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乇賵丕賳卮賳丕爻蹖 丕亘丿丕毓 丿乇 乇蹖丕囟蹖丕鬲
賲賳 亘蹖丕賳 禺賵丕賴賲 讴乇丿 讴賴 丨賱 賮賱丕賳 賯囟蹖賴 鬲丨鬲 亘賴賲丕賳 卮乇丕蹖胤 丕鬲賮丕賯 丕賮鬲丕丿.
丕蹖賳 賯囟蹖賴 蹖讴 賳丕賲 睾蹖乇 賲氐胤賱丨 丿丕乇丿 讴賴 亘乇丕蹖 亘爻蹖丕乇蹖 亘蹖诏丕賳賴 丕爻鬲 丕賲丕 丕蹖賳 賲賵囟賵毓 丕賴賲蹖鬲蹖 賳丿丕乇丿.
丌賳趩賴 亘乇丕蹖 乇賵丕賳卮賳丕爻 噩丕賱亘 丕爻鬲 賳賴 禺賵丿 賯囟蹖賴 亘賱讴賴 丕賵囟丕毓 賵 丕丨賵丕賱蹖 丕爻鬲 讴賴 亘賴 丕亘丿丕毓 丌賳 賲賳噩乇 賲蹖 卮賵丿.
賴丕賳乇蹖 倬賵丌賳讴丕乇賴
丕蹖賳 賯囟蹖賴 蹖讴 賳丕賲 睾蹖乇 賲氐胤賱丨 丿丕乇丿 讴賴 亘乇丕蹖 亘爻蹖丕乇蹖 亘蹖诏丕賳賴 丕爻鬲 丕賲丕 丕蹖賳 賲賵囟賵毓 丕賴賲蹖鬲蹖 賳丿丕乇丿.
丌賳趩賴 亘乇丕蹖 乇賵丕賳卮賳丕爻 噩丕賱亘 丕爻鬲 賳賴 禺賵丿 賯囟蹖賴 亘賱讴賴 丕賵囟丕毓 賵 丕丨賵丕賱蹖 丕爻鬲 讴賴 亘賴 丕亘丿丕毓 丌賳 賲賳噩乇 賲蹖 卮賵丿.
賴丕賳乇蹖 倬賵丌賳讴丕乇賴
171 pages, Paperback
First published January 1, 1945
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About the author
Jacques Hadamard
75听books13听followersJacques Salomon Hadamard was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
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Displaying 1 - 25 of 25 reviews
May 16, 2010
I dove into this book excited to learn how the minds of great scientists churn but instead was reminded of the great danger that accompanies reading old science texts - lengthy discussions of crackpot theories (i.e. phrenology) and passionate defenses of well-accepted ideas (i.e. not all mental activity is conscious). Taken as a survey of late 19th/early 20th century thinking on creativity and thought, the book reveals how stubbornly we humans cling to the mech warrior hypothesis of behavior - that every nugget of our activity stems from the conscious control of a homunculus nested in his HQ and peering out of our eyes like little windows on a spaceship. Many psychologists and philosophers quoted by Hadamard actually deny the existence of nontrivial unconscious processing in creative thought. If this doesn't shock or disgust you and you find yourself sympathizing with this notion, go directly to jail. Do not pass Go. Do not collect $200.
Even so, this short book is worth a skim - the survey questions in the appendix alone are worth the price of admission. Hadamard used these questions to drill all his scientist/mathematician buddies on how they think, imagine, and work. The list is even more than a set of questions though - its a set of suggestions. If you're as narrow-minded and habituated as I am, you'll likely discover entirely new ways to approach problem-solving and find yourself exclaiming, "People think like that?!" I highly recommend reading the survey before the book itself, as this will get you thinking ahead of time about how you think and offer more context for understanding and possibly assimilating the habits of Hadamard's buddies.
The writing (or at least the translation) is also pretty amateurish. Parts of this book read like the dinner table reports of a 4th grader telling his mommy and daddy what all his friends did in class today... except instead of eating boogers and tricking Suzy into thinking she was adopted, Hadamard's friends invent special relativity, bifurcation theory, and cybernetics.
If you can tolerate or skip the many faults of this early thought experiment on thought, however, you're sure to not only learn something about the great minds of the late 19th/early 20th century, but your own feeble brain as well.
Book Notes (Warning: not guaranteed to be interpretable to outside eyes):
Invention is combination followed by selection.
Selection is the more difficult step.
The selection process seems highly emotional. Understanding the emotional character of selection would teach us much about invention.
Two benefits of incubation:
reset (replenishment of mental resources)
restart (retract assumptions and avoid mental ruts)
The incubation paradigm changes the role of the scientist to that of a mental farmer - toil hard in the fields of conscious effort (and failure), then later reap the benefits brought on by subconscious processing.
Ways mathematical minds may differ:
accessibility of thought/depth in unconscious (logical vs. intuitive thinking)
narrowness of thought (logical vs. scattered)
different auxiliary representations (geometric, verbal, auditory, etc)
Two kinds of invention:
Set goal, seek means
Discover means, seek application (more common in mathematics)
I wonder how many grand ideas remain just out of reach in the antechambers of the minds of geniuses, perhaps consciously acknowledged but under-appreciated by them, perhaps tacitly assumed, or perhaps subconscious and nebulous.
The scientist whose aesthetic sense (passion) draws him to discoveries with profound implications is what we call a genius.
Even so, this short book is worth a skim - the survey questions in the appendix alone are worth the price of admission. Hadamard used these questions to drill all his scientist/mathematician buddies on how they think, imagine, and work. The list is even more than a set of questions though - its a set of suggestions. If you're as narrow-minded and habituated as I am, you'll likely discover entirely new ways to approach problem-solving and find yourself exclaiming, "People think like that?!" I highly recommend reading the survey before the book itself, as this will get you thinking ahead of time about how you think and offer more context for understanding and possibly assimilating the habits of Hadamard's buddies.
The writing (or at least the translation) is also pretty amateurish. Parts of this book read like the dinner table reports of a 4th grader telling his mommy and daddy what all his friends did in class today... except instead of eating boogers and tricking Suzy into thinking she was adopted, Hadamard's friends invent special relativity, bifurcation theory, and cybernetics.
If you can tolerate or skip the many faults of this early thought experiment on thought, however, you're sure to not only learn something about the great minds of the late 19th/early 20th century, but your own feeble brain as well.
Book Notes (Warning: not guaranteed to be interpretable to outside eyes):
Invention is combination followed by selection.
Selection is the more difficult step.
The selection process seems highly emotional. Understanding the emotional character of selection would teach us much about invention.
Two benefits of incubation:
reset (replenishment of mental resources)
restart (retract assumptions and avoid mental ruts)
The incubation paradigm changes the role of the scientist to that of a mental farmer - toil hard in the fields of conscious effort (and failure), then later reap the benefits brought on by subconscious processing.
Ways mathematical minds may differ:
accessibility of thought/depth in unconscious (logical vs. intuitive thinking)
narrowness of thought (logical vs. scattered)
different auxiliary representations (geometric, verbal, auditory, etc)
Two kinds of invention:
Set goal, seek means
Discover means, seek application (more common in mathematics)
I wonder how many grand ideas remain just out of reach in the antechambers of the minds of geniuses, perhaps consciously acknowledged but under-appreciated by them, perhaps tacitly assumed, or perhaps subconscious and nebulous.
The scientist whose aesthetic sense (passion) draws him to discoveries with profound implications is what we call a genius.
February 11, 2012
Sitting on the toilet one morning, it suddenly hit me: a Sudoku puzzle is a graph coloring problem in disguise. Such out-of-the-blue moments of mathematical inspiration, which usually come after struggling with a hard problem for days and then engaging in a different activity, are among the topics that Jacques Hadamard explored in this interesting small book.
As P. N. Johnson-Laird notes in the preface of this edition, the book was prescient: when Hadamard set out to explore mathematical invention, he went against the dominant philosophy of psychology of the time, behaviorism, by using introspection and discussing mental processes. Henri Poincar茅's famous lecture before the Soci茅t茅 de Psychologie in Paris inspired Hadamard to undertake this study, so he quoted Poincar茅 extensively, but he also provided his own experiences and insights. More importantly, Hadamard surveyed some of the major mathematicians and scientists of the time, such as George Polya, Norbert Wiener, and Albert Einstein.
In Chapter VIII, Hadamard suggested that under certain circumstances, "even important links of the deduction may remain unknown to the thinker himself who has found them." He cited Pierre de Fermat, Bernhard Riemann, and 脡variste Galois as examples. Each of these mathematicians made a mathematical statement and claimed he had a proof but did not enunciate it due to limitations of space or time. These statements were indeed proved (completely or partially) later using facts and theories that were unknown in the mathematician's time. These facts and theories represent, by themselves, significant discoveries, yet no conception of, or even allusion to them appear in any of the mathematician's writings. It may be speculative, but I find Hadamard's explanation of these "paradoxical cases of intuition" fascinating, and I wish he had elaborated on it more.
Despite its interesting content, I felt something was missing when I finished the book; a sense of closure, perhaps. But take this vague criticism with a grain of salt, because it could be a result of the discontinuity of my reading.
In short, this small book is worth reading, or at least skimming, as it provides a window into the creative processes of some of the great minds of the time. Who knows, perhaps that elusive answer that you were looking for will hit you while you are reading the book, especially if you are sitting on a toilet.
As P. N. Johnson-Laird notes in the preface of this edition, the book was prescient: when Hadamard set out to explore mathematical invention, he went against the dominant philosophy of psychology of the time, behaviorism, by using introspection and discussing mental processes. Henri Poincar茅's famous lecture before the Soci茅t茅 de Psychologie in Paris inspired Hadamard to undertake this study, so he quoted Poincar茅 extensively, but he also provided his own experiences and insights. More importantly, Hadamard surveyed some of the major mathematicians and scientists of the time, such as George Polya, Norbert Wiener, and Albert Einstein.
In Chapter VIII, Hadamard suggested that under certain circumstances, "even important links of the deduction may remain unknown to the thinker himself who has found them." He cited Pierre de Fermat, Bernhard Riemann, and 脡variste Galois as examples. Each of these mathematicians made a mathematical statement and claimed he had a proof but did not enunciate it due to limitations of space or time. These statements were indeed proved (completely or partially) later using facts and theories that were unknown in the mathematician's time. These facts and theories represent, by themselves, significant discoveries, yet no conception of, or even allusion to them appear in any of the mathematician's writings. It may be speculative, but I find Hadamard's explanation of these "paradoxical cases of intuition" fascinating, and I wish he had elaborated on it more.
Despite its interesting content, I felt something was missing when I finished the book; a sense of closure, perhaps. But take this vague criticism with a grain of salt, because it could be a result of the discontinuity of my reading.
In short, this small book is worth reading, or at least skimming, as it provides a window into the creative processes of some of the great minds of the time. Who knows, perhaps that elusive answer that you were looking for will hit you while you are reading the book, especially if you are sitting on a toilet.
June 15, 2016
Mostly of historical interest. Appendix II contains an interesting letter from Einstein, in which he has some intriguing things to say about the nature of his own thought processes.
July 24, 2012
I can appreciate that the author's almost tangential and scattered research of mathematical epiphanies allows me the option to learn many different hypotheses. I can latch onto the theories of many scientists in an attempt to understand and practice the acquisition of mathematical invention and discovery. This is a great blend of psychology and mathematics.
May 21, 2022
Mathematics can rightly lay claim to being the sublimest of the arts, for one must work in that most recalcitrant medium of all, pure unalloyed truth. Hence, its twin ideal properties of absolute certainty and exactitude in its results. Whence comes our knowledge of mathematical truth? Whether from an anamnesis originating in a prior existence or from a laborious process of discovery, in either case we human beings do not enjoy an angelic intellectual intuition, capable of surveying at a glance everything in the intelligible world, but depend upon the piecemeal unfolding of knowledge through the serendipitous working of the individual mathematician鈥檚 mind. In consequence, the psychology of invention in the mathematical field would seem to offer an inexhaustible prospect to the inquisitive investigator. It so happens, nevertheless, despite what one would expect from the magnitude of the topic, there is one only study of any merit in the literature, due to the French mathematician Jacques Hadamard published almost a century ago and now out in a convenient Dover reprint.
Hadamard opens by observing that indirect methods, such as the experimental psychologist might prefer, are not very useful in this case because notable instances of mathematical invention are so few, whereas one can conclude to an experimental finding only upon a sufficiently large sample of repeatable cases 鈥� so we are left with introspection with all its problems. Mathematical intelligence is, of course, correlated with general intelligence but our author does not wish to pursue this subject any further 鈥� topical as it was in his day, when methods of measuring of intelligence were just being devised and the so-called g-factor came to light. Rather, he is interested in intelligence as it pertains specifically to mathematics proper. As he stresses (p. 5), we will see there are several kinds of mathematical mind, for the mathematical faculty is almost surely composite i.e. not attributable to a single feature of the mind.
Everyone will have experienced, on one occasion or another, the puzzling phenomenon of a dream that centers on content of a mathematical nature. In this reviewer鈥檚 case, it almost always turns out to rest upon a misapprehension (thus explaining why, during the dream itself, one can never quite figure out what one is seeking, akin to the sense of inexplicably arrested motion) which dissipates upon awakening, if it ever amounted to anything. Yet, in anecdotal accounts dreams are supposed to be a source of insight not attainable elsewhere. Hadamard, though, dismisses so-called mathematical dreams as not forming a very important part of discovery in any event (p. 7). But he does acknowledge the phenomenon of the sudden appearance of a solution upon awakening, which deserves attention in that it may supply a clue to thought processes in general.
To kick off his analysis, Hadamard quotes extensively the well-known example of Poincar茅鈥檚 discovery of Fuchsian functions, which is significant as it points clearly to the role of the unconsciousness (pp. 12-14). Another instance would be Gauss鈥� hitting upon an arithmetical theorem after years spent searching for it in vain: 鈥楩inally, two days ago, I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible鈥� (p. 15).
Following Helmholtz, then, Hadamard posits an incubation stage followed by illumination, akin to finding the right word afterwards, when no longer thinking about it. He disputes Nicolle鈥檚 chance hypothesis, to the effect that the act of discovery is merely an accident. For, as everyone knows, fortune favors the prepared mind. This prompts Hadamard to undertake a reflection (pp. 29ff) on the various modes by which the mind operates when ruminating upon an unsolved mathematical problem, in which he sees a role for the unconciousness itself, what he calls fringe-consciousness, subconsciousness and several successive layers within the unconsciousness. How he pictures the unconscious mind working: it runs through combinations of ideas, most remaining unknown to us; when it alights upon a seemingly good solution, it precipitates into consciousness.
Yet, the production of the unconscious alone is insufficient, without the step that follows: for invention is discernment, choice, subject to the sieve of aesthetics, including an affective element not just as willed but as means of finding. Poincar茅鈥檚 view of incubation, whom Hadamard quotes here, is that 鈥榯o the unconscious belong not only the complicated task of constructing the bulk of various combinations of ideas, but also the most delicate and essential one of selecting those which satisfy our sense of beauty and, consequently, are very likely to be useful鈥� (p. 32). Hadamard considers some alternatives at this juncture only to reject them: the rest hypothesis, the absence of interference and the forgetting hypothesis (getting rid of false leads).
Now, Hadamard enumerates three kinds of inventive work: fully conscious, illumination preceded by incubation, and a certain peculiar sleepless process (when one stays up at night preoccupied with the problem). The stages are interconnected. One always has to lead off with preparatory conscious work to scope out the lay of the land and to develop one鈥檚 heuristic, even though, in any mathematical problem of substance, the solution does not reveal itself right away:
In all these successive steps, as we see, 鈥榮udden 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鈥� [quoting Poincar茅 again]. (pp. 45)
Poincar茅鈥檚 instructive example prompts a reflection on the respective roles of logic and chance 鈥� as Hadamard himself sees aptly enough, one cannot do without the one or the other. As part of the process, errors (which are frequent even in good mathematicians) are unavoidable and have a role to play: rather than infallibility, the mark of a good mathematician consists in possession of some insight that warns him when his calculations might be off. Finally, from Hadamard鈥檚 own experience: being too closely focused on one aspect of a problem can cause one to miss a significant consequence of one鈥檚 findings (pp. 50-52).
In the major sixth chapter, Hadamard turns to an analysis of mathematical discovery as a process of synthesis. For discovery issues from the adroit combination of thoughts and thoughts ordinarily have to be represented through a word or symbols in order for our minds to manipulate them. This leads to a discussion of the role of signs, in which Hadamard shows some sensitivity to the insights of the celebrated philologist Max M眉ller and the Counter-Enlightenment philosopher Johann Gottfried Herder on language versus wordless thought (pp. 66ff). Hadamard insists that words are entirely absent when he thinks on mathematical problems (like what is the case with the geneticist Francis Galton). Descartes distrusts intervention of the imagination and wants to eliminate it completely from science! To which Hadamard replies, deductions in the realm of numbers may be most generally accompanied by images [germ of a critique of Russellian logical formalism]. Can imagery be educated? As Hadamard remarks, one tends to become more verbal with advancing maturity.
The remaining chapters go on to entertain a few further topics related to the psychology of invention: common sense versus formal deduction say from rules of differential and integral calculus (pp. 100-103); and intuitive versus logical minds (pp. 106-115). There are at least two different definitions of intuition versus logic: see for instance p. 120, Galois was highly intuitive according to definition A but not B. Next, he remarks upon two conceptions of invention itself (pp. 124-125), what one may characterize as goal-oriented versus the free play of the imagination:
We must add, however, that conversely, 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鈥檚 constant relation to theory is the same as that of the leaf to the tree: one supports the other, but the former needs the latter. Not to mention several important physical examples, the first mathematical foundation in Greek science, geometry, was suggested by practical necessity, as can be seen by its very name, which means 鈥榣and-measuring鈥�. But this example is exceptional in the sense that practical questions are most often solved by means of existing theories: practical applications of purely scientific discoveries, as important as they may be, are generally remote in time (though, in recent years, this delay may be considerably shortened, as happened in the case of radio telegraphy, which occurred a few years after the discovery of Hertzian waves). It seldom happens that important mathematical researches are directly undertaken in view of a given practical use: they are inspired by the desire which is the common motive of every scientific work, the desire to know and to understand. Therefore, between the two kinds of invention we have just distinguished from each other, mathematicians are accustomed only to the second one. (pp. 125-126)
A handful of overall observations on this reviewer鈥檚 part.
1) One would wish for a more systematic survey to assess Hadamard鈥檚 contention that among mathematicians known to him except for P贸lya one does not think in words until it comes time to write out and communicate one鈥檚 findings to others. In as much as Hadamard thus frequently draws upon his own experiences having spent a career in the field, the diligent student will profit from reading him only after himself having attained to some degree of mathematical maturity, else one won鈥檛 appreciate very much his illustrations 鈥� though for most of us the stock of experience on which we have to draw will consist mainly in having solved the homework exercises in textbooks and not so much in having done original research!
2) The question as to classification of types of mathematical genius would be worth further investigation, best approached by familiarizing oneself with the history of mathematics and the biographies of the great mathematicians who contributed to it.
3) Why has research become less original and more driven by groupthink in recent decades? There is much more to say about how the researcher goes about selecting his topic. Indeed, as Hadamard points out, only second raters take their research problem from someone else (p. 126). Sociological and pedagogical issues are implicated here. Set aside the obvious pressure to conformity with the community鈥檚 perception of what ought to be its research agenda 鈥� in any educational endeavor, one has to strike a balance between teaching the student new material and leaving him the freedom to explore his own ideas: one cannot invent in a vacuum, after all. For only upon exposure to significant and profound concepts from the past will the student鈥檚 own creative response be elicited. As a rule, though, one must determine that, in doctoral-level training these days, too much stress is placed upon assimilation of the existing store of knowledge and not enough on developing an individual perspective 鈥� hence the phenomenon of vogues, in which the scientific community lurches from one hot topic to another since hardly anyone commands a sturdy, independent identity of his own.
4) Lastly, Hadamard鈥檚 intriguing, stimulating and learned work raises a theoretical question: the conscious mind is ruled by discursiveness, but how does the unconscious operate? It seems to sift through and combine ideas not so much according to logical succession but in some more global fashion, relying on a deeper sense of their import and of how they may fit together (of course, it can err thereby, which is why the stage of verification continues to be indispensable). A most rewarding subject for further reflection for those inclined to psychology!
In summary, a fine start on a fascinating topic but the treatment of its themes remains too sketchy to count as a monograph, 4 stars.
Hadamard opens by observing that indirect methods, such as the experimental psychologist might prefer, are not very useful in this case because notable instances of mathematical invention are so few, whereas one can conclude to an experimental finding only upon a sufficiently large sample of repeatable cases 鈥� so we are left with introspection with all its problems. Mathematical intelligence is, of course, correlated with general intelligence but our author does not wish to pursue this subject any further 鈥� topical as it was in his day, when methods of measuring of intelligence were just being devised and the so-called g-factor came to light. Rather, he is interested in intelligence as it pertains specifically to mathematics proper. As he stresses (p. 5), we will see there are several kinds of mathematical mind, for the mathematical faculty is almost surely composite i.e. not attributable to a single feature of the mind.
Everyone will have experienced, on one occasion or another, the puzzling phenomenon of a dream that centers on content of a mathematical nature. In this reviewer鈥檚 case, it almost always turns out to rest upon a misapprehension (thus explaining why, during the dream itself, one can never quite figure out what one is seeking, akin to the sense of inexplicably arrested motion) which dissipates upon awakening, if it ever amounted to anything. Yet, in anecdotal accounts dreams are supposed to be a source of insight not attainable elsewhere. Hadamard, though, dismisses so-called mathematical dreams as not forming a very important part of discovery in any event (p. 7). But he does acknowledge the phenomenon of the sudden appearance of a solution upon awakening, which deserves attention in that it may supply a clue to thought processes in general.
To kick off his analysis, Hadamard quotes extensively the well-known example of Poincar茅鈥檚 discovery of Fuchsian functions, which is significant as it points clearly to the role of the unconsciousness (pp. 12-14). Another instance would be Gauss鈥� hitting upon an arithmetical theorem after years spent searching for it in vain: 鈥楩inally, two days ago, I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible鈥� (p. 15).
Following Helmholtz, then, Hadamard posits an incubation stage followed by illumination, akin to finding the right word afterwards, when no longer thinking about it. He disputes Nicolle鈥檚 chance hypothesis, to the effect that the act of discovery is merely an accident. For, as everyone knows, fortune favors the prepared mind. This prompts Hadamard to undertake a reflection (pp. 29ff) on the various modes by which the mind operates when ruminating upon an unsolved mathematical problem, in which he sees a role for the unconciousness itself, what he calls fringe-consciousness, subconsciousness and several successive layers within the unconsciousness. How he pictures the unconscious mind working: it runs through combinations of ideas, most remaining unknown to us; when it alights upon a seemingly good solution, it precipitates into consciousness.
Yet, the production of the unconscious alone is insufficient, without the step that follows: for invention is discernment, choice, subject to the sieve of aesthetics, including an affective element not just as willed but as means of finding. Poincar茅鈥檚 view of incubation, whom Hadamard quotes here, is that 鈥榯o the unconscious belong not only the complicated task of constructing the bulk of various combinations of ideas, but also the most delicate and essential one of selecting those which satisfy our sense of beauty and, consequently, are very likely to be useful鈥� (p. 32). Hadamard considers some alternatives at this juncture only to reject them: the rest hypothesis, the absence of interference and the forgetting hypothesis (getting rid of false leads).
Now, Hadamard enumerates three kinds of inventive work: fully conscious, illumination preceded by incubation, and a certain peculiar sleepless process (when one stays up at night preoccupied with the problem). The stages are interconnected. One always has to lead off with preparatory conscious work to scope out the lay of the land and to develop one鈥檚 heuristic, even though, in any mathematical problem of substance, the solution does not reveal itself right away:
In all these successive steps, as we see, 鈥榮udden 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鈥� [quoting Poincar茅 again]. (pp. 45)
Poincar茅鈥檚 instructive example prompts a reflection on the respective roles of logic and chance 鈥� as Hadamard himself sees aptly enough, one cannot do without the one or the other. As part of the process, errors (which are frequent even in good mathematicians) are unavoidable and have a role to play: rather than infallibility, the mark of a good mathematician consists in possession of some insight that warns him when his calculations might be off. Finally, from Hadamard鈥檚 own experience: being too closely focused on one aspect of a problem can cause one to miss a significant consequence of one鈥檚 findings (pp. 50-52).
In the major sixth chapter, Hadamard turns to an analysis of mathematical discovery as a process of synthesis. For discovery issues from the adroit combination of thoughts and thoughts ordinarily have to be represented through a word or symbols in order for our minds to manipulate them. This leads to a discussion of the role of signs, in which Hadamard shows some sensitivity to the insights of the celebrated philologist Max M眉ller and the Counter-Enlightenment philosopher Johann Gottfried Herder on language versus wordless thought (pp. 66ff). Hadamard insists that words are entirely absent when he thinks on mathematical problems (like what is the case with the geneticist Francis Galton). Descartes distrusts intervention of the imagination and wants to eliminate it completely from science! To which Hadamard replies, deductions in the realm of numbers may be most generally accompanied by images [germ of a critique of Russellian logical formalism]. Can imagery be educated? As Hadamard remarks, one tends to become more verbal with advancing maturity.
The remaining chapters go on to entertain a few further topics related to the psychology of invention: common sense versus formal deduction say from rules of differential and integral calculus (pp. 100-103); and intuitive versus logical minds (pp. 106-115). There are at least two different definitions of intuition versus logic: see for instance p. 120, Galois was highly intuitive according to definition A but not B. Next, he remarks upon two conceptions of invention itself (pp. 124-125), what one may characterize as goal-oriented versus the free play of the imagination:
We must add, however, that conversely, 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鈥檚 constant relation to theory is the same as that of the leaf to the tree: one supports the other, but the former needs the latter. Not to mention several important physical examples, the first mathematical foundation in Greek science, geometry, was suggested by practical necessity, as can be seen by its very name, which means 鈥榣and-measuring鈥�. But this example is exceptional in the sense that practical questions are most often solved by means of existing theories: practical applications of purely scientific discoveries, as important as they may be, are generally remote in time (though, in recent years, this delay may be considerably shortened, as happened in the case of radio telegraphy, which occurred a few years after the discovery of Hertzian waves). It seldom happens that important mathematical researches are directly undertaken in view of a given practical use: they are inspired by the desire which is the common motive of every scientific work, the desire to know and to understand. Therefore, between the two kinds of invention we have just distinguished from each other, mathematicians are accustomed only to the second one. (pp. 125-126)
A handful of overall observations on this reviewer鈥檚 part.
1) One would wish for a more systematic survey to assess Hadamard鈥檚 contention that among mathematicians known to him except for P贸lya one does not think in words until it comes time to write out and communicate one鈥檚 findings to others. In as much as Hadamard thus frequently draws upon his own experiences having spent a career in the field, the diligent student will profit from reading him only after himself having attained to some degree of mathematical maturity, else one won鈥檛 appreciate very much his illustrations 鈥� though for most of us the stock of experience on which we have to draw will consist mainly in having solved the homework exercises in textbooks and not so much in having done original research!
2) The question as to classification of types of mathematical genius would be worth further investigation, best approached by familiarizing oneself with the history of mathematics and the biographies of the great mathematicians who contributed to it.
3) Why has research become less original and more driven by groupthink in recent decades? There is much more to say about how the researcher goes about selecting his topic. Indeed, as Hadamard points out, only second raters take their research problem from someone else (p. 126). Sociological and pedagogical issues are implicated here. Set aside the obvious pressure to conformity with the community鈥檚 perception of what ought to be its research agenda 鈥� in any educational endeavor, one has to strike a balance between teaching the student new material and leaving him the freedom to explore his own ideas: one cannot invent in a vacuum, after all. For only upon exposure to significant and profound concepts from the past will the student鈥檚 own creative response be elicited. As a rule, though, one must determine that, in doctoral-level training these days, too much stress is placed upon assimilation of the existing store of knowledge and not enough on developing an individual perspective 鈥� hence the phenomenon of vogues, in which the scientific community lurches from one hot topic to another since hardly anyone commands a sturdy, independent identity of his own.
4) Lastly, Hadamard鈥檚 intriguing, stimulating and learned work raises a theoretical question: the conscious mind is ruled by discursiveness, but how does the unconscious operate? It seems to sift through and combine ideas not so much according to logical succession but in some more global fashion, relying on a deeper sense of their import and of how they may fit together (of course, it can err thereby, which is why the stage of verification continues to be indispensable). A most rewarding subject for further reflection for those inclined to psychology!
In summary, a fine start on a fascinating topic but the treatment of its themes remains too sketchy to count as a monograph, 4 stars.
October 8, 2024
A luminary of complex analysis takes a crack at understanding how the mind invents new ideas. Mostly a lot of (most very charming, and unfamiliar to me) quotes. He bases much of the book as a response to and elaboration upon some comments of Poincar茅, and it鈥檚 clear he holds P鈥檃 account of the invention of fuchsian transformations as basically the prototype, even more than his own (no less important) contributions.
As basically a shape rotator, he takes great umbrage at the suggestion of a wordcel Orientalist that literally all thought takes place in words. Maybe less space could have been devoted to this. But overall a great fun read. By comparison shows how poor, for the most part, our own crop of math popularizers are at seriously engaging with and challenging other sciences.
As basically a shape rotator, he takes great umbrage at the suggestion of a wordcel Orientalist that literally all thought takes place in words. Maybe less space could have been devoted to this. But overall a great fun read. By comparison shows how poor, for the most part, our own crop of math popularizers are at seriously engaging with and challenging other sciences.
June 25, 2021
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May 26, 2022
Deeply edifying introspective look into the mind of a mathematician from a master mathematician, Jacque Hadamard.
The most important thing I learned from the book is the idea of "relay-results"
> To sum up, every stage of the research has to be, so to speak, articulated to the following one by a result in a precise form, which I should propose to call a relay-resuit (or a relay-formula if it is a formula, as in Newton鈥檚 interpretation of Kepler鈥檚 third law). When reaching such a joining, somewhat analogous to railroad bifurcations, the new direction I which further research will follow must be decided, so that they clearly illustrate the directing action of that conscious ego which we were tempted to consider as 鈥渋nferior鈥� to unconsciousness.
> William Hamilton uses an interesting comparison to the process of tunneling through a sand bank. 鈥淚n this operation, it is impossible to succeed unless every foot, nay, almost every inch in our progress be secured by an arch of masonry before we attempt the excavation of another...鈥�
Hadamard describes how he uses the scratch-pad to carry his thinking along, but only for a day. Anything that is not "secured" into a relay-result would be "dead" on the next day.
> I may use algebraic symbols; but, rather often, I do not use them in the usual and regular way. I do not take time to write the equations completely, only caring to see, so to speak, how they look. These equations, or some terms of them, are often disposed in a peculiar and funny order like actors in a scenario, by means of which they 鈥渟peak鈥� to me, as long as I continue to consider them. But if, after having been interrupted in my calculations, I resume them on the following day, what I have written in that way is as if 鈥渄ead鈥� for me. Generally, I can do nothing else than throw the sheet away and begin everything anew, except if, in the first day, I have obtained one or two formulae which I have fully verified and can use as relay-formulae.
There are even some historical gossips for those who are not interested in doing mathematics for a living.
Henry Sidgwick's "symbolic" imagery:
> ... which he himself reported at the International Congress of Experimental Psychology, in 1892. His reasonings on economic questions were almost always accompanied by images, and 鈥渢he image were often curiously arbitrary and sometimes almost undecipherably symbolic. For example, it took him a long time to discover that an odd, symbolic image which accompanied the word 鈥榲alue鈥� was a faint, partial image of a man putting something on a scale.鈥�
Klein's 1893 speculation on race and mathematical styles, an early hint of
> "It would seem as if a strong naive space intuition were an attribute of the Teutonic race, while the critical, purely logical sense is more developed in the Latin and Hebrew races.鈥�... Klein implicitly considers intuition, with its mysterious character, as being superior to the prosaic way of logic (we have already met with such a tendency in Section III) and is evidently happy to claim that superiority for his countrymen. We have heard recently of that special kind of ethnography with Nazism: we see that there was already something of this kind in 1893.
There's this mysterious comment about Hermite, reminding one of the comic "then a miracle occurs".
> ... whose eyes 鈥渟eem to shun contact with the world鈥� and who seeks 鈥渨ithin, not without, the vision of truth.鈥�
> Methods always seemed to be born in his mmd in some mysterious way. In his lectures at the Sorbonne, which we attended with鈥� unfailing enthusiasm, he liked to begin his argument by: 鈥淟et us start from the identity . . .鈥� and here lie was writing a formula the accuracy of which was certain, but whose origin in his brain and way of discovery he did not explain and we could not guess... it seemed quite absurd that [Hermite's calculations] would, that time, lead to the solution; and yet, by a kind of witchcraft, they do.
> Reading one of [Poincare's] great discoveries, I should fancy (evidently a delusion) that, however magnificent, one ought to have found it long before, while such memoirs of Hermite as the one referred to in the text arouse in me the idea: 鈥淲hat magnificent results! How could he dream of such a thing?鈥�
There is this terrifying thought of near-miss: Hadamard almost discovered relativity, twice!
> I was interested in generalizing to hyperspaces the classic notion of curvature of surfaces. I had to deal with Riemann鈥檚 notion of curvature in hyperspaces, which is the generalization of the more elementary notion of the curvature of a surface in ordinary space. What interested me was to obtain that Riemann curvature as the curvature of a certain surface S, drawn in the considered hyperspace, the shape of S being chosen in order to reduce the curvature to a minimum. I succeeded in showing that the minimum thus obtained was precisely Riemann鈥檚 expression; only, thinking of that question, I neglected to take into consideration the circumstances under which the minimum is reached, i.e., the proper way of constructing S in order to reach the minimum. Now, investigating that would have led me to the principle of the so-called 鈥淎bsolute Differential Calculus,鈥� the discovery of which belongs to Ricci and Levi Civita.
> Absolute differential calculus is closely connected with the theory of relativity; and in this connection, I must confess that, having observed that the equation of propagation of light is invariant under a set of transformations (what is now known as Lorentz鈥檚 group) by which space and time are combined together, I added that 鈥渟uch transformations are obviously devoid of physical meaning.鈥� Now, these transformations, supposedly without any physical meaning, are the base of Einstein鈥檚 theory.
Absolute Differential Calculus is currently known as tensor calculus, and is an essential language for general relativity, as well as advanced continuum mechanics.
There's this amazing reference to Allen Poe. Whereas other mathematicians would think he discovered the technique of 鈥渇inite part of infinite integral鈥� by mysterious intuition, Hadamard felt like he was helplessly led to it by a cold, impersonal leash of logic.
> But, in fact, for a long while my mind refused to conceive that idea until positively compelled to. I was led to it step by step as the mathematical reader will easily verify if he takes the trouble to consult my researches on the subject... I could not avoid it any more than the prisoner in Poe鈥檚 tale The Pit and The Pendulum could avoid the hole at the center of his cell.
There's Riemann's "this margin is too narrow" comment about the Riemann Zeta function.
> He proved some important properties of that function, but pointed out several as important ones without giving the proof. At the death of Riemann, a note was found among his papers, saying 鈥淭hese properties of (s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.鈥�
> We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove ail of them but 011e. The question concerning that last one remains unsolved as yet, though, by an immense labor pursued throughout this last half century, some highly interesting discoveries in that direction have been achieved. It seems more and more probable, but still not at ah certain, that the 鈥淩iemann hypothesis鈥� is true.
There is this justification for pure research: if you have a good taste for beauty, then applications will follow.
> When I presented my doctor鈥檚 thesis for examination, Hermite observed that it would be most useful to find applications. At that time, I had none available. Now, between the time my manuscript was handed in and the day when the thesis was sustained, I became aware of an important question... which had been proposed by the French Academy of Sciences as a prize subject; and precisely, the results in my thesis gave the solution of that question. I had been uniquely led by my feeling of the interest of the problem and it led me in the right way.
I hope Hadamard collected that prize. He deserved it.
> A few years later, having, in a further study of the same kind of questions, obtained a very simple result which seemed to me an elegant one, I communicated it to my friend, the physicist Duhem. He asked to what it applied. When I answered that so far I had not thought of that, Duhem, who was a remarkable artist as well as a prominent physicist, compared me to a painter who would begin by painting a landscape without leaving his studio and only then start on a walk to find in nature some landscape suiting his picture... I was right in not worrying about applications: they did come afterwards.
Hadamard is also humble, sometimes amusingly so.
> ... I had been attracted by a question in algebra (on determinants). When solving it, I had no suspicion of any definite use it might have, only feeling that it deserved interest; then in 1900 appeared Fredholm鈥檚 theory, for which the result obtained in 1893 happens to be essential. This is the theory which, as said in Section IV, I failed to discover. It has been a consolation for my self-esteem to have brought a necessary link to Fredholm鈥檚 arguments.
One would think that a master like him would have needed no more ego-boosts.
> Some mathematicians, especially contemporary ones, have improved [Lie group theory] by most beautiful discoveries. Some others鈥擨 confess that I belong to the latter category鈥攖hough being eventually able to use it for simple applications, feel insuperable difficulty in mastering more than a rather elementary and superficial knowledge of it. Psychological reasons for that difference, which seems to me incontestable, would be interesting to find.
The most important thing I learned from the book is the idea of "relay-results"
> To sum up, every stage of the research has to be, so to speak, articulated to the following one by a result in a precise form, which I should propose to call a relay-resuit (or a relay-formula if it is a formula, as in Newton鈥檚 interpretation of Kepler鈥檚 third law). When reaching such a joining, somewhat analogous to railroad bifurcations, the new direction I which further research will follow must be decided, so that they clearly illustrate the directing action of that conscious ego which we were tempted to consider as 鈥渋nferior鈥� to unconsciousness.
> William Hamilton uses an interesting comparison to the process of tunneling through a sand bank. 鈥淚n this operation, it is impossible to succeed unless every foot, nay, almost every inch in our progress be secured by an arch of masonry before we attempt the excavation of another...鈥�
Hadamard describes how he uses the scratch-pad to carry his thinking along, but only for a day. Anything that is not "secured" into a relay-result would be "dead" on the next day.
> I may use algebraic symbols; but, rather often, I do not use them in the usual and regular way. I do not take time to write the equations completely, only caring to see, so to speak, how they look. These equations, or some terms of them, are often disposed in a peculiar and funny order like actors in a scenario, by means of which they 鈥渟peak鈥� to me, as long as I continue to consider them. But if, after having been interrupted in my calculations, I resume them on the following day, what I have written in that way is as if 鈥渄ead鈥� for me. Generally, I can do nothing else than throw the sheet away and begin everything anew, except if, in the first day, I have obtained one or two formulae which I have fully verified and can use as relay-formulae.
There are even some historical gossips for those who are not interested in doing mathematics for a living.
Henry Sidgwick's "symbolic" imagery:
> ... which he himself reported at the International Congress of Experimental Psychology, in 1892. His reasonings on economic questions were almost always accompanied by images, and 鈥渢he image were often curiously arbitrary and sometimes almost undecipherably symbolic. For example, it took him a long time to discover that an odd, symbolic image which accompanied the word 鈥榲alue鈥� was a faint, partial image of a man putting something on a scale.鈥�
Klein's 1893 speculation on race and mathematical styles, an early hint of
> "It would seem as if a strong naive space intuition were an attribute of the Teutonic race, while the critical, purely logical sense is more developed in the Latin and Hebrew races.鈥�... Klein implicitly considers intuition, with its mysterious character, as being superior to the prosaic way of logic (we have already met with such a tendency in Section III) and is evidently happy to claim that superiority for his countrymen. We have heard recently of that special kind of ethnography with Nazism: we see that there was already something of this kind in 1893.
There's this mysterious comment about Hermite, reminding one of the comic "then a miracle occurs".
> ... whose eyes 鈥渟eem to shun contact with the world鈥� and who seeks 鈥渨ithin, not without, the vision of truth.鈥�
> Methods always seemed to be born in his mmd in some mysterious way. In his lectures at the Sorbonne, which we attended with鈥� unfailing enthusiasm, he liked to begin his argument by: 鈥淟et us start from the identity . . .鈥� and here lie was writing a formula the accuracy of which was certain, but whose origin in his brain and way of discovery he did not explain and we could not guess... it seemed quite absurd that [Hermite's calculations] would, that time, lead to the solution; and yet, by a kind of witchcraft, they do.
> Reading one of [Poincare's] great discoveries, I should fancy (evidently a delusion) that, however magnificent, one ought to have found it long before, while such memoirs of Hermite as the one referred to in the text arouse in me the idea: 鈥淲hat magnificent results! How could he dream of such a thing?鈥�
There is this terrifying thought of near-miss: Hadamard almost discovered relativity, twice!
> I was interested in generalizing to hyperspaces the classic notion of curvature of surfaces. I had to deal with Riemann鈥檚 notion of curvature in hyperspaces, which is the generalization of the more elementary notion of the curvature of a surface in ordinary space. What interested me was to obtain that Riemann curvature as the curvature of a certain surface S, drawn in the considered hyperspace, the shape of S being chosen in order to reduce the curvature to a minimum. I succeeded in showing that the minimum thus obtained was precisely Riemann鈥檚 expression; only, thinking of that question, I neglected to take into consideration the circumstances under which the minimum is reached, i.e., the proper way of constructing S in order to reach the minimum. Now, investigating that would have led me to the principle of the so-called 鈥淎bsolute Differential Calculus,鈥� the discovery of which belongs to Ricci and Levi Civita.
> Absolute differential calculus is closely connected with the theory of relativity; and in this connection, I must confess that, having observed that the equation of propagation of light is invariant under a set of transformations (what is now known as Lorentz鈥檚 group) by which space and time are combined together, I added that 鈥渟uch transformations are obviously devoid of physical meaning.鈥� Now, these transformations, supposedly without any physical meaning, are the base of Einstein鈥檚 theory.
Absolute Differential Calculus is currently known as tensor calculus, and is an essential language for general relativity, as well as advanced continuum mechanics.
There's this amazing reference to Allen Poe. Whereas other mathematicians would think he discovered the technique of 鈥渇inite part of infinite integral鈥� by mysterious intuition, Hadamard felt like he was helplessly led to it by a cold, impersonal leash of logic.
> But, in fact, for a long while my mind refused to conceive that idea until positively compelled to. I was led to it step by step as the mathematical reader will easily verify if he takes the trouble to consult my researches on the subject... I could not avoid it any more than the prisoner in Poe鈥檚 tale The Pit and The Pendulum could avoid the hole at the center of his cell.
There's Riemann's "this margin is too narrow" comment about the Riemann Zeta function.
> He proved some important properties of that function, but pointed out several as important ones without giving the proof. At the death of Riemann, a note was found among his papers, saying 鈥淭hese properties of (s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.鈥�
> We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove ail of them but 011e. The question concerning that last one remains unsolved as yet, though, by an immense labor pursued throughout this last half century, some highly interesting discoveries in that direction have been achieved. It seems more and more probable, but still not at ah certain, that the 鈥淩iemann hypothesis鈥� is true.
There is this justification for pure research: if you have a good taste for beauty, then applications will follow.
> When I presented my doctor鈥檚 thesis for examination, Hermite observed that it would be most useful to find applications. At that time, I had none available. Now, between the time my manuscript was handed in and the day when the thesis was sustained, I became aware of an important question... which had been proposed by the French Academy of Sciences as a prize subject; and precisely, the results in my thesis gave the solution of that question. I had been uniquely led by my feeling of the interest of the problem and it led me in the right way.
I hope Hadamard collected that prize. He deserved it.
> A few years later, having, in a further study of the same kind of questions, obtained a very simple result which seemed to me an elegant one, I communicated it to my friend, the physicist Duhem. He asked to what it applied. When I answered that so far I had not thought of that, Duhem, who was a remarkable artist as well as a prominent physicist, compared me to a painter who would begin by painting a landscape without leaving his studio and only then start on a walk to find in nature some landscape suiting his picture... I was right in not worrying about applications: they did come afterwards.
Hadamard is also humble, sometimes amusingly so.
> ... I had been attracted by a question in algebra (on determinants). When solving it, I had no suspicion of any definite use it might have, only feeling that it deserved interest; then in 1900 appeared Fredholm鈥檚 theory, for which the result obtained in 1893 happens to be essential. This is the theory which, as said in Section IV, I failed to discover. It has been a consolation for my self-esteem to have brought a necessary link to Fredholm鈥檚 arguments.
One would think that a master like him would have needed no more ego-boosts.
> Some mathematicians, especially contemporary ones, have improved [Lie group theory] by most beautiful discoveries. Some others鈥擨 confess that I belong to the latter category鈥攖hough being eventually able to use it for simple applications, feel insuperable difficulty in mastering more than a rather elementary and superficial knowledge of it. Psychological reasons for that difference, which seems to me incontestable, would be interesting to find.
January 4, 2020
Czasem ksi膮偶ki, kt贸re uchodz膮 za kultowe czy prze艂omowe, po zapoznaniu si臋 z ich tre艣ci膮, okazuj膮 si臋 rozczarowaniem. Tak si臋 dzieje szczeg贸lnie, gdy p艂odny tematycznie i nowatorski tekst jest przywo艂ywany w innych publikacjach jako 藕r贸d艂o inspiracji. Czasem na艣ladowcy po prostu s膮 lepsi od pierwowzoru, kt贸ry obrasta mitologi膮.(*)
Taki niestety opis pasuje po cz臋艣ci do 鈥濸sychologii odkry膰 matematycznych鈥� wybitnego matematyka Jacquesa Hadamarda. Ta jego praca jest wielokrotnie przytaczana w nowszych publikacjach, w kt贸rych autorzy zastanawiaj膮 si臋 nad prac膮 zawodowego matematyka.
Badacz postanowi艂 przedyskutowa膰 藕r贸d艂a matematycznego natchnienia, inspiracji i sposobu dochodzenia do odkry膰. Publikacja powsta艂a pod koniec drugiej wojny 艣wiatowej i jest konsekwencj膮 d艂ugoletnich zainteresowa艅 Hadamarda. Przygotowany przez niego zestaw pyta艅 ankietowych rozes艂anych do wybitnych koleg贸w, zainspirowa艂 do pozbierania wynik贸w w postaci ksi膮偶kowej. Akurat te jej fragmenty s膮 do艣膰 nieciekawe, bo brak w nich syntetycznego ilo艣ciowego ogl膮du. Pr贸ba obiektywizacji, bez reprezentatywnego odwo艂ania si臋 do odczu膰 badaczy, jest niekonkluzywna. Szczeg贸lnie, 偶e matematyk przytacza stanowiska koleg贸w, kt贸rzy z innych powod贸w s膮 mu bliscy (np. Poincare czy Hermite to jego nauczyciele). Wyczuwa艂em w tym narracj臋 zmierzaj膮c膮 do z g贸ry za艂o偶onych wniosk贸w.
Ciekawsze wyda艂y si臋 fragmenty subiektywne, w kt贸rych Hadamard analizuj膮ce osobiste do艣wiadczenia w dochodzeniu do nowych matematycznych ustale艅. Podzieli艂 typowy proces tw贸rczy na etapy: wyt臋偶onej pracy, niezb臋dnego porzucenia rozmy艣la艅 i czasem ostatecznego ol艣nienia czy prze艂omu jako艣ciowego. Te kolejne stadia odbywaj膮 si臋, wed艂ug autora, na r贸偶nych poziomach 艣wiadomo艣ci i pod艣wiadomo艣ci. W tym kontek艣cie bardzo ciekawie przedstawi艂 relacje mi臋dzy stosowanymi technikami procesu umys艂owego. Wi臋kszo艣膰 鈥樑沜is艂owc贸w鈥� do艣膰 abstrakcyjnie wizualizuje sobie problem, s艂owa nie s膮 istotnym elementem aktywno艣ci. Ta obserwacja Hadamarda prowadzi do kontrowersji z lingwistami, kt贸rzy akcentuj膮 przemo偶ny wp艂yw s艂owa na ka偶dy typ intelektualnych wyzwa艅. Odpowied藕 Einsteina na ankiet臋, umieszczona w dodatku, jest dobrym przyk艂adem my艣lenia obrazami, bez etykiet j臋zykowych.
鈥濸sychologi臋 odkry膰 matematycznych鈥� powinien przeczyta膰 czytelnik zainteresowany prac膮 matematyk贸w, od strony teorii umys艂u. Mnie ksi膮偶ka jednak troch臋 zwiod艂a.
Polecam wytrwa艂ym.
========================
(*) Polecam pozycje, kt贸re pe艂niej, ja艣niej i ciekawiej dotykaj膮 podobnej tematyki:
鈥濧pologia matematyka鈥�, G.H. Hardy
鈥濩zy fizyka i matematyka to nauki humanistyczne?鈥�, M. Heller, S. Krajewski
鈥濸i razy drzwi. Szkice o liczeniu, my艣leniu i istnieniu鈥�, J.D. Barrow
鈥濽mys艂 matematyczny鈥�, B. Bro偶ek, M. Hohol
艢REDNIA - 6/10
Taki niestety opis pasuje po cz臋艣ci do 鈥濸sychologii odkry膰 matematycznych鈥� wybitnego matematyka Jacquesa Hadamarda. Ta jego praca jest wielokrotnie przytaczana w nowszych publikacjach, w kt贸rych autorzy zastanawiaj膮 si臋 nad prac膮 zawodowego matematyka.
Badacz postanowi艂 przedyskutowa膰 藕r贸d艂a matematycznego natchnienia, inspiracji i sposobu dochodzenia do odkry膰. Publikacja powsta艂a pod koniec drugiej wojny 艣wiatowej i jest konsekwencj膮 d艂ugoletnich zainteresowa艅 Hadamarda. Przygotowany przez niego zestaw pyta艅 ankietowych rozes艂anych do wybitnych koleg贸w, zainspirowa艂 do pozbierania wynik贸w w postaci ksi膮偶kowej. Akurat te jej fragmenty s膮 do艣膰 nieciekawe, bo brak w nich syntetycznego ilo艣ciowego ogl膮du. Pr贸ba obiektywizacji, bez reprezentatywnego odwo艂ania si臋 do odczu膰 badaczy, jest niekonkluzywna. Szczeg贸lnie, 偶e matematyk przytacza stanowiska koleg贸w, kt贸rzy z innych powod贸w s膮 mu bliscy (np. Poincare czy Hermite to jego nauczyciele). Wyczuwa艂em w tym narracj臋 zmierzaj膮c膮 do z g贸ry za艂o偶onych wniosk贸w.
Ciekawsze wyda艂y si臋 fragmenty subiektywne, w kt贸rych Hadamard analizuj膮ce osobiste do艣wiadczenia w dochodzeniu do nowych matematycznych ustale艅. Podzieli艂 typowy proces tw贸rczy na etapy: wyt臋偶onej pracy, niezb臋dnego porzucenia rozmy艣la艅 i czasem ostatecznego ol艣nienia czy prze艂omu jako艣ciowego. Te kolejne stadia odbywaj膮 si臋, wed艂ug autora, na r贸偶nych poziomach 艣wiadomo艣ci i pod艣wiadomo艣ci. W tym kontek艣cie bardzo ciekawie przedstawi艂 relacje mi臋dzy stosowanymi technikami procesu umys艂owego. Wi臋kszo艣膰 鈥樑沜is艂owc贸w鈥� do艣膰 abstrakcyjnie wizualizuje sobie problem, s艂owa nie s膮 istotnym elementem aktywno艣ci. Ta obserwacja Hadamarda prowadzi do kontrowersji z lingwistami, kt贸rzy akcentuj膮 przemo偶ny wp艂yw s艂owa na ka偶dy typ intelektualnych wyzwa艅. Odpowied藕 Einsteina na ankiet臋, umieszczona w dodatku, jest dobrym przyk艂adem my艣lenia obrazami, bez etykiet j臋zykowych.
鈥濸sychologi臋 odkry膰 matematycznych鈥� powinien przeczyta膰 czytelnik zainteresowany prac膮 matematyk贸w, od strony teorii umys艂u. Mnie ksi膮偶ka jednak troch臋 zwiod艂a.
Polecam wytrwa艂ym.
========================
(*) Polecam pozycje, kt贸re pe艂niej, ja艣niej i ciekawiej dotykaj膮 podobnej tematyki:
鈥濧pologia matematyka鈥�, G.H. Hardy
鈥濩zy fizyka i matematyka to nauki humanistyczne?鈥�, M. Heller, S. Krajewski
鈥濸i razy drzwi. Szkice o liczeniu, my艣leniu i istnieniu鈥�, J.D. Barrow
鈥濽mys艂 matematyczny鈥�, B. Bro偶ek, M. Hohol
艢REDNIA - 6/10
April 8, 2020
Fantastic title
Fantastic title. Probably why this work has lingered so long. Otherwise quite pedestrian. Almost old fashioned and quaint by modern standards. Psychoanalysis. The 鈥榰nconscious鈥� re the supernatural. Come on. His psychological observations are worn and trite. With none of the sophistication of modern cognitive science. Fair enough it was mid century or earlier but not even a glimmer. Non conscious processes process material in the background. Well duh!! And that鈥檚 about it. The whole content.
Except his claims to have almost anticipated tensor calculus and even special relativity. Yes his actual productive mathematical work was by then that far in the past. And .. anticipated!! Really Jacques. Such an easy claim to make.
There are a few good riffs/ accounts on mathematicians doing their creative work. Hardly makes up for all the dribble.
Fantastic title. Probably why this work has lingered so long. Otherwise quite pedestrian. Almost old fashioned and quaint by modern standards. Psychoanalysis. The 鈥榰nconscious鈥� re the supernatural. Come on. His psychological observations are worn and trite. With none of the sophistication of modern cognitive science. Fair enough it was mid century or earlier but not even a glimmer. Non conscious processes process material in the background. Well duh!! And that鈥檚 about it. The whole content.
Except his claims to have almost anticipated tensor calculus and even special relativity. Yes his actual productive mathematical work was by then that far in the past. And .. anticipated!! Really Jacques. Such an easy claim to make.
There are a few good riffs/ accounts on mathematicians doing their creative work. Hardly makes up for all the dribble.
October 2, 2019
Some nice facts about and research approaches from first rate mathematicians (Hadamard included), but gets totally bogged down in tedious speculation based off Freud and other in-vogue-at-the-time psychologists.
August 2, 2019
The shortest and best way between two truths of the real domain often passes through the imaginary one.
~ Jacques Hadamard
An interesting read.
~ Jacques Hadamard
An interesting read.
August 6, 2020
A really great read at the intersection of philosophy and psychology, and with no mathematical experience required. I highly recommend it to anyone interested in human thought.
March 13, 2021
They say you can't judge a book by its cover, but this book was every bit as boring as its cover.
Want to read
November 8, 2021Ben goertzel
February 27, 2022
Critical point of view possibly held and still sustained amongst certain mathematicians concerning Poincar茅鈥檚 originalities.
January 2, 2025
Our perceptions of the world are captured by language. For mathematicians this language extends to numbers and the creative problem solving process facilitated by mathematical language is immensely useful to us. The problem Hadamard investigates is how come the unconsciousness is able to present striking ingenious solutions to us? And how is it able to do that without even using language?
The work done by the unconsciousness is a deep puzzle that we do not understand and Hadamard sheds light on the common observation among mathematicians that the unconsciousness does mathematics completely differently to the conscious mind and often simply presents the solution to us. Don DeLillo put poetically 鈥淚n the dark the mind runs on like a devouring machine鈥�. The unconscious constantly monitors us and acts as a guide and yet the ideas the unconscious has exist independently of language and we have little to no idea how that works.
If the mind was working on the solution, and figured it out already, why does it use images and dreams to tell us? Why is it that when you are gripped by a problem (mathematical or not), that the unconsciousness shows you solutions like portents in a fire and not in language? It may not be too radical to question whether at some level, language may in fact be a barrier between us and the unconscious. Wittgenstein, no doubt inspired by Nietzsche, famously said 鈥淭he limits of my language mean the limits of my world.鈥� This of course is an observation implicit in Hamlet. Though Hadamard doesn't present any solutions to this puzzle, he makes insightful comments nonetheless.
Fundamentally, this is a book about the relationship between the unconscious and language and requires zero mathematical knowledge to appreciate what is being discussed. It should be read by anyone with any serious interest in the unconsciousness and the philosophy of mind.
The work done by the unconsciousness is a deep puzzle that we do not understand and Hadamard sheds light on the common observation among mathematicians that the unconsciousness does mathematics completely differently to the conscious mind and often simply presents the solution to us. Don DeLillo put poetically 鈥淚n the dark the mind runs on like a devouring machine鈥�. The unconscious constantly monitors us and acts as a guide and yet the ideas the unconscious has exist independently of language and we have little to no idea how that works.
If the mind was working on the solution, and figured it out already, why does it use images and dreams to tell us? Why is it that when you are gripped by a problem (mathematical or not), that the unconsciousness shows you solutions like portents in a fire and not in language? It may not be too radical to question whether at some level, language may in fact be a barrier between us and the unconscious. Wittgenstein, no doubt inspired by Nietzsche, famously said 鈥淭he limits of my language mean the limits of my world.鈥� This of course is an observation implicit in Hamlet. Though Hadamard doesn't present any solutions to this puzzle, he makes insightful comments nonetheless.
Fundamentally, this is a book about the relationship between the unconscious and language and requires zero mathematical knowledge to appreciate what is being discussed. It should be read by anyone with any serious interest in the unconsciousness and the philosophy of mind.
July 4, 2015
In it's early days, artificial intelligence naively sought to model human reasoning using combinatorial syllogism engines. This went nowhere: No one knew how to constrain the growth in discovery space within workable limits.
Another class of attempts was based on strong reductionism; modelling neuron function.
It occurs to me in reading this book that I know of no attempts at the mid-level, i.e. modelling Hadamard's fringe consciousness, Poincar茅's idea fragments, activated by association, etc.
Another class of attempts was based on strong reductionism; modelling neuron function.
It occurs to me in reading this book that I know of no attempts at the mid-level, i.e. modelling Hadamard's fringe consciousness, Poincar茅's idea fragments, activated by association, etc.
August 2, 2013
The rapid development of neuroscience and psychology in the last several decades makes some of the ideas in this book a bit outdated, although I don't think it renders them completely useless.
This book is a bold interdisciplinary move in trying to understand what goes on in the mind of mathematicians and other scientists engaging in mathematical thinking. I think it lacks something, but I'm not quite sure what... other than that, it's pretty good!
This book is a bold interdisciplinary move in trying to understand what goes on in the mind of mathematicians and other scientists engaging in mathematical thinking. I think it lacks something, but I'm not quite sure what... other than that, it's pretty good!
Read
August 27, 2014Uz si to nepamatuju do detail暖, ale doporu膷il mi to Hedrlin a bylo to prvni 膷ten铆 o matematice. kde se otev艡en臎 p铆拧e o my拧len铆. Zaj铆malo by m臎, jak茅 druhy motivac铆 a postoj暖 m臎ly matematici kolem Poincareho a Hadamarda.
October 20, 2011
a bit dated, but still has a lot of interesting stuff.
February 20, 2012
The mostly inventions happen in the sub conscious space rather than in the conscious space.
February 16, 2016
Fascinating. Like a character study of mathematical creativity.
December 12, 2014
Reminds me of How to solve it and A mathematician's apology, different in content but similar in style.
March 3, 2016
For some reason didn't learn much.
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