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Mathematical
QuotesFrom Frank Wikstrom's Mathematical Quotes:
Mathematicians are like Frenchmen: whatever you say to them they translate
into their own language and forthwith it is something entirely different.
-- Johann Wolfgang von Goethe (Maxims and Reflexions, 1829)
Mathematicians are a species of Frenchmen: if you say something to them they translate it into their own language and presto! it is something entirely different -- Goethe [Source of quote unknown, quoted in Chapter 3 of Pi in the Sky by John D. Barrow]
From Raven's Memorable Quotes from Alt.Sysadmin.Recovery:
... More quotes from alt.sysadmin.recovery in Microsoft Forlorn and Computing Quotes
Q: I'd like some kind of gift suggestion for the "guy who has it all". Any
ideas?
A: Give him the set of guys who don't have it all.
-- Exchange on
Forum 3000
"I think Whitehead and Russell probably win the prize for the most
notation-intensive non-machine-generated piece of work that's ever been done.
"
-- Stephen Wolfram,
www.stephenwolfram.com/publications/talks/mathml/mathml2.html
From Robert Kaplan, The Nothing That Is - A Natural History of Zero:
Quotes from The Penguin Book of Curious and Interesting Mathematics by David Wells:
"In your otherwise beautiful poem, there is a verse which reads:
'Every moment dies a man,
Every moment one is born.'
"It must be manifest that, were this true, the population of the world would be at a standstill: In truth the rate of birth is slightly in excess of that of death. I would suggest that in the next edition of your poem you have it read:
'Every moment dies a man,
Every moment 1 1/6 is born.'
"Strictly speaking this is not correct. The actual figure is a decimal so long that I cannot get it in the line, bit I believe 1 1/6 will be sufficiently accurate for poetry. I am, etc."
-- from Clifton Fadiman, Fantasia Mathematica, Simon & Schuster, New Yor, 1958, p.293
All mathematicians admire the great geometer Bolyai, whose eccentricities were of an insane character; thus he provoked thirteen officials to duels and fought them, and between each duel he played the violin, the only piece of furniture in his house [...]
-- Cesare Lombroso, The Man of Genius, Walter Scott, 1891, p. 73.
From The Times [London] Wednesday 3 January 2001 in an article titled Pupils sum up maths teachers as fat nerds by Simon de Bruxelles [http://www.thetimes.co.uk/article/0,,2-61352,00.html]:
From Alice's Adventures in Wonderland by Lewis Carroll:
From Simon Singh, Fermat's Enigma:
While Shimura was fastidious, Taniyama was sloppy to the point of laziness. Surprisingly this was a trait that Shimura admired: "He was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this and tried in vain to imitate him, but found it quite difficult to make good mistakes."
From Charles Seife, Zero - The Biography of a Dangerous Idea:
From John D. Barrow, The Book of Nothing:
From Eli Maor, Trigonometric Delights, Chapter 5, Measuring Heaven and Earth:
From Morris Kline, Mathematics - The Loss of Certainty:
From David Ruelle, Chance and Chaos:
From Wilfrid Hodges, An Editor Recalls Some Hopeless Papers, The Bulletic on Symbolic Logic, Vol 4, No 1, March 1998 [PostScript version available here]:
From The Book of Imaginary Beings by Jorge Luis Borges:
In Alexandria over five hundred years later, Origen, one of the Fathers of the Church, taught that the blessed would come back to life in the form of spheres and would enter rolling into heaven.
[p. 21-22]
From The Number Devil, by Hans Magnus Enzensberger:
"I don't know what's got into the boy lately," said Robert's mother, shaking her head. "Here," she added, handing him a cup of hot chocolate, "maybe this will help. You say the oddest things."
Robert drank his hot chocolate in silence. There are some things you can't tell your mother, he thought.
[p. 46 -- The Second Night]
From At Home In The Universe - The Search for Laws of Complexity by Stuart Kauffman (Penguin Books, 1995):
From The Artful Universe by John D. Barrow (Oxford University Press, 1995):
Newsgroups: sci.math Subject: Re: I'm looking for axioms and proof in math texts Date: 6 Aug 2003 02:53:12 -0700 From: euclid@softcom.net (prometheus666) [...] and being passing familiar with the number line -- I'm sure if I don't say passing familiar somebody here will say, "You have to know vector tensor shmelaculus in 15 triad synergies to really understand the number line. It's not even called that, it's called the real torticular space." or something to that effect -- [...]
From Pi in the Sky by John Barrow (Oxford University Press, 1992):
[Contrast this with a quote by Steven Weinberg in his book Dreams of a Final Theory, p. 259: "[...] all logical arguments can be defeated by the simple refusal to reason logically"]
[Footnote: One Hungarian physicist once remarked in the course of writing a textbook that, although he would often be referring to the motions and collisions of billiard balls to illustrate the laws of mechanics, he had neither seen nor played this game and his knowledge of it was derived entirely from the study of physics books.]
Near the turn of the [twentieth] century this practice provoked the French mathematician, Henri Poincaré, to criticize it as a very 'English' approach to the study of Nature, typified by the work of Lord Kelvin and his collaborators, who were never content with an abstract mathematical theory of the world but instead sought always to reduce that mathematical abstraction to a picture involving simple mechanical concepts with which they had an intuitive familiarity. Kelvin's preference for simple mechanical pictures of what the equations were saying, in terms of rolling wheels, strings, and pulleys, has come to characterize the popularization of science in the English language. But if we look more closely at what scientists do, it is possible to see their descriptions as the search for analogies that differ from those used as a popularising device only by the degree of sophistication and precision with which they can be endowed. Just as our picture of elementary particles of matter as little billiard balls, or atoms as mini solar systems, breaks down if pushed far enough, so our more sophisticated scientific description in terms of particles, fields or strings may well break down as well if pushed too far.
Mathematics is also seen by many as an analogy. But it is implicitly assumed to be the analogy that never breaks down. Our experience of the world has failed to reveal any physical phenomenon that cannot be described mathematically. That is not to say that there are not things for which such a description is wholly inappropriate or pointless. Rather, there has yet to be found any system in Nature so unusual that it cannot be fitted into one of the strait-jackets that mathematics provides.
This state of affairs leads us to the overwhelming question: Is mathematics just an analogy or is it the real stuff of which the physical realities are but particular reflections?
[Chapter 1, p. 21]
"In practice whatever they may possess in their language, they certainly use no greater number than three. When they wish to express four they take to their fingers, which are to them formidable instruments of calculation as a sliding rule is to an English schoolboy. They puzzle very much after five, because no spare hand remains to grasp and secure the fingers that are required for units. Yet they seldom lose oxen; the way in which they discover the loss of one is not by the number of the herd being diminished, but by the absence of a face they know. When bartering is going on each sheep must be paid for separately. Thus suppose two stick of tobacco to be the rate of exchange for one sheep, it would sorely puzzle a Damara to take two sheep and give him four sticks. I have done so, and seen a man take two of the sticks apart and take a sight over them at one of the sheep he was about to sell. Having satisfied himself that one was honestly paid for, and finding to his surprise that exactly two sticks remained in hand to settle the account for the other sheep, he would be afflicted with doubts; the transaction seemed to him to come too 'pat' to be correct, and he would refer back to the first couple of sticks, and then his mind got hazy and confused, and wandered from one sheep to the other, and he broke off the transaction until two sticks were placed in his hand and one sheep driven away, and then the other two sticks given him and the second sheep driven away."
[Chapter 2, p. 35-36]
In each of these four languages the words for 'one' and 'first' are quite distinct in form and emphasize the distinction between solitariness (one) and priority (being first). In Italian and the more old-fashioned German and French usage of ander and second, there is also a clear difference between the words used for 'two' and 'second', just as there is in English. This reflects the Latin root sense in English, French, and Italian of being second, this is, coming next in line, and this does not necessarily have an immediate association with two quantities. But when we get to three and beyond, there is a clear and simple relationship between the cardinal and ordinal words. Presumably this indicates that the dual aspect of number was appreciated by the time the concepts of 'threeness' and 'fourness' had emerged linguistically, following a period when only words describing 'oneness' and 'twoness' existed with greater quantities described by joining those words together as we described above.
In all the known languages of Indo-European origin, numbers larger than four are never treated as adjectives, changing their form according to the thing they are describing. But, numbers up to and including four are: we say they are 'inflected'. [...] a rather antiquated structure that barely survives in the modern forms of many Indo-European languages. For example, in French we find two words un and une corresponding to the English 'one' and they are used according to the gender of what is being counted. An analogous feature of language that certainly survives in English is the way in which different adjectives are associated with the same quantities of different things. We speak of a pair of shoes, a brace of pheasants, a yoke of oxen, or a couple of people, but we would never speak of a brace of chickens or a couple of shoes. [...]
We have seen that the distinction between cardinal and ordinal aspects of number and the use of inflected adjectives is clear up to the number four but conflated beyond that. [Footnote: In Finnish there are still two kinds of plural, as in classical Greek, Biblical Hebrew and Arabic: one for two things and another for more than two. Also interesting in this respect is the fact that there is no connection between the words for '2' and '½' in the Romance and Slavic languages (nor in Hungarian which is not an Indo-European language) but in all the European languages the words for '3' and '1/3', '4' and '1/4' and so on, are closely related, just as they are in English. This may indicate that the concept of a fraction, or the relation between a number and the concept of a ratio, only emerged after counting beyond 'two'.]
[...]
A curious speculation arises [...] to give special status to the number 8 - the total number of fingers excluding the thumbs - that many known languages originally possessed a base-8 system (which they later replaced by something better), because the word for the number 'nine' appears closely related to the word for 'new' suggesting that nine was a new number added to a traditional system. There are about twenty examples of this link, including Sanskrit, Persian, and the more familiar Latin, where we can see novus = 'new' and novem = 'nine'.
[Chapter 2, p. 37-38]
[Footnote: This state of affairs is not entirely unknown today and its manifestations in some subjects are tellingly documented by the sociologist Stanislav Andreski in his book Social Sciences as Sorcery.]
An amusing example of the power of this approach is the famous occasion on which Leonhard Euler, the great Swiss mathematician who was sometimes tutor to Catherine the Great of Russia during the eighteenth century, decided to bamboozle the Voltarean philosophers at Court in an argument about the existence of God. Calling for a blackboard, he wrote:
(x+y)2 = x2+2xy+y2
therefore God exists.[Presumably in a language other than English -- Fred.]
Unwilling to confess their ignorance of the formula or unable to question its relevance to the question at hand, his opponents accepted his argument with a nod of profound approval.
[Chapter 3, p. 107-108]
[Chapter 3, p. 133-134]
[Chapter 4, p. 164]
"But what if beings were even found whose laws of thought flatly contradicted ours and therefore frequently led to contrary results even in practice? The psychological logician could only acknowledge the fact and say simply: those laws hold for them, these laws hold for us. I should say: we have a hitherto unknown type of madness."
[Chapter 4, p. 176]
[Footnote: The Reader may recall that the essence of the plot in Agatha Christie's first novel The Mysterious Affair at Styles was that a villain who, having committed a murder by some means, deliberately laid false clues to make it appear that he had committed the murder in a different way. He had a cast-iron alibi against having committed the murder in the manner suggested by this trail of false clues, and so if he could be charged and tried for the murder on the basis of the false evidence then the revelation of the alibi at the last moment in court would result in his acquittal with no possibility of him being retried even if the true means of the murder were to be discovered in the future.]
[Chapter 5, p. 186]
[Chapter 6, p. 215]
[Chapter 5, p. 216]
Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer exists, he should know how to find it. If God has mathematics of his own that needs to be done, let him do it himself.[Chapter 5, p. 220]
He invites a colloquium on classical analysis,
for my participation, in the absence of the contructivists;
He frequently and approvingly abstracts me
in
Mathematical Reviews;
my reputation flourishes internationally.
Surely, honours and grants shall follow me
all the days of my career,
And I shall rise in the ranks of the Department,
to Emeritus.
Amen.
[From a fable by the American mathematician John Hays, quoted in Chapter 5, p. 226]
[...] So, 'somewhere', Plato maintained, there must exist perfect straight lines, perfect circles and triangles, or exactly parallel lines; and these perfect forms would exist even if there were no particular examples of them for us to see. This 'somewhere' was not simply the human mind. Strikingly, Plato maintained that we discover the truths and theorems of mathematics; we do not simply invent them. [...]
The first problem with Plato's picture of reality is to clarify the relationship between universals and particular examples of them. Plato's idea that there exist perfect blueprints of which the particulars are imperfect approximations does not seem very helpful when one gives it a second thought: for, as far as our minds are concerned, the universal blueprint is just another particular. So, whilst we could say that Plato maintained that universals would exist even in the absence of particulars, this statement does not really have any clear meaning. If all the particulars vanished, so would all those mental images of concrete perfect blueprints together with all the blueprints themselves. Aristotle [...] also picked upon this weakness in Plato's doctrine of Ideas, arguing that it leads to an endless regress:
"If all men are alike because they have something in common with Man, the ideal and eternal archetype, how can we explain the fact that one man and Man are alike without assuming another archetype? And will not the same reasoning demand a third, fourth and fifth archetype, and so on into the regress of more and more ideal worlds?"
[...] Plato wants to relate the universal abstract blueprint of a perfect circle to the approximate circles that we see in the real world. But why should we regard the 'approximate' circles, or 'almost parallel lines', or 'nearly triangles' as imperfect examples of perfect blueprints? Why not regard them as perfect exhibits of universals of 'approximate circles', 'almost parallel lines' and 'nearly triangles'? When viewed in this light the distinction between universals and particulars seems to be eroded.
[Chapter 6, p. 253-255]
"outwardly it does not seem to hamper our daily work, yet I for one confess that it has had a considerable practical influence on my mathematical life. It directed my interests to fields I considered relatively 'safe', and has been a constant drain on the enthusiasm and determination with which I pursued my research work."
Here we see a viewpoint that [...] hints that even if Platonism is not true, it is most effective for the working mathematician to act as if it were true. The French mathematician Emile Borel, writing a century ago, tends a little closer to this attitude:
"Many do, however, have a vague feeling that mathematics exists somewhere, even though, when they think about it, they cannot escape the conclusion that mathematics is exclusively a human creation. Such questions can be asked of many other concepts such as state, moral values, religion, ... we tend to posit existence on all those things which belong to a civilization or culture in that we share them with other people and can exchange thoughts about them. Something becomes objective (as opposed to 'subjective') as soon as we are convinced that it exists in the minds of others in the same form as it does in ours, and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence regardless of whether it has another origin."
If we look back to the end of the nineteenth century when Borel was writing then we find some of the most emphatic Platonists. Charles Hermite regarded mathematics as an experimental science:
"I believe that the numbers and functions of analysis are not the arbitrary product of our spirits: I believe that they exist outside us with the same character of necessity as the objects of objective reality; and we find or discover them and study them as do the physicists, chemists and zoologists."
Henri Poincaré highlights Hermite's approach as especially remarkable in that it was not just a philosophy of mathematics, more a psychological attitude. Poincaré had been Hermite's student and was also much taken to careful introspective analysis of his mathematical thought processes and creativity.
Unlike many subscribers to the Platonic philosophy, Hermite did not regard all mathematical discovery as the unveiling of the true Platonic world of mathematical forms; he seems to have drawn a distinction between mathematical discoveries of this pristine sort and others, like Cantor's development of different orders of infinity [...]that he regarded as mere human inventions. Poincaré writes, in 1913, twelve years after Hermite's death, that
"I have never known a more realistic mathematician in the Platonist sense then Hermite ... He accused Cantor of creating objects instead of merely discovering them. Doubtless because of his religious convictions he considered it a kind of impiety to with to penetrate a domain which God alone can encompass [i.e. the infinite], without waiting for Him to reveal its mysteries one by one. He compared the mathematical sciences with the physical sciences. A natural scientist who sought to divine the secret of God, instead of studying experience, would have seemed to him not only presumptuous but also lacking in respect for the divine majesty: the Cantorians seemed to him to want to act in the same way in mathematics. And this is why, a realist in theory, he was an idealist in practice. There is a reality to be known, and it is external to and independent of us; but all we can know of it depends on us, and is no more than a gradual development, a sort of stratification of successive conquests. The rest is real but eternally unknowable."
[...] G. H. Hardy [...] maintained a solid Platonic realism about the nature of mathematics:
"I believe that mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations ... 317 is a prime number, not because we think it so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."
[...]
Nevertheless [...] there are many dissenters to such a confident Platonic view [...] the subtle change of direction in the titles of applied mathematics books which highlight changing attitudes to the status of mathematical descriptions of the world. They emphasize the use of mathematics as a tool for deriving approximate descriptions ('models') of the real thing. There is no implication that the mathematics being presented is the reality.
[Chapter 6, p. 259-262]
Platonism allows freedom of thought, but only in the sense that it is your fault if you want to think the wrong thoughts. Whereas the formalist is free to create any logical system he chooses, a Platonist like Gödel maintained that only one system of axioms captured the truths that existed in the Platonic world. Although a conjecture like the continuum hypothesis was undecidable from the axioms of standard set theory, Gödel believed it was either true or false in reality and this would be decided by adding appropriate axioms to those we were in the habit of using for set theory. He writes of the undecidability of Cantor's continuum hypothesis:
"Only someone who (like the intuitionist) denies that the concepts and axioms of classical set theory have any meaning ... could be satisfied with such a solution, not someone who believes them to describe some well-determined reality. For in reality Cantor's conjecture must be either true or false, and its undecidability from the axioms as known today can only mean that these axioms do not contain a complete description of reality."
This seems a strange view because there would always exist other undecidable statements no matter what additional axioms are prescribed.
[Chapter 6, p. 263-264]
When studying the simultaneous discovery of mathematical concepts one must be careful not to generalize over the whole of mathematics. The motivations for ancient mathematical discoveries are rather different to those which inspire developments in modern times. The most primitive notions of geometry and counting are tied to practical applications in obvious ways and there was clear motivation for their adoption by less developed cultures who wanted to trade and converse with more sophisticated neighbours. But in modern times mathematics need not be tied to practical applications and one cannot argue that notions of pure mathematics are developed in response to social or utilitarian pressures. Modern communications mean that mathematicians and scientists are in effect a single intellectual society that transcends national and cultural borders. [...] it is hard for mathematicians to be independent in the deepest sense. They grow up in the body of mathematical knowledge that is taught to mathematicians of every colour and creed. They attend conferences, or read reports of them, and sense the direction of the subject, they see the great unsolved problems being laid out and conjectures as to their solution being proposed. [...] they have common predecessors, common inherited intuitions and methods. They are fish that swim in the same big pond.
Another intriguing aspect of mathematics that seems to distinguish it from the arts and humanities is the extent to which mathematicians, like scientists, collaborate in their work.
[Chapter 6, p. 266-267]
From Words Fail Me by Philip Howard (Hamish Hamilton Ltd, 1980) [Chapter 19, Millennium, p. 109]:
Human kind, especially the English human kind, cannot bear very much mathematics. Plato said that he had hardly ever known a mathematician who was capable of reasoning. The passion for anniversaries, decades, centuries, and all dates with noughts in them is deeply engrained in the human attitude to time And yet it is an irrationally tidy way to measure life, which does not conform to the decimal system. The seventeenth century ought to begin in 1603. The death of Oscar Wilde in 1900 and the Dreyfus case mark the end of an intellectual and moral epoch. [...] But the nineteenth century properly ends in 1914. You cannot wrap men and ideas up in parcels of centuries in order to make literary and historical generalizations about them with the appearance of mathematical exactitude. You should not, but we all do. Since A.D. 1 there have been not nineteen but 1979 complete centuries. The real world is regardless of our systems of reckoning; events and men slip over years with noughts in their dates, with as little shudder as is felt on a liner passing over a tropic or a car crossing a county boundary.
From The Science Show, ABC Radio National (Australia), 1 Nov 2003:
Mark Lythgoe: [...] it was about 4 or 5 years ago - and I read an article by Simon Baron-Cohen. And he'd looked at a thousand students from Cambridge and assessed them on what he called "the autistic spectrum".Now these were normal, functioning intelligent people that had tried to decide whether they had particular autistic traits or not. And he found that the scientists, as opposed to the arts and humanists, came significantly higher on this autistic spectrum, they had more autistic traits. Then the mathematicians were virtually off the top of the spectrum, they were very autistic.
Robyn Williams: And the engineers.
MP3 audio of quote (143 Kbyte)When I turned two I was really anxious, because I'd doubled my age in a year. I thought, if this keeps up, by the time I'm six I'll be ninety. -- Steven Wright
From Teri Perl, Math Equals: Biographies of Women Mathematicians (Addison Wesley, 1978):
Sonya Kovalevskaya had been an extraordinarily versatile and talented woman. With the greatest of ease she could turn from a lecture on Abel's functions, to research on Saturn's rings, to the writing of verse in French or a novel in Russian or a play in Swedish, to sewing a lace collar for her little daughter Fufi. In reply to a friend's surprise at her involvement in literature as well as mathematics, she wrote,
Many who have never had an opportunity of knowing any more about mathematics confound it with arithmetic and consider it an arid science. In reality however, it is a science which requires a great amount of imagination, and one of the leading mathematicians of our century states the case quite correctly when he says that it is impossible to be a mathematician without being a poet in soul.... one must renounce the ancient prejudice that a poet must invent something that does not exist, that imagination and invention are identical. It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing. [Source of quote: Sonya Kovalévsky, Her Recollections of Childhood, translated by Isabel F. Hapgood, The Century Co., New York, 1895, p. 316]
At the time of her death, Sonya Kovalevskaya was at the very height of her fame. By penetrating deeply into the methods of mathematical research, she had made brilliant discoveries. Her contributions are considered equal to those of any mathematician of her day by any of her colleagues who are qualified to judge.
[p. 136]
From How to Write Mathematics by Norman E. Steenrod, Paul R. Halmos, Menahem M. Schiffer, Jean R. Dieudonné (American Mathematical Society, 1973):
The arguments against having an introductory chapter are: (1) they are difficult to write, (2) it is a waste of effort to say imprecisely what is said precisely later on, (3) a reader who completes the book will forget that there was an introduction, and (4) a sales-promotion job should be beneath the dignity of a mathematician. I have no sympathy for reason (4); the direction of a young mathematician's career is largely determined by interests that have been aroused. It is absurd to suppose that a graduate student will learn just enough of all areas to be able to make a logical choice of his research topic. I am in sympathy with reason (1), but a purpose of this essay is to ease the task.
Reasons (2) and (3) go together. The fact that a reader forgets the introduction is no objection if the introduction helps him grasp the formal structure more quickly. At stake here is the question of how a student learns best. The first of two contending procedures is to ask him to examine first the lumber, bricks, and small structural members out of which the building is to be made, then to make subassemblies, and finally to erect the building from these. The second procedure is first to describe the building roughly but globally and provide a framework for viewing it, and then examine the construction of the building in detail. The first procedure would appeal to a student with a leisurely attitude who enjoys successive revelations. The second procedure, which I espouse, has the advantage that motivation is present at every stage; the student knows where each item belongs when he examines it.
The second procedure can be elaborated by inserting between the first rough scan and the final detailed examination a series of scannings revealing successively finer details. Max Eastman in his book The Enjoyment of Laughter advocates and exemplifies this procedure in an amusing and convincing fashion. An argument favoring these successive approximations goes as follows. It has been observed that one learns a subject best not when first exposed to it but later when using the material in another study, or else when required to teach the subject. This can be paraphrased by saying that the nth scan is fixed in the memory by making the (n + 1)st. Stated otherwise, when a reader has finished a book, he will retain in his memory only a more or less rough picture of the formal structure. This being so, why shouldn't the author assist the reader in formulating this rough picture? Surely the author's condensed version of the overall picture will be better balanced and more nearly accurate than one formed by an average reader.
[Norman E. Steenrod, p. 10-12]
Just as there are two ways for a sequence not to have a limit (no cluster points or too many), there are two ways for a piece of writing not to have a subject (no ideas or too many).
The first disease is the harder one to catch. It is hard to write many words about nothing, especially in mathematics, but it can be done, and the result is bound to be hard to read. There is a classic crank book by Carl Theodore Heisel [The Circle Squared Beyond Refutation] that serves as an example. It is full of correctly spelled words strung together in grammatical sentences, but after three decades of looking at it every now and then I still cannot read two consecutive pages and make a one-paragraph abstract of what they say; the reason is, I think, that they don't say anything.
The second disease is very common: there are many books that violate the principle of having something to say by trying to say too many things. Teachers of elementary mathematics in the U.S.A. frequently complain that all calculus books are bad. That is a case in point. Calculus books are bad because there is no such subject as calculus; it is not a subject because it is many Subjects. What we call calculus nowadays is the union of a dab of logic and set theory, some axiomatic theory of complete ordered fields, analytic geometry and topology, the latter in both the "general" sense (limits and continuous functions) and the algebraic sense (orientation), real-variable theory properly so called (differentiation), the combinatoric symbol manipulation called formal integration, the first steps of low-dimensional measure theory, some differential geometry, the first steps of the classical analysis of the trigonometric, exponential, and logarithmic functions, and, depending on the space available and the personal inclinations of the author, some cook-book differential equations, elementary mechanics, and a small assortment of applied mathematics. Any one of these is hard to write a good book on; the mixture is impossible.
[Paul R. Halmos, p. 20-21]
From Adventures of a Mathematician by Stanislaw M. Ulam (Charles Scribner, NY, 1976):
[...]
In mathematical discussions, or in short remarks he made on general subjects, one could feel almost at once the great power of his mind. He worked in periods of great intensity separated by stretches of apparent inactivity. During the latter his mind kept working on selecting the statements, the sort of alchemist's probe stones that would best serve as focal theorems in the next field of study.
He enjoyed long mathematical discussions with friends and students. I recall a session with [Stanisław] Mazur and Banach at the Scottish Cafe which lasted seventeen hours without interruption except for meals. What impressed me most was the way he could discuss mathematics, reason about mathematics, and find proofs in these conversations.
[Chapter 2, Student Years 1927-1933, p. 33]
[...]
[Kazimierz] Kuratowski and [Hugo] Steinhaus, each in a different way, represented elegance, rigor, and intelligence in mathematics. [...] His mathematics was characterized by what I would call a Latin clarity. In the proliferation of mathematical definitions and interests (now even more bewildering than at that time), Kuratowski's measured choice of problems had the quality of what is hard to define -- common sense in the abstractions.
[...]
[Hugo Steinhaus] had a talent for applying mathematical formulations to matters as common as problems of daily life. Certainly his inclinations were to single out problems of geometry that could be treated from a combinatorial point of view -- actually anything that presented the visual, palpable challenge of a mathematical treatment.
He had great feeling for linguistics, almost pedantic at times. He would insist on absolutely correct language when treating mathematics or domains of science susceptible to mathematical analysis.
[Chapter 2, Student Years 1927-1933, p. 40-42]
The ceremony was a rather formal affair. It took place in a large Institute hall with family and friends attending. I had to wear a white tie and gloves. My sponsors Stozek and Kuratowski each gave a little speech describing my work and the papers I had written. After a few words about the thesis, they handed me a parchment document.
The "aula" -- the large hall in which the ceremony took place -- was decorated with traditional frescoes. These were very much like some I saw twenty years later on the walls of the MIT cafeteria. The MIT frescoes depict scantily dressed women in postures of flight, symbolizing sciences and arts, and a large female figure of a goddess hovering over a recoiling old man. I used to joke that it represented the Air Force giving a contract to physicists and mathematicians. In Fuld Hall, the Institute Ruilding in Princeton, there is also an old painting in the tea room where people assemble for conversation in the afternoon. There again one sees an old man who seems to be shying away from an angel coming down from the clouds. When I was told that nobody knew what it was supposed to represent, I suggested that it might be a representation of Minna Ries, the lady mathematician who directed the Office of Naval Research at the time, proposing a Navy contract to Einstein, who is recoiling in horror.
[Chapter 2, Student Years 1927-1933, p. 47]
[Chapter 4, Princeton Days 1935-1936, p. 70]
[Chapter 4, Princeton Days 1935-1936, p. 71]
In those days he still called me "Mr. Ulam." Once he said to me as we were driving in the rain and were caught in a traffic jam, "Mr. Ulam, cars are no good for transportation anymore, but they make marvelous umbrellas." I often remember this when I am caught in the traffic jams of today. Johnny always loved cars but he drove somewhat carelessly.
[Chapter 4, Princeton Days 1935-1936, p. 71]
[Chapter 4, Princeton Days 1935-1936, p. 73]
[Chapter 5, Harvard Years 1936-1939, p. 87-88]
The story of the Athenian expedition to the island of Melos, the atrocities and killings that followed, and the lengthy debates between the opposing parties fascinated him for reasons which I never quite understood. He seemed to take a perverse pleasure in the brutality of a civilized people like the ancient Greeks. For him, I think it threw a certain not-too-complimentary light on human nature in general. Perhaps he thought it illustrated the fact that once embarked on a certain course, it is fated that ambition and pride will prevent a people from swerving from a chosen course and inexorably it may lead to awful ends, as in the Greek tragedies. Needless to say this prophetically anticipated the vaster and more terrible madness of the Nazis. Johnny was very much aware of the worsening political situation. In a Pythian manner, he foresaw the coming catastrophe.
[Chapter 5, Harvard Years 1936-1939, p. 102]
We both thought about ways to do it. Suddenly, while we were in a motel I found a combinatorial trick showing that it could not be done. It was, if I say so myself, rather ingenious. I explained it to Johnny. As we drove Johnny later simplified this proof in the sense that he found an example of a continuum group which is even Abelian (commutative) and yet unable to assume a separable topology. In other words, there exist abstract groups of power continuum in which there is no possible continuous separable topology. What is more, there exist such groups that are Abelian. Johnny, who liked verbal games and to play on words, asked me what to call such a group. I said, "nonseparabilizable." It is a difficult word to pronounce; on and off during the car ride we played at repeating it.
Mathematicians have their own brand of "in" humor like this. Generally speaking, they are amused by stories involving triviality of identity of two definitions or "tautologies." They also like jokes involving vacuous sets. If you say something which is true "in vacuo," that is to say, the conditions of the statement are never satisfied, it will strike them as humorous. They appreciate a certain type of logical non sequitur or logical puzzle. For instance, the story of the Jewish mother who gives a present of two ties to her son-in-law. The next time she sees him, he is wearing one of them, and she asks, "You don't like the other one?"
Some of von Neumann's remarks could be devastating, even though the sarcasm was of an abstract nature. Ed Condon told me in Boulder of a time he was sitting next to Johnny at a physics lecture in Princeton. The lecturer produced a slide with many experimental points and, although they were badly scattered, he showed how they lay on a curve. According to Condon, von Neumann murmured, "At least they lie on a plane."
[Chapter 5, Harvard Years 1936-1939, p. 103-104]
[Chapter 5, Harvard Years 1936-1939, p. 104]
I have heard mathematicians sneer at the special theory of relativity, calling it nothing but a technically trivial quadratic equation and a few consequences. Yet it is one of the monuments of human thought. So what is "trivial"? Simple arithmetic? It may be trivial to us, but is it to the third-grade child?
Let us consider some other words mathematicians use: what about the adjective "continuous"? Out of this one word came all of topology. Topology may be considered as a big essay on the word "continuous" in all its ramifications, generalizations, and applications. Try to define logically or combinatorially an adverb like "even" or "nevertheless." Or take an ordinary word like "key," a simple object. Yet it is an object far from easy to define quasi-mathematically. "Billowing" is a motion of smoke, for example, in which puffs are emitted from puffs. It is almost as common in nature as wave motion. Such a word may give rise to a whole theory of transformations and hydrodynamics. I once tried to write an essay on the mathematics of three-dimensional space that would imitate it.
Were I thirty years younger I might try to write a mathematical dictionary about the origins of mathematical expressions and concepts from commonly used words, imitating the manner of Voltaire's Dictionnaire Philosophique.
[Chapter 5, Harvard Years 1936-1939, p. 104-105]
[Chapter 6, Transition and Crisis 1936-1940, p. 106]
[Chapter 6, Transition and Crisis 1936-1940, p. 120]
[Chapter 6, Transition and Crisis 1936-1940, p. 120]
[Chapter 7, The University of Wisconsin 1941-1943, p. 123-124]
[Chapter 7, The University of Wisconsin 1941-1943, p. 125]
[Chapter 8, Los Alamos 1943-1945, p. 160]
[Chapter 8, Los Alamos 1943-1945, p. 164-165]
[Chapter 8, Los Alamos 1943-1945, p. 165]
In their lectures to students or scientific gatherings, they demonstrated their different approaches. Johnny did not mind showing off brilliancy or special ingenuity; Fermi, on the contrary, always strived for the utmost simplicity, and when he talked everything appeared in a most natural, direct, bright, clear light. After students had gone home, they were often unable to reconstruct Fermi's dazzlingly simple explanation of some phenomenon or his deceptively simple-looking idea on how to treat a physical problem mathematically. In contrast, von Neumann showed the effects of his sojourns at German universities. He was absolutely devoid of pomposity, but in his language structure he could be complicated, though perfect logic always gave a unique interpretation to his words.
They held high opinions of each other. I remember a discussion of some hydrodynamical problem Fermi had been thinking about. Von Neumann showed a way to consider it, using a formal mathematical technique. Fermi told me later with admiration, "He is really a professional, isn't he!" As for von Neumann, he always took external evidences of success seriously; he was quite impressed by Fermi's Nobel Prize. He also appreciated wistfully other people's ability to get results by intuition or seemingly pure luck, especially by the apparent effortlessness of Fermi's fundamental physics discoveries. After all, Fermi was perhaps the last all-around physicist in the sense that he knew the theory, did original work in many branches, and knew what experiments to suggest and even do himself; he was the last to be great both in theory and as an experimenter.
[Chapter 8, Los Alamos 1943-1945, p. 166-167]
On the road to Santa Fe, each time we drove by a place called Totavi (really more a name than a place), I would launch into Latin and recite, "Toto, totare, totavi, totatum," and he would add some form of the future. This was one of our nonsensical verbal games. [...]
In the early years after the war, the AEC started to build an elegant, permanent structure for its offices and those of the security services, even before new and more comfortable housing was ready for the residents. Johnny remarked that this was entirely in the tradition of all government administrations through the ages, and he decided to call the building "El Palacio de Securita." This was a good enough mixture of Spanish, Latin and Italian. So to go him one better, I immediately named a newly built church "San Giovanni delle Bombe."
This is about the time we made up a "Nebech index." Johnny had told me the classic story of the little boy who came home from school in pre-World-War-I Budapest and told his father that he had failed his final examination. The father asked him, "Why? What happened?" The boy replied, "We had to write an essay. The teacher gave us a theme: the past, the present, and the future of the Austro-Hungarian Empire." The father asked, "So, what did you write?" and the boy answered, "I wrote: Nebech, nebech, nebech." "That is correct," his father said. "Why did you receive an F?" "I spelled nebech with two bb's," was the answer.
This gave me the idea of defining the nebech index of a sentence as the number of times the word nebech could be inserted in it and still be appropriate, though giving a different flavor to the meaning of the sentence according to the word it qualifies. For instance, one could argue that the most perfect "nebech three" sentence is Descartes' statement: Cogito, ergo sum. One can say, Cogito nebech, ergo sum. Or Cogito, ergo nebech sum. Or Cogito, ergo sum nebech. Unfortunately this elegant example occurred to me only after Johnny's death. Johnny and I used this index frequently during mathematical talks, physics meetings or political discussions. We would nudge one another, whisper "Nebech two" at a particular statement, and enjoy this greatly.
Now, if the reader is sufficiently mystified, I will explain that "nebech" is an untranslatable Yiddish expression, a combination of commiseration, scorn, drama, ridicule.
To try to give the flavor of the word, imagine the William Tell story as acted out in a Jewish school. In the scene where William Tell waits in hiding to shoot Gessler, an actor says, in Yiddish: "Through this street the Nebech must come." It is obvious that Gessler is a Nebech since he will be the victim of William Tell. But if nebech had been in front of the word street, then the accent would be on street, indicating that it was not much of a street. To appreciate this may take years of apprenticeship.
[Chapter 10, Back at Los Alamos 1946-1949, p. 193-195]
[Chapter 10, Back at Los Alamos 1946-1949, p. 205]
[Chapter 10, Back at Los Alamos 1946-1949, p. 206]
[Chapter 10, Back at Los Alamos 1946-1949, p. 206]
[Chapter 10, Back at Los Alamos 1946-1949, p. 207]
[Chapter 11, The "Super" 1949-1952, p. 218]
[Chapter 13, Government Science 1957-1967, p. 251-252]
[Chapter 13, Government Science 1957-1967, p. 261]
Self-portrait of Mr. S. U.His expression is usually ironic and quizzical. In truth he is very much affected by all that is ridiculous. Perhaps he has some talent to recognize and feel it at once, so it is not surprising that this is reflected in his facial expression.
His conversation is very uneven, sometimes serious, sometimes gay, but never tiring or pedantic. He only tries to amuse and distract the people he likes. With the exception of the exact sciences, there is nothing which appears so certain or obvious to him that he would not allow for differing opinions: on almost any subject one can say almost anything.
He brought to the study of mathematics a certain talent and facility which allowed him to make a name for himself at an early age. Dedicated to work and solitude until he was twenty-five, he became more worldly rather late. Nevertheless he is never rude because he is neither coarse nor hard. If he sometimes offends it is through inattention or ignorance. In speech he is neither gallant nor graceful. When he says kind things it is because he means them. Therefore the essence of his character is a frankness and truthfulness which are sometimes a little strong but never really shocking.
Impatient and choleric to the point of violence, everything that contradicts or wounds him affects him in an uncontrollable way, but this usually disappears when he has vented his feelings.
He is easy to influence or govern provided he is unaware that this is intended.
Some people think that he is malicious because he makes merciless fun of pretentious bores. His temperament is naturally sensitive and renders him subject to delicate moods. This makes him at once gay and melancholy.
Mr. U. behaves according to this general rule: he says a lot of foolish things, seldom writes them and never does any.
When Francoise read this description she felt it agreed well with what she then knew of me but was very surprised at the quality of my French, until she came to the last paragraph:
And now I shall change from my text which I came upon by chance yesterday. The above are verbatim extracts from a letter of d'Alembert to Mademoiselle de Lespinasse written some two hundred years ago!
(D'Alembert was a famous French mathematician and encyclopedist of the eighteenth century.) Francoise was very amused.
Some thirty years have elapsed since I copied this little text. I will now add as a finishing touch that I don't think I have changed much, but that there is one trait which d'Alembert did not mention that I possess -- all this merely si parva magnis comparare licet -- it is a certain impatience. I have been afflicted with this all my life. It may be increasing with advancing years. (If Einstein or Cantor came to lecture here today I would have the split reaction of a schoolboy -- wanting to learn on the one hand and to skip class on the other.) While I still feel quite happy giving lectures, talks, or discussions I am becoming less and less able to sit through hours of such given by others. I am, I told some colleagues, "like an old boxer who can still dish it out but can't take it any more." This amused them no end.
[Chapter 14, Professor Again 1967-1972, p. 270-272]
"Mathematics is a language in which one cannot express unprecise or nebulous thoughts," said Poincare, I believe in a speech on world science which he gave at the St. Louis fair many years ago. And he gave as an example of the influence of language on thought a description of how differently he felt using English instead of French.
I tend to agree with him. It is a truism to say that there is a clarity to French which is not there in other tongues, and I suppose this makes a difference in the mathematical and scientific literature. Thoughts are steered in different ways. In French generalizations come to my mind and stimulate me toward conciseness and simplification. In English one sees the practical sense; German tends to make one go for a depth which is not always there.
In Polish and Russian, the language lends itself to a sort of brewing, a development of thought like tea growing stronger and stronger. Slavic languages tend to be pensive, soulful, expansive, more psychological than philosophical, but not nebulous or carried by words as much as German, where words and syllables concatenate. They concatenate thoughts which sometimes do not go very well together. Latin is something else again. It is orderly; clarity is always there; words are separated; they do not glue together as in German; it is like well-cooked rice compared to overcooked.
Generally speaking, my own impressions of languages are the following: When I speak German everything I say seems overstated, in English on the contrary it feels like an understatement. Only in French does it seem just right, and in Polish, too, since it is my native language and feels so natural.
Some French mathematicians used to manage to write in a more fluent style without stating too many definite theorems. This was more agreeable than the present style of the research papers or books which have so much symbolism and formulae on every page. I am turned off when I see only formulas and symbols, and little text. It is too laborious for me to look at such pages not knowing what to concentrate on. I wonder how many other mathematicians really read them in detail and enjoy them.
There do exist, though, important, laborious and inelegant theorems. For example some of the work connected with partial differential equations tends to be less "beautiful" in form and style, but it may have "depth," and may be pregnant with consequences for interpretations in physics.
[Chapter 15, Random Reflections on Mathematics and Science, p. 274-276]
Yet every formalism, every algorithm, has a certain magic in it. The Jewish Talmud, or even the Kabbalah, contains material which does not appear particularly enlightening intellectually, being a vast collection of grammatical or culinary recipes, some perhaps poetic, others mystical, all rather arbitrary. Over centuries thousands of minds have pored over, memorized, dissected, and classified these works. In so doing people may have sharpened their memories and deductive practice. As one sharpens a knife on a whetstone, the brain can be sharpened on dull objects of thought. Every form of assiduous thinking has its value.
[Chapter 15, Random Reflections on Mathematics and Science, p. 277-278]
[Chapter 15, Random Reflections on Mathematics and Science, p. 279]
With all its grandiose vistas, appreciation of beauty, and vision of new realities, mathematics has an addictive property which is less obvious or healthy. It is perhaps akin to the action of some chemical drugs. The smallest puzzle, immediately recognizable as trivial or repetitive can exert such an addictive influence. One can get drawn in by starting to solve such puzzles. I remember when the Mathematical Monthly occasionally published problems sent in by a French geometer concerning banal arrangements of circles, lines and triangles on the plane. "Belanglos," as the Germans say, but nevertheless these figures could draw you in once you started to think about how to solve them, even when realizing all the time that a solution could hardly lead to more exciting or more general topics. This is much in contrast to what I said about the history of Fermat's theorem, which led to the creation of vast new algebraical concepts. The difference lies perhaps in that little problems can be solved with a moderate effort whereas Fermat's is still unsolved and a continuing challenge. Nevertheless both types of mathematical curiosities have a strongly addictive quality for the would-be mathematician which exists on all levels from trivia to the most inspiring aspects.
[Chapter 15, Random Reflections on Mathematics and Science, p. 290-291]
It is natural in experimental physics that investigators work together on the different phases of instrumentation. By now every experiment is really a class of technical projects, especially on the great machines which require hundreds of engineers and specialists for their construction and operation. In theoretical physics this is perhaps not as evident, but it exists, and strangely enough in mathematics also. We have seen that the creative effort in mathematics requires intense concentration and constant thinking in depth for hours on end, and that it is often shared by two individuals who just look at each other and occasionally make a few remarks when they collaborate. It is now definitely so that even in the most abstruse mathematical questions two or more persons work together on trying to find a proof. Many papers have now two, sometimes three or more authors. The exchange of conjectures, suggesting tentative approaches, helps to build up partial results along the way. It is easier to talk than to write down every thought.
[...]
Other variations in the working habits of scientists have been slower. The mode of life in the ivory tower world of science now includes more scientific meetings, more involvement in governmental work.
A simple but important thing like letter-writing has also undergone a noticeable change. It used to be an art, not only in the world of literature. Mathematicians were voluminous letter writers. They wrote in longhand and communicated at length intimate and personal details as well as mathematical thoughts. Today the availability of secretarial help renders such personal exchanges more awkward, and as it is difficult to dictate technical material scientists in general and mathematicians in particular exchange fewer letters. In my file of letters from all the scientists I have known, a collection extending over more than forty years, one can see the gradual, and after the war accelerated, change from long, personal, handwritten letters to more official, dry, typewritten notes. In my correspondence of recent years, only two persons have continued to write in longhand: George Gamow and Paul Erdős.
[Chapter 15, Random Reflections on Mathematics and Science, p. 292-294]
One evening a group of men came to a town. They needed to have their laundry done so they walked around the city streets trying to find a laundry. They found a place with the sign in the window, "Laundry Taken in Here." One of them asked: "May we leave our laundry with you?" The proprietor said: "No. We don't do laundry here." "How come?" the visitor asked. "There is such a sign in your window." "Here we make signs," was the reply. This is somewhat the case with mathematicians. They are the makers of signs which they hope will fit all contingencies. Yet physicists have created a lot of mathematics.
[Chapter 15, Random Reflections on Mathematics and Science, p. 294]
From Euclid's Window - The Story of Geometry from Parallel Lines to Hyperspace by Leonard Mlodinow (Free Press [Simon & Schuster], 2001):
[Chapter 2, The Geometry of Taxation, p. 7]
There is also a distinction between intelligent talk and blather, a distinction that Pythagoras did not always make. Pythagoras' awe of numerical relations swept him into forming many mystical numerological beliefs. He was the first to divide numbers into the categories "odd" and "even", but he took the extra step of personifying them: the odd he called "masculine", the even, "feminine". He associated specific numbers with ideas, such as the number 1 with reason, 2 with opinion, 4 with justice. Since 4 in his system was represented by a square, the square was associated with justice, the origin of the expression we still use today, "a square deal". In the interests of giving Pythagoras a square deal, one must recognise that it is easier to judge the brilliant from the blather with the perspective of a couple of thousand years.
[Chapter 4, The Secret Society, p. 21-23]
Pythagoras could have pushed up the invention of the real number system by many centuries, had he done a simple thing: given the diagonal a name, say, d, or even better, [symbol for the square-root of 2]. Had he done that, he might have pre-empted Descartes's coordinate revolution, for, absent a numerical representation, the need to describe this new number begged for the invention of the number line. Instead, Pythagoras retreated from his promising practice of associating geometric figures with numbers, and proclaimed that some lengths cannot be expressed as a number. The Pythagoreans called such lengths alogon, "not a ratio", which we today translate as "irrational". The word alogon had a double meaning, though: it also meant "not to be spoken". Pythagoras had solved his dilemma with a doctrine that would have been hard to defend, so, in keeping with his overall doctrine of secrecy, he banned his followers from revealing the embarrassing paradox. Not obeyed. According to legend, one of his followers, Hippasus, did reveal the paradox. Today people are murdered for many reasons -- love, politics, money, religion -- but not because somebody squealed about the square root of 2. To the Pythagoreans, though, mathematics was a religion, so when Hippasus broke the oath of silence, he was assassinated.
Resistance to irrationals continued for thousands of years. In the late nineteenth century, when the gifted German mathematician Georg Cantor did groundbreaking work to put them on firmer footing, his former teacher, a crab named Leopold Kronecker who "opposed" the irrationals, violently disagreed with Cantor and sabotaged his career at every turn. Cantor, unable to tolerate this, had a breakdown and spent his last days in a mental institution.
[Chapter 4, The Secret Society, p. 26-27]
College provided no haven from these miserable conditions. The concept of a college campus did not yet exist. Typically, a university had no buildings at all. Students lived in cooperative housing. Professors lectured in rented rooms, rooming houses, churches, even brothels. The classrooms, like the dwellings, were poorly lit and heated. Some universities employed a system that sounds, well, medieval: professors were paid directly by the students. At Bologna, students hired and fired professors, fined them for unexcused absence or tardiness, or for not answering difficult questions. If the lecture was not interesting, going too slow, too fast, or simply not loud enough, they would jeer and throw things. In Leipzig, the university eventually found it necessary to promulgate a rule against throwing stones at professors. As late as 1495, a German statute explicitly forbade anyone associated with the university from drenching freshmen with urine. In numerous cities, students rioted and fought with the townspeople. Across Europe, it was the fate of college professors to deal with behaviour that would make Animal House seem like an instructional video for good manners.
[Chapter 9, The Legacy of the Rotten Romans, p. 64-65]
[Chapter 9, The Legacy of the Rotten Romans, p. 66]
[...] It wasn't the church Gauss feared, it was its remnant, the secular philosophers.
In Gauss's day, science and philosophy hadn't completely separated. Physics wasn't yet known as "physics" but "natural philosophy". Scientific reasoning was no longer punishable by death, yet ideas arising from faith or simply intuition were often considered equally valid. [...]
The philosopher whose followers Gauss feared most was Immanual Kant, who had died in 1804. [...] Kant, noting that geometers of the day appealed to common sense and graphical figures in their "proofs", believed that the pretense of rigour ought to be dispensed with, and intuition embraced. Gauss held the opposite view -- that rigour was necessary, and most mathematicians were incompetent.
In Critique of Pure Reason, Kant calls Euclidean space "an inevitable necessity of thought." Gauss did not dismiss Kant's work out of hand. He read it first, then dismissed it. In fact, Gauss is said to have read Critique of Pure Reason five times attempting to understand it, a lot of effort for a fellow who picked up Russian and Greek with less effort than it would take most of us to find the Χωριάτικη Σαλάτα on an Athens menu. Gauss's struggle becomes more understandable when you consider the clarity of writing that led to passages of Kant's such as this one, on the distinction between analytic and synthetic judgements:
In all judgements in which the relations of a subject to the predicate is thought (I take into consideration affirmative judgements only, the subsequent application to negative judgements being easily made), this relation is possible in two different ways. Either the predicate to the subject A, as something which is (covertly) contained in the concept A; or outside the concept A, although it does indeed stand in connection with it. In the one case I entitle the judgement analytic, in the other synthetic.
Today, mathematicians and physicists worry little about what a philosopher would think of their theories. Famous American physicist Richard Feynman, when asked what he thought of the field of philosophy, gave a concise answer consisting of two letters, a "b", and [an "s"]. But Gauss took Kant's work seriously. He wrote that the above distinction between analytic and synthetic theories "is such a one that either peters out in triviality or is false". Yet he would divulge these thoughts, like his theories on non-Euclidean space, only to those he trusted. [...]
[Chapter 16, The Fall of the Fifth Postulate, p. 115-118]
From My Brain is Open - The Mathematical Journeys of Paul Erdős by Bruce Schechter (Simon & Schuster, 1998):
[Chapter 2, Proof, p. 19-20]
[Chapter 4, The Happy End Problem, p. 67]
[Chapter 6, Paradise Lost, p. 104-105]
[Chapter 8, The Primes of Dr. Paul Erdős, p. 142]
[Chapter 10, Six Degrees of Collaboration, p. 177-178]
From The Mathematical Universe by William Dunham (John Wiley & Sons, 1994):
Unfortunately, Father Mersenne's assertion contained sins of both commission and omission. For instance, he missed the fact that the number 261 - 1 is prime. On the other hand, 267 - 1 turned out not to be prime at all. This latter fact was established in 1876 by Edouard Lucas (1842-1891), who demonstrated that the number was composite using an argument so indirect that it did not explicitly exhibit any of the factors.
[Arithmetic, p. 5]
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