Ursprung (Origin) is a wonderfully enthusiastic piece of writing. I like to imagine it as welling up in Husserl all his life, as something he desperately wanted to say. It has the excitement of adolescence about it more than the precision of the mature scholar. Or rather the nostalgia of an old man in carpet slippers recalling the delights of youth, a bit like the second movement of Mahler’s second symphony. That is called the Resurrection symphony. After the first movement buries the young hero, the second is all recollected lyricism and love tunes, before we carry on with the serious business of resurrection in the remaining three movements. Mahler sometimes but only sometimes called the second movement irrelevant. One might sometimes but only sometimes also call Ursprung irrelevant. It demands careful reflection but not carping criticism. I hope that my remarks below, although in a style and tradition entirely different from Husserl’s, can partake of Husserl’s sense of sheer joy and wonder. Naive astonishment should set the tone for all serious discussion.
“What Mathematics Has Done to Some and Only Some Philosophers”: Under that title I argued that some philosophers have been overwhelmed by the phenomenon of mathematical demonstration (Hacking, 2000, 83–138). Plato, Leibniz, Kant, Bertrand Russell, Ludwig Wittgenstein, and Imre Lakatos figure prominently in my discussion. Each, in his way, allowed a concern for mathematical proof to infect his whole philosophy. Other philosophers are notable for their lack of interest, and perhaps they are right to ignore the phenomenon of mathematical knowledge, even to disdain it. But I have always suspected that some and only some people are bowled over by proofs, while others simply fail to grasp their power. Proof brings with it the idea of a priori knowledge and necessary truth.
The feeling of astonishment, even of awe, that comes with understanding a notably deep and perspicuous demonstration is one of the things that has moved the philosophers. In my opinion, it is a mark of profound philosophical sensibility to be so moved and distinguishes the important thinkers from lesser minds. My list of philosophers struck by mathematical proof did not include Husserl, but he was among the awed.
I shall be discussing what mathematics did to Husserl. I shall not claim that it invaded his whole philosophy, but it certainly moved him deeply from the time of his early interest in the foundations of arithmetic to his 1935 lecture published in 1939 as “Die Frage nach dem Ursprung der Geometrie als intentionalhistorisches Problem,” which henceforth I shall call Ursprung. And it certainly did infect his thinking about the exact sciences in the Crisis. He focused not on proof but on geometry as the science of ideal objects. But as I shall explain, you cannot have ideal objects unless you also have the practice of making and telling proofs. It is proofs that ensure—or create?—the objectivity of mathematics that was so much the concern of the Crisis. By proof, of course I do not mean Leibnizian, combinatorial, line-by-line proofs, but what I have already called perspicuous proofs, proofs that one can in the Cartesian way take in whole at once, by what Husserl would have called a direct intuition of its validity.
I share Husserl’s awe in the face of geometry. Perhaps like Plato, he is more impressed by idealized shapes, while I am more impressed by perspicuous proofs. But Husserl was after far more than the origins of geometry. Even the second half of the title that was retrospectively given to his lecture—“as intentional-historical problem”—does not get at the half of it. The inquiry is to be “exemplary.” It aims at carrying out, “in the form of historical meditations, self-reflections about our own present philosophical situation in the hope that in this way we can finally take possession of the meaning, method, and beginning of philosophy, the one philosophy to which our life seeks to be and ought to be devoted” (Origin of Geometry, p. 354). I do not share this project, which is not to say that I dismiss it, only that I cannot participate in it.
In addition, Ursprung was written at the same time as the Crisis and was probably intended to be included in it. That is a book written by a German nationalist proud to claim intellectual descent from the grand line of German philosophers, indeed the “European” philosophers, the tradition that begins in Greece and is fully transmitted only to Germany. Husserl was dismayed by the irrationalism that was sweeping his nation. He was wholly German, with a son who had died for the Fatherland at Verdun. But he was also an elderly Jew (he had retired from his chair in 1928). His best pupil had overtaken his own role as the most prominent German philosopher—with a philosophy that, in his judgement, was another symptom of the pervasive irrationalism of the time. The abiding theme of Crisis was to be a return to (what Husserl thought was) the greatest European contribution to world history, namely the discovery of rationality and objectivity.
Looking back, as we do now, the book is a work of tragedy. And that in a strong sense. It was doomed to fail in its attempt to restore the state of reason. First, because any purely intellectual endeavor would, no matter how impassioned, be impotent against the raging politics of violence and hate. Worse, that politics had its rival vision of the greatness that was Europe. Second, the book was doomed because its analysis of the deep ground of pending disaster was just wrong. Whatever be the merits of phenomenology, it was not the case that Europe fell into the abyss of selfdestruction because European science had lost touch with something deep and original in the human spirit, a depth that underlay the sciences and made them possible. I shall not further discuss these matters, but it is important to recall that Ursprung aims at more than an analysis of the origins of geometry.
Over and over we read in the essay words that are translated as primal sources, the primal beginnings (for example, Origin of Geometry, p. 367). These are the Ur words. When it is beginnings in time, Husserl tended to use the word Urstiftung. The title word Ursprung means not simply origin but source. In the case of knowledge, it is not only the source of the knowledge but also the ground, the evidence, for that knowledge. I have called the present chapter “Husserl on the Origins [plural] of Geometry.” It would be misleading to call it “Husserl on the Origin [singular] of Geometry.” Yes, this is a chapter about Husserl’s Ursprung, the lecture, but I have no commitment to the notion that geometry has a singular origin, or to the idea of a primal evidence.
I distrust the lust for things primal. There is, for sure, an ineluctable drive in Western consciousness to find the first moment. To find the skeleton of the first human being. To scan the first three seconds of the universe. To reveal the primal scene, or to relive the primal scream. Each of those programs makes sense, although some may prove to be wrongheaded or illusory.
I do believe that geometry had beginnings. Undoubtedly, there were several beginnings. In these days of postcolonial history, one notes that Husserl lived in the confident world where civilization mostly meant European civilization and its Mediterranean predecessors. The history of ancient mathematics had been codified in the mid-nineteenth century by mostly German and a few French and British scholars. It was a cumulative story that led from Babylonia via Athens and Alexandria to Göttingen. Today the texts of that original history are being reread with new eyes. Other civilizations are being opened up. Karine Chemla and Shuchun Guo have just published the most classic of Chinese texts, The Nine Chapters. To quote from Geoffrey Lloyd’s preface:
One can thus say that the classical mathematics of ancient China, and Euclid’s Elements, represent two radically different styles of mathematical reasoning. This comparison teaches a crucial lesson: that there is no unique pattern of development that mathematics has to follow. (Chemla and Guo, 2004, xi)
Lloyd goes on to say that the existence of different styles of mathematical reasoning shows that there was no one thing at which mathematicians were aiming and that it is important for Westerners to grasp the existence of alternative styles.
Perhaps Lloyd took the so-called axiomatic method of Euclid as definitive of the Greek style. Husserl states that axioms are not what is fundamental in Greek mathematical reasoning. I agree, although perhaps for a reason different from Husserl’s. In my opinion the making of proofs was fundamental.
The ability to make and grasp proofs must be a human universal, although a few people in any culture are much better at it than others. It is utterly immaterial whether the discovery of this human faculty occurred only once, among some peoples on the eastern shores of the Mediterranean, or many times in many places. My problem is that there is nothing metaphysically deep to learn about this discovery, nothing primal, nothing that leads to a depth problem or a depth solution bearing on that philosophy to which our life ought to be devoted.
Perhaps all quests for the primal will turn out to be a mistake. The idea of the primal scene has been trashed, most effectively not by the Freud bashers but by the Wolfman himself. The primal scream has echoes, now, only in the murmurings of psychobabble. You may think this reference is flippant, but no. Husserl’s idea of the primal, of the Ur, is curiously close to that of Freud’s primal scene. Both, of course, have roots in Hegel. Freud’s thought was that if only the Wolfman could fully recover the scene of his parents’ copulation in certain positions—and the reworking of it after latency—then he would understand and overcome the neuroses that swamped him later in life. Now whatever might be the imagined mechanism by which the scene worked on the Wolfman’s unconscious, there was no doubt that it was supposed to be deep, that is, of profound significance. I find it interesting but not deep, and have exactly the same respectful skepticism about Husserl’s primal sources.
The classic statement of naive astonishment at geometry is to be found in one of the passionate passages to be found in Kant. I suppose everyone knows it, but please allow me to quote the entire paragraph in translation. I shall repeat certain words in square brackets as a sort of running tally of the points being made.
In the earliest times of which history affords us a record, mathematics had already entered on the sure course of science, among that wonderful nation, the Greeks [the Greeks]. Still it is not to be supposed that it was as easy for this science to strike into, or rather to construct for itself, that royal road, as it was for logic, in which reason has only to deal with itself [easy for logic, hard for maths]. On the contrary, I believe that it must have remained long—chiefly among the Egyptians—in the stage of blind groping after its true aims and destination, and that it was revolutionized by the happy idea of one man, who struck out and determined for all time the path which this science must follow, and which admits of an infinite advancement [revolution, one man]. The history of this intellectual revolution—much more important than the discovery of the passage round the celebrated Cape of Good Hope—and of its author has not been preserved [a revolution whose importance can hardly be overestimated]. But Diogenes Laertius, in naming the supposed discoverer of some of the simplest elements of geometrical demonstration—elements which, according to ordinary opinion, do not even require to be proved—makes it apparent that the change introduced by the first indication of this new path, must have seemed of the utmost importance to the mathematicians of that age, and it has thus been obscured against the chance of oblivion. A new light must have flashed on the mind of the first man (Thales, or whatever may have been his name) who demonstrated the properties of the isosceles triangle. [A new light flashes on a single mind! ] For he found that it was not sufficient to meditate on the figure, as it lay before his eyes, or the conception of it, as it existed in his mind, and thus endeavour to get at the knowledge of its properties, but that it was necessary to produce these properties, as it were, by a positive a priori construction [a proof, by constructing these properties from that concept]; and that in order to arrive with certainty at an a priori cognition, he must not attribute to the object any other properties than those which necessarily followed from that which he had himself, in accordance with his conception, placed in the object. [It is not just observing the properties of the triangle, but demonstrating it by a construction—one that is a priori and needs no measurement.] (Kant, 1924, Bxi–xii; translated by Meiklejohn)
Husserl agreed with the content of Kant’s prehistory of geometry and shared in its spirit. This does not go without saying, and that for several reasons. First, although he comes back to the legend of Thales, Husserl is evidently thinking of communal activity: “the first oral cooperation of the beginning geometers” (Origin of Geometry, p. 368). The emphasis on community is an essential supplement to Kant. But there is a beginning all right. “Clearly, then, geometry must have arisen out of a first acquisition, out of first creative activities” (Origin of Geometry, p. 355). The recent disciplines known as science studies or sociology of scientific knowledge insist that all science is social. Husserl realized that long ago, while Kant seems not to have done so.
There is a second consideration. Husserl is only imperfectly aligned with Kant. Kant emphasizes proof, a construction out of the concept that yet does not go beyond the concept. Husserl accentuates the ideal objects of geometry, the ideal objects that fall under the idealized concept. But in any “principle-of-charity” reading of the two philosophers, the difference must be superficial. For only by demonstrating properties of the object (say Kant’s isosceles triangle) is there any sense of the ideal object, an object that of necessity has these properties. Some readers of Husserl would insist that Husserl wants us to form a direct acquaintance with the ideal object, but that is to miss what Husserl saw: It is perspicuous proof that allows us to directly intuit the objects.
Conversely, if there is certainty about the properties, apodictic certainty (to use Kant’s word apodictic, which Husserl makes his own), then we are not speaking of the empirical isosceles, but of an ideal object.
After the long passage just quoted, Kant turns to seventeenth-century physics. It matters little to the sequel, but we should record a fundamental difference between Kant’s attitude and that of Husserl. Kant does say that he is confining himself “to the empirical side of natural science,” yet it is clear that he believes that “the wise Bacon gave a new direction to physical studies.” Galileo figures because he “experimented with balls of a definite weight on the inclined plane.” After the achievements of Galileo and Torricelli and Stahl, “a light broke upon all natural philosophers.” Kant’s hero was Newton. Kant’s Galileo introduces the “experimental method.” He added in a footnote that “the first steps” of that method “are involved in some obscurity.” Indeed the contributions of Galileo and Stahl are separated by about a century!
Husserl, finishing his book in 1935, never mentioned Bacon. The name of Newton occurs three times in the Crisis, but only as a name. Galileo is Husserl’s hero. It is a Galileo of the 1930s, perhaps best represented a little later in the work of the great French historian of science, Alexandre Koyré. That is, a Platonist Galileo who, to exaggerate, never did an experiment in his life but worked out the mathematical form of nature. No coincidence that Husserl and Koyré shared a common attitude to Galileo. Koyré, like so many other French humanists, was trained as a philosopher, and as a young man went to study with the best, namely in Göttingen with Husserl. He was also an active member of Husserl’s phenomenological circle in Munich.
Galileo the experimenter made a comeback in the 1970s. An indefatigable amateur, Stillman Drake, found Galileo’s autograph records of experimental observations and confirmed that you would observe those results with Galileo’s modest apparatus. Which is not to say that the vision of Koyré and his immediate predecessors was wrong, only that it was incomplete, although not as incomplete as that of Kant. Husserl’s Galileo was above all the mastermind behind the mathematization of nature. What Husserl named “the Galilean style” of reasoning was the mode of reasoning by which, in Kant’s words, “physics entered on the highway of science.” The implied notion of a relatively small number of fundamentally distinct styles of reasoning has attracted a number of twentieth-century writers, from at least the time of Oswald Spengler. The cosmologist Steven Weinberg picked up the phrase Galilean style from Husserl, and Noam Chomsky picked it up from Weinberg.2 For those readers, the Galilean style was the method of making abstract mathematical models of phenomena, and only later, when possible, testing them against experience. Husserl certainly meant that, but he also meant something more specific. He thought that Galileo not only made mathematical models of nature: He also transformed nature by making her geometrical.
No one now reads Husserl for a literal history of science. His is a rational reconstruction, and his Galileo is less the historical Galileo than an emblem of what happened to the sciences in the late Renaissance. The world became mathematical, or rather, in his opinion, geometrical. In Galileo’s day, and for some time afterward, geometrical pretty well meant what we would call mathematical. But Husserl had a stronger reference, for he thought of Galileo as planting a geometrical structure on the world. And this is a matter of a vision of nature as composed of the ideal objects, the ideal structures, of geometry. This leads directly to the project of the Crisis. Physical scientists have lost touch with the way in which the ideal objects of geometry have come into being. Hence they have what is in effect a false image of what they are doing. I hope it is not inapt, or an abuse of overused phrases, to say that they work unwittingly in bad faith; theirs is the unhappy consciousness. Only a return to primal sources, to the Ursprung, will liberate them. In particular, we shall regain a full sense of the objectivity of the sciences by comprehending the source, the originary evidence, for this objectivity.
I have already expressed skepticism about this desire to uncover the origins as the deep source of objectivity. But there is a more specific worry. Galileo (the emblem that we call Galileo) did mathematize the world—by which I think Husserl meant that the new science caused the scientists of that and all later epochs to conceive of, and indeed to perceive, the world as having a structure described by and conforming to mathematics. But the world was not geometrized in the sense in which we now think of geometry. Galileo mathematized motion, a mathematics that culminated in the differential calculus of Leibniz and Newton. I am not at all sure that it makes sense to say that the calculus has ideal objects. Perhaps the Newtonian version does—the infinitesimals of which the good Berkeley made so much fun. Does the Leibnizian version have ideal objects? I think not.
This observation is not an instance of the carping that is to be avoided. Husserl was out to give a sense of the objectivity of mathematical reasoning in all its richness. In my opinion it would have served him well to attend to the new proofs that arose in the Galilean era and on into the epoch of rational mechanics. Readers more close to the spirit of Husserl will retort that I am thinking in too limited a way of the objects of geometry as circles and isosceles triangles. No, what was meant by “ideal objects” was ideal formal structures. Excellent. Let us compromise by saying that living proofs that can be grasped are the mathematics-in-action side of the Galilean style, and that the structures are what it produces.
Kant saw physics as entering the royal road of science with the empirical methods of Bacon, Galileo, and Stahl. Husserl saw it as mathematizing nature. Husserl’s vision of the new physics was a profound advance on that of Kant. Yet at another level their pictures of ancient mathematics are close kin. We have long been under the sway of the great German historians, and a few French and English ones, for whom the Greeks were indeed “that wonderful nation” that created geometry. Nowadays, as already stated, a critical rereading and rethinking of ancient texts is under way. There was more to ancient Babylon than the old historians taught, and there is a great tradition of Chinese mathematics. Alas, I have not yet been reconstructed. I think that the possibility of making proofs, the discovery of mathematical demonstration, really did take place on the shores of West Asia, by people who numbered among them Thales, whatever was his name. And I have no more problem in using “Thales,” without the usual qualification, as the name of one such man. So I am closer to Kant and Husserl than more contemporary minds.
I am not so unreconstructed as to imagine that “a new light flashed upon the mind” of just one man, that man whom we call Thales. Maybe it did, but he had to create a school that grasped what he did. It is more likely that there was an emerging awareness of the possibility of proof, of gaining knowledge by thinking and arguing, of an independence of certain facts from the confirmations of measurement. Of course there could have been just one amazing figure who perceived the first proof, and who by talk and toil created a community in which mathematical proving went on. What we do know is that there was a long, slow development of proof ideas, and that during that process, proof was experienced as something extraordinary. Even very late in the day, Plato used the possibility of proof, illustrated by the slave boy in Meno, in an astonishing argument for the immortality of the soul. He insisted that decades spent studying mathematics was an absolute precondition for joining the administrative elite of a wise republic. Call Pythagoras and Plato math-obsessed freaks: Even so, no other civilization left traces of freaks like those who were also cultural icons. So I agree, in principle, with the vision of Kant and Husserl. Yes, there was a revolution as profound as any in the history of the human race, and it had a profound impact on all future Western philosophizing.
One hopes that postcolonial history of mathematics will teach us more about an emblem other than Thales, namely Al-Khwarizmi (c. 785–850). That name gave us the word algorithm. His book Hisab al-jabr w’almuqabala gave us the word algebra. Among my as yet unreconstructed beliefs is that in the House of Wisdom in Baghdad, sometime around 850, say, they understood what an algorithm is. It is a procedure that by routine application of rules definitively solves any chosen member of a given class of problems. The Nine Chapters of Chinese mathematics mentioned above had a not unrelated concept six centuries earlier than the Arabians. But the Chinese commentaries immediately introduce “corrections”—that is to say, these are not strictly algorithms in the Arabic sense of the word, but systems of approximation. The Chinese rules are brilliant, but also exactly what an algorithm isn’t.
A simplistic reader of Husserl might go on to urge that the Chinese rules therefore lack the objectivity of the mathematics descended from Greek and the Arab sources. Not so: They are systematic objective approximations. They are closer to the actual practice of physics as we know it than our official methodology that wrongly teaches that approximation is always a mere approximation towards the real truth. If world-historical mathematics had developed from the Chinese model rather than by a fusion of the Greek and Arab models, we might have had far healthier philosophies than the ones we have been saddled with. I do not just mean a richer philosophy of mathematics or of the sciences: I mean philosophy. But it would not have been a philosophy so curiously populated by those in awe of mathematics—Plato, Leibniz, Russell, and the rest whom I have mentioned. But note that, pace Husserl, our mathematics and our sciences would have been just as objective as the actual ones that that took the royal road of ideal objects.
We should now consider Husserl’s phrase “historical a priori.” It is easy to be misled here, for this very phrase was to be lifted out of Husserl by a couple of generations of French philosophers. Ursprung was first published in the Revue internationale de philosophie in 1939. Ideas related to it and other aspects of Husserl’s phenomenology of the sciences were analyzed and criticized by the brilliant young philosopher of mathematics Jean Cavaillès, since canonized for his execution as a resistance fighter. His work sparked the French revolt against phenomenology. It was in many ways at the heart of the French abandonment of their own Jean-Paul Sartre and Maurice Merleau-Ponty. Jacques Derrida translated Ursprung , but what he took from it was the sensible emphasis that one aspect of the transmission and indeed sedimentation of mathematics is the transcription of arguments in sentences. After all, we know about Euclid because it is written down. Derrida turned that elementary fact into the amazing doctrine that the sentence is primary and trumps the spoken word. The promotion of the pure sentence, stripped of speech and of both speaker and author, accompanied—may even have led to—the “death of the subject,” rumors of which have been greatly exaggerated. Nevertheless, some of Husserl’s phrases stuck.
Notable among them was historical a priori, used, with a mixture of high seriousness and mordant irony, by Georges Canguilhem and Michel Foucault. The passage from Husserl through Cavaillès and Canguilhem to Foucault has been well chronicled and analyzed by David Hyder.3 I myself have traveled on the edges of this awesome field of force. Hence you may find in my discussion some of what Howard Bloom called the anxiety of influence.
What did he mean by historical a priori? I have heard it said that he meant no more than truths about history that we can know a priori—for example, that geometry must have originated in such and such a way. Given the present tenor of history of mathematics, that is a rather uncharitable understanding of Husserl. We have also heard more complex explanations, which tie it in to the entire project of transcendental phenomenology. I would like to propose a simpler account.
The phrase occurs only in the Ursprung. So it is plausible to suppose that it has a specific connection with geometry and its origins. I suggest first that Husserl is speaking of a priori knowledge in exactly the sense of Kant. Except that Kant thought that structures knowable a priori were preconditions for possible experience, at all times and places. Those structures are out there, to speak crudely, and the man whom we call Thales just happened to pull the light switch that was always there on the wall, always available to illumine some mind. In Husserl, the light that flashed upon the minds of the first mathematicians was a historical event before which there was no mathematical, geometrical, a priori. Thus I propose that historical a priori denotes an a priori structure that comes into being at a historical time. Perhaps the chief merit of this reading is simplicity. It is not so relativistic a thought as might appear: We should recall that a priori is an adjective whose first job is to characterize a type of knowledge. Even if there are eternal truths, knowledge of them comes into being in history.
To classify geometrical knowledge as historical a priori is not to imply that all a priori knowledge is historical in the same way. One imagines that a priori knowledge about geometrical objects was preceded by other types of a priori knowledge dependent upon analytical connections; perhaps also there is a priori knowledge dependent upon aspects of experience that phenomenology can reveal.
My understanding is a “French” one. When Canguilhem and Foucault spoke of the historical a priori, with a touch of irony I am sure, they meant the structures of systems of thought that determine the space of possibilities of what can be said. Not exactly preconditions, but conditions of possibility and exclusion that come into being, and may go out of existence, at canonical moments. In Foucault’s case, those moments tend to coincide extensionally with 1619, namely the year of Descartes’ night in the poêle, or 1789, the revolution. Yes, such historical a prioris can mutate or cease to exist, but they are otherwise Husserlian in that they are (a) historical, coming into being in historical or near-historical time, and (b) a priori determinants of all possible surface knowledge in a domain.
Husserl gives a very vivid account of the ways in which the original insights and achievements of the first mathematical communities might (or must?) have been developed into written texts. They evolved into Euclid’s Elements and grew and grew, into Galileo and beyond, an everopening horizon of new mathematics. As layer upon layer of proofs and texts evolve, so we have more and more sedimentation. And it becomes increasingly time-consuming to dig down right to the original evidence of what we call Thales’s generation. Descartes had something like the same thought. He held that you should some time in your life become convinced of every foundation (recall that Descartes was “foundational” in a way that Husserl never was), but you cannot do that every day of your life. Be content to remember that you once had a clear and distinct proof before you. The era of Descartes—that of Galileo—was just the beginning of the Husserlian mathematization of nature. Endless layers of proofs and postulates have piled up since. The metaphor of sedimentation is the most natural one in the world. And whereas at the time of Descartes and Galileo, one really could reach down to bottom at least once in one’s life, that is no longer practicable for the working mathematician.
There is a related, and perhaps more primal, kind of sedimentation. Husserl insisted that long before the man whom we call Thales, there must have been experience of approximations to ideal objects. Boards had to be made straight, planed, in order to be useful lumber for building. Lumber? Better perhaps to say that blocks of stone were hewn, and bricks baked as rectangular solids, so that they would fit together and pile up. Thus the empirical straightedge came into being. We should be morally certain that the intuition of the first circles came from the potter’s wheel.
There was also measuring and surveying. There were lines of sight. You might think that the line of sight is an abstraction, a taken-for-granted that would never be experienced, intuited, or named. That is armchair speculation; anyone who has ever surveyed in hot and sandy country knows of a terrible problem: Heat makes things shimmer, so you cannot get a decent sight line. And so you get the idea of an ideal line of sight during the cool of the clearing dawn, and finally idealize that into the straight line.
Husserl seems to have believed that the sedimentation of both concepts and proofs, so necessary for any practical purposes, hindered anyone in the modern world who wanted to delve for the Ursprung of geometry, and hence to grasp the origin of all science. That is both origin in the sense of beginning and origin in the sense of primal evidence. For those who take this to be a serious problem—for those who lust after the primal as I do not—I propose a reality check, first into pure mathematics and then into mathematical physics.
Real, creative, pure mathematics is far less sedimented than Husserl seems to have thought. It is true indeed that textbooks looks like middens. But go to a research talk by a mathematician. No notes. No sentences tediously displayed by PowerPoint in an authorized Microsoft format. Quite often some drawings, perhaps with a felt pen on a transparency to save time—only the material differs from the drawings that Socrates made in the sand for the benefit of Meno’s boy slave. Since Husserl speaks of geometry, I shall give two geometrical examples, one personal, one public.
I wanted to understand the construction by ruler and compass of the seventeen-sided regular polygon. Gauss discovered how to do it when he was eighteen or so. He also showed that the regular heptagon cannot be constructed with ruler and compass. A truly Euclidean problem solved. I obtained a rough idea by consulting sedimented sentences, including virtual ones on the Web. But I did not fully grasp the proof idea. I asked a mathematical colleague for help—it was just before the beginning of a rather dull, but very long business meeting. I saw him across the table, beavering away for almost two hours. At the lunch break, he had a couple of pages of scribbles every which way, to which he occasionally referred when explaining the idea. He had literally gone back to “first principles.” Not by going back through a classic text (Hardy and Wright’s 1938 Theory of Numbers would still be a natural, if sedimented, choice). There were certain things he took for granted—the idea of the circle and the fact that it is defined by a quadratic equation. So anything you do with a ruler and compass can be described by quadratics ... now start thinking of all the possibilities ... a certain group of transformations looms up before you....
My friend said with a certain feeling of awe, “You see, Gauss hit upon the rudiments of the theory of groups before he was eighteen.” I suggest this personal story as a parable for pure mathematics in action. If as a philosophical outsider you examine textbooks, the notion of heavy sedimentation is immediately forced upon you. But if, even as such an outsider, you become a participant observer in mathematical activity, you will find very little sedimentation. There is the living experience of the very evidence, the primal evidence, that Husserl sought.
My second example comes from the popular science press, accessible to anyone.4 It is also geometrical. It is the story of the late David Huffman (1925–99), a distinguished computer scientist who became interested in a more sophisticated version of the construct-with-ruler-and-compass problem. What shapes are constructable by folding paper? That is, what is the class of all possible origami? People who know about the calculus of variations will know there is indeed a history here. One stage goes back to the blind Belgian mathematician and physicist Joseph-Antoine Plateau (1801–83). He determined the surfaces of least area subtended by a given closed curve in space by telling sighted assistants to dip wire shapes in a soap solution and tell him the resulting soap surface. Then he went on to demonstrate the solutions for special cases.5 Aspects of Plateau’s problems continue to excite interest, and there are still things to find out. That is where Huffman started his career, but he moved on to folding paper, also allowing himself curved paper not used in classic origami. And so he went on to develop remarkable abstract theorems, from scratch, as we say. (For example, the pi theorem: If you have a point surrounded by creases, and you want the form to fold into a plane, all alternate angles must sum to pi radians, or 180 degrees.) Popular science accounts, such as the one cited, include photographs of quite amazing shapes that David Huffman showed how to construct from a single sheet of paper. Even what Husserl would call the ideal shapes, the ideal objects that Huffman created, were new. Eternal possibilities, you may say, but historical a priori realities that came into being fairly late in the last century and exist on a horizon of possibilities that are being investigated at this very moment. Amount of sedimentation? Practically zero. Another parable.
Throughout the week of June 28 through July 3, 2004, all the string theorists in the world met in the amphitheater two floors below my office in Paris. I know even less string theory than I know Husserl, but I took the opportunity to sit in on a few talks early in the week. It was a fascinating experience in the anthropology of science, but here I restrict myself to one observation. It was astonishing to see that a great deal of the mathematics being used was classical mathematical physics of the sort that firmed up in the nineteenth century: Lagrangians, Fourier transforms, tensors, slightly more recent matrix theory. It was pretty much what I learned in my third or fourth year as an undergraduate at a mediocre department of mathematical physics nearly fifty years ago. The applications were radically new speculations and structures, for sure. I am referring to the tool kit employed by these (mostly very young) string theorists.6 It is very natural to use the metaphor of sedimentation to apply to what, using another metaphor, is the tool kit of cutting-edge mathematical physicists.
In the Crisis Husserl was indeed addressing the most abstract sciences of his day, above all mathematical physics. If you did not pause in your philosophizing to take a look, you would only suppose that there has been more sedimentation since 1935. But here we have a natural science whose sedimentation appears to the amateur ethnographer to be not much different from that of Husserl’s time. String theory a natural science? Well, a speculative science that aims at a deeper understanding of the physical world than ever was before.
All right, it will be replied, but still Husserl was correct. Even if the sediment used in the tool kit of the string theorist was mostly known to Max Planck, it was still sediment! Yes—but sediment not only easily excavated, but excavated before your very eyes when you are a youth learning the trade. One of the reasons that the tools used in these research reports were immediately recognizable to me was that once upon a time, I was shown where they come from, I was taught the ideal version of the fundamental evidence on which they were based. No, I was not literally given a course in then history of mathematics, which would tell me just how Stokes and Green hit on their theorems. Not an Ur-experience like that, thank goodness. But when I got home, I checked in my still-saved textbook of long ago (sedimentation once again) that I had been shown, more or less from scratch, the evidence for Stokes’s and Green’s theorems and could (in a manner more reminiscent of Descartes than Husserl) pretty well relive that experience right now. And that despite the loss of the relevant bits of brain capacity over the years. Happy coincidence: The text was written by Richard Courant, who had been Hilbert’s assistant in Göttingen in 1907. Right alongside Husserl who worked there 1901–16 (Courant, 1937). And Courant also wrote that classic exposition of Plateau on soap bubbles.
In short, I suggest in this section and in the previous one that there is far less sedimentation than armchair inspection of mathematics and physics textbooks would suggest. Oddly, I think that Jacques Derrida was faithful to an aspect of Husserl when he took away the thought that the sentence is primary, trumping speech. The trouble is that Husserl himself, when he had become elderly, seems no longer to have taken a look at reason in action. He thought we were separated from the original evidence by almost undiggable layers of sediment. I said in the Primal Beginnings section above that the primal does not seem to be so important as Husserl thought. But even if it were, I have argued in these last two sections that we are far closer to innumerable primal Ursprünge of geometrical evidence than Husserl noticed.
I will seem to have distanced myself so far from Husserl, that, contrary to the praise expressed in the Husserl aux pantoufles section above, I have come to bury Husserl on geometry. Not so. I resonated with his (what shall we call it?) attempt to weld together historical speculation and speculative philosophy. My own such ongoing attempt, of which only a threadbare sketch was published over a decade ago,7 holds that mathematical reasoning and demonstrative proof are a discovery of human civilization at the dawn of history. The emblematic, mythical origins of what we now call mathematics include the man whom we call Thales, truly lost in the mists of the backward-looking horizon named prehistory, and the man whom we call Al-Khwarizmi, whose historical past will probably be reconstituted with far more detail in the next few years as scholars turn their eyes from the Greek to the Islamic mathematical tradition. Both the geometrical and the algorithmic (algebraic, combinatorial, what you will) styles of reasoning—which united but did not merge in the European Renaissance—originate in the history of particular human societies, perhaps of the Ionian coast and of Baghdad, but they rely on universal human capacities. That is, they are a phenomenon that emerged in culture, but which are possible only because of cognitive universals. In Kant’s myth, Husserl’s myth, my myth, Thales experienced the human potential to produce that engine of discovery, mathematical proof and the associated (“ideal”) structures.
I hold the subsidiary thesis that every style of reasoning (in the specific sense that I have developed elsewhere [1992]) introduces its own domain of new objects. That is why there are ontological debates—over the reality of mathematical objects, over the reality of unobservable theoretical entities, over the reality of the taxa of systematic biology, and so forth. So my view of the relation between ideal objects and their production by demonstration is Kantian, not Husserlian. But even though I hold that ontological thesis to be of fundamental importance in diagnosing the source of realism/antirealism debates of the sort that continue to flourish among philosophers, it is of no great importance for this chapter. What does matter is that the legend of Thales meets a different precondition than that upon which Husserl focused. The precondition is not a historical a priori but rather a cognitive one. It is part of our human genetic envelope (a phrase I take from my colleague Jean-Pierre Changeux). It is the kind of thing about which some cognitive scientists like to speculate, when they imagine that we have one or more innate cognitive modules that enable us to reason mathematically. Much such theorizing consists in hand waving—we know a good deal less about cognitive modules for reasoning than we do about the day when the “new light ... flashed on the mind of the first man (Thales, or whatever may have been his name).” But I want to encourage more hand waving, not less. Could it, for instance, be that diagrams were not mere adventitious tools that helped in explaining geometrical arguments? Could it be that only through the use of diagrams were humans able to “access” those regions of the brain that are essential for constructing proofs (Netz, 1999)? There opens up a horizon of research, part historical and cultural, part neurological and cognitive. It continues the most important of the reflections that Husserl began in his lecture of 1935, on the Origin of Geometry.