Einstein on Geometrical Intuition

Thursday — Thanksgiving Day

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Studies in Formal Thought:

Albert Einstein (14 March 1879 – 18 April 1955)

Einstein’s Philosophy of Mathematics

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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

2. Epistemic Hubris

3. Parsimonious Formulations

4. Foucault’s Formalism

5. Cartesian Formalism

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

7. The Church-Turing Thesis and the Asymmetry of Intuition

8. Unpacking an Einstein Aphorism

9. The Overview Effect in Formal Thought

10. 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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The Study of Civilization as Formal Science and as Adventure Science

Tuesday

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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The Overview Effect in Formal Thought

Monday

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Studies in Formalism:

The Synoptic Perspective in Formal Thought

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

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

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

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

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

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

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

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

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

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

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

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

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

And of mathematical depth Hardy hazarded:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

…and Russell elaborated…

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

Loc. cit.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In P or not-P, cited above, I quoted French mathematician Alain Connes:

“Constructivism may be compared to mountain climbers who proudly scale a peak with their bare hands, and formalists to climbers who permit themselves the luxury of hiring a helicopter to fly over the summit …the uncountable axiom of choice gives an aerial view of mathematical reality — inevitably, therefore, a simplified view.”

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

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

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

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

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

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

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Studies in Formalism

1. The Ethos of Formal Thought

2. Epistemic Hubris

3. Parsimonious Formulations

4. Foucault’s Formalism

5. Cartesian Formalism

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

7. The Church-Turing Thesis and the Asymmetry of Intuition

8. Unpacking an Einstein Aphorism

9. The Overview Effect in Formal Thought

10. Einstein on Geometrical intuition

11. Methodological and Ontological Parsimony (in preparation)

12. The Spirit of Formalism (in preparation)

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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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Philosophy of History in Our Time, Revisited

Friday

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Much of what I write here, whether commenting on current affairs to delving into the depths of prehistory, could be classed under the general rubric of philosophy of history. One of my early posts to this forum was Of What Use is Philosophy of History in Our Time? (An echo of the title of Hans Meyerhoff’s widely available anthology Philosophy of History in Our Time.) It could be argued that my subsequent posts have been attempts to answer this question (that is to say, to answer the question what is the use of philosophy of history in our time), to demonstrate the usefulness of bringing a philosophical perspective to history, contemporary and otherwise. The reader is left to judge whether this attempt has been a success (partial or otherwise) or a failure (partial or otherwise).

In several recent posts — as, for example in The Science of Time, Addendum on Big History as the Science of Time, and Human Agency and the Exaptation of Selection, inter alia — I have been writing a lot about the philosophy of history from the perspective of big history, which is a contemporary historiographical school that comes to history from the perspective of the big picture and primarily proceeds according to scientific naturalism. This latter condition makes of big history a particular species of naturalism.

In many posts to this forum I have emphasized my own naturalistic perspective both in philosophy generally speaking as well as more specifically in the philosophy of history. For example, in posts such as Natural History and Human History, The Continuity of Civilization and Natural History, and An Existentialist Philosophy of History, I have emphasized the continuity of human history and natural history, especially making the attempt to place civilization in a natural historical context.

This emphasis on big history and naturalism has meant that I have spent very little time writing about alternatives to naturalistic historical thought — with a certain exception, which the reader may well not immediately recognize, so I will point it out explicitly. In several posts — The Ethos of Formal Thought, Foucault’s Formalism, Cartesian Formalism, and Formal Strategy and Philosophical Logic: Work in Progress among them — I have discussed the possibility of formal thought in relation to historical understanding, i.e., topics not usually discussed from a formal perspective (which is usually confined to logic, mathematics, and some branches of science). Formalism represents a certain kind of countervailing intellectual influence to naturalism, and it has probably served roughly that function in my thought.

I have previously mentioned Darren Staloff’s lectures on the philosophy of history, The Search for a Meaningful Past: Philosophies, Theories and Interpretations of Human History. One of the motifs running through Staloff’s lectures is a contrast between what he calls naturalism and idealism. He sums up this motif in the final lecture, in which he adopts the perspectives of naturalism and idealism in turn, trying give the listener a sense of the claims of each tradition. I found Staloff’s exposition of idealism less persuasive that his exposition of naturalism, and so I found the motif of a contrast between naturalism and idealism a bit strained, since it seemed to me that idealism really couldn’t carry its own weight in the way that it might have been able to in the past.

Recently I’ve encountered an approach to the philosophy of history that could be called “idealist” (at least in a certain sense), and this is much more persuasive to me that Staloff’s analytical representatives of the idealist tradition, like R. G. Collingwood. I have found this idealist perspective in the work of Ludwig Landgrebe, who was one of Husserl’s research assistants.

The casual reader of this blog might well have picked up on the amount of contemporary continental philosophy that I have read, but it unlikely to have realized the extent to which Edmund Husserl and phenomenology have been an influence on my thought. Nevertheless, that influence has been profound, to the point that many of Husserl’s expositors and commentators have also influenced my thinking. Recently I have been reading some essays by Ludwig Landgrebe, and this has started to give me another perspective on the philosophy of history.

Landgrebe wrote at least two papers on the philosophy of history, as well as one chapter of his book, Major Problems in Contemporary European Philosophy, from Dilthey to Heidegger. No doubt there is more material, but this is what I have found translated into English. (Landgrebe wrote an entire book on the phenomenological philosophy of history, Phänomenologie und Geschichte, but this has not been translated into English.) The two papers are “Phenomenology as Transcendental Theory of History” (which can be found in the collection of essays Husserl: Expositions and Appraisals, edited by Elliston and McCormick, University of Notre Dame Press, 1977. pp. 101-113) and “A Meditation on Husserl’s Statement: ‘History is the grand fact of absolute Being'” (The Southwestern Journal of Philosophy, Vol. 5, Issue 3, Fall 1974, pp. 111-125).

It is well known that Husserl’s last work, The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Philosophy, assembled posthumously from his papers, is the work in which Husserl placed phenomenology in historical context (for all practical purposes, for the first time), and considered the emergence of Western scientific thought in historical context. As such, this has been the point of departure of much historically-oriented phenomenological research, and the Crisis (as it has come to be known) and its supplementary texts were clearly influential for Landgrebe.

Landgrebe, however, as Husserl’s research assistant, was more than conversant with Husserl’s logical thought also. Husserl’s Experience and Judgment: Investigations in a Genealogy of Logic was a text assembled by Landgrebe from Husserl’s notes. Landgrebe consulted with Husserl throughout this project, and the original texts are all due to Husserl, but the structure of the book is entirely Landgrebe’s doing. Landgrebe brings the kind of rigor one learns in studying logic to his very compact essays on the philosophy of history. In this way, Landgrebe’s formulations have a formal character that makes them very congenial to me. Landgrebe’s approach is essentially that of a formal phenomenological theory of history, and this perspective allows me to assimilate Landgrebe’s insights both to idealistic historiography as well as my long-standing interest in formal thought.

If I were now to revise my speculative syllabus If I Lectured on the Philosophy of History (lecture 13 of which I had already assigned to phenomenology), I would definitely showcase Landgrebe’s philosophy of history as the most sophisticated phenomenological contribution to the philosophy of history.

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The Church-Turing Thesis and the Asymmetry of Intuition

Friday

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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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Theoretical Geopolitics: History and Geography

Monday

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Geopolitics and Geostrategy

as a formal sciences

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

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

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

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

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

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

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

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

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

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

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

“It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved. And the same holds in other cases where self-evident things are to be proved.”

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

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

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

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

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

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

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

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

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

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

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Benoît Mandelbrot, R.I.P.

Sunday

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Benoît B. Mandelbrot, 20 November 1924 to 14 October 2010

Famed French mathematician Benoît Mandelbrot has passed away a few days ago at the age of 85. When someone dies who has lived such a productive life, it would sound a little odd to say that we have “lost” him. We haven’t lost Benoît Mandelbrot. His contribution is permanent. The Mandelbrot set will take its place in history beside Euclidean geometry and Cantorian transfinite numbers. And when I say “history” you might think that I am consigning it to the dead past, but what I mean by “history” is an ongoing tradition of which we are a part, and which spills out into the future, informing lives yet to be lived in unpredictable and unprecedented ways.

The Mandelbrot set is a mathematical set of points in the complex plane, the boundary of which forms a fractal. The Mandelbrot set is the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. (from Wikipedia)

Mandelbrot will be best remembered for having invented (discovered? formulated?) fractals and fractal geometry. By an odd coincidence, I have been thinking quite a bit about fractals lately. A few days ago I wrote The Fractal Structure of Exponential Growth, and I had recently obtained from the library the NOVA documentary Fractals: Hunting the Hidden Dimension. As with many NOVA documentaries, I have watched this through repeatedly to try to get all that I can out of it, much as I typically listen through recorded books multiple times.

There are a couple of intuitive definitions of fractals to which I often refer when I think about them. You can say that a fractal is an object that retains its properties under magnification, or you can say that a fractal is an object that possesses self-similarity across orders of magnitude. But while these definitions informally capture some important properties of fractals, the really intuitive aspect of fractals is their astonishing appearance. Mandelbrot was unapologetically interested in the appearance of fractals. In the NOVA documentary he says in an interview, “I don’t play with formulas, I play with pictures. And that is what I’ve been doing all my life.”

The equation for generating the Mandelbrot set, the later (and more interesting) iterations of which were only made possible by the sheer calculating power of computers.

It is often said that mathematicians are platonists during the week and formalists on the weekend. In other words, while actively working with mathematics they feel themselves to be engaged with objects as real as themselves, but when engaged in philosophical banter over the weekend, the mathematician defends the idea of mathematics as a purely formal activity. This formulation reduced mathematical formalism to a mere rhetorical device. But if formalism is the mathematician’s rhetoric while platonism is the philosophy that he lives by in the day-to-day practice of his work, the rhetorical flourish of formalism has proved to be decisive in the direction that mathematics has taken. Perhaps we could say that mathematicians are strategic formalists and tactical platonists. In this case, we can see how the mathematician’s grand strategy of formalism has shaped the discipline.

Julia sets, predecessors of the Mandelbrot set, found within the Mandelbrot set.

It was in the name of such formalism that “geometrical intuition” began to be seriously questioned at the turn of the nineteenth to the twentieth century, and several generations of mathematicians pursued the rigorization of analysis by way of taking arithmetization as a research program. (Gödel’s limitative theorems were an outgrowth of this arithmetization of analysis; Gödel produced his paradox by an arithmetization of mathematical syntax.) It is to this tacit research program that Mandelbrot implicitly referred when we said, “The eye had been banished out of science. The eye had been excommunicated.” (in an interview in the same documentary mentioned above) What Mandelbrot did was to rehabilitate the eye, to recall the eye from its scientific banishment, and for many this was liberating. To feel free to once again trust one’s geometrical intuition, to “run with it,” as it were, or — to take a platonic figure of thought — to follow the argument where it leads, set a new generation of mathematicians free to explore the visceral feeling of mathematical ideas that had gone underground for a hundred years.

The self-similarity of fractals means that one can find smaller interations of the Mandelbrot set within itself, i.e., the Mandelbrot set microcosm within the Mandelbrot set macrocosm.

It is the astonishing appearance of fractals that made Mandelbrot famous. Since one could probably count the number of famous mathematicians in any one generation on one hand, this is an accomplishment of the first order. Mandelbrot more-or-less singlehandedly created a new branch of mathematics, working against institutionalized resistance, and this new branch of mathematics has led to a rethinking of physics as well as to concrete technological applications (such as cell phone antennas) in widespread use within a few years of fractals coming to the attention to scientists. Beyond this, fractals became a pop culture phenomenon, to the degree that a representation of the Mandelbrot set even appeared as a crop circle.

While the crop circle representation of the Mandelbrot set would seem to have brought us to the point of mere silliness, so much so that we seem to have passed from the sublime to the ridiculous, it does not take much thought to bring us back around to understanding the intellectual, mathematical, scientific, and technological significance of fractal geometry. This is Mandelbrot’s contribution to history, to civilization, to humankind.

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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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The Ethos of Formal Thought

Tuesday

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Aristotle as portrayed by Raphael

Aristotle claimed that mathematics has no ethos (Metaphysics, Book III, Chap. 2, 996^a). Aristotle, of course, was more interested in the empirical sciences than his master Plato, whose Academy presumed and demanded familiarity with geometry — and we must understand that for the ancients, long before the emergence of analytical geometry in the work of Descartes (allowing us to formulate geometry algebraically, hence arithmetically), that geometry was always axiomatic thought, rigorously conceived in terms of demonstration. For the Greeks, this was the model and exemplar of all rigorous thought, and for Aristotle this was a mode of thought that lacked an ethos.

Euclid provided the model of formal thought with his axiomatization of geometry. Legend has it that there was a sign over the door of Plato's Academy stating, 'Let no one enter here who has not studied geometry.'

In this, I think, Aristotle was wrong, and I think that Plato would have agree on this point. But the intuition behind Aristotle’s denial of a mathematical ethos is, I think, a common one. And indeed it has even become a rhetorical trope to appeal to rigorous mathematics as an objective standard free from axiological accretions.

In his famous story within a story about the Grand Inquisitor, Dostoyevsky has the Grand Inquisitor explain how “miracles, mystery, and authority” are used to addle the wits of others.

Our human, all-too-human faculties conspire to confuse us, to addle our wits, when we begin talking about morality, so that the purity and rigor of mathematical and logical thought seem to be called into question if we acknowledge that there is an ethos of formal thought. We easily confuse ourselves with religious, mystical, and ethical ideas, and since the great monument of mathematical thought has been mostly free of this particular species of confusion, to deny an ethos of formal thought can be understood as a strategy to protect and defend of the honor of mathematics and logic by preserving it from the morass that envelops most human attempts to think clearly, however heroically undertaken.

Kant famously said that he had to limit knowledge to make room for faith.

Kant famously stated in the Critique of Pure Reason that, “I have found it necessary to deny knowledge in order to make room for faith.” I should rather limit faith to make room for rigorous reasoning. Indeed, I would squeeze out faith altogether, and find myself among the most rigorous of the intuitionists, one of whom has said: “The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof…”

Western asceticism can be portrayed as demonic torment or as divine illumination; the same diversity of interpretation can be given to ascetic forms of reason.

Though here again, with intuitionism (and various species of constructivism generally), we have rigor, denial, asceticism — intuitionistic logic is no joyful wisdom. (An ethos of formal thought need not be an inspiring and edifying ethos.) It is logic with a frown, disapproving, censorious — a bitter medicine justified only because it offers hope of curing the disease of contradiction, contracted when mathematics was shown to be reducible to set theory, and the latter shown to be infected with paradox (as if the infinite hubris of set theory were not alone enough for its condemnation). Is the intuitionist’s hope justified? In so far as it is hope — i.e., hope and not proof, the expectation that things will go better for the intuitionistic program than for logicism — it is not justified.

Dummett has said that intuitionistic logic and mathematics are to wear their justification on their face:

“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

The hope that contradiction will not arise from intuitionistic methods clearly is no such evident justification. As a matter of fact, empirically and historically verifiable, we know that intuitionism has resulted in no contradictions, but this could change tomorrow. Intuitionism stands in need of a consistency proof even more than formalism. There is, in its approach, a faith invested in the assumption that infinite totalities caused the paradoxes, and once we have disallowed reference to them all will go well. This is a perfectly reasonable assumption, but one, in so far as it is an article of faith, which is at variance with the aims and methods of intuitionism.

And what is a feasible proof, which our ultra-intuitionist would allow? Have we not with “feasible proof” abandoned proof altogether in favor of probability? Again, we will allow them their inconsistencies and meet them on their own ground. But we shall note that the critics of the logicist paradigm fix their gaze only upon consistency, and in so doing reveal again their stingy, miserly conception of the whole enterprise.

“The Ultra-Intuitionistic Criticism and the Antitraditional program for the foundations of Mathematics” by A. S. Yessenin-Volpin (who was arguing for intellectual freedom in the Soviet Union at the same time that he was arguing for a censorious conception of reason), in Intuitionism and Proof Theory, quoted briefly above, is worth quoting more fully:

The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof, in spite of the traditional incompleteness theorem, which deals only with a very narrow kinds [sic] of proofs (which I call ‘formal proofs’). I define proof as any fair way of making a sentence incontestable. Of course this explication is related to ethics — the notion fair means ‘free from any coercion or fraud’ — and to the theory of disputes, indicating the cases in which a sentence is to be considered as incontestable. Of course the methods of traditional mathematical logic are not sufficient for this program: and I have to enlarge the domain of means explicitly studied in logic. I shall work in a domain wherein are to be found only special notions of proof satisfying the mentioned explication. In this domain I shall allow as a means of proof only the strict following of definitions and other rules or principles of using signs.

Intuitionism and proof theory: Proceedings of the summer conference at Buffalo, N.Y., 1968, p. 3

What is coercion or fraud in argumentation? We find something of an illustration of this in Gregory Vlastos’ portrait of Socrates: “Plato’s Socrates is not persuasive at all. He wins every argument, but never manages to win over an opponent. He has to fight every inch of the way for any assent he gets, and gets it, so to speak, at the point of a dagger.” (The Philosophy of Socrates, Ed. by Gregory Vlastos, page 2)

According to Gregory Vlastos, Socrates used the kind of 'coercive' argumentation that the intuitionists abhor.

What appeal to logic does not invoke logical compulsion? Is logical compulsion unique to non-constructive mathematical thought? Is there not an element of logical compulsion present also in constructivism? Might it not indeed be the more coercive form of compulsion that is recognized alike by constructivists and non-constructivists?

The breadth of the conception outlined by Yessenin-Volpin is impressive, but the essay goes on to stipulate the harshest measures of finitude and constructivism. One can imagine these Goldwaterite logicians proclaiming: “Extremism in the defense of intuition is no vice, and moderation in the pursuit of constructivist rigor is no virtue.” Brouwer, the spiritual father of intuitionism, even appeals to the Law-and-Order mentality, saying that a criminal who has not been caught is still a criminal. Logic and mathematics, it seems, must be brought into line. They verge on criminality, deviancy, perversion.

Quine was no intuitionist by a long shot, but as a logician he brought a quasi-disciplinary attitude to reason and adopted a tone of disapproval not unlike Brouwer.

The same righteous, narrow, anathamatizing attitude is at work among the defenders of what is sometimes called the “first-order thesis” in logic. Quine sees a similar deviancy in modal logic (which can be shown to be equivalent to intuitionistic logic), which he says was “conceived in sin” — the sin of confusing use and mention. These accusations do little to help us understand logic. We would do well to adopt Foucault’s attitude on these matters: “leave it to our bureaucrats and our police to see that our papers are in order. At least spare us their morality when we write.” (The Archaeology of Knowledge, p. 17)

Foucault had little patience for the kind of philosophical reason that seemed to be asking if our papers are in order, a function he thought best left to the police.

The philosophical legacy of intuitionism has been profound yet mixed; its influence has been deeply ambiguous. (Far from the intuitive certainty, immediacy, clarity, and evident justification that it would like to propagate.) There is in inuitionism much in harmony with contemporary philosophy of mathematics and its emphasis on practices, the demand for finite constructivity, its anti-philosophical tenor, its opposition to platonism. The Father of Intuitionism, Brouwer, was, like many philosophers, anti-philosophical even while propounding a philosophy. No doubt his quasi-Kantianism put his conscience at rest in the Kantian tradition of decrying metaphysics while practicing it, and his mysticism gave a comforting halo (which softens and obscures the hard edges of intuitionist rigor in proof theory) to mathematics which some have found in the excesses of platonism.

L. E. J. Brouwer: philosopher of mathematics, mystic, and pessimistic social theorist

In any case, few followers of Brouwer followed him in his Kantianism and mysticism. The constructivist tradition which grew from intuitionism has proved to be philosophically rich, begetting a variety of constructive techniques and as many justifications for them. Even if few mathematicians actually do intuitionistic mathematics, controversies over the significance of constructivism have a great deal of currency in philosophy. And Dummett is explicit about the place of philosophy in intuitionistic logic and mathematics.

The light of reason serves as an inspiration to us as it shines down from above, and it remains an inspiration even when we are not equal to all that it might ideally demand of us.

Intuitionism and constructivism command our respect in the same way that Euclidean geometry commanded the respect of the ancients: we might not demand that all reasoning conform to this model, but it is valuable to know that rigorous standards can be formulated, as an ideal to which we might aspire if nothing else. And and ideal of reason is itself an ethos of reason, a norm to which formal thought aspires, and which it hopes to approximate even if it cannot always live up the the most exacting standard that it can recognize for itself.

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Studies in Formalism

1. The Ethos of Formal Thought

2. Epistemic Hubris

3. Parsimonious Formulations

4. Foucault’s Formalism

5. Cartesian Formalism

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

7. The Church-Turing Thesis and the Asymmetry of Intuition

8. Unpacking an Einstein Aphorism

9. Methodological and Ontological Parsimony (in preparation)

10. The Spirit of Formalism (in preparation)

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