Thursday — Thanksgiving Day


Studies in Formal Thought:

Albert Einstein (14 March 1879 – 18 April 1955)

Albert Einstein (14 March 1879 – 18 April 1955)

Einstein’s Philosophy of Mathematics


For some time I have had it on my mind to return to a post I wrote about a line from Einstein’s writing, Unpacking an Einstein Aphorism. The “aphorism” in question is this sentence:

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

…which, in the original German, was…

“Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nict sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.”

Although this sentence has been widely quoted out of context until it has achieved the de facto status of an aphorism, I was wrong to call it an aphorism. This sentence, and the idea it expresses, is entirely integral with the essay in which it appears, and should not be treated in isolation from that context. I can offer in mitigation that a full philosophical commentary on Einstein’s essay would run to the length of a volume, or several volumes, but this post will be something of a mea culpa and an effort toward mitigation of the incorrect impression I previously gave that Einstein formulated this idea as an aphorism.

The first few paragraphs of Einstein’s lecture, which includes the passage quoted above, constitute a preamble on the philosophy of mathematics. Einstein wrote this sententious survey of his philosophy of mathematics in order to give the listener (or reader) enough of a methodological background that they would be able to follow Einstein’s reasoning as he approaches the central idea he wanted to get across: Einstein’s lecture was an exercise in the cultivation of geometrical intuition. Unless one has some familiarity with formal thought — usually mathematics or logic — one is not likely to have an appreciation of the tension between intuition and formalization in formal thought, nor of how mathematicians use the term “intuition.” In ordinary language, “intuition” usually means arriving at a conclusion on some matter too subtle to be made fully explicit. For mathematicians, in contrast, intuition is a faculty of the mind that is analogous to perception. Indeed, Kant made this distinction, implying its underlying parallelism, by using the terms “sensible intuition” and “intellectual intuition” (which can also be called “outer” and “inner” intuition).

Intuition as employed in this formal sense has been, through most of the history of formal thought, understood sub specie aeternitatis, i.e., possessing many of the properties once reserved for divinity: eternity, immutability, impassibility, and so on. In the twentieth century this began to change, and the formal conception of intuition came to be more understood in naturalistic terms as a faculty of the human mind, and, as such, subject to change. Here is a passage from Gödel that I have quoted many times (e.g., in Transcendental Humors), in which Gödel delineates a dynamic and changing conception of intuition:

“Turing… gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but is constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure.”

“Some remarks on the undecidability results” (Italics in original) in Gödel, Kurt, Collected Works, Volume II, Publications 1938-1974, New York and Oxford: Oxford University Press, 1990, p. 306.

If geometrical intuition (or mathematical intuition more generally) is subject to change, it is also subject to improvement (or degradation). A logician like Gödel, acutely aware of the cognitive mechanisms by which he has come to grasp logic and mathematics, might devote himself to consciously developing intuitions, always refining and improving his conceptual framework, and straining toward developing new intuitions that would allow for the extension of mathematical rigor to regions of knowledge previously given over to Chaos and Old Night. Einstein did not make this as explicit as did Gödel, but he clearly had the same idea, and Einstein’s lecture was an attempt to demonstrate to his audience the cultivation of geometrical intuitions consistent with the cosmology of general relativity.

Einstein’s revolutionary work in physics represented at the time a new level of sophistication of the mathematical representation of physical phenomena. Mathematicized physics began with Galileo, and might be said to coincide with the advent of the scientific revolution, and the mathematization of physics reached a level of mature sophistication with Newton, who invented the calculus in order to be able to express his thought in mathematical form. The Newtonian paradigm in physics was elaborated as the “classical physics” of which Einstein and Infeld, like mathematical parallels of Edward Gibbon, recorded the decline and fall.

Between Einstein and Newton a philosophical revolution in mathematics occurred. The philosophy of mathematics formulated by Kant is taken by many philosophers to express the conception of mathematics to be found in Newton; I do not agree with this judgment, as much for historiographical reasons as for philosophical reasons. But perhaps if we scrape away the Kantian idealism and subjectivism there might well be a core of Newtonian philosophy of mathematics in Kant, or, if you prefer, a core of Kantian philosophy of mathematics intimated in Newton. For present purposes, this is neither here nor there.

The revolution that occurred between Newton and Einstein was the change to hypothetico-deductivism from that which preceded it. So what was it that preceded the hypothetico-deductive conception of formal systems in mathematics? I call this earlier form of mathematics, i.e., I call pre-hypothetico-deductive mathematics, categorico-deductive mathematics, because the principles or axioms now asserted hypothetically were once asserted categorically, in the belief that the truths of formal thought, i.e., of logic and mathematics, were eternal, immutable, unchanging truths, recognized by the mind’s eye as incontrovertible, indubitable, necessary truths as soon as they were glimpsed. It was often said (and is sometimes still today said), that to understand an axiom is ipso facto to see that it must be true; this is the categorico-deductive perspective.

In mathematics as it is pursued today, as an exercise in hypothetico-deductive reasoning, axioms are posited not because they are held to be necessarily true, or self-evidently true, or undeniably true; axioms need not be true at all. Axioms are posited because they are an economical point of origin for the fruitful derivation of consequences. This revolution in mathematical rigor transformed the landscape of mathematical thought so completely that Bertrand Russell, writing in the early twentieth century could write, “…mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Here formal, logical truth is entirely insulated from empirical, intuitive truth. It is at least arguable that the new formalisms made possible by the hypothetico-deductive method are at least partially responsible for Einstein’s innovations in physics. (I have earlier touched on Einstein’s conception of formalism in A Century of General Relativity and Constructive Moments within Non-Constructive Thought.)

If you are familiar with Einstein’s lecture, and especially with the opening summary of Einstein’s philosophy of mathematics, you will immediately recognize that Einstein formulates his position around the distinction between the categorico-deductive (which Einstein calls the “older interpretation” of axiomatics) and the hypothetico-deductive (which Einstein calls the “modern interpretation” of axiomatics). Drawing upon this distinction, Einstein gives us a somewhat subtler and more flexible formulation of the fundamental disconnect between the formal and the material than that which Russell paradoxically thrusts in our face. By formulating his distinction in terms of “as far as,” Einstein implies that there is a continuum of the dissociation between what Einstein called the “logical-formal” and “objective or intuitive content.”

Einstein then goes on to assert that a purely logical-formal account of mathematics joined together with the totality of physical laws allows us to say something, “about the behavior of real things.” The logical-formal alone can can say nothing about the real world; in its isolated formal purity it is perfectly rigorous and certain, but also impotent. This marvelous structure must be supplemented with empirical laws of nature, empirically discovered, empirically defined, empirically applied, and empirically tested, in order to further our knowledge of the world. Here we see Einstein making use of the hypothetico-deductive method, and supplementing it with contemporary physical theory; significantly, in order to establish a relationship between the formalisms of general relativity and the actual world he didn’t try to turn back the clock by returning to categorico-deductivism, but took up hypothetic-deductivism and ran with it.

But all of this is mere prologue. The question that Einstein wants to discuss in his lecture is the spatial extension of the universe, which Einstein distills to two alternatives:

1. The universe is spatially infinite. This is possible only if in the universe the average spatial density of matter, concentrated in the stars, vanishes, i.e., if the ratio of the total mass of the stars to the volume of the space through which they are scattered indefinitely approaches zero as greater and greater volumes are considered.

2. The universe is spatially finite. This must be so, if there exists an average density of the ponderable matter in the universe that is different from zero. The smaller that average density, the greater is the volume of the universe.

Albert Einstein, Geometry and Experience, Lecture before the Prussian Academy of Sciences, January 27, 1921. The last part appeared first in a reprint by Springer, Berlin, 1921

It is interesting to note, especially in light of the Kantian distinction noted above between sensible and intellectual intuition, that one of Kant’s four antinomies of pure reason was whether or not the universe was finite or infinite in extent. Einstein has taken this Kantian antimony of pure reason and has cast it in a light in which it is no longer exclusively the province of pure reason, and so may be answered by the methods of science. To this end, Einstein presents the distinction between a finite universe and an infinite universe in the context of the density of matter — “the ratio of the total mass of the stars to the volume of the space through which they are scattered” — which is a question that may be determined by science, whereas the purely abstract terms of the Kantian antimony allowed for no scientific contribution to the solution of the question. For Kant, pure reason could gain no traction on this paralogism of pure reason; Einstein gains traction by making the question less pure, and moves toward more engagement with reality and therefore less certainty.

It was the shift from categorico-deductivism to hypothetico-deductivism, followed by Einstein’s “completion” of geometry by the superaddition of empirical laws, that allows Einstein to adopt a methodology that is both rigorous and scientifically fruitful. (“We will call this completed geometry ‘practical geometry,’ and shall distinguish it in what follows from ‘purely axiomatic geometry’.”) Where the simplicity of Euclidean geometry allows for the straight-forward application of empirical laws to “practically-rigid bodies” then the simplest solution of Euclidean geometry is preferred, but where this fails, other geometries may be employed to resolve the apparent contradiction between mathematics and empirical laws. Ultimately, the latter is found to be the case — “the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction” — and so it is Riemannian geometry rather than Euclidean geometry that is the mathematical setting of general relativity.

Einstein’s use of Riemannian geometry is significant. The philosophical shift from categorico-deductivism to hypothetico-deductivism could be reasonably attributed to (or, at least, to follow from) the nineteenth century discovery of non-Euclidean geometries, and this discovery is an interesting and complex story in itself. Gauss (sometimes called the “Prince of Mathematicians”) discovered non-Euclidean geometry, but published none of it in his lifetime. It was independently discovered by the Hungarian János Bolyai (the son of a colleague of Gauss) and the Russian Nikolai Ivanovich Lobachevsky. Both Bolyai and Lobachevsky arrived at non-Euclidean geometry by adopting the axioms of Euclid but altering the axiom of parallels. The axioms of parallels had long been a sore spot in mathematics; generation after generation of mathematicians had sought to prove the axiom of parallels from the other axioms, to no avail. Bolyai and Lobachevsky found that they could replace the axiom of parallels with another axiom and derive perfectly consistent but strange and unfamiliar geometrical theorems. This was the beginning of the disconnect between the logical-formal and objective or intuitive content.

Riemann also independently arrived at non-Euclidean geometry, but by a different route than that taken by Bolyai and Lobachevsky. Whereas the latter employed the axiomatic method — hence its immediate relevance to the shift from the categorico-deductive to the hypothetico-deductive — Riemann employed a metrical method. That is to say, Riemann’s method involved measurements of line segments in space defined by the distance between two points. In Euclidean space, the distance between two points is given by a formula derived from the Pythagorean theorem — d = √(x2x1)2 + (y2y1)2 — so that in non-Euclidean space the distance between two points could be given by some different equation.

Whereas the approach of Bolyai and Lobachevsky could be characterized as variations on a theme of axiomatics, Riemann’s approach could be characterized as variations on a theme of analytical geometry. The applicability to general relativity becomes clear when we reflect how, already in antiquity, Eratosthenes was able to determine that the Earth is a sphere by taking measurements on the surface of the Earth. By the same token, although we are embedded in the spacetime continuum, if we take careful measurements we can determine the curvature of space, and perhaps also the overall geometry of the universe.

From a philosophical standpoint, it is interesting to ask if there is an essential relationship between the method of a non-Euclidean geometry and the geometrical intuitions engaged by these methods. Both Bolyai and Lobachevsky arrived at hyperbolic non-Euclidean geometry (an infinitude of parallel lines) whereas Riemann arrived at elliptic non-Euclidean geometry (no parallel lines). I will not attempt to analyze this question here, though I find it interesting and potentially relevant. The non-Euclidean structure of Einstein’s general relativity is more-or-less a three dimensional extrapolation of the elliptic two dimensional surface of a sphere. Our minds cannot conceive this (at least, my mind can’t conceive of it, but there may be mathematicians who, having spent their lives thinking in these terms, are able to visualize three dimensional spatial curvature), but we can formally work with the mathematics, and if the measurements we take of the universe match the mathematics of Riemannian elliptical space, then space is curved in a way such that most human beings cannot form any geometrical intuition of it.

Einstein’s lecture culminates in an attempt to gently herd his listeners toward achieving such an impossible geometrical intuition. After a short discussion of the apparent distribution of mass in the universe (in accord with Einstein’s formulation of the dichotomy between an infinite or a finite universe), Einstein suggests that these considerations point to a finite universe, and then explicitly asks in his lecture, “Can we visualize a three-dimensional universe which is finite, yet unbounded?” Einstein offers a visualization of this by showing how an infinite Euclidean plane can be mapped onto a finite surface of a sphere, and then suggesting an extrapolation from this mapping of an infinite two dimensional Euclidean space to a finite but unbounded two dimensional elliptic space as a mapping from an infinite three dimensional Euclidean space to a finite but unbounded three dimensional elliptic space. Einstein explicitly acknowledges that, “…this is the place where the reader’s imagination boggles.”

Given my own limitations when it comes to geometrical intuition, it is no surprise that I cannot achieve any facility in the use of Einstein’s intuitive method, though I have tried to perform it as a thought experiment many times. I have no doubt that Einstein was able to do this, and much more besides, and that it was, at least in part, his mastery of sophisticated forms of geometrical intuition that had much to do with his seminal discoveries in physics and cosmology. Einstein concluded his lecture by saying, “My only aim today has been to show that the human faculty of visualization is by no means bound to capitulate to non-Euclidean geometry.”

Above I said it would be an interesting question to pursue whether there is an essential relationship between formalisms and the intuitions engaged by them. This problem returns to us in a particularly compelling way when we think of Einstein’s effort in this lecture to guide his readers toward conceiving of the universe as finite and unbounded. When Einstein gave this lecture in 1922 he maintained a steady-state conception of the universe. About the same time the Russian mathematician Alexander Friedmann was formulating solutions to Einstein’s field equations that employed expanding and contracting universes, of which Einstein himself did not approve. It wasn’t until Einstein met with Georges Lemaître in 1927 that we know something about Einstein’s engagement with Lemaître’s cosmology, which would become the big bang hypothesis. (Cf. the interesting sketch of their relationship, Einstein and Lemaître: two friends, two cosmologies… by Dominique Lambert.)

Ten years after Einstein delivered his “Geometry and Experience” lecture he was hesitantly beginning to accept the expansion of the universe, though he still had reservations about the initial singularity in Lemaître’s cosmology. Nevertheless, Einstein’s long-standing defense of the counter-intuitive idea (which he attempted to make intuitively palatable) of a finite and unbounded universe would seem to have prepared his mind for Lemaître’s cosmology, as Einstein’s finite and unbounded universe is a natural fit with the big bang hypothesis: if the universe began from an initial singularity at a finite point of time in the past, then the universe derived from the initial singularity would still be finite any finite period of time after the initial singularity. Just as we find ourselves on the surface of the Earth (i.e., our planetary endemism), which is a finite and unbounded surface, so we seem to find ourselves within a finite and unbounded universe. Simply connected surfaces of these kinds possess a topological parsimony and hence would presumably be favored as an explanation for the structure of the world in which we find ourselves. Whether our formalisms (i.e., those formalisms accessible to the human mind, i.e., intuitively tractable formalisms) are conductive to this conception, however, is another question for another time.

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An illustration from Einstein’s lecture Geometry and Experience

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Studies in Formalism

1. The Ethos of Formal Thought

2. Epistemic Hubris

3. Parsimonious Formulations

4. Foucault’s Formalism

5. Cartesian Formalism

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

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

8. Unpacking an Einstein Aphorism

9. The Overview Effect in Formal Thought

10. Einstein on Geometrical intuition

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Wittgenstein's Tractatus Logico-Philosophicus was part of the efflourescence of formal thinking focused on logic and mathematics.

Wittgenstein’s Tractatus Logico-Philosophicus was part of an early twentieth century efflorescence of formal thinking focused on logic and mathematics.

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Monday


Edmund Husserl wanted philosophy to become a rigorous science.

Edmund Husserl wanted philosophy to become a rigorous science.

In several posts I have discussed the need for a science of civilization (cf., e.g., The Future Science of Civilizations), and this is a theme I intended to continue to pursue in future posts. It is no small matter to constitute a new science where none has existed, and to constitute a new science for an object of knowledge as complex as civilization is a daunting task.

The problem of constituting a science of civilization, de novo for all intents and purposes, may be seen in the light of Husserl’s attempt to constitute (or re-constitute) philosophy as a rigorous science, which was a touchstone of Husserl’s work. Here is a passage from Husserl’s programmatic essay, “Philosophy as Strict Science” (variously translated) in which Husserl distinguishes between profundity and intelligibility:

“Profundity is the symptom of a chaos which true science must strive to resolve into a cosmos, i.e., into a simple, unequivocal, pellucid order. True science, insofar as it has become definable doctrine, knows no profundity. Every science, or part of a science, which has attained finality, is a coherent system of reasoning operations each of which is immediately intelligible; thus, not profound at all. Profundity is the concern of wisdom; that of methodical theory is conceptual clarity and distinctness. To reshape and transform the dark gropings of profundity into unequivocal, rational propositions: that is the essential act in methodically constituting a new science.”

Edmund Husserl, “Philosophy as Rigorous Science” in Phenomenology and the Crisis of Philosophy, edited by Quentin Lauer, New York: Harper, 1965 (originally “Philosophie als strenge Wissenschaft,” Logos, vol. I, 1911)

Recently re-reading this passage from Husserl’s essay I realized that much of what I have attempted in the way of “methodically constituting a new science” of civilization has taken the form of attempting to follow Husserl’s pursuit of “unequivocal, rational propositions” that eschew “the dark gropings of profundity.” I think much of the study of civilization, immersed as it is in history and historiography, has been subject more often to profound meditations (in the sense that Husserl gives to “profound”) than conceptual clarity and distinctness.

The Cartesian demand for clarity and distinctness is especially interesting in the context of constituting a science of civilization given Descartes’ famous disavowal of history (on which cf. the quote from Descartes in Big History and Scientific Historiography); if an historical inquiry is the basis of the study of civilization, and history consists of little more than fables, then a science of civilization becomes rather dubious. The emergence of scientific historiography, however, is relevant in this context.

The structure of Husserl’s essay is strikingly similar to the first lecture in Russell’s Our Knowledge of the External World. Both Russell and Husserl take up major philosophical movements of their time (and although the two were contemporaries, each took different examples — Husserl, naturalism, historicism, and Weltanschauung philosophy; Russell, idealism, which he calls “the classical tradition,” and evolutionism), primarily, it seems, to show how philosophy had gotten off on the wrong track. The two works can profitably be read side-by-side, as Russell is close to being an exemplar of the naturalism Husserl criticized, while Husserl is close to being an exemplar of the idealism that Russell criticized.

Despite the fundamental difference between Husserl and Russell, each had an idea of rigor and each attempted to realize in their philosophical work, and each thought of that rigor as bringing the scientific spirit into philosophy. (In Kierkegaard and Russell on Rigor I discussed Russell’s conception of rigor and its surprising similarity to Kierkegaard’s thought.) Interestingly, however, the two did not criticize each other directly, though they were contemporaries and each knew of the other’s work.

The new science Russell was involved in constituting was mathematical logic, which Roman Ingarden explicitly tells us that Husserl found inadequate for the task of a scientific philosophy:

“It is maybe unexpected and surprising that Husserl who was trained as a mathematician did not seek salvation for philosophy in the mathematical method which had from time to time stood out like a beacon as an ideal worthy of imitation by philosophers. But mathematical logic could not satisfy him… above all he fought for responsibility in philosophical research and devoted many years to the elaboration of a method which, according to him, was to secure for philosophy the status of a science.”

Roman Ingarden, On the Motives which Led Husserl to Transcendental Idealism, Translated from the Polish by Arnor Hannibalsson, Den Haag: Martinus Nijhoff, 1975, p. 9.

Ingarden’s discussion of Husserl is instructive, in so far as he notes the influence of mathematical method upon Husserl’s thought, but also that Husserl did not try to employ a mathematical method directly in philosophy. Rather, Husserl invested his philosophical career in the formulation of a new methodology that would allow the values of rigorous scientific practice to be expressed in philosophy and through a philosophical method — a method that might be said to be parallel to or mirroring the mathematical method, or derived from the same thematic motives as those that inform mathematical methodology.

The same question is posed in considering the possibility of a rigorously scientific method in the study of civilization. If civilization is sui generis, is a sui generis methodology necessary to the formulation of a rigorous theory of civilization? Even if that methodology is not what we today know as the methodology of science, or even if that methodology does not precisely mirror the rigorous method of mathematics, there may be a way to reason rigorously about civilization, though it has yet to be given an explicit form.

The need to think rigorously about civilization I took up implicitly in Thinking about Civilization, Suboptimal Civilizations, and Addendum on Suboptimal Civilizations. (I considered the possibility of thinking rigorously about the human condition in The Human Condition Made Rigorous.) Ultimately I would like to make my implicit methodology explicit and so to provide a theoretical framework for the study of civilization.

Since theories of civilization have been, for the most part, either implicit or vague or both, there has been little theoretical framework to give shape or direction to the historical studies that have been central to the study of civilization to date. Thus the study of civilization has been a discipline adrift, without a proper research program, and without an explicit methodology.

There are at least two sides to the rigorous study of civilization: theoretical and empirical. The empirical study of civilization is familiar to us all in the form of history, but history studied as history, as opposed to history studied for what it can contribute to the theory of civilization, are two different things. One of the initial fundamental problems of the study of civilization is to disentangle civilization from history, which involves a formal rather than a material distinction, because both the study of civilization and the study of history draw from the same material resources.

How do we begin to formulate a science of civlization? It is often said that, while science begins with definitions, philosophy culminates in definitions. There is some truth to this, but when one is attempting to create a new discipline one must be both philosopher and scientist simultaneously, practicing a philosophical science or a scientific philosophy that approaches a definition even as it assumes a definition (admittedly vague) in order for the inquiry to begin. Husserl, clearly, and Russell also, could be counted among those striving for a scientific philosophy, while Einstein and Gödel could be counted as among those practicing a philosophical science. All were engaged in the task of formulating new and unprecedented disciplines.

This division of labor between philosophy and science points to what Kant would have called the architectonic of knowledge. Husserl conceived this architectonic categorically, while we would now formulate the architectonic in hypothetico-deductive terms, and it is Husserl’s categorical conception of knowledge that ties him to the past and at times gives his thought an antiquated cast, but this is merely an historical contingency. Many of Husserl’s formulations are dated and openly appeal to a conception of science that no longer accords with what we would likely today think of as science, but in some respects Husserl grasps the perennial nature of science and what distinguishes the scientific mode of thought from non-scientific modes of thought.

Husserl’s conception of science is rooted in the conception of science already emergent in the ancient world in the work of Aristotle, Euclid, and Ptolemy, and which I described in Addendum on the Agrarian-Ecclesiastical Thesis. Russell’s conception science is that of industrial-technological civilization, jointly emergent from the scientific revolution, the political revolutions of the eighteenth century, and the industrial revolution. With the overthrow of scholasticism as the basis of university curricula (which took hundreds of years following the scientific revolution before the process was complete), a new paradigm of science was to emerge and take shape. It was in this context that Husserl and Russell, Einstein and Gödel, pursued their research, employing a mixture of established traditional ideas and radically new ideas.

In a thorough re-reading of Husserl we could treat his conception of science as an exercise to be updated as we went along, substituting an hypothetico-deductive formulation for each and every one of Husserl’s categorical formulations, ultimately converging upon a scientific conception of knowledge more in accord with contemporary conceptions of scientific knowledge. At the end of this exercise, Husserl’s observation about the different between science and profundity would still be intact, and would still be a valuable guide to the transformation of a profound chaos into a pellucid cosmos.

This ideal, and ever more so the realization of this ideal, ultimately may not prove to be possible. Husserl himself in his later writings famously said, “Philosophy as science, as serious, rigorous, indeed apodictically rigorous, science — the dream is over.”(It is interesting to compare this metaphor of a dream to Kant’s claim that he was awoken from his dogmatic slumbers by Hume.) The impulse to science returns, eventually, even if the idea of an apodictically rigorous science has come to seem a mere dream. And once the impulse to science returns, the impulse to make that science rigorous will reassert itself in time. Our rational nature asserts itself in and through this impulse, which is complementary to, rather than contradictory of, our animal nature. To pursue a rigorous science of civilization is ultimately as human as the satisfaction of any other impulse characteristic of our species.

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