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| 0156028719
| 9780156028714
| 0156028719
| 3.96
| 2,139
| 2003
| Dec 01, 2003
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liked it
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The neuroscientist Antonio Damasio writes pleasant, elegant prose. Unfortunately, aside from that, this book, first published in 2003, is somewhat of
The neuroscientist Antonio Damasio writes pleasant, elegant prose. Unfortunately, aside from that, this book, first published in 2003, is somewhat of a disappointment. The main concern of his scientific career has been to understand the mechanisms underlying "emotions" and "feelings". He has given good accounts of this subject in two previous books: Descartes' Error (1994) and The Feeling of What Happens (1999). What is good about Damasio's writing, especially in the earlier books, is that he doesn't do much dumbing down of the material, by avoiding technical terms, to appeal to the "general reader", as too many "science writers" do. The book reviewed here, however, doesn't cover the subject in as much depth as the previous books, and in particular it doesn't very well illuminate the distinction - which the author insists upon - between "emotion" and "feeling". It appears that Damasio wanted to write on what interested him about Spinoza, but didn't have enough to fill a whole book. So the first five chapters (about 3/4 of the total text) are devoted mostly to the neuropsychological issues, while the final two chapters are on Spinoza, and are connected only tenuously with the rest of the book. Damasio has championed the idea that human consciousness and other psychological phenomena - emotions and feelings especially - aren't rooted primarily in the brain, but instead are shaped by physiological processes going on throughout the whole body. This may be surprising to some, but it's not an especially radical idea. It makes good evolutionary sense. An animal's main evolutionary objective is to be good at survival and reproduction. Emotions (at least in animals with more than a rudimentary nervous system) exist to motivate an individual to seek things that favor survival and reproduction (shelter, food, sex), and to avoid threatening things (excessive heat or cold, predators, reproductive rivals). They seem to form a bridge between the sensory and motor systems. In animals with a developed cerebral cortex, like humans, emotions work partly through cognition. Note that the words "emotion" and "motivation" share the same linguistic root: the Indo-European MEUh-. Emotions, whether conscious or not, are what motivates animal behavior. Emotions in general and feelings in particular allow humans to make critical decisions quickly, when the situation requires that. It seems unlikely that inhabitants of the planet Vulcan, like Mr. Spock of Star Trek, could have successfully evolved without the help of emotions. (Though perhaps they became able to suppress them at a later stage.) I wish Damasio had been clearer in this book about his distinction between emotions and feelings. Are things like "fear", "pleasure", "shame", etc. emotions or feelings? Most people, I think, might use either term for them. But for Damasio, it seems, an emotion is represented in the brain only in certain specific regions, and may or may not appear in consciousness. For instance, a person (who is capable of consciousness) may have a "je ne sais quoi" sensation of fear on encountering an animal or object or situation with which the individual has had a negative experience in the past, even if that has been forgotten. The person will still avoid the particular stimulus without giving much thought as to why. A feeling, on the other hand, enters consciousness and additionally involves parts of the brain related to deliberate behavior. ("I like (or don't like) this whatever and want to remain (or not remain) exposed to it.") Naturally, if an animal doesn't have "consciousness" in the human sense - a worm, say - the animal can still be said to have "emotions" if it is motivated to approach or avoid certain things, for its own benefit. At any rate, that's how I interpret Damasio's thesis, and if I've misinterpreted it, a lack of clarity may be the reason. As far as the two chapters on Spinoza are concerned, they may be the most interesting part of the book in spite of their brevity. He lived from 1632 to 1677, entirely in Holland. This was mostly before what historians consider the "Age of Enlightenment", which flowered in the 18th century. Spinoza, however, is generally considered one of its earliest avatars. He was born into a moderately prosperous Jewish family, but eventually renounced both his material and religious heritage. Temperamentally he was reclusive, yet congenial with others in his limited social sphere. He came to reject both Judaism and Christianity, evidently for both philosophical reasons (of which see below) as well as revulsion at the irrationality and cruelty of both religious traditions. Fortunately for Spinoza, he lived in Holland, which at the time featured the least intolerant variety of Christianity. Nevertheless, his main philosophical work, the Ethics, was published only posthumously - and was almost immediately banned by both secular and religious (Jewish, Catholic, and Calvinist) authorities because of its "heretical" philosophy. Later leading philosophers of the Enlightenment (e. g. Locke, Hume, Leibniz, and Kant) apparently studied the Ethics - but were fearful of acknowledging its influence on them. At least Spinoza managed to escape the fates of other "heretics" like Giordano Bruno and Galileo. If you're interested in much discussion of Spinoza's philosophy, the present book is disappointing on this too, for at least three reasons. First, Damasio alludes in passing only to a few places in Spinoza's writing that deal with the psychology of emotions and feelings. Although he suggests that Spinoza foreshadowed current research findings, Spinoza's musings on these issues, however prescient, can't be much more than lucky guesses about what neuroscience now knows. Second, Damasio is wise not to deal at length with Spinoza's take on philosophical questions like "free will" and the "mind-body" problem. That's because the occupation of philosophers is to argue endlessly about issues that can only be satisfactorily resolved by scientific investigation. Third, Spinoza's opinions on religion aren't crystal clear. It's true that Spinoza was perhaps the most noteworthy Western philosopher of the preceding 1500 or so years to flatly reject dogma of the polluted swamp of traditional religion. However, arguments (among philosophers who care about such things) are still going on as to whether Spinoza's opinions actually represented atheism, agnosticism, "panentheism", or "pantheism" (which has generally been attributed to Spinoza). ...more |
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Feb 03, 2018
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Feb 15, 2018
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| 0802139868
| 9780802139863
| 0802139868
| 2.83
| 3,636
| 2000
| Apr 01, 2003
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it was amazing
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This book is a memoir - that is, not necessarily a complete autobiography, possibly not more than an account written to memorialize a particular event
This book is a memoir - that is, not necessarily a complete autobiography, possibly not more than an account written to memorialize a particular event or phase or aspect of one's life. In Catherine Millet's case that is specifically her sexual life. In her telling, which we have no reason to suppose isn't accurate, that part of her life was unusually active and varied. Perhaps it was no more active and varied than that of, say, a talented porn star or fille de joie. But we have no way to guess, since there are few if any other women who've published such an explicit account. In any case, as far as can be told from the account, Millet was never a sex worker and never sought remuneration for her sexual activities. (She, however, at one point refers to "my only failed attempt at prostitution", simply to explain "I would have been completely incapable of negotiating the least exchange of this sort" (p. 69).) Instead she has had a very respectable career as a writer, art critic, and founder/editor of a notable fine art magazine. According to her report the recompense for her sexual activity has consisted entirely in her own pleasure and that of her many - and not infrequently anonymous - partners. The account is quite explicit. I'm not going to quote or paraphrase examples - though they're quite numerous - since if that's what you're interested in you should just buy the book. (You won't be disappointed, unless you have limited imagination.) Millet's language is blunt and eschews euphemisms. This isn't for shock effect or titillation. The language isn't exactly clinical either, but rather uses just the normal vocabulary most people would use if they're not embarrassed to talk about sexual anatomy and activities. Millet had no intention of writing pornography (which, etymologically, is "writing about prostitutes"), and she hasn't, even though much the same vocabulary is used. So, what was her purpose in writing the book? In an Afterword published soon following the book itself Millet addresses that question, and she struggles to descry an answer. Initially she says "The aim here was merely to describe one person's sexuality, the sexuality of Catherine M." A little later she writes "I should try neither to understand nor explain, and even less to justify. There is no trial, no case to be made, because there is nothing more than a laying-out of facts." I would say her main urge was simply to document this aspect of her life. Further, however, it appears she wanted to prove, by example, that sexuality is an important part of life, and it can be discussed matter-of-factly - without trivializing it - just as one can discuss other fine things like enjoying good food or raising children. Some people may find the account dull and boring - but they were probably expecting erotica, not a serious account of one person's sexuality. Her concluding words: Surely the circulation of this book and of conversation around it mean we can envision, within the realm of possibility, a realization of this easing of human relations, an easing facilitated by an acceptance and tolerance of sexual desire, and which some passages in my book represent in a clearly utopic, and fantastical way. And surely we should take pleasure and rejoice in this vision. Millet's book was first published (in French) in 2001. I think it's a good thing that since then (thougn not necessarily because of this book) other women - as well as men - have found it increasingly easy to write publicly about their sexuality and sexual experiences. And this isn't just women as "liberated" as Catherine M., but people in the whole spectrum from herself to people either unpaired or in traditional, monogamous relationships. Sadly, many people obviously have very negative opinions about the author and her book. I feel sorry for them. Note: All quotations here are from the English translation. ...more |
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Feb 01, 2018
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| 3319255665
| 9783319255668
| 3319255665
| 4.21
| 107
| Feb 22, 2016
| Apr 12, 2016
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really liked it
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Very few top mathematicians have written autobiographies or personal memoirs. Other than the work under review, the names that come readily to mind in
Very few top mathematicians have written autobiographies or personal memoirs. Other than the work under review, the names that come readily to mind include Paul Halmos (I Want to Be a Mathematician), Laurent Schwartz (A Mathematician Grappling with His Century), Stanislaw Ulam (Adventures of a Mathematician), André Weil (The Apprenticeship of a Mathematician), and Norbert Wiener (Ex-Prodigy, I Am a Mathematician). In all these cases the individuals were near or past retirement age when their book appeared (69, 86, 67, 85, and 59, respectively). Ken Ono, however, was only 48 and still in the midst of a very successful career. Other recent personal memoirs by younger mathematicians – Edward Frenkel (Love and Math; my review), Michael Harris (Mathematics without Apologies; my review), Cédric Villani (Birth of a Theorem; my review) – have appeared. But these books deal largely or mostly with mathematics itself, and the personal details, while very interesting, don't seem to be the main point. Ken Ono's book is different. It is concerned mostly with details of his personal life, and somewhat with the life of his mathematical idol, Srinivasa Ramanujan. The latter's life has been celebrated in a number of books (and a major motion picture). Ono's life, of course, has not, though it has a variety of poignant details. These details should be of great interest to young mathematicians and mathematical students, so the book should be especially recommended for them, as that audience will probably garner much encouragement towards the achievement of a successful mathematical career. In fact, any young person struggling for success in almost any difficult effort should also derive considerable encouragement from Ono's story. The details of Ramanujan's life were similar in a number of respects, which is one reason (but hardly the only one) that Ono was so inspired by Ramanujan's story. I'm not going to summarize Ono's story here. If the foregoing description is intriguing to you, by all means read the book. It's fairly short. Instead I'll register a couple of disappointments with the book, which account for giving it only 4 stars. The main disappointment is that the book doesn't go into more detail on the mathematical work of both Ramanujan and Ono, in which there are many things in common. This commonality is no accident, since Ono made extending and clarifying Ramanujan's amazing mathematical insights his principal professional objective. And that's why the omission of more mathematical detail is so unfortunate. Ramanujan's main work consisted of an astonishing variety of explicit but often abstruse mathematical formulas and numerical examples. Even today, many of these are not well understood. However, many of the formulas involve only "simple" algebraic relations, without any calculus or more advanced concepts. These should be intelligible to most readers who did OK in high school math classes, even if their real import is more obscure. For instance, one interest that Ramanujan and Ono have in common is "partitions" of integers – a very elementary concept, which is simply the number of different ways a positive integer can be written as a sum of smaller positive integers. Ramanujan recorded only a few simple examples and was otherwise tantalizingly vague. In spite of the quite elementary nature of the question, answers are surprizingly difficult to come by. Ono (and several collaborators) have gone much, much farther. Although the methods are difficult, it would have been most welcome if the general outline could at least have been sketched. It turns out that the underlying explanation for many of the strange results Ramanujan recorded involve mathematical functions ("modular forms") that defy clear description for general readers, but which codify certain symmetry relationships possessed by relatively simple geometric objects (such as a 2-dimensional plane). Although Ramanujan wrote nothing explicitly about modular forms, he apparently had a vague intuition of their importance. Today modular forms have a very central place in contemporary mathematical research. They play an absolutely crucial role, for instance, in Wiles' proof of Fermat's Last Theorem. Ono is an acknowledged expert in this field. Consequently, one wishes he had devoted perhaps 10 or 20 pages to at least attempting to give general readers at least some general idea of what modular forms are about. It shouldn't have been that hard, and could have been relegated to an appendix. One other minor criticism is that the book really needs an index. There are so many individuals, events, and concepts mentioned throughout the book that one really wants the ability to locate all references. To conclude, the following quote from Ono perfectly captures what it means to really "do mathematics": Doing mathematics is a mental voyage in which clarity of thought and openness to insight make it possible to see the deeper beauty of a mathematical structure, to enter a world where triumph over a problem depends less on conscious effort than on confidence, creativity, determination, and intellectual rigor....more |
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Sep 18, 2017
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| 0387903488
| 9780387903484
| 0387903488
| 3.91
| 70
| 1895
| Dec 19, 1978
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really liked it
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Sofya Kovalevskaya is arguably the most important female mathematician of the nineteenth century. Unfortunately, there wasn't a lot of competition. An
Sofya Kovalevskaya is arguably the most important female mathematician of the nineteenth century. Unfortunately, there wasn't a lot of competition. And, sadly, Sofya died at the age of 41 of influenza. The common belief that mathematicians seldom do important work after the age of 40 isn't really true - especially with respect to really creative mathematicians. (Paul Erdős was going strong almost up until his death at 83, Karl Weierstrass, Sofya's most important mentor, was still teaching at a junior high school when he was 40; most of his seminal work in mathematical analysis was done after that.) So there's no telling what Kovalevskya might have done if she hadn't died so young. Sofya's memoir of her childhood begins with her earliest memories and concludes with a chapter on the friendship that developed between herself and her older sister Anna with Fyodor Dostoevsky. Sofya was only 15 at that time (1865), and Anna was 23. Dostoevsky seems to have had a romantic interest in Anna, but the younger sister was present at many of the encounters with Dostoevsky. The latter was about 44 at the time, had been a published author for almost 20 years, had just divorced his first wife, and was on the verge of publishing novels for which he is now best known. Obviously, to have come to the attention of a man like Dostoevsky, Sofya and her sister weren't from a family of "ordinary" people. Their father was a retired general of the Russian army and the owner of an impressive country estate. So the childhood of both Sofya and Anna was hardly a typical one. However, the estate was isolated and remote from important Russian cities like Petersburg. Although the children were supervised by several governesses and tutors, it doesn't seem, based on Sofya's memoir, that they were especially "spoiled" (except by comparison with children in much more impoverished circumstances). The basic details of Sofya's life are laid out in a 40-page introduction by the translator, Beatrice Stillman. Sofya herself has nothing to say in the memoir about her early interest in mathematics, let alone the details of her later accomplishments. The introduction doesn't really say much about the mathematics either. We do learn that "At thirteen Sofya began to exhibit an aptitude and avidity for algebra." Since access to higher education was completely unavailable to women in 1870's Russia (or most other countries), Sofya's burgeoning interest in advanced math was first noticed when she was in Heidelberg with Anna. After "enormous effort" Sofya managed to gain permission to attend lectures (but certainly not to enroll as a regular student). Yet it was enough that Sofya's mathematical abilities quickly came to the attention of her teachers. According to Stillman, "Sofya had come to a momentous decision for herself: that her true vocation was mathematics and that there was one mathematician in the entire world she wanted to study with - Professor Karl Weierstrass, of the University of Berlin." Sofya certainly wasn't daunted by eminent men - Weierstrass has a position in the history of mathematics comparable to that of Dostoevsky in the history of literature. Weierstrass did take her under his wing, and wasted little time ensuring that she received the mathematical education she deserved. For readers interested primarily in mathematics, it must be understood that Kovalevskaya's memoir is entirely about her childhood, up to the age of 15 - and only about scattered incidents at that. Don't pick it up expecting to learn much about mathematical prodigies. Even so, it has interesting and charming stories. There is in the present volume, quite separate from the memoir, a 15-page "Autobiographical Sketch" that Sofya wrote in 1890. There are some nice tidbits in there, such as "In the field of mathematics in general, it is mostly by reading the works of other scholars that one comes upon ideas for one's independent research." There is, also, a 20-page appendix "On the Scientific Work of Sofya Kovalevsky" by a (modern) Russian mathematician. Its focus is, first, on the "Cauchy-Kovalevsky Theorem", which deals with partial differential equations. Sofya's far-reaching generalization of Cauchy's work was presented by Weierstrass in 1874 as Sofya's PhD thesis. Secondly, Sofya's comprehensive solution of a problem concerning "the motion of a heavy rigid body near a fixed point is described. This is an important result in classical mechanics. Anybody seriously interested in the history of mathematics should find the present volume a very worthwhile read. ...more |
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Jun 13, 2017
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Jun 19, 2017
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| 0801885876
| 9780801885877
| 0801885876
| 2.91
| 33
| May 18, 2007
| Jul 16, 2007
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really liked it
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The book is a welcome attempt to use insights from psychology and related fields - together with biographical examples - to explain how the minds of o
The book is a welcome attempt to use insights from psychology and related fields - together with biographical examples - to explain how the minds of outstanding mathematicians work in order to come up with important mathematical breakthroughs. The first author, Michael Fitzgerald, is a psychoanalyst and professor of psychiatry. The second author, Ioan James, is a mathematician who's been an important contributor in the fields of geometry and topology. There's a lot of good information in their book, but it still falls somewhat short of illuminating the central questions. Here are three of the key questions. (1) How did the minds of exceptional mathematicians like Gauss, Poincaré, and Hilbert function in order to produce their extraordinary results? (2) Were there specific mental methods, techniques, habits, or practices these people used? (3) Are there specific and identifiable positive or negative psychological traits or biographical details that these historical masters have in common? The book offers some answers to each of these questions. In a scant 160 pages the authors don't seriously attempt to provide new or better answers beyond what has been discussed among mathematicians for hundreds of years, without a lot of definitive conclusions. But the book does provide a decent survey of some of the proposed answers. The first part of the book, which is not quite half by page count and may have been written mostly by Fitzgerald, is a "tour of the literature" that deals with three topics: (1) the nature of mathematics as a discipline and the milieu in which research mathematicians operate; (2) the nature of "mathematical ability" and the specific skills it comprises; (3) the "dynamics of mathematical creation" - how creativity in mathematics has both similarities and differences with creativity in other pursuits, such as art, music, and literature. Numerous entire books have been written on each of these topics. The discussion in this book occupies all of 60 pages, so it's necessarily a very compressed and selective summary. The second part of the book, which was probably written mostly by James, offers very brief biographical summaries of 20 historically outstanding mathematicians. That works out to an average of about 5 pages per person. The subjects are highly exceptional individuals who worked mostly between 1750 and 1950 and whose lives, for the most part, were far more varied and eventful than average. So the material presented on each can hardly scratch the surface of personal lives that are more unique than 99% of the population might imagine. Not only that, but readers interested in mathematics -likely to be the vast majority of the book's audience - will find almost no details of the most noteworthy contributions of each person described. (Ioan James a few years earlier authored another book (Remarkable Mathematicians: From Euler to Von Neumann) that profiles 60 outstanding mathematicians from roughly the same time period. That's 7 pages per person, so it's almost equally sketchy. In both books the descriptions, despite their brevity, are mostly interesting, lively, colorful, and well-written. But they're probably not too helpful for deriving useful general conclusions - especially since little reliable biographical information is available for most of the subjects who worked in the first half of the time period. Of the 20 mathematicians profiled in the book reviewed here, all but 4 are also in the second book. (The exceptions are Ada (Byron) Lovelace, R. A Fisher, Paul Dirac, and Kurt Gödel. What's common to these 4 is having contributed somewhat less to pure mathematics despite outstanding contributions in somewhat more peripheral fields.) The slightly longer profiles in the second book have more mathematical details.) So, in spite of the brevity of the book under review, are there interesting general conclusions that can be drawn? Yes, of course. Firstly, almost all the individuals profiled are extremely unusual and atypical of the general population. But this is to be expected because of the selection bias inherent in dealing with people who've made contributions of historic proportions to the difficult, abstruse field of mathematics. Most contemporary professional mathematicians have certain peculiarities too, but hardly to the same extent. Unsurprisingly, almost all the profiled mathematicians seem to possess exceptionally high general intelligence. This, again, is to be expected from the selection bias, even though the high intelligence is not simply in the mathematical sphere. Many of the individuals also had exceptional memories and ability to concentrate. Many were "geniuses" or "prodigies", in that they were recognized as unusually intelligent at a young age. Many entered college (or equivalent) when unusually young, and entered a professional mathematical career also quite young. Other indications of high general intelligence were things like mastery of a number of foreign languages and noteworthy talent in non-mathematical areas, such as teaching, music, or other scientific fields. (Some also completely lacked such talents - especially teaching.) However, few individuals also had success in certain other fields, such as law, politics, business, or philosophy. This is understandable, since notable success generally requires devotion of a considerable portion of one's time, which would then be unavailable for mathematics. In earlier eras, people like Descartes, Fermat, Pascal, Newton, and Leibniz had great accomplishment in fields outside of mathematics. But increasing specialization is certainly the historical trend. In a few cases, some of the profiled mathematicians had only mediocre achievements, or even disastrous failures, in other aspects of their lives. Galois couldn't stay out of trouble as a political radical, and managed to get himself killed in a duel (possibly more of a suicide?) before his 21st birthday. Ramanujan had difficulty finding employment in India and could hardly cope with life in England. Both Cantor and Gödel had distinct episodes of mental illness that left them unable to do mathematics for long periods of time. Indeed, most of the individuals profiled had significant difficulties or abnormalities in dealing with other people. Skillfulness in handling normal human interactions is generally not something that outstanding mathematicians are known for, though there are exceptions to this too. Cauchy was known for arrogance and religious zealotry, Gauss for aloofness, Hardy for evidence of insecurities, Riemann for shyness and difficulties relating to people, Wiener for strange behavior, and Dirac for general strangeness (The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom). On the other hand, a few were quite socially adept, such as Jacques Hadamard and Emmy Noether. There seem to be two types of psychological dysfunction that are often present, to some degree or other, in the examples presented. Fitzgerald, as a psychiatrist, evidently took special notice of these. One is cyclothymia (mild bipolar disorder), which involves mood swings between depression and mania. It's impossible to figure out from the examples presented whether this is more prevalent or less among outstanding mathematicians compared with the general public. Determining that requires an extensive study of living examples, and the sample size of top mathematicians is likely to be rather small. Additionally, it isn't clear whether or not phases of either depression or mania could actually be helpful or harmful to mathematical productivity. The other dysfunction that may be relevant is the now famous Asperger syndrome. There are a number of different diagnostic indicators of AS, and in most individual cases not all will be present. Most of the individuals considered in the book have at least some of the symptoms. But it's quite hard to say whether specific individuals "really" have AS, especially without a clinical evaluation. The lack of much first-hand evidence for most of the earlier mathematicians makes the determination essentially impossible. AS disorder in a person generally manifests as difficulty in social interaction with others. That seemingly should be detrimental to outstanding mathematical achievement - and yet it seems to be rather common in the individuals profiled. Interestingly, 20th century examples (Hardy, Ramanujan, Dirac) seem to be especially rich in symptoms. The book's co-author Fitzgerald has argued (in another work) to include Gödel too. Indeed, he argues elsewhere for a significant connection between AS and creativity. The other author, James, seems to agree in another book of his own. Final conclusion? It may be impossible to find enough evidence regarding earlier mathematicians. But the more recent examples, based on what's in this book, do suggest that some degree of psychological dysfunction goes along with high achievement. ...more |
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May 07, 2017
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May 07, 2017
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| 0684192594
| 9780684192598
| 0684192594
| 4.02
| 8,574
| 1991
| Jan 01, 1991
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it was amazing
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Dec 20, 2002
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Oct 29, 2016
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| 0312426941
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| 0312426941
| 4.24
| 1,344
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| Jun 12, 2007
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it was amazing
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May 30, 2013
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Oct 23, 2016
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| 0786884061
| 9780786884063
| 0786884061
| 4.05
| 9,763
| 1998
| Jan 01, 1998
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really liked it
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When Paul Erdős died in 1996 Paul Hoffman had known him for about 10 years and interviewed him a number of times. Hoffman had also known and interview
When Paul Erdős died in 1996 Paul Hoffman had known him for about 10 years and interviewed him a number of times. Hoffman had also known and interviewed many of Erdős' friends and associates. So it's fair to say that Hoffman had a lot more knowledge of the subject of his biography than most biographers (unless they're family or close friends) ever do. The biography is indeed quite good, and provides a clear and informative portrait of the very unique, appealing, and colorful individual that Erdős was. However, what defined Paul Erdős even more than his idiosyncrasies and his humanity was that he was quintessentially a mathematician – a singularly talented and prolific one. Another biography of Erdős, by PhD physicist Bruce Schechter (My Brain is Open: The Mathematical Journeys of Paul Erdos), was published at almost the same time as Hoffman's. (I have reviewed that as well, and I'll try not to duplicate much from that other review.) Hoffman freely concedes that he is not a mathematician. While Schechter isn't a professional mathematician, he is more able to give a complete and accurate account of Erdős' work than Hoffman. The picture of Erdős that comes across in both biographies is much the same, since both books are based on a lot of the same source material. Hoffman's book has the advantage that the source of all Erdős quotes (except those Hoffman got directly) are documented in footnotes. But because Hoffman also can report many anecdotes from his interviews with Erdős himself and with his associates, this biography offers a somewhat clearer picture of the subject as a person, apart from his work. One anecdote is especially revealing. On an occasion in the late 1960s Erdős (who was about 55 at the time) was staying with an old friend from Hungary in southern California, very near the beach. One day Erdős went out to walk on an esplanade above the beach. Only ten minutes later his hosts received a phone call, from someone who lived close to the esplanade about five blocks away, to report that Erdős turned up on their doorstep saying he was lost and needed help finding his way back to his friend's place. So Erdős, in spite of his prodigious memory for details of his mathematics and his mathematical collaborators, couldn't even remember how to retrace his steps. The natural explanation is that he was so absorbed in his mathematical thoughts that no vestige of his short walk had registered in his memory. As other anecdotes made clear, Erdős was fully capable of recalling details of mathematical conversations he'd engaged in years before, and he could also keep track of more or less simultaneous conversations he carried on with several different mathematicians at the same meeting. He could also recall details of perhaps thousands of technical papers he'd read decades before. It seems reasonable to conclude that his ability to concentrate and to recall mathematical detail had a great deal to do with his singular power as a mathematician – as exemplified by his ability either to solve quickly new mathematical problems or at least to judge accurately their level of difficulty almost effortlessly. In contrast to this clear portrait of Erdős as a person, Hoffman's lack of mathematical background means he must rely on the testimony of others he interviewed to describe Erdős' mathematics. The result is a somewhat less satisfactory account. One example is what Hoffman calls "friendly numbers". He says this means a pair of numbers (a and b) where the sum of proper divisors of a is equal to b and the sum of proper divisors of b is equal to a. This is actually the definition of "amicable numbers". The example given is the pair 220 and 284, which was known to Pythagoras. Those are indeed "amicable numbers". The accepted definition of "friendly numbers" is numbers that have the same ratio between themselves and their own sum of divisors. By this definition, 220 and 284 aren't "friendly". An even more serious problem is the discussion of Bernhard Riemann's non-Euclidean geometry. Hoffman writes "He [Riemann] builds a seemingly ridiculous assumption that it's not possible to draw two lines parallel to each other. His non-Euclidean geometry replaces Euclid's plane with a bizarre abstraction called curved space." It's not actually bizarre at all, since the surface of any sphere is one example. Straight lines on the surface of a sphere are "great circles" (which are by definition the largest circles that can be drawn on a sphere, like the Earth's equator). Great circles are never "parallel", since they always intersect. Apart from these and a few other mathematical glitches, Hoffman's book is almost free of trivial proofreading errors. But there's one glaring exception (at least in the paperback edition). Sixteen pages of pictures are included in the middle of the book, and given appropriate page numbers. However, the footnotes at the end of the book are all keyed to page numbers, and they haven't been corrected to account for the picture pages, so that all references to pages after the pictures are off by 16 – a very annoying problem if one wants to actually check these references. In spite of these problems, Hoffman's book provides a fine portrait, based on personal experiences, of Erdős the man. ...more |
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| 0684859807
| 4.21
| 602
| Sep 01, 1998
| Feb 28, 2000
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it was amazing
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It's probably fair to say that a large majority of the general public in the U. S. could not name a single important mathematician who was active in t
It's probably fair to say that a large majority of the general public in the U. S. could not name a single important mathematician who was active in the period 1933 to 1996 (the years of Paul Erdős' adult life). A few people might think of Andrew Wiles, John von Neumann, or perhaps Kurt Gödel. It would be surprising if more than a percent or two could mention Paul Erdős or even recognize the name. Yet Erdős is probably one of the dozen or so most important mathematicians of that period. For instance, nobody in the history of the world published more new mathematical results (almost 1500), except for Leonhard Euler (who died in 1783). Erdős created himself or was among the creators of a number of new mathematical specialties, including combinatorics, Ramsey theory, probabalistic number theory, and combinatorial geometry. He also made ground-breaking advances in the early stages of set theory and graph theory. And he added very substantially to classical number theory (one of Euler's specialties). In addition to his astonishing productivity, he remained active and prolific until he died at the age of 83 – something almost unheard of among mathematicians. Nevertheless he had little or no interest in many other branches of modern mathematics, such as topology, abstract algebra, or mathematical physics. Schechter's book is very good at explaining in general terms what each of those mathematical topics are about and the contribution made by Erdős. Unlike most modern mathematics, the things that interested Erdős the most were "simple" things, like numbers and geometrical figures that are familiar to most people. So it's easy enough to understand most of what Erdős worked on. It would be nearly impossible, however, to explain to non-mathematicians the techniques by which the results were obtained. (After all, his results had eluded all earlier mathematicians.) And even mathematicians are mystified by the mental processes that led to the results. Erdős was quite an unusual person in other respects as well. He never married or had any apparent sexual interests. During most of his adult life he had no permanent home and almost no possessions other than a couple of suitcases, some notebooks, and a few changes of clothes. He was almost perpetually on the go from one place to another after his welcome at one host's abode started to wear thin. And his memory was phenomenal – though only for details that were important to him, namely anything relevant to mathematics he cared about and his vast network of mathematical collaborators. He co-authored papers with almost 500 other mathematicians, and personally discussed mathematics with hundreds of others. He generally knew the phone numbers and other personal details of most in his network. (Yet he was often unable to associate their names and details with their faces when he encountered them at meetings.) He was also able to recall technical details and publication information of thousands of mathematical papers, which may have been published decades previously. This fact probably helps explain his ability in many cases to solve new problems within minutes, because he could recall such a vast number of earlier results and techniques. Schechter's biography is generally quite good. The author has a PhD in physics and is obviously conversant with the mathematics that interested his subject. There are just a few minor issues. Although Schechter never actually met Erdős, many personal anecdotes are reported. Those must have come from conversations with associates of Erdős or articles about him, but there's only a two-page "Note on Sources" instead of footnotes with specific details. There is a good bibliography, however. The book is only about 200 pages, so it's a quick read. However, if it had been a little longer, it could have gone into somewhat more detail about Erdős' mathematics – such as set theory and probability theory. Another Erdős biography (The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth) was published about the same time, less than 2 years after the subject's death. (I've reviewed that one too.) So both biographers most likely wanted to get published as quickly as possible. There are clear signs of haste in both cases. In the present book there are few things that a proofreading should have caught, but only one more serious error: in Chapter 4 there are incorrect references to two of the graph diagrams. All in all, this biography can be highly recommended for an overview of Erdős' mathematics, a fine portrait of a very unique and colorful individual, and the opportunity to gain a little understanding of the social process in which mathematics of the highest caliber is actually created. ...more |
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| 4.09
| 85
| Jul 28, 1989
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it was amazing
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Many people probably think that a famous scientist's life is rather unlikely to be as interesting and exciting as, say, that of a famous artist or cel
Many people probably think that a famous scientist's life is rather unlikely to be as interesting and exciting as, say, that of a famous artist or celebrity. They would be wrong, at least about Erwin Schrödinger. Of course, there are a few other notable exceptions, such as Albert Einstein and Richard Feynman. However, the fame of these two is, at least in small part, based on their personal lives as well as their science. Einstein, for instance, is known for many famous quotations on all sorts of topics, as well as a turbulent personal life (which included two failed marriages, one of which was with a first cousin). Schrödinger's personal life was, if anything, rather more colorful and turbulent than Einstein's, These two men, who were personal friends for many years, had much more in common, as well. Though Schrödinger fell slightly short of Einstein's - and also Feynman's - eminence as a physicist, all three men certainly occupied similar positions at the top of their field - all won Nobel prizes in physics. Both Einstein and Schrödinger were born and lived many years in German-speaking countries - Germany and Austria, respectively. The lives of both were severely disrupted by the two World Wars - Schrödinger's far more than Einstein's. Even though Schrödinger made what was probably the most important contribution of anyone to the new science of quantum mechanics (the Schrödinger equation), both were in a small but prominent minority of physicists regarding how the theory should be interpreted. And both spent much of their last years in futile pursuit of a "unified field theory" based on the same mathematics (Einstein's general relativity). Both also thought deeply about philosophical questions that ranged far beyond physics and science, and they both distanced themselves from conventional monotheistic religions - quite strongly in Schrödinger's case. Walter Moore's biography of Schrödinger, first published in 1989, is the definitive work, and probably among the best biographies of a scientist ever. Although it doesn't give a general technical presentation of quantum mechanics, there is much technical material. A reader who doesn't know a lot of both mathematics and physics shouldn't expect to follow the technical discussions. Simply having read popular, nonmathematical accounts of quantum mechanics won't suffice for the task. Even the underlying philosophical nuances will not be well appreciated. Fortunately, however, it's not at all necessary to understand the science in order to enjoy the rest of the biography. (Moore has also published an abridged version of the book (A Life of Erwin Schrödinger), which nonscientific readers will probably prefer, that leaves out most of the technical science.) In addition to the extensive personal information on Schrödinger, there's much about his philosophical beliefs. These have little direct relevance to his scientific interests and in fact he disclaimed a connection. But the philosophy is itself somewhat esoteric and abstruse. Why would one expect any less from a person of Schrödinger's intellect? Or the likes of Einstein, Feynman, etc., for that matter? Scientifically, Schrödinger was something of a late-bloomer. He was just 8 years younger than Einstein. But he was 14 years older than Werner Heisenberg, the other principal contributor to "modern" quantum mechanics. Heisenberg and Schrödinger published their independent original discoveries almost simultaneously, in 1925-6. Schrödinger was close to 40 at the time. Their mathematical approaches were quite different, but quickly found to be equivalent. However, Schrödinger's approach has proven to be the most useful for practical computation. Schrödinger received his graduate degree in physics in 1910 from the University of Vienna, having been a top student from his earliest years. But in the following 15 or so years he drifted through various scientific pursuits -including statistical mechanics, meteorology, and the physics of color vision - without publishing anything of really "breakthrough" quality. Consequently, he was unable to obtain a satisfactory faculty position at a first-rate university - even after receiving a Nobel Prize for his physics work. Therefore he and his first (and only official) wife drifted from place to place, including Vienna, Jena, Zürich, Breslau (Poland), and Berlin. On top of that, he served in the Austrian military during W.W. I - far from the worst fighting, fortunately. All the moving around, of course, made for a rather unsettled life. From 1927 to 1934 Schrödinger did have a good faculty position in Berlin. However, although he was not Jewish, he was quite upset by Hitler's rise in Germany. While that's hardly surprising, in fact Schrödinger despised politics in all forms. That, too, is quite understandable, but he was also dangerously naive about politics. As a result he was quite outspoken about his distaste for German politics, which eventually made him persona non grata to the Nazis. The more immediate result of his political feelings was that he chose to leave Berlin for Graz (Austria), yet another disruptive move. His political naïveté soon put him in an even more perilous position after the Anschluss in Austria in 1938, which his disdain for politics had prevented him from foreseeing. He, at the age of 51, and his wife were forced to flee, with neither money nor possessions, winding up eventually in Dublin. From 1938 to 1956 Schrödinger and his wife lived comparatively peacefully in Dublin. Although he found some aspects of Irish life agreeable, the Irish climate was not among them, and besides Ireland was just too unlike his native Austria, so he was finally able to return for the last 5 years of his life. But he and his wife were both then in precarious health, so he was unable to fully enjoy some of what he liked most about Austria, such as the spectacular scenery and the opportunity for hiking and skiing. Because of the recurrent need to relocate and his age when he made his first great scientific achievement, Schrödinger (unlike Einstein) never surpassed the first achievement. Yet in spite of the turmoil in his life, he managed to cope pretty well. This is because of an aspect of his life not yet mentioned. In the words of his biographer "He believed that everything beautiful in life and art is a consequent of sex." And he lived that belief to the fullest, in a way that would make almost any prominent celebrity envious. Over the course of his adult life until his early 60s Schrödinger engaged in a continual series of sexual relationships with women other than his wife of 41 years Anny. It seems that both Erwin (primarily) and Anny (to some extent) were (or became) believers in the ideas now called "open marriage" and "polyamory". The most important of these relationships were not at all secret. Close friends of both the Schrödingers were quite aware of the goings-on. In fact, this was, apparently, not considered particularly scandalous in the intellectual circles of society to which Erwin and Anny belonged. (According to Moore, "Extramarital affairs were not only condoned, they were expected, and they seemed to occasion little jealous anxiety.") One of Anny's boyfriends was the eminent mathematician and good friend of Erwin, Hermann Weyl, whose wife, in turn, had a lover of her own. These were not brief affairs, either. In some cases, the relationships lasted many years. It's not entirely clear how Anny felt about all this. However, she and Erwin continued to live together except for brief periods until his death, and she took affectionate care of him, as much as her health permitted, in his last years. We know a lot of the facts, in part, because those involved were so open about things with their friends and even casual acquaintances. Erwin often went on long trips with his girlfriends, with Anny's full knowledge and (apparent) approval. Erwin fathered (at least, as far as known) only three children during his life, all girls. None of these were with Anny, though she was fully aware of them and even sometimes became the child's primary caregiver. In one case the mother was actually the wife of a very close friend of Erwin's, and she even lived for years in Dublin with Erwin and Anny some of the time the child was growing up. As though such an arrangement were perfectly normal. Another reason we know a lot about Erwin's amorous adventures is that he kept detailed journals of what was happening in his life. A lot of his sex life was recorded in the journals. In addition to the long-term relationships, other liaisons - one-night stands or filles de joie - were noted, usually without the partner being named. However, according to Moore, who studied the journals, "Erwin was not, or not usually, a libertine. He speaks the authentic language of romantic love, seeking transcendence in the person of his beloved." As already mentioned, Schrödinger also had deep philosophical beliefs. When he was 38, just before his groundbreaking results in quantum mechanics, he wrote, but did not publish, an account of his philosophy of life. He firmly rejected the worldview of Western monotheistic religion, being greatly influenced by the works of Arthur Schopenhauer, all of which he'd read. Consequently he warmly embraced Eastern philosophy and religion, especially Vedanta. At the very end of his life, as described in his book, My View of the World, his beliefs had certainly evolved and deepened, but not substantially changed. In a nutshell, this worldview rejected any real distinction between "self" and "the real world". Instead there is just the "oneness" of everything - physical reality as well as individual selves. In this, Schrödinger shared not only a sexual lifestyle resembling that of 1960s hippies, but also their kind of mysticism and "spirituality". Of course, both the lifestyle and the beliefs of the hippies existed in the West long before the 1960s. Certainly before Schrödinger too. Interestingly, although various new-agey physicists in the last few decades have proclaimed that quantum mechanics embodies and justifies Eastern philosophical ideas, Schrödinger himself denied any influence either way between quantum mechanics and Eastern mysticism. Moore's excellent biography of Schrödinger lays out in great detail much of the expectable complexity of its highly intelligent subject: The brilliance of his mind and the naïveté of his politics. The rigor of his science and the other-worldliness of his mysticism. His suitably honored intellectual accomplishments and the unorthodoxy of his lifestyle. ...more |
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| 0552777595
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| 0552777595
| 3.93
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| Mar 01, 2012
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it was amazing
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I'm giving this a high rating because I think it deserves it for what the book seems intended to deliver: not so much a full biography of Schrödinger
I'm giving this a high rating because I think it deserves it for what the book seems intended to deliver: not so much a full biography of Schrödinger as a quick history of the "quantum revolution" that's reasonably up-to-date as of 2010 or so and the part Schrödinger played in it. If you don't care about the science (and shame on you) or instead were looking for a more detailed treatment of either the science or Schrödinger, you won't be fully satisfied. There's no question that the book is a birds-eye view of its topics. But the value of that is you can learn about the broader context of QM and the personalities of the people who developed it. That's very helpful, because it's quite a vast subject. If you happen to find it interesting and want to go further, there are lots of ways to proceed, such as several online courses in QM, introductory textbooks, or more detailed books for general readers who like science. Likewise, there are more detailed biographies of Schrödinger and other quantum pioneers. What you'll learn from this book, if you know little about Schrödinger, is that he had a fairly colorful life (unlike many scientists), and that you'd probably like to learn a lot more about it. (E. g. how his life was "entangled" with that of Hermann Weyl, a mathematician whose total contributions to both physics and mathematics outshone even Schrödinger's. That's something I'm sorry Gribbin didn't say more about.) You'll learn that he also had various quite human flaws (like everyone else). And you'll also learn that the whole "Schrödinger's cat" business was intended as a mockery of the "Copenhagen interpretation" of QM, and not a serious intellectual argument. ...more |
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