for human culture, society, and civilization. The works of the human mind, like the human mind itself, are not static, but are constantly developing.”
In a similar vein I also wrote about this unlimited and incremental development of ideas and the consciousness that embodies ideas in The Growth of Historical Consciousness. Our historical experience grows with the passage of history, so that later history is experienced against a different intellectual background, which changes the nature of history and its experience. This is a social instance of an argument that Bergson formulated almost a hundred years ago, when he argued that the individual’s experience of life is cumulative and therefore plays a role in later experience, which makes life non-deterministic.
It is fallacy to suppose that we are stuck with a finite stock of ideas, or a finite number of states (or forms) of consciousness (what Gödel, following Turing, calls distinguishable states of mind), just as it is a fallacy to suppose that there are only a finite number of possibilities for political society, for economic organization, for the administration of justice, for social institutions, for science, for mathematics, for philosophy, or or any other human activity. It is a fallacy, and it is a familiar fallacy that we have previously encountered in Comte de Maistre’s Finitistic Political Theory. We do not need to settle for a static, stationary conception of the human future; our aspirations can be as dynamic as our imagination is free to conceive as-yet-unactualized possibilities.
But what is the mechanism by which incremental change comes about in the mind and its ideas? How is this even possible? In my recent post The Size of the World I quoted Colin McGinn’s book Problems in philosophy: The Limits of Inquiry and his formulation he calls Transcendental Naturalism:
“Philosophy is an attempt to get outside the constitutive structure of our minds. Reality itself is everywhere flatly natural, but because of our cognitive limits we are unable to make good on this general ontological principle. Our epistemic architecture obstructs knowledge of the real nature of the objective world. I shall call this thesis transcendental naturalism, TN for short.”
Colin McGinn, Problems in philosophy: The Limits of Inquiry, pp. 2-3
Such a position makes change in the mind and change in ideas virtually impossible, but it is an impossibility predicated upon a presumption of the unchanging nature of the mind. As Gödel pointed out, mind, in its use, is not static, but is constantly developing, and because the mind and its ideas are developing, our cognitive limits are developing, i.e., changing, at times increasing, while at times knowledge pushes these limits outward and our minds expand, mastering previously inconceivable ideas. What McGinn has called “our epistemic architecture” makes us think of the static architecture of a grand edifice, like a cathedral, but we need to think of our cognitive architecture as being something more like the familiar metaphors of rebuilding a ship at sea, as famously formulated by Otto Neurath:
“There is no way of taking conclusively established pure protocol sentences as the starting point of the sciences. No tabula rasa exists. We are like sailors who must rebuild their ship on the open sea, never able to dismantle it in dry-dock and to reconstruct it there out of the best materials. Only the metaphysical elements can be allowed to vanish without trace.”
Otto Neurath, “Protocol sentences,” in Logical Positivism, edited by A.J. Ayer, Free Press, Glencoe, IL, 1959, p. 201.
Quine employed the same metaphor:
“We are like sailors who on the open sea must reconstruct their ship but are never able to start afresh from the bottom. Where a beam is taken away a new one must at once be put there, and for this the rest of the ship is used as support. In this way, by using the old beams and driftwood the ship can be shaped entirely anew, but only by gradual reconstruction.”
W. V. O. Quine, Word and Object
Naval architecture, then, under the constraint of rebuilding at sea while staying afloat, is a better model for our epistemic architecture, than the idea of a grand and permanent structure, a cathedral of the mind. Under these conditions, we can gradually bring within the sphere of the mind’s capability ever more refined and comprehensive conceptions that better reflect the structure of the world itself. Such conceptions are never perfect, but they are also never so inadequate as to leave us with an absolute mystery. Scientific mysteries are subject to revision, even as the formerly inconceivable comes within the scope of the mind and reveals new mysteries for the mind to tackle in turn.
In The Size of the World I further wrote:
“…while our cognitive abilities are admittedly limited (for all the reasons discussed above, as well as other reasons not discussed), these limits are not absolute, but rather admit of revision. McGinn’s position as stated above implies a false dichotomy between staying within the constitutive structure of our minds and getting outside it. This is a classic case of facing the sheer cliff of Mount Improbable: while it is impossible to get outside our cognitive architecture in one fell swoop, we can little by little transgress the boundaries of our cognitive architecture, each time ever-so-slightly expanding our capacities. Incrementally over time we improve our ability to stand outside those limits that once marked the boundaries of our cognitive architecture. Thus in an ironic twist of intellectual history, the evolutionary argument, rather than demonstrating metaphysical modesty, is rather the key to limiting the limitations on the human mind.”
Evolutionary transcendence comes about gradually, incrementally, bit-by-bit, reconfiguring our epistemic architecture just enough with each development that we can understand a little bit more than we understood before. Gödel’s “understand[ing] abstract terms more and more precisely as we go on using them,” such that, “more and more abstract terms enter the sphere of our understanding” is the gradual mechanism by which we “get outside the constitutive structure of our minds” — only on the margins, at the far edge of reason, and to a limited extent. Better, we revise the constitutive structure of our minds, and thereby get outside the constitutive structure of our mind as it was in the past.
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Gödel’s Lesson for Geopolitics
10 February 2012
Friday
Kurt Gödel 1906-1978
Kurt Gödel was possibly the greatest logician of the twentieth century, and certainly among the handful of greatest logicians of all time. Tarski called himself the “greatest living sane logician,” implicitly conceding Gödel first place if the qualifier “sane” is removed. Gödel’s greatest contributions were his incompleteness theorems, which have subsequently been extrapolated to an entire class of limitative theorems that formally demonstrate that which formal systems cannot prove. I just mentioned in The Clausewitzean Conception of Civilization that Gödel’s results were widely interpreted as the death-knell of Hilbert’s program to provide a finite axiomatization for all mathematics.
Gödel’s incompleteness theorems, however, were not his only contribution. Over the past few years his correspondence and unpublished papers have been published, giving a better idea of the full scope of Gödel’s thought, which ranged widely across logic, mathematics, cosmology, and even theology. Hao Wang in his Reflections on Kurt Gödel called Gödel’s, “A life of fundamental theoretical work,” and this is an apt characterization.
It strikes me as fitting and appropriate, then, to apply Gödel’s fundamental theoretical work whenever and wherever it might be applicable, and I will suggest that Gödel’s work has implications for theoretical geopolitics (and even, if there were such a discipline, for theoretical biopolitics).
Now, allow me to back up for a moment and mention Francis Fukuyama again, since I have mentioned him and the “end of history” thesis in several recent posts: Addendum on Marxist Eschatology, Another Future: The New Agriculturalism, Addendum on Neo-Agriculturalism, Geopolitics and Biopolitics, and Addendum on Geopolitics and Biopolitics to name a few. Should the reader think that I am beating a dead horse, I would submit to you that Fukuyama himself is still thinking through the consequences of his thesis. In his book The End of History and the Last Man, the idea of a “struggle for recognition” plays an important role, and Fukuyama has mentioned this again quite recently in his recent Foreign Policy essay, The Drive for Dignity. And this is the way it should be: our impatient society may frown upon spending ten or twenty years thinking through an idea, but this is what philosophers do.
In the aforementioned The End of History and the Last Man Fukuyama poses this question, related to his “end of history” thesis:
“Whether, at the end of the twentieth century, it makes sense for us to once again to speak of a coherent and directional History of of mankind that will eventually lead the greater part of humanity to liberal democracy?”
Fukuyama answers “yes” to this question, giving economics and the “struggle for recognition” as his reasons for so arguing. Although Fukuyama seems to avoid the tendentious formulation he employed earlier, yes, history is, after all, coming to an end. But wait. There is more. In his later book Our Posthuman Future and in some occasional articles, Fukuyama has argued that history can’t quite come to and end yet because science hasn’t come to an end. Moreover, the biotechnology revolution holds out either the promise or the threat of altering human nature itself, and if human nature is altered, the possibilities for our future history are more or less wide open.
From these two lines of argument I conclude that Fukuyama still thinks today that the ideological evolution of humanity has come to an end in so far as humanity is what it is today, but that this could all change if we alter ourselves. In other words, our ideological life supervenes upon our physical structure and the mode of life dictated by that physical structure. We only have a new ideological future if we change what human beings are on an essential level. Now, this is a very interesting position, and there is much to say about it, but here I am only going to say a single reason why I disagree with it.
Human moral evolution has not come to an end, and although it would probably be given a spur to further and faster growth by biotechnological interventions in human life (and most especially by human-induced human speciation, which would certainly be a major event in the history of our species), human moral evolution, and the ideological changes that supervene upon human moral evolution, will continue with or without biotechnological intervention in human life.
To suppose that human moral evolution had come to an end with the advent of the idea and implementation of liberal democracy, however admirable this condition is (or would be), is to suppose that we had tried all possible ideas for human society and that there will be no new ideas (at least, there will be no new moral ideas unless we change human nature through biotechnological intervention). I do not accept either that all ideas for society have been tried and rejected or that there will be no fundamentally new ideas.
The denial of future conceptual innovation is interesting in its own right, and constitutes a particular tradition of thought that one runs into from time to time. This is the position made famous by Ecclesiastes who said that, “The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun.” Politicians, geopoliticians, geostrategists, and strategists simpliciter have been as vulnerable to “group think” (i.e., intellectual conformity) as any other group of people, and they tend to think that if every idea has been pretty much discussed and exhausted among their circle of friends, that ideas in general have been pretty much exhausted. The idea that there are no new ideological ideas forthcoming represents group think at the nation-state level, and in part accounts for the increasing ossification of the nation-state system as it exists today. I have mentioned elsewhere the need for nothing less than a revolution to conduct a political experiment. It is no wonder, then, that new ideas don’t get much of a hearing.
To the position of Ecclesiastes we can oppose the position of Gödel, who saw clearly that some have argued and will argue for the end of the evolution of the human mind and its moral life. In a brief but characteristically pregnant lecture Gödel made the following argument:
“Turing . . . gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but is constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure.”
“Some remarks on the undecidability results” (Italics in original) in Gödel, Kurt, Collected Works, Volume II, Publications 1938-1974, New York and Oxford: Oxford University Press, 1990, p. 306
Since we are, today, living in the Age of Turing (as I write this, the entire current year of 2012 has been declared The Alan Turing Year), ushered in by the pervasive prevalence of computers in contemporary life, it is to be expected that those who follow Turing in his conception of the mind are at or near the flood-tide of their influence, and this conception might well be as pervasively prevalent as the computers that Turing made possible by his own fundamental theoretical work. And in fact, in contemporary philosophers of mind, we find a great many expressions of the essentially mechanical nature of the mind, sometimes called the computational model of the mind. It has become a commonplace to see the mind as the “software” installed in the body’s “hardware,” despite the fact that most of the advocates of a computational theory of mind also argue strongly against Cartesian dualism.
Gödel is right. The human mind is always developing and changing. Because the mind is not static, it formulates novel ideas on a regular basis. It is a fallacy to conflate the failure of new ideas of achieve widespread socio-political currency with the absence of novel ideas. Among the novel ideas constantly pioneered by the dynamism of human cognition are moral and political ideas. In so far as there are new moral and political ideas, there are new possibilities for human culture, society, and civilization. The works of the human mind, like the human mind itself, are not static, but are constantly developing.
I have recently argued that biopolitics potentially represents a fundamentally novel moral and political idea. An entire future history of humanity might be derived from what is implicit in biopolitics, and this future history would be distinct from the future history of humanity based on the idea of liberal democracy and its geopolitical theoreticians. I wrote about biopolitics because I could cite several examples and go into the idea in some level of detail (although much more detail is required — I mean a level of detail relative to the context), but there are many ideas that are similarly distinct from the conventions of contemporary statesmen and which might well be elaborated in a future that would come as a surprise to us all.
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Adventures in Geometrical Intuition
31 October 2010
Sunday
Euclid provided the model of formal thought with his axiomatization of geometry, but Euclid also, if perhaps unwittingly, provided the model of intuitive mathematical thought by his appeals to geometrical intuition.
Over the past few days I’ve posted several strictly theoretical pieces that have touched on geometrical intuition and what I have elsewhere called thinking against the grain — A Question for Philosophically Inclined Mathematicians, Fractals and the Banach-Tarski Paradox, and A visceral feeling for epsilon zero.
Benoît Mandelbrot rehabilitated geometrical intuition.
Not long previously, in my post commemorating the passing of Benoît Mandelbrot, I discussed the rehabilitation of geometrical intuition in the wake of Mandelbrot’s work. The late nineteenth and early twentieth century work in the foundations of mathematics largely made the progress that it did by consciously forswearing geometrical intuition and seeking instead logically rigorous foundations that made no appeal to our ability to visualize or conceive particular spatial relationships. Mandelbrot said that, “The eye had been banished out of science. The eye had been excommunicated.” He was right, but the logically motivated foundationalists were right also: we are misled by geometrical intuition at least as often as we are led rightly by it.
Kurt Gödel was part of the tradition of logically rigorous foundationalism, but he did not reject geometrical intuition on that account.
Geometrical intuition, while it suffered during a period of relative neglect, was never entirely banished, never excommunicated to the extent of being beyond rehabilitation. Even Gödel, who formulated his paradoxical theorems employing the formal machinery of arithmetization, therefore deeply indebted to the implicit critique of geometrical intuition, wrote: “I only wanted to show that an innate Euclidean geometrical intuition which refers to reality and is a priori valid is logically possible and compatible with the existence of non-Euclidean geometry and with relativity theory.” (Collected Papers, Vol. III, p. 255) This is, of course, to damn geometrical intuition by way of faint praise, but being damned by faint praise is not the same as being condemned (or excommunicated). Geometrical intuition was down, but not out.
As Gödel observed, even non-Euclidean geometries are compatible with Euclidean geometrical intuition. When non-Euclidean geometries were first formulated by Bolyai, Lobachevski, and Riemann (I suppose I should mention Gauss too), they were interpreted as a death-blow to geometrical intuition, but it became apparent as these discoveries were integrated into the body of mathematical knowledge that what the non-Euclidean geometries had done was not to falsify geometrical intuition by way of counter-example, but to extend geometrical intuition through further (and unexpected) examples. The development of mathematics here exhibits not Aristotelian logic but Hegelian dialectical logic: Euclidean geometry was the thesis, non-Euclidean geometry was the antithesis, and contemporary geometry, incorporating all of these discoveries, is the synthesis.
Bertrand Russell was a major player in extending the arithmetization of analysis by pursing the logicization of arithmetic.
Bertrand Russell, who was central in the philosophical struggle to find rigorous logical formulations for mathematical theories that had previously rested on geometrical intuition, wrote: “A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science.” (from the famous “On Denoting” paper) Though Russell thought of this as a test of logical theories, it is also a wholesome plan to stock the mind with counter-intuitive geometrical examples. Non-Euclidean geometry greatly contributed to the expansion and extrapolation of geometrical intuition by providing novel examples toward which intuition can expand.
In the interest of offering exercises and examples for geometrical intuition, In Fractals and the Banach-Tarski Paradox I suggested the construction of a fractal by raising a cube on each side of a cube. I realized that if instead of raising a cube we sink a cube inside, it would make for an interesting pattern. With a cube of the length of 3, six cubes indented into this cube, each of length 1, would meet the other interior cubes at a single line.
If we continue this iteration the smaller cubes inside (in the same proportion) would continue to meet along a single line. Iterated to infinity, I suspect that this would look interesting. I’m sure it’s already been done, but I don’t know the literature well enough to cite its previous incarnations.
The two dimensional version of this fractal looks like a square version of the well-known Sierpinski triangle, and the pattern of fractal division is quite similar.
One particularly interesting counter-intuitive curiosity is the ability to construct a figure of infinite length starting with an area of finite volume. If we take a finite square, cut it in half, and put the halves end-to-end, and then cut one of the halves again, and again put them end-to-end, and iterate this process to infinity (as with a fractal construction, though this is not a fractal), we take the original finite volume and stretch it out to an infinite length.
With a little cleverness we can make this infinite line constructed from a finite volume extend infinitely in both directions by cutting up the square and distributing it differently. Notice that, with these constructions, the area remains exactly the same, unlike Banach-Tarski constructions in which additional space is “extracted” from a mathematical continuum (which could be of any dimension).
Thinking of these above two constructions, it occurred to me that we might construct an interesting fractal from the second infinite line of finite area. This is unusual, because fractals usually aren’t constructed from rearranging areas in quite this way, but it is doable. We could take the middle third of each segment, cut it into three pieces, and assemble a “U” shaped construction in the middle of the segment. This process can be iterated with every segment, and the result would be a line that is infinite two over: it would be infinite in extent, and it would be infinite between any two arbitrary points. This constitutes another sense in which we might construct an infinite fractal.
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Fractals and Geometrical Intuition
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
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