On Extrema I

"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of the great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." (Herbert G Wells, "Mankind in the Making", 1903)

"According to Leibniz our world is the best possible. That is why its laws can be described by extremal principles." (Carl L Siegel)

"In Continuity, it is impossible to distinguish phenomena at their merging-points, so we look for them at their extremes." (Charles Fort, "The Book of the Damned", 1919)

"Change is most sluggish at the extremes precisely because the derivative is zero there." (Steven Strogatz, The Joy of X: A Guided Tour of Mathematics, from One to Infinity, 2012)

"Most practical questions can be reduced to problems of largest and smallest magnitudes […] and it is only by solving these problems that we can satisfy the requirements of practice which always seeks the best, the most convenient." (Pafnuty L Chebyshev)

"[…] nothing takes place in the world whose meaning is not that of some maximum or minimum." (Leonhard Euler)

"The world is not dialectical - it is sworn to extremes, not to equilibrium, sworn to radical antagonism, not to reconciliation or synthesis." (Jean Baudrillard)

"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented." (Leonhard Euler)

"We shall consider the simplest maximum and minimum problem that points to a natural transition from functions of a finite number of variables to magnitudes that depend on an infinite number of variables." (Vito Volterra)

"When a quantity is greatest or least, at that moment its flow neither increases nor decreases." (Isaac Newton)

On Topology IV (More on Topology)

"The young mathematical disciple 'topology' might be of some help in making psychology a real science." (Kurt Lewin, Principles of topological psychology, 1936)

"Topology provides the synergetic means of ascertaining the values of any system of experiences. Topology is the science of fundamental pattern and structural relationships of event constellations." (R Buckminster Fuller, "Operating Manual for Spaceship Earth", 1969)

"Topology is not ‘designed to guide us’ in structure. It is this structure." (Jacques Lacan, "L’Étourdit", 1972)

"No other theory known to science [other than superstring theory] uses such powerful mathematics at such a fundamental level. […] because any unified field theory first must absorb the Riemannian geometry of Einstein’s theory and the Lie groups coming from quantum field theory. […] The new mathematics, which is responsible for the merger of these two theories, is topology, and it is responsible for accomplishing the seemingly impossible task of abolishing the infinities of a quantum theory of gravity." (Michio Kaku, "Hyperspace", 1995)

"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I. Monastyrsky, "Riemann, Topology, and Physics" 2nd Ed., 2008)

"At the basis of the distance concept lies, for example, the concept of convergent point sequence and their defined limits, and one can, by choosing these ideas as those fundamental to point set theory, eliminate the notions of distance." (Felix Hausdorff)

"In every subject one looks for the topological and algebraic structures involved, since these structures form a unifying core for the most varied branches of mathematics." (K Weise and H Noack, "Aspects of Topology")

"Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only." (Henri Poincaré)

"Poetry and code - and mathematics - make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations." (Stephanie Strickland)

"Topology is the study of the modal relations of spatial figures and the laws of connectivity, mutual position, and ordering of points, lines, surfaces, and solids and their parts independently of measure and magnitude relations." (Johann B Listing)

On Topology III (Topology with a Twist)

"The connection of topology with physics is no passing interlude but rather represents a length affair." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language, whose beauty eventually - in retrospect - compensates for all the drudgery." (Volker Runde, "A Taste of Topology", 2005)

"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)

"A child[’s …] first geometrical discoveries are topological…If you ask him to copy a square or a triangle, he draws a closed circle." (Jean Piaget)

"If you wear glasses, and you wake up in the morning and you’re not wearing your glasses, and everything is blurred together, that’s what the indiscrete topology is like." (Anonymous)

"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl)

"Point set topology is a disease from which the human race will soon recover." (Henri Poincaré)

"The true traditional doughnut has the topology of a sphere. It is a matter of taste whether one regards this as having separate internal and external surfaces. The important point is that the inner space should be filled with good raspberry jam. This is also a matter of taste." (Peter B Fellgett)

"Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything." (Edward Witten)

On Topology II (Definitions)

"I believe that we need another analysis properly geometric or linear, which treats PLACE directly the way that algebra treats MAGNITUDE." (Gottfried W Leibniz, 1670s)

"The branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that has been almost unknown up to now; Leibniz spoke of it first, calling it the ‘geometry of position’ (geometria situs). This branch of geometry deals with relations dependent on position; it does not take magnitudes into considerations, nor does it involve calculation with quantities. But as yet no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them." (Leonhard Euler, 1735) 

"Topology is the study of the modal relations of spatial figures and the laws of connectivity, mutual position, and ordering of points, lines, surfaces, and solids and their parts independently of measure and magnitude relations." (Johann B Listing)

"The use of figures is, above all, then, for the purpose of making known certain relations between the objects that we study, and these relations are those which occupy the branch of geometry that we have called Analysis Situs [that is, topology], and which describes the relative situation of points and lines on surfaces, without consideration of their magnitude." (Henri Poincaré, "Analysis Situs", Journal de l'Ecole Polytechnique 1, 1895)

"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)

"Topology begins where sets are implemented with some cohesive properties enabling one to define continuity." (Solomon Lefschetz, "Introduction to Topology", 1949)

"In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure." (David Hilbert, "Geometry and Imagination", 1952)

"Topology is precisely that mathematical discipline which allows a passage from the local to the global." (René Thom)

"Topology studies the properties of geometrical objects that remain unchanged under transformations called homeomorphisms and deformations." (Victor V Prasolov, "Intuitive Topology", 1995)

"Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings but not through breaks or cuts." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table". 2005)

"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I Monastyrsky, "Riemann, Topology, and Physics" 2nd Ed., 2008)

"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)

5 Books 10 Quotes III: Beauty and Symmetry III

James R Newman, "The World of Mathematics Vol. I", 1956
	"In the everyday sense symmetry carries the meaning of balance, proportion, harmony, regularity of form. Beauty is sometimes linked with symmetry, but the relationship is not very illuminating since beauty is an even vaguer quality than symmetry."
"Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection." (Herman Weyl, "Symmetry")
James R Newman, "The World of Mathematics Vol. II", 1956
	"Mathematicians study their problems on account of their intrinsic interest, and develop their theories on account of their beauty." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?")
"If we seek a cause wherever we perceive symmetry, it is not that we regard a symmetrical event as less possible than the others, but, since this event ought to be the effect of a regular cause or that of chance, the first of these suppositions is more probable than the second." (Pierre-Simon de Laplace, "Concerning Probability")
James R Newman, "The World of Mathematics Vol III", 1956
	"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess – it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old")
"The world of ideas which it discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance." (James J Sylvester, "The Study That Knows Nothing of Observation")
James R Newman, "The World of Mathematics Vol IV", 1956
	"[...] what are the mathematic entities to which we attribute this character of beauty and elegance, and which are capable of developing in us a sort of esthetic emotion? They are those whose elements are harmoniously disposed so that the mind without effort can embrace their totality while realizing the details. This harmony 'is at once a satisfaction of our esthetic needs and an aid to the mind, sustaining and guiding." (Henri Poincare, "Mathematical Creation")
"When, for instance, I see a symmetrical object, I feel its pleasurable quality, but do not need to assert explicitly to myself, ‘How symmetrical!’. This characteristic feature may be explained as follows. In the course of individual experience it is found generally that symmetrical objects possess exceptional and desirable qualities. Thus our own bodies are not regarded as perfectly formed unless they are symmetrical. Furthermore, the visual and tactual technique by which we perceive the symmetry of various objects is uniform, highly developed, and almost instantaneously applied. It is this technique which forms the associative 'pointer.' In consequence of it, the perception of any symmetrical object is accompanied by an intuitive aesthetic feeling of positive tone." (George D Birkhoff, "Mathematics of Aesthetics")
K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997
	"Math has its own inherent logic, its own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It’s clean, neat, above it all. It lives in an ideal universe built on the geometer’s perfect circles and polygons, the number theorist’s perfect sets. It matters not that these objects don’t exist in the real world. They are articles of faith."
"How deep truths can be defined as invariants – things that do not change no matter what; how invariants are defined by symmetries, which in turn define which properties of nature are conserved, no matter what. These are the selfsame symmetries that appeal to the senses in art and music and natural forms like snowflakes and galaxies. The fundamental truths are based on symmetry, and there’s a deep kind of beauty in that." Previous Post <<||>> Next Post

On Algebra: Definitions I

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved." (Omar Khayyam [quoted by J.J. Winter and W. Arafat, "The Algebra of ‘Umar Khayyam’", Journal of the Royal Asiatic Society of Bengal, Volume 41, 1950)

 "By the help of God and with His precious assistance I say that algebra is a scientific art. The objects with which it deals are absolute numbers and (geometrical) magnitudes which, though themselves unknown, are related to things which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. […] What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of algebra as stated above." (Omar Khayyam [quoted by Daoud Suleiman Kasir in "The Algebra of Omar Khayyam", 1931)

"[…] the sciences that are expressed by numbers or by other small signs, are easily learned; and without doubt this facility rather than its demonstrability is what has made the fortune of algebra." (Julien Offray de La Mettrie, "Man a Machine", 1747)

"Algebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose, and found convenient. It is called an Universal Arithmetic, and proceeds by Operations and Rules similar to those in Common Arithmetic, founded upon the same Principles." (Colin Maclaurin, "A Treatise on Algebra", 1748)

"Algebra is a way to bring us to certainty in mathematics; but must it be presently condemned as an ill way, because there are some questions in mathematics, which a man cannot come to certainty in by the way of Algebra?" (John Locke)

"[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values." (Auguste Comte, "Philosophy of Mathematics", 1851)

“Algebra is but written geometry and geometry is but figured algebra.” (Sophie Germain, "Mémoire sur les Surfaces Élastiques", 1880)

"Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghost-like character of the real variable." (Alfred N Whitehead, "Essays in Science and Philosophy", 1948)

"[…] algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world." (Simone Weil, "The Organization of Thought", 1974)

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." (Michael F Atiyah, 2004)

On Complex Numbers II

“I have obtained these values by a singular analogy based on the passages from the real to the imaginary, passages that can be considered as a means of discovery.” (Pierre-Simon Laplace)

“I did not understand how such a quantity could be real, when imaginary or impossible numbers were used to express it.” (Gottfried W Leibniz) 

“But it is just that the Roots of Equations should be often impossible (complex), lest they should exhibit the cases of Problems that are impossible as if they were possible." (Isaac Newton, “Universal Mathematic” 2nd Ed., 1728)

"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers." (Carl F Gauss, cca. 1831)

“[…] such numbers, which by their natures are impossible, are ordinarily called imaginary or fanciful numbers, because they exist only in the imagination.”  (Leohnard Euler, 1732)

“We completely repudiate the symbol √-1, abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.” (Augustin-Louis Cauchy, 1847)

“Analysis […] would lose immensely in beauty and balance and would be forced to add very hampering restrictions to truths which would hold generally otherwise, if […] imaginary quantities were to be neglected.” (Garrett Birkhoff, 1973)

"It is a curious fact that the first introduction of the imaginaries occurred in the theory of cubic equations, in the case where it was clear that real solutions existed though in an unrecognizable form, and not in the theory of quadratic equations, where our present textbooks introduce them." (Dirk J Struik, “A Concise History of Mathematics” Vol. I, 1948)

"We have shown the symbol √-1 to be void of meaning, or rather self-contradictory and absurd. Nevertheless, by means of such symbols, a part of algebra is established which is of great utility. It depends upon the fact, which must be verified by experience, that the common rules of algebra may be applied to these expressions without leading to any false results." (Augustus De Morgan)

"The word ‘imaginary’ is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used. Books on elementary algebra give a simple interpretation of imaginary numbers in terms of rotations. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’." (Philip E B Jourdain, "The Nature of Mathematics" in [James R Newman, “The World of Mathematics” Vol. I, 1956])

See also:
5 Books 10 Quotes: Complex Numbers V
Complex Numbers IV
Complex Numbers III
Complex Numbers I

On Learning: Aphorisms

"For the things we have to learn before we can do, we learn by doing." (Aristotle, "Nicomachean Ethics", Book II, 349 BC)

"A little learning is a dangerous thing." (Alexander Pope)

"A few moments to learn, a lifetime to master." (proverb)

"Poor is the pupil who does not surpass his master." (Leonardo da Vinci)

"Much learning does not teach understanding." (Heraclitus, "Fragments", 6th c. BC)

"The learning of many things does not teach intelligence […]." (Pythagoras of Samos)

"Much learning does not teach a man to have intelligence." (Heraclitus of Ephesus)

"Curiosity is the wick in the candle of learning." (William A Ward)

"Learning is a treasure which accompanies its owner everywhere." (proverb)

"Learning is its own exceeding great reward." (William Hazlitt, "The Plain Speaker", 1826)

"What we learn with pleasure we never forget." (Louis Mercier)

"Whatever is good to know is difficult to learn." (Greek proverb)

"We learn to walk by stumbling." (Bulgarian proverb)

"He who is afraid to ask is ashamed of learning." (Danish proverb)

"It takes ten pounds of common sense to carry one pound of learning." (Persian proverb)

"He who has imagination without learning has wings but no feet." (Joseph Joubert)

"Learning is not attained by chance. It must be sought for with ardor and attended to with diligence." (Abigail Adams)

"Learning hath gained most by those books by which the printers have lost." (Thomas Fuller)

"[…] education is not something which the teacher does, but that it is a natural process which develops spontaneously in the human being." (Maria Montessori)

On Chess I: Chess and Mathematics I

"A chess problem is genuine mathematics, but it is in some way ‘trivial’ mathematics. However, ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful –‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men." (Friedrich Waismann & Karl Menger, "Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics", 1951)

"It [mathematics] is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst." (Norbert Wiener, "Ex-prodigy: My Childhood and Youth", 1953)

"The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation." (Norbert Wiener, "Ex-Prodigy: My Childhood and Youth", 1953)

"Chess combines the beauty of mathematical structure with the recreational delights of a competitive game." (Martin Gardner, "Mathematics, Magic, and Mystery", 1956)

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess - it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1956)

"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"[…] mathematics can never prove anything. No mathematics has any content. All any mathematics can do is – sometimes – turn out to be useful in describing some aspects of our so-called ‘physical universe’. That is a bonus; most forms of mathematics are as meaning-free as chess." (Robert A Heinlein, "The Number of the Beast", 1980)

"[…] mathematics is not best learned passively; you don’t sop it up like a romance novel. You’ve got to go out to it, aggressive, and alert, like a chess master pursuing checkmate." (Robert Kanigel, "The Man Who Knew Infinity: A Life of the Genius Ramanujan", 1991)

"Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, "The Mathematical Experience", 1995)

Beyond the History of Mathematics I

"The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store." (Florian Cajori, "A History of Mathematics", 1893)

 "The whole history of the development of mathematics has been a history of the destruction of old definitions, old hobbies, old idols." (David E Smith, American Mathematical Monthly, Vol. 1, No 1, 1894)

"The history of mathematics is the mirror of civilization." (Lancelot Hogben, "Mathematics for the Million", 1917)

"[…] a history of mathematics is largely a history of discoveries which no longer exist as separate items, but are merged into some more modern generalization, these discoveries have not been forgotten or made valueless. They are not dead, but transmuted." (John W N Sullivan, "The History of Mathematics in Europe", 1925)

"In the history of mathematics, the ‘how’ always preceded the ‘why’, the technique of the subject preceded its philosophy." (Tobias Dantzig, "Number: The Language of Science", 1930)

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the […] development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes." (Julian L Coolidge, "A History of Geometrical Methods", 1940)

"The study of the history of mathematics shows clearly enough that after each period of research and extension there follows a period of review and synthesis during which more general methods are evolved and the foundation of mathematics consolidated." (Gustave Choquet, "What is Modern Mathematics", 1963)

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"Mathematics is a vast adventure of ideas; its history reflects some of the noblest thoughts of countless generations." (Dirk J Struik, "A Concise History of Mathematics", 1967)

"Under the present dominance of formalism, one is tempted to paraphrase Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the of mathematics, has become empty." (Imre Lakatos, "Proofs and Refutations: The Logic of Mathematical Discovery", 1976)

Previous Post <<||>> Next Post