## Einstein on Geometrical Intuition

### 23 November 2017

**Thursday — Thanksgiving Day **

**Studies in Formal Thought: **

**Einstein’s Philosophy of Mathematics **

**F**or 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.”

**A**lthough 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.

**T**he 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).

**I**ntuition 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.

**I**f 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.

**E**instein’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.

**B**etween 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.

**T**he 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.

**I**n 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**.)

**I**f 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.”

**E**instein 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.

**B**ut 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

**I**t 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.

**I**t 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.

**E**instein’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.

**R**iemann 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* = √(*x*_{2} – *x*_{1})^{2} + (*y*_{2} – *y*_{1})^{2} — so that in non-Euclidean space the distance between two points could be given by some different equation.

**W**hereas 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.

**F**rom 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.

**E**instein’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.”

**G**iven 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.”

**A**bove 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.)

**T**en 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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Studies in Formalism

1. The Ethos of Formal Thought

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

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

8. Unpacking an Einstein Aphorism

9. The Overview Effect in Formal Thought

10. Einstein on Geometrical intuition

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## The Sun a Star, and Every Star a Sun

### 7 December 2016

**Wednesday **

**T**he full awareness of our sun being a star, and the stars being suns in their own right, was a development nearly coextensive with the entire history of science, from its earliest stirrings in ancient Greece to its modern form at the present time. During the Enlightenment there was already a growing realization of this, as can be seen in a number of scientific works of the period, but scientific proof had to wait for a few generations more until new technologies made available by the industrial revolution produced scientific instruments equal to the task.

**T**he scientific confirmation of this understanding of cosmology, which is, in a sense, the affirmation of Copernicanism (as distinct from heliocentrism) came with two scientific discoveries of the nineteenth century: the parallax of **61 Cygni**, measured by **Friedrich Wilhelm Bessel** and published in 1838, which was the first accurate distance measured to a star other than the sun, and the spectroscopy work of several scientists — Fraunhofer, Bunsen, Kirchhoff, Huggins, and Secchi, *inter alia* (cf. **Spectroscopy and the Birth of Astrophysics**) — which demonstrated the precise chemical composition of the stars, and therefore showed them to be made of the same chemical elements found on Earth. The stars were no longer immeasurable or unknowable; they were now open to scientific study.

**T**he Ptolemaic conception of the universe that preceded this Copernican conception painted a very different picture of the universe, and of the place of human beings within that universe. According to the Ptolemaic cosmology, the heavens were made of a different material than the Earth and its denizens (viz. *quintessence* — the *fifth* element, i.e., the element other than earth, air, fire, and water). Everything below the sphere of the moon — *sublunary* — was ephemeral and subject to decay. Everything beyond the sphere of the moon — *superlunary* — was imperishable and perfect. Astronomical bodies were perfectly spherical, and moved in perfectly circular lines (except for the epicycles). Comets were a problem (i.e., an *anomaly*), because their elliptical orbits ought to send them crashing through the perfect celestial spheres.

**T**his Ptolemaic cosmology largely satisfied the scientific, philosophical, moral, and spiritual needs of western thought from classical antiquity to the end of the Middle Ages, and this satisfaction presumably follows from a deep consonance between this conception of the cosmos and a metaphysical vision of what the world ought to be. Ptolemaic cosmology is the intellectual fulfillment of a certain kind of heart’s desire. But this was not the *only* metaphysical vision of the world having its origins (or, at least, its initial expression) in classical antiquity. Another intellectual tradition that pointed in a different direction was mathematics.

**M**athematics was the first science to attain anything like the rigor that we demand of science today. It remains an open question to this day — an open *philosophical* question — whether mathematics is *a* science, *one* of the sciences (a science among sciences), or whether it is something else entirely, which happens to be useful in the sciences, as, for example, the formal propaedeutic to the empirical sciences, in need of formal structure in order to organize their empirical content. The sciences, in fact, get their rigor from mathematics, so that if there were no mathematical rigor, there would be no possibility of scientific rigor.

**M**athematics has been known since antiquity as the paradigm of exact thought, of precision, the model for all sciences to follow (remembering what science meant to the ancients, which is *not* what it means today: a demonstrative science based on first principles), and this precision has been seen as a function of its formalism, which is to say its definiteness, it boundedness, its participation in the *peras*. Despite this there was yet a recognition of the infinite (*apeiron*) in mathematics. I would go further, and assert that, while mathematics as a rigorous science has its origins in the *peras*, it has its *telos* in the *apeiron*. This is a dialectical development, as we will see below in Proclus.

**P**roclus expresses the negative character of the infinite in his commentary on Euclid’s *Elements*:

“…the infinite is altogether incomprehensible to knowledge; rather it takes it hypothetically and uses only the finite for demonstration; that is, it assumes the infinite not for the sake of the infinite, but for the sake the infinite.”

Proclus,

A Commentary on the First Book of Euclid’s Elements, translated, with an introduction and notes, by Glenn R. Morrow, Princeton: Princeton University Press, 1992, Propositions: Part One, XII, p. 223. This whole section is relevant, but I have quoted only a brief portion.

**T**here is no question that the *apeiron* appeared on the inferior side of the Pythagorean table of opposites, but it is also interesting to note what Proclus says earlier on:

“The objects of Nous, by virtue of their inherent simplicity, are the first partakers of the Limit (*περας*) and the Unlimited (*ἄπειρον*). Their unity, their identity, and their stable and abiding existence they derive from the Limit; but for their variety, their generative fertility, and their divine otherness and progression they draw upon the Unlimited. Mathematicals are the offspring of the Limit and the Unlimited…”

Proclus,

Commentary on the First Book of Euclid, Prologue: Part One, Chap. II

**H**ere the *apeiron* appears on an equal footing with the *peras*, both being necessary to mathematical being. “Mathematicals” are born of the dialectic of the finite and the infinite. Both of these elements are also found (hundreds of years earlier) in the foundations of geometry. As the philosophers produced proofs that there could be no infinite number or infinite space, Euclid spoke of lines and planes extended “indefinitely” (as “apeiron” is usually translated in Euclid). Even later when the Stoics held that the material world was surrounded by an infinite void, this void had special properties which distinguished it from the material world, and indeed which kept the material world from having any relation with the void. The use of infinities in geometry, however, even though in an abstract context, force one to maintain that space locally, directly before one, is essentially of the same kind as space anywhere else along the infinite extent of a line, and indeed the same as space infinitely distant. All spaces are of the same kind, and all are related to each other. This constitutes a purely formal conception of the uniformity and continuity of nature. One might interpret the subsequent history of science as redeeming, through empirical evidence, this formal insight.

**T**he infinite is the “internal horizon” (to use a Husserlian phrase) and the *telos* of mathematical objects. Given this conception of mathematics, the question that I find myself asking is this: what was the mathematical horizon of the Greeks? Did the idea of a line or a plane immediately suggest to them an infinite extension, and did the idea of number immediately suggest the infinite progression of the series, or were the Greeks able to contain these conceptions within the *peras*, using them not unlike we use them, but allowing them to remain limited? Did ancient mathematical imagination encompass the infinite, or must such a conception of mathematical objects (as embedded in the infinite) wait for the infinite to be disassociated from the *apeiron*?

**T**he wait was not long. While the explicit formulation of the mathematical infinite had to wait until Cantor in the nineteenth century, Greek thought was dialectical, so regardless of the nature of mathematical concepts as initially conceived, these concepts inevitably passed into their opposite numbers and grew in depth and comprehensiveness as a result of the development of this dialectic. Greek thought may have begun with an intellectual commitment to the *peras*, and a desire to contain mathematics within the *peras*, consequently an almost ideological effort to avoid the mathematical infinite, but a commitment to dialectic confounds the demand for limitation. It is, then, this dialectical character of Greek thought that gives us the transition from purely local concepts to a formal concept of the uniformity of nature, and then the transition from a formal conception of uniformity to an empirical conception of uniformity, and this latter is the cosmological principle that is central to contemporary cosmology.

**T**he cosmological principle brings us back to where we started: To say that the sun is a star, and every star a sun, is to say that *the sun is a star among stars*. Earth is a planet among planets. The Milky Way is a galaxy among galaxies. This is not only a Copernican idea, it is also a formal idea, like the formal conception of the uniformity of nature. (In **A Being Among Beings** I made a similar about biological beings.) To be one among others of the same kind is to be a member of a class, and to be a member of a class is to be the value of a variable. Quine, we recall, said that *to be is to be the value of a variable*. This is a highly abstract and formal conception of ontology, and that is precisely the importance of the formulation. This is the point beyond which we can begin to reason rigorously about our place in the universe.

**W**e require a class of instances before we can draw inductive inferences, generalize from all members of this class, or formalize the concept represented by any individual member of that class. This is one of the formal presuppositions of scientific thought never made explicit in the methodology of science. We could not formulate the cosmological principle if we did not have a concept of “essentially the same,” because the “same” view that we see looking in any direction in the universe is not *identically* the same, but rather *essentially* the same. Of any two views of the universe, *every* detail is different, but the overview is the same. The cosmological principle is not a generalization, not an inductive inference from empirical evidence; it is a formal idea, a regulative idea that makes a certain kind of cosmological thought possible.

**F**ormal principles like this are present throughout the sciences, though not often recognized for what they are. Bessel’s observations of 61 Cygni not only required industrialized technology to produce the appropriate scientific instruments, these observations also presupposed the mathematics originating in classical antiquity, so that the nineteenth century scientific work that proved the stars to be like our sun (and vice versa) was predicated upon parallel formal conceptions of universality structured into mathematical thought since its inception as a theoretical discipline (in contradistinction to the practical use of mathematics as a tool of engineering). *Formal Copernicanism preceded empirical Copernicanism*. Without that formal component of scientific knowledge, that scientific knowledge would never have come into being.

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## Kierkegaard and Russell on Rigor

### 27 December 2014

**Saturday **

**T**he human mind is a strange and complex entity, and while the mind possesses unappreciated subtlety (of the kind I attempted to describe in **The Human Overview**), rigorous thinking does not come naturally to it. Rigor is a hard-won achievement, not a gift. If we want to achieve some measure of conceptual clarity we must make a particular effort to think rigorously. This is not easy. If you let the mind do what comes naturally and easily to it, you will probably not be thinking rigorously, and you will probably not attain conceptual clarity.

**B**ut what is rigor? To ask this question puts us in a position not unlike Saint Augustine who asked, “What, then, is time?” If no one asks me, I know what rigor is. If I wish to explain it to him who asks, I do not know. What distinguishes rigorous thinking from ordinary thinking? And what distinguishes a rigorous life from an ordinary life? Is there any relation between the formal and existential senses of rigor?

**A**s a first and rough approximation, we could say that rigor is the implementation of a **precise idea of precision**. Whether or not a precise idea of precision can be applied to the **human condition**, a question that I have addressed in **The Human Condition Made Rigorous**, is a question of whether the formal sense of rigor is basic, and existential rigor is an implementation of formal rigor in life.

**K**ierkegaard concerned himself with what I am here calling *existential rigor*, i.e., the idea of living a rigorous life. One of the central themes that runs through Kierkegaard’s substantial corpus is the question of how one becomes an authentic Christian in an inauthentic Christian society (though this is not how Kierkegaard himself expressed the problem that preoccupied him). Kierkegaard expresses himself in the traditional Christian idiom of suffering for the truth, but Kierkegaard’s suffering is not pointless or meaningless: it is conducive to existential rigor:

“My purpose is to make it difficult to become a Christian, yet not more difficult than it is, nor to make it difficult for stupid people, and easy for clever pates, but qualitatively difficult, and essentially difficult for every man equally, for essentially it is equally difficult for every man to relinquish his understanding and his thinking, and to keep his soul fixed upon the absurd; it is comparatively more difficult for a man if he has much understanding — if one will keep in mind that not everyone who has lost his understanding over Christianity thereby proves that he has any.”

KIERKEGAARD’S CONCLUDING UNSCIENTIFIC POSTSCRIPT, Translated from the Danish by DAVID F. SWENSON, PROFESSOR OF PHILOSOPHY AT THE UNIVERSITY OF MINNESOTA, Completed after his death and provided with Introduction and Notes by WALTER LOWRIE, PRINCETON: PRINCETON UNIVERSITY PRESS, p. 495

**T**he whole of Kierkegaard’s book ** Attack Upon Christendom** is an explicit attack upon “official” Christianity, which he saw as too safe, too comfortable, too well-connected to the machinery of the state. In Kierkegaard’s Denmark, no one was suffering in order to bear witness to the truth of Christianity:

“…hundreds of men are introduced who instead of following Christ are snugly and comfortably settled, with family and steady promotion, under the guise that their activity is the Christianity of the New Testament, and who live off the fact that others have had to suffer for the truth (which precisely is Christianity), so that the relationship is completely inverted, and Christianity, which came into the world as the truth men die for, has now become the truth upon which they live, with family and steady promotion — ‘Rejoice then in life while thy springtime lasts’.”

Søren Kierkegaard,

Attack Upon Christendom, Princeton: Princeton University Press, 1946, p. 42

**A**nd from Kierkegaard’s journals…

“Could you not discover some way in which you too could help the age? Then I thought, what if I sat down and made everything difficult? For one must try to be useful in every possible way. Even if the age does not need ballast I must be loved by all those who make everything easy; for if no one is prepared it difficult it becomes all too easy — to make things easy.”

Søren Kierkegaard,

The Soul of Kierkegaard: Selections from His Journals, 1845, p. 93

**K**ierkegaard is full of such passages, and if you read him through you will probably find more compelling instances of this idea than the quotes I have plucked out above.

**K**ierkegaard called into question the easy habits of belief that we follow mostly without questioning them; Russell called into question the intuitions that come naturally to us, to the human mind, and which we mostly do not question. Both Kierkegaard and Russell thought there was value in doing things the hard way, not in order to court difficulty for its own sake, but rather for the different perspective it affords us by not simply doing what comes naturally, but having to think things through for ourselves.

**R**ussell’s approach to rigor is superficially antithetical to that of Kierkegaard. While Kierkegaard was interested in the individual and his individual existence, Russell was interested in universal logical principles that had nothing to do with individual existence. William James once wrote to Russell, “My dying words to you are ‘Say good-by to mathematical logic if you wish to preserve your relations with concrete realities!'” Russell’s response was perfect deadpan: “As for the advice to say goodbye to mathematical logic if I wish to preserve my relation with concrete realities, I am not wholly inclined to dispute its wisdom. But I should push it farther, & say that it would be well to give up all philosophy, & abandon the student’s life altogether. Ten days of standing for Parliament gave me more relations to concrete realities than a lifetime of thought.”

**N**evertheless, beyond these superficial differences, both Kierkegaard and Russell understood, each in his own way, that the easy impulse must be resisted. A passage from Bertrand Russell that I previously quoted in **The Overview Effect in Formal Thought** makes this point for formal rigor:

“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”

**A**nd elsewhere…

“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

**F**or Russell, the use of symbols in reasoning constitutes a reformulation of the intuitive in a counter-intuitive form, and this makes it possible for us to struggle toward the truth without being distracted by matters that seem so obvious that our cognitive biases lead us toward deceptive obviousness instead of toward the truth. There is another name for this, *defamailiarization* (which I previously discussed in **Reversing the Process of Defamiliarization**). Great art *defamiliarizes* the familiar in order to present it to us again, anew, in unfamiliar terms. In this way we see the world with new eyes. Just so, the reformulation of intuitive thought in counter-intuitive forms presents the familiar to us in unfamiliar terms and we see our reasoning anew with the mind’s eye.

**I**ntuitions have their place in **formal thought**. I have in the past written of the tension between intuition and formalization that characterizes formal thought, as well as of the place of intuition in philosophical argument (cf. **Doing Justice to Our Intuitions: A 10 Step Method**). But if intuitions have their place, they also have their limitations, and the making of easy things difficult is a struggle against the limitations of intuition. What Kierkegaard and Russell have in common in their conception of rigor is that of making something ordinarily easy into something difficult in order to overcome the limitations of the natural and the intuitive. All of this may sound rather arcane and confined to academic squabbles, but it is in fact quite directly related to the world situation today.

**I** have often written about the anonymity and anomie of life in **industrial-technological civilization**; this is a familiar theme that has been worked through quite extensively in twentieth century sociology, and one could argue that it is also a prominent element in existentialism. But the human condition in the context of our civilization today is not only marked by anonymity and anomie, but also by high and rising standards of living, which usually translates directly into comfort. While we are perhaps more bereft of meaning than ever, we are also more comfortable than ever before in history. This has also been studied in some detail. Occasionally this combination of a comfortable but listless life is called “affluenza.”

**K**ierkegaard’s defamiliarization of (institutionalized and inauthentic) Christianity was intended to make Christianity difficult for bourgeois worldlings; the militant Islamists of our time want to make Islam difficult and demanding for those who would count themselves Muslims. It is the same demand for existential rigor in each that is the motivation. If it is difficult to understand why young men at the height of their prowess and physical powers can be seduced into extremist militancy, one need only reflect for a moment on the attraction of difficult things and the earned honors of existential rigor. The west has almost completely forgotten the attraction of difficult things. What remains is perhaps the interest in “extreme” sports, in which individuals test themselves against contrived physical challenges, which provides a kind of existential rigor along with bragging rights.

**E**xtremist ideologies offer precisely the two things for which the individual hungers but cannot find in contemporary industrialized society: meaning, and a challenge to his complacency. An elaborately worked out eschatological conception of history shows the individual his special place within the grand scheme of things (this is the familiar ground of **cosmic warfare** and the **eschatological conception of history**), but this eschatological vision is not simply handed for free to the new communicant. He must work for it, strive for it, sacrifice for it. And when he has proved himself equal to the demands placed upon him, then he is rewarded with the profoundly satisfying gift of an *earned* honor: membership in a community of the elect.

**T**his view is not confined to violent extremists. We meet with this whenever someone makes the commonplace remark that we don’t value that which is given away for free, and Spinoza expressed the thought with more eloquence: “All noble things are as difficult as they are rare.” Anyone who feels this pull of difficult things, who desires a challenge, who wants to be tested in order to prove their worth in the only way that truly counts, is an existentialist in action, if not in thought, because it is the existentialist conception of authenticity that is operative in this conception of existential rigor.

**W**e have tended to think of pre-modern societies, mostly **agrarian-ecclesiastical civilization**, with their rigid social hierarchies and inherited social positions, as paradigmatic examples of inauthentic societies, but we have managed to create a thoroughly inauthentic society in the midst of our **industrial-technological civilization**. This civilization and its **social order** may have its origins in the overturning of the inauthentic social order of earlier ages, but, after an initial period of social experimentation, the present social order ossified and re-created many of the inauthentic and hierarchical forms that characterized the overthrown social order.

**I**nauthentic societies are awash in unearned unearned advantages. I wrote about this earlier in discussing the urban austerity of **Simone Weil**, the wilderness austerity of **Christopher McCandless** (also known as Alexander Supertramp), and comparing the two in **Weil and McCandless: Another Parallel**:

“…the accomplishments of the elite and the privileged are always tainted by the fact that what they have attained has not been earned. But it is apparent that there are always a few honest individuals among the privileged who are acutely aware that their position has not been earned, that it is tainted, and the only way to prove that one can make it on one’s own is to cut one’s ties to one’s privileged background and strike out on one’s own.”

**T**here is a certain sense in which the available and ample comforts of industrial-technological civilization transformed the greater part of the global population into complacent consumers who accept an inauthentic life. There is another name of this too; Nietzsche called such individuals **Last Men**.

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## The Overview Effect in Formal Thought

### 20 January 2014

**Monday **

**Studies in Formalism: **

**The Synoptic Perspective in Formal Thought **

**I**n 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.

**T**he 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**.

**A**nd 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.

**L**ate 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

**H**ardy’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.

**I**t 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.

**I**f 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

**A**lthough 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)

**A**nd 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)

**T**his 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.

**H**ardy’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.

**F**ew 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

**T**his 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.

**K**ant, 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*).

**T**here 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.

**T**he 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*.

**T**here 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,”inFoundations of Science, 14 (1-2), 2009

**F**or 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.

**T**he 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,

, Volume I, second edition, Cambridge: Cambridge University Press, 1963, p. 2Principia Mathematica

…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.

**T**hinking 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”

**R**ussell 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

**A**lfred 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,

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

**T**his 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,

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

**I**n 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.

**R**ecent 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

**W**hile 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.

**K**ahneman’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).

**F**ormal 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.

**I**n **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.

**I**n **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.”

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

**I**n 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**.

**I**f 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*.

**I**t 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 inBrief Addendum on the Overview Effect in Formal Thought.

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Studies in Formalism

1. The Ethos of Formal Thought

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

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

8. Unpacking an Einstein Aphorism

9. The Overview Effect in Formal Thought

10. Einstein on Geometrical intuition

11. Methodological and Ontological Parsimony (in preparation)

12. The Spirit of Formalism (in preparation)

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## Comte de Maistre’s Finitistic Political Theory

### 14 June 2013

**Friday **

**I**t was the Romanian expatriate writer E. M. Cioran writing in French (and translated into English by the indefatigable Richard Howard) who first made me aware of Joseph-Marie, Comte de Maistre. Cioran’s ** Anathemas and Admirations** has a chapter on de Maistre, the latter himself an intemperate expatriate gifted with a literary style so powerful that it wins the reader’s attention for doctrines so marginal as to be laughable — if only they had not been taken so deadly seriously by men who have died for them. But not everything in de Maistre is as trivial or marginal as his monarchism and his defense of the

**.**

*Ancien Régime***A**long with Edmund Burke, de Maistre (when he is remembered today) is remembered as a proto-conservative, staking out positions that would later become doctrinaire among conservative thinkers. Both were great stylists, but Burke was really a poet — did he not write one of the eighteenth-century tracts on the sublime that gentlemen of good taste wrote in those times? — while de Maistre was an original, ruthless, and brutal thinker, i.e., he was everything that a philosopher *ought* to be. But today de Maistre is held in low opinion because of his at times virulent racism (as though this were worse than virulent monarchism, or virulent sexism, etc.).

**T**here are two sides of the coin of *ad hominem*** arguments**: either love or hatred of a man can lead us to embrace or reject his ideas. We need to try to see beyond both de Maistre’s fearsome if not untouchable reputation *and* the beauty of this style, if we are to engage with de Maistre the thinker — and this is a task worth the effort, because de Maistre has some interesting ideas that deserve exposition. His low reputation today might lead us to ignore these ideas, or his literary style might lead us to assent to ideas that, while interesting, certainly do not deserve our assent.

**T**he intransigence of de Maistre invites the reader to shout back at him, even to shout him down, with a long and detailed catalog of the absurdities that have been perpetrated upon the world by men who believed in the doctrines that de Maistre defends. I doubt any of this would have made the slightest impression on de Maistre, whose own obvious contempt for such an approach comes across in every dismissive formulation that is presented as though no counter-veiling principle were even possible, even *thinkable*. With such a mind it would be utterly irrelevant to debate details; I have no doubt that de Maistre would have dismissed every challenge to his examples and instances with a contemptuous wave of the hand and a disapproving expression. In reading de Maistre, therefore, it behooves us to think only in terms of principles.

**W**hat are de Maistre’s principles? What is the essence of de Maistre’s thought? It is easy to take the wrong lesson from such a vigorous and expressive writer. The least imaginative and least creative among us read the likes of Burke and de Maistre and believe that they have found the whole meaning in a blueprint for contemporary society. But this is a mere detail, an accident of historical circumstances that might be construed in dramatically different ways in different periods of human history. What is of the essence of de Maistre’s thought is something not at all obvious, and it is his finitistic perspective.

**I** have previously quoted from de Maistre’s ** An Essay on the Generative Principle of Constitutions** — a short, incisive, and suggestive work, i.e., everything that a philosophical work should be — in

**Fairness and the Social Contract**and

**Why Revolutions Happen**. Comte de Maistre begins his

*Essay*by recounting the counter-intuitive nature of political science, citing several examples of putative political “common sense” and how experience has shown these to be “disastrous.” This points to an unexpected empiricism in de Maistre’s thought. Echoing but altering Thucydides’ famous aphorism,

**, de Maistre wrote that**

*history is philosophy teaching by example**history is experimental politics*.

**I**n the preface to his *Essay*, de Maistre anonymously quotes his own ** Considerations on France**, Chap. VI. Following is how the two passages appear, first in

**:**

*Considerations on France*1. No government results from a deliberation; popular rights are never written, or at least constitutive acts or written fundamental laws are always only declaratory statements of anterior rights, of which nothing can be said other than that they exist because they exist.

2. God, not having judged it proper to employ supernatural means in this field, has limited himself to human means of action, so that in the formation of constitutions circumstances are all and men are only part of the circumstances. Fairly often, even, in pursuing one object they achieve another, as we have seen in the English constitution.

**A**nd this is how they appear, in a slightly revised form, in de Maistre’s ** Essay**:

1. No constitution arises from deliberation. The rights of the people are never written, except as simple restatements of previous, unwritten rights.

2. [In the formation of constitutions] human action is so far circumscribed that the men who act become only circumstances. [It is even very common that in pursuing a certain end they attain another.] 3. The rights of the PEOPLE, properly so called, proceed almost always from the concessions of sovereigns and thus may be definitely fixed in history, but no one can ascertain the date or the authors of the rights of the monarch and the aristocracy.

**T**his in itself, in its most tightly circumscribed formulation, I cannot reject — human action is most certainly circumscribed, and unintended consequences often outweigh intended consequences. Indeed, de Maistre’s thought here closely echoes my own formulations in terms of the **permutations of human agency**, and in so doing de Maistre reveals his **eschatological conception of history**, affirming non-human agency as the source of political constitutions.

**F**urther to this eschatological conception, Comte de Maistre quotes the theologian Bergier:

Law is only truly sanctioned, and properly *law*, when assumed to emanate from a higher will, so that its essential quality is to be *not the will of all* [*la volonte de tous*]. Otherwise, laws would be *mere ordinances*. As the author just quoted states, “those who were free to make these conventions have not deprived themselves of the power of revocation, and their descendants, with no share in making these regulations, are bound even less to observe them.”

Essay on the Generative Principle of Political Constitutions and other Human Institutions, M. the Count de Maistre, the citation is from Bergier,Traite historique et dogmatique de la Religion, III, ch. 4 (after Tertullian,Apologeticus, 45)

**B**ergier has here put his finger on something important, though of course the lesson I take from it is rather different than the lesson that de Maistre takes from it. The same idea finds a very different expression in Gibbon, and I have quoted this several times:

“In earthly affairs, it is not easy to conceive how an assembly equal of legislators can bind their successors invested with powers equal to their own.”

Edward Gibbon,

History of the Decline and Fall of the Roman Empire, Vol. VI, Chapter LXVI, “Union Of The Greek And Latin Churches.–Part III.

**I** have called this *Gibbon’s Principle of Inalienable Autonomy for Political Entities*, or, more briefly, ** Gibbon’s Principle**. Bergier and de Maistre invoke a distinction between laws and ordinances, with ordinances being mere human things subject to change, while laws are laid up in heaven. This is de Maistre’s realism.

**T**he political theologizing of de Maistre is what is most predictable and *least* interesting in his thought; it only *becomes* interesting as a consequence of his finitism. The implications of de Maistre’s finitism, once extrapolated to its logical conclusion throughout his political thought, converges upon a radical finitism in political science, and this I cannot accept or endorse. More interesting than his theologizing is de Maistre’s political realism — and by “realism” I do not mean that “political realism” used in discussions of policy, that prides itself on its rejection of humanitarianism and of moral and political ideals, but de Maistre’s **Platonic realism** in politics that, on the contrary, raises up moral and political ideals as the only *true* reality.

**T**he strong position de Maistre takes on ineffability is related to his Platonic realism: constitutions are real in a Platonic sense, but our knowledge of them is imperfect, and if we try to write them down we will only get it wrong, much as a mathematician using a compass to draw a circle inscribes only an imperfect image of a circle that represents, for pedagogical reasons, the “real” and “true” circle to which the imperfect drawing refers. The harder we try to inscribe a perfect circle, the more we are going to depart from the Platonic form of a circle, and the more we try to write down the perfect constitution, the more it departs from the Platonic form of a constitution. In de Maistre, written law is not only derivative of unwritten law, i.e., the mere appearance or a more fundamental reality, but it is, moreover, always wrong because the unwritten fundamental reality is essentially ineffable.

**T**his is how de Maistre himself formulates it in his *Essay*:

1.The fundamental principles of political constitutions exist prior to all written law.

2.Constititional law is and can only be the development or sanction of a pre-existing and unwritten law.

3.What is most essential, most inherently constitutional and truly fundamental law is never written, and could not be, without endangering the State.

4.The weakness and fragility of a constitution are actually in direct proportion to the number of written constitutional articles.

**T**his is really quite close to **Brouwer’s intuitionism**; indeed, we might call de Maistre’s thought intuitionistic political science. Both Brouwer and de Maistre place a strong emphasis on the ineffability of experience, and the ways in which language misleads and falsifies, but de Maistre’s ineffability is predicated upon realism while Brouwer was what we might call a proto-anti-realist. Intuitionism after Brouwer went on to inspire a generation of philosophers to formulate anti-realist positions that owe much to Brouwer’s inspiration.

**T**hus de Maistre’s realism coupled with finitism and an eschatological conception of history stake out a unique (or nearly unique) position in the history of thought. It would be entirely possible to formulate this Platonic realism in politics in an infinitistic context (just as de Maistre could have justified his finitism according to other conceptions but in fact chose to justify it in theological terms, invoking an eschatological conception of history), but de Maistre is thoroughly finitistic in his orientation.

**C**omte de Maistre uses an eschatological conception of history to provide the ideological superstructure of justify his theological exposition of finite human agency, but he could make the same point invoking a cataclysmic conception of history or a naturalistic conception of history. Even a modified and qualified formulation of the political conception of history, which makes human agency fundamental and central to history, would be consistent with de Maistre’s finitism, so that the theological justification, however much weight de Maistre himself might have attached to it, is of little intrinsic interest. The point I am making is that de Maistre’s theology is dispensable in defining his theory, while de Maistre’s finitism is *in*dispensable.

**J**oseph de Maistre’s finitistic political theory represents something of an antithesis to an infinitistic conception of political society such as I outlined in what I called **Gödel’s Lesson for Geopolitics** (and something I touched upon again in **Addendum on Technological Unemployment**).

**I** hope to return to this idea in future posts, and to be able to show why this is important, because I know that this sounds rather recondite and marginal, but it is neither. One of the most persistent themes of Western historiography in the modern period is the idea of progress, which is attacked at least as often as it is put forward as an interpretation of history (not long ago in **Progress, Stagnation, and Retrogression** I mentioned my surprise that Kevin Kelly offered an explicit defense of historical progress in his book ** What Technology Wants**). A finitistic conception of history knows nothing of progress; we must have an infinitistic conception of history before the idea of progress can even have meaning for us. This, however, is a complex idea that requires many qualifications and therefore independent exposition. I will leave that for another day.

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## Theoretical Geopolitics: History and Geography

### 9 April 2012

**Monday **

**Geopolitics and Geostrategy **

**as a formal sciences **

**I**n 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.

**T**here 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.

**M**ost 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**.)

**G**eopolitics, 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.

**T**here 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.

**T**he 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.

**W**e 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.

**T**he 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.

**G**eopolitics, 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).

**G**eopolitics 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,

, “Mathematics and the Metaphysicians”Mysticism and Logic

**R**ussell 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.

**R**ussell 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).

**A**s 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.

**E**cological 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.

**A**nother 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.

**S**ometimes, 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.

**I**f 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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