## 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 Three Revolutions

### 12 November 2017

**Sunday **

**Three Revolutions that Shaped the Modern World **

**T**he world as we know it today, *civilization* as we know it today (because, for us, civilization *is* the world, *our* world, the world we have constructed for ourselves), is the result of three revolutions. What was civilization like before these revolutions? Humanity began with the development of an agricultural or pastoral economy subsequently given ritual expression in a religious **central project** that defined independently emergent civilizations. Though widely scattered across the planet, these early agricultural civilizations had important features in common, with most of the pristine civilizations beginning to emerge shortly after the Holocene warming period of the current Quaternary glaciation.

**A**lthough independently originating, these early civilizations had much in common — arguably, each had more in common with the others emergent about the same time than they have in common with contemporary industrialized civilization. How, then, did this very different industrialized civilization emerge from its agricultural civilization precursors? This was the function of the three revolutions: to revolutionize the conceptual framework, the political framework, and the economic framework from its previous traditional form into a changed modern form.

**T**he institutions bequeathed to us by our agricultural past (the era of exclusively **biocentric civilization**) were either utterly destroyed and replaced with *de novo* institutions, or traditional institutions were transformed beyond recognition to serve the needs of a changed human world. There are, of course, subtle survivals from the ten thousand years of agricultural civilization, and historians love to point out some of the quirky traditions we continue to follow, though they make no sense in a modern context. But this is peripheral to the bulk of contemporary civilization, which is organized by the institutions changed or created by the three revolutions.

**The Scientific Revolution **

**T**he scientific revolution begins as the earliest of the three revolutions, in the early modern period, and more specifically with Copernicus in the sixteenth century. The work of Copernicus was elaborated and built upon by Kepler, Galileo, Huygens, and a growing number of scientists in western Europe, who began with physics, astronomy, and cosmology, but, in framing a scientific method applicable to the pursuit of knowledge in any field of inquiry, created an epistemic tool that would be universally applied.

**T**he application of the scientific method had the *de facto* consequence of stigmatizing pre-modern knowledge as superstition, and the attitude emerged that it was necessary to extirpate the superstitions of the past in order to build anew on solid foundations of the new epistemic order of science. This was perceived as an attack on traditional institutions, especially traditional cultural and social institutions. It was this process of the clearing away of old knowledge, dismissed as irrational superstition, and replacing it with new scientific knowledge, that gave us the conflict between science and religion that still simmers in contemporary civilization.

**T**he scientific revolution is ongoing, and continues to revolutionize our conceptual framework. For example, four hundred years into the scientific revolution, in the twentieth century, the Earth sciences were revolutionized by plate tectonics and geomorphology, while cosmology was revolutionized by general relativity and physics was revolutionized by quantum theory. The world we understood at the end of the twentieth century was a radically different place from the world we understood at the beginning of the twentieth century. This is due to the iterative character of the scientific method, which we can continue to apply not only to bodies of knowledge not yet transformed by the scientific method, but also to earlier bodies of scientific knowledge that, while revolutionary in their time, were not fully comprehensive in their conception and formulation. Einstein recognized this character of scientific thought when he wrote that, “There could be no fairer destiny for any physical theory than that it should point the way to a more comprehensive theory, in which it lives on as a limiting case.”

**The Political Revolutions **

**T**he political revolutions that began in the last quarter of the eighteenth century, beginning with the American Revolution in 1776, followed by the French Revolution in 1789, and then a series of revolutions across South America that displaced Spain and the Spanish Empire from the continent and the western hemisphere (in a kind of revolutionary contagion), ushered in an age of representative government and popular sovereignty that remains the dominant paradigm of political organization today. The consequences of these political revolutions have been raised to the status of a dogma, so that it no longer considered socially acceptable to propose forms of government not based upon representative institutions and popular sovereignty, however dismally or frequently these institutions disappoint.

**W**e are all aware of the experiment with democracy in classical antiquity in Athens, and spread (sometimes by force) by the Delian League under Athenian leadership until the defeat of Athens by the Spartans and their allies. The ancient experiment with democracy ended with the Peloponnesian War, but there were quasi-democratic institutions throughout the history of western civilization that fell short of perfectly representative institutions, and which especially fell short of the ideal of popular sovereignty implemented as universal franchise. Aristotle, after the Peloponnesian War, had already converged on the idea of a *mixed* constitution (a constitution neither purely aristocratic nor purely democratic) and the Roman political system over time incorporated institutions of popular participation, such as the Tribune of the People (*Tribunus plebis*).

**M**edieval Europe, which Kenneth Clark once called a, “conveniently loose political organization,” frequently involved self-determination through the devolution of political institutions to local control, which meant that free cities might be run in an essentially democratic way, even if there were no elections in the contemporary sense. Also, medieval Europe dispensed with slavery, which had been nearly universal in the ancient world, and in so doing was responsible for one of the great moral revolutions of human civilization.

**T**he political revolutions that broke over Europe and the Americas with such force starting in the late eighteenth century, then, had had the way prepared for them by literally thousands of years of western political philosophy, which frequently formulated social ideals long before there was any possibility of putting them into practice. Like the scientific revolution, the political revolutions had deep roots in history, so that we should rightly see them as the inflection points of processes long operating in history, but almost imperceptible in their earliest expression.

**The Industrial Revolution **

**T**he industrial revolution began in England with the invention of James Watt’s steam engine, which was, in turn, an improvement upon the Newcomen atmospheric engine, which in turn built upon a long history of an improving industrial technology and industrial infrastructure such as was recorded in Adam Smith’s famous example of a pin factory, and which might be traced back in time to the **British Agricultural Revolution**, if not before. The industrial revolution rapidly crossed the English channel and was as successful in transforming the continent as it had transformed England. The Germans especially understood that it was the scientific method as applied to industry that drove the industrial revolution forward, as it still does today. It is science rather than the steam engine that truly drove the industrial revolution.

**A**s the scientific revolution drove epistemic reorganization and the political revolutions drove sociopolitical reorganization, the industrial revolution drove economic reorganization. Today, we are all living with the consequences of that reorganization, with more human beings than ever before (both in terms of absolute numbers and in terms of rates) living in cities, earning a living through employment (whether compensated by wages or salary is indifferent; the invariant today is that of being an employee), and organizing our personal time on the basis of clock times that have little to do with the sun and the moon, and schedules that have little or no relationship to the agricultural calendar.

**T**he emergence of these institutions that facilitated the concentration of labor (what Marx would have called “industrial armies”) where it was most needed for economic development indirectly meant the dissolution of multi-generational households, the dissolution of the feeling of being rooted in a particular landscape, the dissolution of the feeling of belonging to a local community, and the dissolution of the way of life that was embodied in these local communities of multi-generational households, bound to the soil and the climate and the particular mix of cultivars that were dietary staples. As science dismissed traditional beliefs as superstition, the industrial revolution dismissed traditional ways of life as impractical and even as unhealthy. Le Courbusier, a great prophet of the industrial city, possessed of revolutionary zeal, forcefully rejected **pre-modern technologies of living**, asserting, “We must fight against the old-world house, which made a bad use of space. We must look upon the house as a machine for living in or as a tool.”

**Revolutionary Permutations **

**T**errestrial civilization as we know it today is the product of these three revolutions, but must these three revolutions occur, and must they occur in this specific order, for any civilization whatever that would constitute a peer technological civilization with which we might hope to engage in communication? That is to say, if there are other civilizations in the universe (or even in a counterfactual alternative history for terrestrial civilization), would they have to arrive at radio telescopes and spacecraft by this same sequence of revolutions in the same order, or would some other sequence (or some other revolutions) be equally productive of technological civilizations?

**T**his may well sound like a strange question, perhaps an arbitrary question, but this is the sort of question that formal historiography asks. In several posts I have started to outline a conception of formal historiography in which our interest is not only in what has happened on Earth, or what *might* yet happen on Earth, but what can happen with any civilization whatsoever, whether on Earth or elsewhere (cf. **Big History and Scientific Historiography**, **History in an Extended Sense**, **Rational Reconstructions of Time**, **An Alternative Formulation of Rational Reconstructions of Time**, and **Placeholders for Null-Valued Time**). While this conception is not formulated for the express purpose of investigating questions like the Fermi paradox, I hope that the reader can see how such an investigation bears upon the Fermi paradox, the Drake equation, and other “big picture” conceptions that force us to think not in terms of terrestrial civilization, but rather in terms of any civilization whatever.

**F**rom a purely formal conception of social institutions, it could be argued that something like these revolutions would have to take place in something like the terrestrial order. The epistemic reorganization of society made it possible to think scientifically about politics, and thus to examine traditional political institutions rationally in a spirit of inquiry characteristic of the Enlightenment. Even if these early forays into political science fall short of contemporary standards of rigor in political science, traditional ideas like the divine right of kings appeared transparently as little better than political superstitions and were dismissed as such. The social reorganization following from the rational examination the political institutions utterly transformed the context in which industrial innovations occurred. If the steam engine or the power loom had been introduced in a time of rigid feudal institutions, no one would have known what to do with them. Consumer goods were not a function of production or general prosperity (as today), but rather were controlled by sumptuary laws, much as the right to engage in certain forms of commerce was granted as a royal favor. These feudal political institutions would not likely have presided over an industrial revolution, but once these institutions were either reformed or eliminated, the seeds of the industrial revolution could take root.

**I**n this interpretation, the epistemic reorganization of the scientific revolution, the social reorganization of the political revolutions, and the economic reorganization of the industrial revolution are all tightly-coupled both synchronically (in terms of the structure of society) and diachronically (in terms of the historical succession of this sequence of events). I am, however, suspicious of this argument because of its implicit anthropocentrism as well as its teleological character. Rather than seeking to justify or to confirm the world we know, framing the historical problem in this formal way gives us a method for seeking variations on the theme of civilization as we know it; alternative sequences could be the basis of thought experiments that would point to different kinds of civilization. Even if we don’t insist that this sequence of revolutions is *necessary* in order to develop a technological civilization, we can see how each development fed into subsequent developments, acting as a social equivalent of **directional selection**. If the sequence were different, presumably the directional selection would be different, and the development of civilization taken in a different direction.

**I** will not here attempt a detailed analysis of the permutations of sequences laid out in the graphic above, though the reader may wish to think through some of the implications of civilizations differently structured by different revolutions at different times in their respective development. For example, many science fiction stories imagine technological civilizations with feudal institutions, whether these feudal institutions are retained unchanged from a distant agricultural past, or whether they were restored after some kind of political revolution analogous to those of terrestrial history, so one could say that, *prima facie*, political revolution might be entirely left out, i.e., that political reorganization is dispensable in the development of technological civilization. I would not myself make this argument, but I can see that the argument can be made. Such arguments could be the basis of thought experiments that would present civilization-as-we-do-*not*-know-it, but which nevertheless inhabit the same parameter space of civilization-as-we-know-it.

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