# The Mathematician's Mind  ## Metadata - Author: [[Jacques Hadamard]] - Full Title: The Mathematician's Mind - Category: #books ## Highlights - Hadamard proposed four chronological stages in the process of creation: (1) Preparation. You work hard on a problem, giving your conscious attention to it. (2) Incubation. Your conscious preparation sets going an unconscious mechanism that searches for the solution. Poincaré wrote that ideas are like the hooked atoms of Epicurus: preparation sets them in motion and they continue their dance during incubation. The unconscious mechanism evaluates the resulting combinations on aesthetic criteria, but most of them are useless. (3) Illumination. An idea that satisfies your unconscious criteria suddenly emerges into your consciousness. (4) Verification. You carry out further conscious work in order to verify your illumination, to formulate it more precisely, and perhaps to follow up on its consequences. ([Location 57](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=57)) - Tags: [[aqua]] - Hadamard rejected the view that thinking is possible only with the use of language, and he argued that many mathematicians, like Einstein, make use of images and “mental pictures.” ([Location 77](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=77)) - Tags: [[aqua]] - If creation is a computational process, then a case can be made that there are only three sorts of creative processes.4 First, processes that mimic the neo-Darwinian account of the origin of species: a generative stage in which there is a random combination or modification of existing ideas and a critical stage in which knowledge is used to select the more viable possibilities. Evolution depends on repeated iterations of these two stages. Second, neo-Lamarckian processes that use all their knowledge to constrain the generative stage and make a random selection when knowledge fails to select among equally viable alternatives. Such processes seem particularly appropriate for creation in “real time,” such as musical improvisation or poetic extemporization. Third, and most plausible for mathematical invention, processes that use knowledge both to generate ideas and to evaluate viable possibilities. The first stage is presumably unconscious, and the second is conscious. ([Location 89](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=89)) - Tags: [[aqua]] - But is mathematical invention a computable process? Hadamard did not address the issue, but one of his successors, the mathematician and physicist Roger Penrose, gives a negative answer. Penrose argues that consciousness and visual images depend on noncomputational processes.5 Certain physical processes are not computational—for example, the bleaching of visual pigment in retinal cells when light falls on them—but such processes can at least be modeled in a computer program. According to Penrose, however, the mental processes of mathematical invention cannot even be modeled computationally. ([Location 97](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=97)) - Tags: [[aqua]] - the roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. ([Location 103](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=103)) - Tags: [[aqua]] - As always in psychology, two kinds of methods are available: the “subjective” and the “objective” methods.1 Subjective (or “introspective”) methods are those which could be called “observing from the inside,” that is, those where information about the ways of thought is directly obtained by the thinker himself who, looking inwards, reports on his own mental process. ([Location 202](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=202)) - Tags: [[aqua]] - Objective methods—observing from the outside—are those in which the experimenter is other than the thinker. Observation and thought do not interfere with each other ; but on the other hand, only indirect information is thus obtained, the significance of which is not easily seen. ([Location 212](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=212)) - Tags: [[aqua]] - There is every reason to think that the mathematical faculty must be at least as composite as has been found for the faculty of language. ([Location 252](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=252)) - Tags: [[aqua]] - “I wanted to represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate; the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and succeeded without difficulty in forming the series I have called thetafuchsian. “Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the result at my leisure. “Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty, that the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidian geometry.” ([Location 341](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=341)) - Tags: [[aqua]] - “Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable.” ([Location 358](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=358)) - Tags: [[aqua]] - “One evening,” Poincaré says, “contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination.” That strange phenomenon is perhaps the more interesting for the psychologist because it is more exceptional. Poincaré lets us know that it is rather frequent as concerns himself : “It seems, in such cases, that one is present at his own unconscious work, made partially perceptible to the over-excited consciousness, yet without having changed its nature. Then we vaguely comprehend what distinguishes the two mechanisms or, if you wish, the working methods of the two egos.” ([Location 362](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=362)) - Tags: [[aqua]] - a celebrated letter of Mozart : “When I feel well and in a good humor, or when I am taking a drive or walking after a good meal, or in the night when I cannot sleep, thoughts crowd into my mind as easily as you could wish. Whence and how do they come? I do not know and I have nothing to do with it. Those which please me, I keep in my head and hum them ; at least others have told me that I do so. Once I have my theme, another melody comes, linking itself to the first one, in accordance with the needs of the composition as a whole : the counterpoint, the part of each instrument, and all these melodic fragments at last produce the entire work. Then my soul is on fire with inspiration, if however nothing occurs to distract my attention. The work grows ; I keep expanding it, conceiving it more and more clearly until I have the entire composition finished in my head though it may be long. Then my mind seizes it as a glance of my eye a beautiful picture or a handsome youth. It does not come to me successively, with its various parts worked out in detail, as they will be later on, but it is in its entirety that my imagination lets me hear it. “Now, how does it happen, that, while I am at work, my compositions assume the form or the style which characterize Mozart and are not like anybody else’s? Just as it happens that my nose is big and hooked, Mozart’s nose and not another man’s. I do not aim at originality and I should be much at a loss to describe my style. It is quite natural that people who really have something particular about them should be different from each other on the outside as well as on the inside.” ([Location 382](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=382)) - Tags: [[aqua]] - very ordinary facts illustrate with full evidence not only the intervention of unconscious phenomena, but one of their important properties : I allude to the familiar —which does not mean simple—fact of recognizing a human face. Identifying a person you know requires the help of hundreds of features, not a single one of which you could explicitly mention (if not especially gifted or trained for drawing). Nevertheless, all these characters of the face of your friend must be present in your mind—in your unconscious mind, of course—and all of them must be present at the same instant. Therefore, we see that the unconscious has the important property of being manifold ; several and probably many things can and do occur in it simultaneously. This contrasts with the conscious ego which is unique. We also see that this multiplicity of the unconscious enables it to carry out a work of synthesis. In the above case, the numerous details of a physiognomy result, for our consciousness, in only one sensation, viz., recognition.2 ([Location 509](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=509)) - Tags: [[aqua]] - It seems that we can identify this with what Francis Galton3 calls the “ante-chamber” of consciousness, beautifully describing it as follows: “When I am engaged in trying to think anything out, the process of doing so appears to me to be this : The ideas that lie at any moment within my full consciousness seem to attract of their own accord the most appropriate out of a number of other ideas that are lying close at hand, but imperfectly within the range of my consciousness. There seems to be a presence-chamber in my mind where full consciousness holds court, and where two or three ideas are at the same time in audience, and an ante-chamber full of more or less allied ideas, which is situated just beyond the full ken of consciousness. Out of this ante-chamber the ideas most nearly allied to those in the presence-chamber appear to be summoned in a mechanically logical way, and to have their turn of audience.” ([Location 523](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=523)) - Tags: [[aqua]] - There seems to be a kind of continuity between full consciousness and more and more hidden levels of the unconscious : a succession which seems to be especially well described in Taine’s book On Intelligence, when he writes :6 “You may compare the mind of a man to the stage of a theatre, very narrow at the footlights but constantly broadening as it goes back. At the footlights, there is hardly room for more than one actor. . . . As one goes further and further away from the footlights, there are other figures less and less distinct as they are more distant from the lights. And beyond these groups, in the wings and altogether in the background, are innumerable obscure shapes that a sudden call may bring forward and even within direct range of the footlights. Undefined evolutions constantly take place throughout this seething mass of actors of all kinds, to furnish the chorus leaders who in turn, as in a magic lantern picture, pass before our eyes.” ([Location 550](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=550)) - Tags: [[aqua]] - it is obvious that invention or discovery, be it in mathematics or anywhere else, takes place by combining ideas.1 Now, there is an extremely great number of such combinations, most of which are devoid of interest, while, on the contrary, very few of them can be fruitful. Which ones does our mind—I mean our conscious mind—perceive? Only the fruitful ones, or exceptionally, some which could be fruitful. However, to find these, it has been necessary to construct the very numerous possible combinations, among which the useful ones are to be found. It cannot be avoided that this first operation take place, to a certain extent, at random, so that the role of chance is hardly doubtful in this first step of the mental process. But we see that that intervention of chance occurs inside the unconscious: for most of these combinations—more exactly, all those which are useless—remain unknown to us. Moreover, this shows us again the manifold character of the unconscious, which is necessary to construct those numerous combinations and to compare them with each other. ([Location 597](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=597)) - Tags: [[aqua]] - As we just saw, and as Poincaré observes, to create consists precisely in not making useless combinations and in examining only those which are useful and which are only a small minority. Invention is discernment, choice. ([Location 608](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=608)) - Tags: [[aqua]] - “It takes two to invent anything. The one makes up combinations ; the other one chooses, recognizes what he wishes and what is important to him in the mass of the things which the former has imparted to him. “What we call genius is much less the work of the first one than the readiness of the second one to grasp the value of what has been laid before him and to choose it.” ([Location 611](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=611)) - Tags: [[aqua]] - “The privileged unconscious phenomena, those susceptible of becoming conscious, are those which, directly or indirectly, affect most profoundly our emotional sensibility. “It may be surprising to see emotional sensibility invoked à propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility.” ([Location 620](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=620)) - Tags: [[aqua]] - In all these successive steps, as we see, “sudden inspirations (and the examples already cited sufficiently prove this), never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come, where the way taken seems totally astray. These efforts then have not been as sterile as one thinks. They have set going the unconscious machine and without them it would not have moved and would have produced nothing.” Helmholtz had similarly observed that what we have called incubation and illumination must be preceded by this stage of preparation. Its existence has been, after Helmholtz and Poincaré, recognized by psychologists as a general fact, and probably it exists even when it is not apparent, as in the case of Mozart (who does not mention incubation either). ([Location 833](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=833)) - Tags: [[aqua]] - Discovery necessarily depends on preliminary and more or less intense action of the conscious. ([Location 843](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=843)) - Tags: [[aqua]] - Having recognized this, we cannot any longer think of the conscious as being subordinated to the unconscious. On the contrary, it starts its action and defines, to a greater or lesser extent, the general direction in which that unconscious has to work. ([Location 849](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=849)) - Tags: [[aqua]] - To illustrate that directing action, Poincaré uses a striking and remarkably fruitful comparison. He imagines that the ideas which are the future elements of our combinations are “something like the hooked atoms of Epicurus. During the complete repose of the mind, these atoms are motionless ; they are, so to speak, hooked to the wall ; so this complete rest may be indefinitely prolonged without the atoms meeting, and consequently without any combination between them.” The act of studying a question consists of mobilizing ideas, not just any ones, but those from which we might reasonably expect the desired solution. It may happen that that work has no immediate result. “We think we have done no good, because we have moved these elements a thousand different ways in seeking to assemble them and have found no satisfactory aggregate.” But, as a matter of fact, it seems as though these atoms are thus launched, so to speak, like so many projectiles and flash in various directions through space. “After this shaking-up imposed upon them by our will, these atoms do not return to their primitive rest. They freely continue their dance.” Consequences can now be foreseen. “The mobilized atoms undergo impacts which make them enter into combinations among themselves or with other atoms at rest, which they struck against in their course.” In those new combinations, in those indirect results of the original conscious work, lie the possibilities of apparently spontaneous inspiration. ([Location 851](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=851)) - Tags: [[aqua]] - Every scientist can probably record similar failures. In my own case, I have several times happened to overlook results which ought to have struck me blind, as being immediate consequences of other ones which I had obtained. Most of these failures proceed from the cause which we have just mentioned, viz., from attention too narrowly directed. ([Location 902](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=902)) - Tags: [[aqua]] - Such instances show us that, in research, it may be detrimental to scatter our attention too much, while overstraining it too strongly in one particular direction may also be harmful to discovery. ([Location 951](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=951)) - Tags: [[aqua]] - The feeling of absolute certitude which accompanies the inspiration generally corresponds to reality; but it may happen that it has deceived us.1 Whether such has been the case or not must be ascertained by our properly so-called reason, a task which belongs to our conscious self. ([Location 986](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=986)) - Tags: [[aqua]] - It never happens, as Poincaré observes, that the unconscious work gives us the results of a somewhat long calculation already solved in its entirety. If we should, as concerns the unconscious, retain the original idea which is suggested by the quality “automatic” imputed to it, we should suppose that, thinking of an algebraic calculation before falling asleep, we might hope to find its result ready made upon our awakening; but nothing of the sort ever happens, and indeed we begin to understand that automatism of the unconscious must not be understood in that way. On the contrary, effective calculations which require discipline, attention and volition, and, therefore, consciousness, depend on the second period of conscious work which follows the inspiration. Thus, we come to the paradoxical-looking conclusion— to which, besides, we shall have to bring a correction as we already have done in Newton’s case—that this intervention of our will, i.e., of one of the highest faculties of our soul, happens in a rather mechanical part of the work, where it is in some way subordinated to the unconscious, though supervising it. The second operation is inseparable from the first, from verification. The conscious mind performs them both at the same time. ([Location 990](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=990)) - Tags: [[aqua]] - any mathematical argument, however complicated, must appear to me as a unique thing. I do not feel that I have understood it as long as I do not succeed in grasping it in one global idea and, unhappily, as with Rodin, this often requires a more or less painful exertion of thought. ([Location 1115](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1115)) - Tags: [[aqua]] - Later experiments by Dwelshauvers (Les Mécanismes Subconscients), carried on with students, led to the same main conclusions as Binet’s about the conditions of the apparition of images. He finds that images appear only if we give our ideas uncontrolled freedom, i.e., when we are dreaming while awake. As soon as full consciousness, voluntary consciousness, returns, images weaken, darken; they seem to withdraw into some unknown region. ([Location 1212](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1212)) - Tags: [[aqua]] - About the mathematicians born or resident in America, whom I asked, phenomena are mostly analogous to those which I have noticed in my own case.19 Practically all of them—contrary to what occasional inquiries had suggested to Galton as to the man in the street—avoid not only the use of mental words but also, just as I do, the mental use of algebraic or any other precise signs ; also as in my case, they use vague images. ([Location 1344](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1344)) - Tags: [[aqua]] - G. Pólya’s case—I intend to speak only of men who have made quite significant discoveries—is different. He does make an eventual use of words. “I believe,” he writes to me, “that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the. situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately ; perhaps, it arises at the same time as the decisive idea. . . . The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. . . . It may contribute to fix it in the mind.” Moreover, he finds that a proper notation—that is, a properly chosen letter to denote a mathematical quantity—can give him similar help ; and some kind of puns, whether of good or poor quality, may be useful for that purpose. For instance, Pólya, teaching in German at a Swiss university, usually made his junior students observe that z and w are the initials of the German words “Zahl” and “Wert,” which precisely denote the respective roles which z and w had to play in the theory which he was explaining. ([Location 1353](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1353)) - Tags: [[aqua]] - “Signs are a necessary support of thought. For socialized thought (stage of communication) and for the thought which is being socialized (stage of formulation), the most usual system of signs is language properly called ; but internal thought, especially when creative, willingly uses other systems of signs which are more flexible, less standardized than language and leave more liberty, more dynamism to creative thought. . . . Amongst all these signs or symbols, one must distinguish between conventional signs, borrowed from social convention and, on the other hand, personal signs which, in their turn, can be subdivided into constant signs, belonging to general habits, to the individual pattern of the person considered and into episodical signs, which are established ad hoc and only participate in a single creative act.” ([Location 1504](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1504)) - Tags: [[aqua]] - at least in a certain class of questions relating to principles,1 we cannot surely rely on our ordinary space-intuition : as geometrical properties can always be reconducted to numerical ones, thanks to the invention of analytical geometry, arguments should always be fully arithmetized, or, at least, it must be ascertained that this arithmetization, if not given at length for brevity’s sake, is possible. Pascal’s word “Tout ce qui passe la Géométrie nous passe” is replaced, for the modern mathematician, by “Tout ce qui passe l’Arithmétique nous passe.” ([Location 1678](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1678)) - Tags: [[aqua]] - “To understand the demonstration of a theorem, is that to examine successively each syllogism composing it and ascertain its correctness, its conformity to the rules of the game? . . . For some, yes; when they have done this, they will say, I understand. “For the majority, no. Almost all are much more exacting; they wish to know not merely whether all the syllogisms of a demonstration are correct, but why they link together in this order rather than another. In so far as to them they seem engendered by caprice and not by an intelligence always conscious of the end to be attained, they do not believe they understand. ([Location 1698](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1698)) - Tags: [[aqua]] - some mathematicians are “intuitive’’ and others “logical.” Poincaré has dealt with that distinction and so has the German mathematician Klein. Poincaré’s lecture on the subject begins as follows: “The one sort are above all preoccupied by logic ; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke, make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.” ([Location 1725](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1725)) - Tags: [[aqua]] - Summing up the results of that analysis, let us remember that every mental work and especially the work of discovery implies the cooperation of the unconscious, be it the superficial or (fairly often) the more or less remote one; that, inside of that unconscious (resulting from a preliminary conscious work), there is that starting of ideas which Poincaré has compared to a projection of atoms and which can be more or less scattered ; that concrete representations are generally used by the mind for the maintenance and synthesis of combinations. This carries, in the first place, the consequence that, strictly speaking, there is hardly any completely logical discovery. Some intervention of intuition issuing from the unconscious is necessary at least to initiate the logical work. ([Location 1795](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1795)) - Tags: [[aqua]] - (A) More or Less Depth in the Unconscious. As we know that there must be several layers in the unconscious, some quite near consciousness while some may lie more and more remote, it is clear that the levels at which ideas meet and combine may be deeper or, on the contrary, more superficial; and it is not unreasonable to admit that there is a usual behavior of every single mind from that point of view. ([Location 1802](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1802)) - Tags: [[aqua]] - (B) More or Less Narrowly Directed Thought. In the second place, we have seen that the projection of Poincaré’s atoms—the starting of ideas, to use a less metaphorical language—can be more or less scattered. This is another reason why we can have the sensation of an intuitive mind (which will happen if there is much scattering) or (in the contrary case) of a logical mind; and the second reason can, at least a priori, be without any connection with the first one: the direction of thought may be narrower or wider, be it at one level of unconsciousness or at another. ([Location 1816](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1816)) - Tags: [[aqua]] - (C) Different Auxiliary Representations. We have seen how differently scientists behave as to the way in which their thought is helped by mental pictures or other concrete representations : differences can bear either on the nature of representations or on the way they influence the work of the mind. It is evident that some of these kinds of representations may give the thought a rather logical course, some others a rather intuitive one. But this side of the question is much less accessible to study, precisely because phenomena are not always comparable in different minds. ([Location 1821](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1821)) - Tags: [[aqua]] - IF, IN SOME exceptionally intuitive minds, ideas may evolve and combine in still deeper unconscious layers than in the above-mentioned cases, then even important links of the deduction may remain unknown to the thinker himself who has found them. The history of science offers some remarkable examples. ([Location 1873](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1873)) - Tags: [[aqua]] - Claparède, in his introductory lecture before the above-mentioned meeting at the Centre de Synthèse, observes that there are two kinds of invention : one consists, a goal being given, in finding the means to reach it, so that the mind goes from the goal to the means, from the question to the solution ; the other consists, on the contrary, in discovering a fact, then imagining what it could be useful for, so that, this time, mind goes from the means to the goal ; the answer appears to us before the question. Now, paradoxical as it seems, that second kind of invention is the more general one and becomes more and more so as science advances. Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle. ([Location 1984](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=1984)) - Tags: [[aqua]] - application is useful and eventually essential to theory by the very fact that it opens new questions for the latter. One could say that application’s constant relation to theory is the same as that of the leaf to the tree : one supports the other, but the former feeds the latter. ([Location 2000](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=2000)) - Tags: [[aqua]] - As Renan also curiously notices,1 there is a scientific taste just as there is a literary or artistic one; and that taste, according to individuals, may be more or less sure. ([Location 2022](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=2022)) - Tags: [[aqua]] - whether it be in the choice of questions or in their treatment, a man without some love of science could not be successful, because he would be unable to choose. ([Location 2104](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=2104)) - Tags: [[aqua]] - The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined. ([Location 2253](https://readwise.io/to_kindle?action=open&asin=B086RB31Q2&location=2253)) - Tags: [[aqua]]