mathematicians-from-the-netherlands

"[Einstein's cosmological constant] is a name without any meaning. ...We have, in fact, not the slightest inkling of what it's real significance is. It is put in the equations in order to give the greatest possible degree of mathematical generality."

"Our own galaxy system is only one of a great many, and observations made from any of the others would show exactly the same thing: all systems are receding, not from any particular centre, but from each other: the whole system of galaxies is expanding."

"Gradually... during the second half of the nineteenth century, the uncomfortable feeling of dislike of the action at a distance, which had been so strong in Huygens and other contemporaries of Newton, but had subsided during the eighteenth century, began to emerge again, and gained strength rapidly. This was favoured by the purely mathematical transformation (which can be compared in a sense with that from the Ptolemaic to the Copernican system), replacing Newton's finite equations by the differential equations, the potential becoming the primary concept, instead of the force, which is only the gradient of the potential. These ideas, of course, arose first in the theory of electricity and magnetism or perhaps one should say in the brain of Faraday.<!--"

"No doubt once it was real progress when developers and teachers offered learners tangible material in order to teach them arithmetic of whole number... The best palpable material you can give the child is its own body."

"Space and the bodies around us are early mental objects... Name-giving is a first step towards consciousness."

"The present book is not a methodology of mathematics in the sense that I will systematically show how some teaching matter should taught; it is not even a systematic analysis of subject matter. I hardly ever refer to well-organized classroom experiments evaluated by statistical methods, nor do I cite experimental results of developmental psychology or the psychology of learning. Maybe the most striking feature is that this book contains few quotations. I will try to justify all these features."

"The case of methodology is analogous though less clear. Nobody would object to the subject of methodology being science, or some pseudo science. On closer inspection, however, this agreement is no more than a verbal coincidence. It rests on what is meant by science, as reported as a subject of methodology. In fact the subject methodologists call science is more often than not different from what scientists call science. Methodologists are inclined to consider a science as a linguistic system whereas the scientist would only admit that his science has a language, not that it is a language."

"The subject of a science is never well circumscribed and there is little use sharpening its definition. However, nobody will deny that physics deals with nature and sociology with human society in some of their aspects. With logic, it is another matter. Logic is usually understood nowadays as a study of certain formal systems, though in former times there were philosophers who held that the subject matter of logic was the formal rules of human thought. In the latter sense it would be an empirical rather than a formal science, though its empirical subject matter would still be fundamentally different from that of psychology of thinking. One interpretation of logic does not exclude the other. Formal approaches are often easier than empirical ones, and for this reason one can understand why logic as a study of formal systems has till now made more progress than logic as a study of the formal rules of thought, even if restricted to scientific thought."

"The urn model is to be the expression of three postulates: (1) the constancy of a probability distribution, ensured by the solidity of the vessel, (2) the randomcharacter of the choice, ensured by the narrowness of the mouth, which is to prevent visibility of the contents and any consciously selective choice, (3) the independence of successive choices, whenever the drawn balls are put back into the urn. Of course in abstract probability and statistics the word " choice " can be avoided and all can be done without any reference to such a model. But as soon as the abstract theory is to be applied, random choice plays an essential role."

"No statistician present at this moment will have been in doubt about the meaning of my words when I mentioned the common statistical model. It must be a stochastic device producing random results. Tossing coins or a dice or playing at cards are not flexible enough. The most general chance instrument is the urn filled with balls of different colours or with tickets bearing some ciphers or letters. This model is continuously used in our courses as a didactic tool, and in our statistical analyses as a means of translating realistic problems into mathematical ones. In statistical language " urn model " is a standard expression."

"No scientist is as model minded as is the statistician; in no other branch of science is the word model as often and consciously used as in statistics."

"Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world. Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created, and to which it has extended in the learning process of mankind, and, as far as this description is concerned with the learning process of the young generation, it is didactical phenomenology, a way to show the teacher the places where the learner might step into the learning process of mankind."

"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical. Rather than behaving anti-didactically, one should recognise that the learner is entitled to recapitulate in a fashion of mankind. Not in the trivial matter of an abridged version, but equally we cannot require the new generation to start at the point where their predecessors left off."

"[The goal of developmental research is to] consciously experience, describe and justify the cyclic process of development and research so that it can be passed on to others in such a way that they can witness and relive the experience."

"Science should be distinguished from technique and its scientific instrumentation, technology. Science is practised by scientists, and techniques by ‘engineers’ — a term that in our terminology includes physicians, lawyers, and teachers. If for the scientist knowledge and cognition are primary, it is action and construction that characterises the work of the engineer, though in fact his activity may be based on science. In history, technique often preceded science."

"A true aphorism legitimates itself; whoever feels the need to legitimate an aphorism, admits that it is illegal. The surface of an aphorism should conceal profound truth. The claim that everybody can learn everything is superficial, but is as wrong as it can be. As a matter of fact, it is no aphorism but an advertising slogan, and the excuse that it is an aphorism, is a mere wink: in advertising you cannot do without exaggerating. But even as a wink it does not become more true."

"Educational technique needs a philosophy, which is a matter of faith rather than of science."

"Geometry is grasping space. And since it is about the education of children, it is grasping that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it."

"Learners should be allowed to find their own levels and explore the paths leading there with as much and as little guidance as each particular case requires."

"Horizontal mathematising leads from the world of life to the world of symbols."

"[Guided reinvention is] striking a subtle balance between the freedom of inventing and the force of guiding, between allowing the learner to please himself and asking him to please the teacher. Moreover, the learner’s free choice is already restricted by the “re” of “reinvention”. The learner shall invent something that is new to him but well-known to the guide."

"Vertical mathematising is the most likely part of the learning process for the bonds with reality to be loosened and eventually cut."

"In appearance and behaviour, Norbert Wiener was a baroque figure, short, rotund, and myopic, combining these and many qualities in extreme degree. His conversation was a curious mixture of pomposity and wantonness. He was a poor listener. His self-praise was playful, convincing and never offensive. He spoke many languages but was not easy to understand in any of them. He was a famously bad lecturer."

"While studying antiaircraft fire control, Wiener may have conceived the idea of considering the operator as part of the steering mechanism and of applying to him such notions as feedback and stability, which had been devised for mechanical systems and electrical circuits. No doubt this kind of analogy had been operative in Wiener’s mathematical work from the beginning and sometimes had even been productive. As time passed, such flashes of insight were more consciously put to use in a sort of biological research for which Wiener consulted all kinds of people, except mathematicians, whether or not they had anything to do with it. Cybernetics, or Control and Communication in the Animal and the Machine (1948) is a rather eloquent report of these abortive attempts, in the sense that it shows there is not much to be reported. The value and influence of Cybernetics, and other publications of this kind, should not, however, be belittled. It has contributed to popularizing a way of thinking in communication theory terms, such as feedback, information, control, input, output, stability, homeostasis, prediction, and filtering . On the other hand, it also has contributed to spreading mistaken ideas of what mathematics really means"

"Even measured by Wiener's standards Cybernetics is a badly organised work — a collection of misprints, wrong mathematical statements, mistaken formulas, splendid but unrelated ideas, and logical absurdities. It is sad that this work earned Wiener the greater part of his public renown, but this is an afterthought. At that time mathematical readers were more fascinated by the richness of its ideas than by its shortcomings."

"The classic instrument to measure drawn angles and to draw angles of a given measure is the — essentially half a circular ring, subdivided by ray segments into 180 degrees. For reasons I was unable to find out, this instrument has recently been superseded by an isosceles right triangle — called geo-triangle, solid, transparant, made of plastic — with an angular division radiating from the midpoint of the hypotenuse to the other sides. Well, inside the triangle half a circle with the midpoint of the hypotenuse as its centre is indicated, and from the position of the degree numbers it becomes clear that it is the semicircle that really matters. One is inclined to say "an outrageously misleading instrument"..."

"Euclid defines the angle as an inclination of lines…he meant halflines, because otherwise he would not be able to distinguish adjacent angles from each other… Euclid does not know zero angles, nor straight and bigger than straight angles…Euclid takes the liberty of adding angles beyond two and even four right angles; the result cannot be angles according to the original definitions…Nevertheless one feels that Euclid’s angle concept is consistent."

"Grasping spatial gestalts as figures is mathematizing of space. Arranging the properties of a parallelogram such that a particular one pops up to base the others on it in order to arrive at a definition of parallelogram, that is mathematizing the conceptual field of the parallelogram."

"Angles are measured by arcs, such that 360° and 2π correspond to each other."

"Ich vermeinte, man verlange physische Determinationen und nicht abstracte integrationes. Es fängt sich ein verderblicher goût an einzuschleichen, durch welchen die wahren Wissenschaften viel mehr leiden, als sie avancirt werden, und wäre es oft besser für die realem physicam, wenn keine Mathematik auf der Welt wäre."

"Strategy requires thought, tactics require observation."

"I honestly feel very humble when I study Capablanca's games."

"Alekhine is a poet who creates a work of art out of something that would hardly inspire another man to send home a picture post card."

"The academicians—especially in the person of Picard—were carrying through a revolution in observational astronomy made possible by Huygens' astronomical pendulum clock, the filar micrometer perfected (if not invented) by Auzout, and the application of telescopes to large-scale graduated instruments appropriate for the measure of small angles. It was with this equipment that Picard undertook to measure the distance between two localities approximately on the meridian of Paris, to determine the differences in their latitudes, and to deduce from those results the length of degree of meridian. The eminently successful arc measure, marked by a precision thirty to forty times greater than any previously achieved..."

"...the power of this line [the cycloid] to measure time."

"Having converted Galileo's discovery of the isochronism of the pendulum into an accurate timepiece in 1656, Huygens had, in 1662, developed a marine variation employing a short pendulum which had subsequently been subjected to tests at sea with the aid of the English. News of the device having come to Colbert... the new director of France's economic life was determined to secure its advantages for his nation. Accordingly, Huygens was lured to Paris in 1665."

"Like Hooke, Huygens made fundamental improvements to the clock as a time-keeping mechanism; and Hooke invented the first passable for the same purpose. ...Huygens discovered the rings of Saturn, and the formula for centrifugal force. He did important work in mechanics and optics, and one of his merits was that he made young Leibnitz enthusiastic for these subjects."

"The world is my country, to promote science is my religion."

"What a wonderful and amazing Scheme have we here of the magnificent Vastness of the Universe! So many Suns, so many Earths, and every one of them stock’d with so many Herbs, Trees and Animals, and adorn’d with so many Seas and Mountains! And how must our wonder and admiration be encreased when we consider the prodigious distance and multitude of the Stars?"

"Since 'tis certain that Earth and Jupiter have their Water and Clouds, there is no reason why the other Planets should be without them. I can't say that they are exactly of the same nature with our Water; but that they should be liquid their use requires, as their beauty does that they be clear. This Water of ours, in Jupiter or Saturn, would be frozen up instantly by reason of the vast distance of the Sun. Every Planet therefore must have its own Waters of such a temper not liable to Frost."

"Now since in so many Things they... agree, what can be more probable than that in others they agree too; and that the other Planets are as beautiful and as well stock'd with Inhabitants as the Earth? Or what shadow of Reason can there be why they should not?"

"We shall be less apt to admire what this World calls Great, shall nobly despise those Trifles the generality of Men set their Affections on, when we know that there are a multitude of such Earths inhabited and adorned as Well as our own."

"Here we may mount from this dull Earth, and viewing it from on high, consider whether Nature has laid out all her Cost and Finery upon this small Speck of Dirt."

"There are many degrees of Probable, some nearer Truth than others, in the determining of which lies the chief exercise of our Judgment."

"These Gentlemen must be told, that they take too much upon themselves when they pretend to appoint how far and no farther Men shall go in their Searches, and to set bounds to other Mens Industry; as if they knew the Marks that God has placed to Knowledge..."

"It's evident God had no design to make a particular Enumeration in the Holy Scriptures, of all the Works of his Creation."

"A Man that is of Copernicus’s Opinion, that this Earth of ours is a Planet, carry’d round and enlighten’d by the Sun, like the rest of the Planets, cannot but sometimes think, that it’s not improbable that the rest of the Planets have their Dress and Furniture, and perhaps their Inhabitants too as well as this Earth of ours..."

"I esteem his [Newton's] understanding and subtlety highly, but I consider that they have been put to ill use in the greater part of this work, where the author studies things of little use or when he builds on the improbable principle of attraction."

"I had not thought of this regular decrease of gravity, namely that it is as the inverse square of the distance; this is a new and highly remarkable property of gravity."

"One finds in this subject a kind of demonstration which does not carry with it so high a degree of certainty as that employed in geometry; and which differs distinctly from the method employed by geometers in that they prove their propositions by well-established and incontrovertible principles, while here principles are tested by inferences which are derivable from them. The nature of the subject permits of no other treatment. It is possible, however, in this way to establish a probability which is little short of certainty. This is the case when the consequences of the assumed principles are in perfect accord with the observed phenomena, and especially when these verifications are numerous; but above all when one employs the hypothesis to predict new phenomena and finds his expectations realized."