Mathematics

"Ancient Indian culture has regarded the science of numbers as the noblest of its arts … A thousand years ahead of Europeans, Indian savants knew that zero and infinity were mutually inverse notions. In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit word used to describe numbers and the science of numbers bears witn The early passion which Indian civilization had for high numbers was a significant factor contributing to the discovery of the place-value system, and not only offered the Indians the incentive to go beyond the calculable physical world, but also led to an understanding much earlier than in our civilization of the notion of mathematical infinity itself."

"It is clear how much we owe to this brilliant civilization, and not only in the field of arithmetic; by opening the way to the generalization of the concept of the number, the Indian scholars enabled the rapid development of mathematics and exact sciences. The discoveries of these men doubtless required much time and imagination and, above all, a great ability for abstract thinking. These major discoveries took place within an environment which was at once mystical, philosophical, religious, cosmological, mythological and metaphysical."

"Because the Greeks were ‘ingenious and inventive’, they must have developed mathematics, and that is what we must assume even in the absence of evidence. And because the Hindus were backward and primitive, they could not have created mathematics, and that is what we must assume even in the presence of evidence."

"In the whole history of Mathematics, there has been no more revolutionary step than the one which the Indian made when they invented the sign ‘0’ for the empty column of the counting frame."

"In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”"

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"C'est de l'Inde que nous vient l'ingénieuse méthode d'exprimer tous les nombres avec dix caractères, en leur donnant à la fois, une valeur absolue et une valeur de position; idée fine et importante, qui nous paraît maîntenant si simple, que nous en sentons à peine, le mérite. Mais cette simplicité même, et l'extrême facilité qui en résulte pour tous les calculs, placent notre système d'arithmétique au premier rang des inventions utiles; et l'on appréciera la difficulté d'y parvenir, si l'on considère qu'il a échappé au génie d'Archimède et d'Apollonius, deux des plus grands hommes dont l'antiquité s'honore."

"In the Vedic Age, India was very religious, but it was also ahead of the rest in mathematics and astronomy. Thus, the geometry of the Shulba Sutras, geometrical appendices to the manuals of ritual (Shrauta Sutras), include the oldest known formulation of the theorem named after Pythagoras, developed in the context of Vedic altar-building. Modern Hindus are fond of recalling this scientific element in their tradition, e.g. by quoting Carl Sagan: “Hindu cosmology gives a time-scale for the earth and the universe which is consonant with that of modern scientific cosmology”, as opposed to the limited Biblical-Quranic cosmology, which was protected against more far-sighted alternatives by a vigilant religious orthodoxy."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"My confidence in our shared future is grounded in my respect for India’s treasured past—a civilization that has been shaping the world for thousands of years. Indians unlocked the intricacies of the human body and the vastness of our universe. And it is no exaggeration to say that our information age is rooted in Indian innovations—including the number zero."

"Through the necessity of accurately laying out the open-air site of a sacrifice Indians very early evolved a simple system of geometry, but in the sphere of practical knowledge the-world owes most to India in the realm of mathematics, which were developed in Gupta times to a stage more advanced than that reached by any other nation of antiquity. ‘The success of Indian mathematics was mainly due to the fact that the Indians had a clear conception of abstract number, as distinct from numerical quantity of objects or spatial extension. While Greek mathematical science was largely based on mensuration and geometry, India transcended these conceptions quite early, and, with the aid of a simple numeral notation, devised a rudimentary algebra which allowed more complicated calculations than were possible to the Greeks, and led to the study of number for its own sake."

"Medieval Indian mathematicians, such as Brahmagupta (7th century), Mahavira (9th century) and Bhüskara (19th century), made several discoveries which in Europe were not known until the Renaissance or later, They understood the import of positive and negative quantities, evolved sound systems of extracting square and cube roots, and could solve quadratic and certain types of indeterminate equations."

"The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately"

"It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit."

"The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings. It has been universally adopted. ...It is the nearest thing we have to a universal language."

"In the Surya Siddhanta is contained a system of trigonometry which not only goes beyond anything known to the Greeks, but involves theorem which were not discovered in Europe till two centuries ago."

"We owe a lot to the Indians who taught us how to count, without which no worthwhile scientific discovery could have been made."

"In order to instill a proper and well-founded pride in Hindus, it is (once more) most important to restore the truth about Hindu history, especially about Hindu society's glorious achievements. In technology, it cannot match China, which was the world leader until a mere three, four centuries ago. But in abstract sciences like linguistics, logic, mathematics, Hindu culture has been the chief pioneer."

"Because the links between a convolution integral and a Laplace or are so important... we briefly present Borel's (1899) work on a "Laplace like" transform. Note Mellin's work (1896)... was unknown to Borel... Borel defined two functions f(z) and g(z) by their following Laplace integrals...:f(z) = \int\limits_{0}^{+\infty}F(u)e^{-u/z}\, \frac{du}{z}; \quad g(z) = \int\limits_{0}^{+\infty}G(v)e^{-v/z}\, \frac{dv}{z}and then showed that the convolution integral is H(x) = \int\limits_{0}^{x}F(t)G(x-t)\,dt. The Laplace transform of the convolution integral H(x) reduced to a simple product of the two separate transforms f(z) and g(z). Borel failed to see all the possibilities of his theorem. Volterra... also did not see the possible uses... But... in 1920, Doetsch produced a doctoral thesis... on Borel's summability theory of diverging series. Doetsch knew Borel's proof and was able to introduce modern, proper mathematical ideas on convolution integrals and Laplace transforms. The word Faltung was first introduced by Doetsch and Bernstein in 1920. The Laplace transform... and the Fourier transform... are both adequate tools for evaluating a convolution integral. ...Doetsch would introduce the convolution integral by analogy with a between two ..."

"[T]he so-called ... finds applications in finding the sum of infinite series, the asymptotic value of an integral involving a large parameter, signal analysis, and imaging technique. ...In , [it] is an important tool in studying the distributions of products of two s. In particular, the Mellin transform of the product of two independent random variables equals the product of the Mellin transforms of the two variables. The Mellin transform is closely related to the two-sided Laplace transform. The so-called Mellin transform has been considered by Laplace and used by Riemann in his study of the zeta function. It was, however, Mellin who provided a systematic formation of the transform and its application to solve ODEs and to estimate the value of integrals. ... ...was a student of Mittang-Leffler and Weierstrass. The kernel for the Mellin transform is K(s,t) = t^{s-1}The Mellin transform and its inversion are defined as:F(s) = M[f(x)] = \int\limits_{0}^{\infty}x^{s-1} f(x)\, dx, f(x) = M^{-1}[F(s)] = \frac{1}{2\pi i} \int\limits_{c-i \infty}^{c+i \infty}x^{-s} F(s)\, dswhere c is a constant that lies on the right of all singularities of the kernel function. With the proper change of variables, the Mellin transform can be converted to a two-sided Laplace transform. In particular, a two-sided Laplace transform can be written asL[g(t)] = \int\limits_{-\infty}^{+\infty} g(t)e^{-st}\, dt"

"Doetsch and Bernstein, beginning from the early 1920s, worked together on the subject of Laplace transformation, integral equations and s. They published several papers together, in which the connection between the Laplace transformation and convolution, i.e., Faltung, is discussed often. ...[T]he Laplace transformation of a function f(t), denoted by \mathcal{L}(f), where f is defined for all real numbers t > 0, is the following complex function of F:F(t) = \mathcal{L}(f) = \int\limits_{0}^{\infty}e^{-tu} f(u)\, du.The relation between the Laplace transformation and convolution is...:\mathcal{L}(f*g) = \mathcal{L}(f) \cdot \mathcal{L}(g). ...In 1922, they remark, regarding the Laplace transformation: "[w]e distinguish the functions of a subfield and a field [Oberkörper], which are connected by a certain process. The operations in the subfield are actual, proper [eigentliche] ones, which are only symbolic in the field, but which in certain cases are capable of an actual analytical representation.""

"A characterization... of the main features of the maverick tradition could be..; a. antifoudationalism, i.e. there is no certain foundation... mathematics is... fallible... b. anti-logicism, i.e. mathematical logic cannot provide the tools for an adequate analysis of mathematics and its development; c. attention to mathematical practice: only detailed analysis and reconstruction of large and significant parts of mathematical practice can provide a philosophy of mathematics..."

"Already in the 1960s, first with Lakatos and later through a group of 'maverick' philosophers of mathematics (Kitcher, Tymoczko, and others), a strong reaction set in against the philosophy of mathematics conceived as foundation of mathematics. ...What these philosophers called for was an analysis of mathematics that was more faithful to its historical development."

"There is an interesting analogy... with the philosophy of the natural sciences, which has flourished under the combined influence of both general methodology and classical metaphysical questions (realism vs. antirealism, space, time, causation, etc.) interacting with detailed case studies in... (physics, biology, chemistry, etc.)... [C]ase studies both historical (studies of Einstein's relativity, Maxwell's electromagnetic theory, , etc.). By contrast, with few exceptions, philosophy of mathematics has developed without the corresponding detailed case studies."

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed... were made explicit when logic was formalized early in the this century... These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a... conjecture. ... arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. ...Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof."

"The dialogue takes place... The class gets interested in a Problem: Is there a relation between the number of vertices V, the number of edges E and the number of faces F of a polyhedra—particularly of regular polyhedra—analogous to the trivial relation between the number of vertices and edges of polygons, namely, that there are as many edges as vertices: V = E? ...After much tiral and error they notice that for all regular polyhedra V - E + F = 2. Somebody guesses that this may apply for any polyhedron whatever. Others try to falsify [test] this conjecture... it holds good. The results corroborate the conjecture, and suggest that it could be proved. It is at this point—after the stages problem and conjecture—that we... offer a proof."

"By "philosophy of mathematics" I mean the specific set of concepts, categories, and theories employed, implicitly or explicitly, by philosophers and mathematicians in their discourse about mathematics. Understood in this way, philosophy of mathematics would include, among other things, some rather ethereal discussions on the nature of numbers by several hermetic philosophers, and the status of various notions, including number, space, infinity, according to the philosophers and mathematicians operating in the seventeenth century, as well as several other areas of investigation. Therefore I must introduce a qualification contained in the concept of "mathematical practice." ...I use this term as it is used today in mathematical logic and philosophy of mathematics, simply to indicate mathematics as it is done, not as it should be done according to some preconceived philosophical viewpoint. ...[F]ar from eliminating the philosophical questions, an interest in mathematical practice has actually extended their range. Addressing the issue of mathematical practice requires a detailed knowledge of the mathematical literature of the period."

"[M]athematical apriorism... has not gone completely unquestioned. J. S. Mill attempted to argue that mathematics is an empirical science, thereby making himself the subject of Frege's biting criticism. More recently, W. V. Quine, , and Imre Lakatos have... challenged the... thesis. However, none of these have offered a systematic account of our mathematical knowledge. ...[T]he alternative...—mathematical empiricism—has never been given a detailed articulation. I shall try... I have gained much from insights of Quine and Putnam... [and] learned from Mill... My quarrel with earlier empiricists is, for the most part, that they have been incomplete rather than mistaken."

"Philosophy of mathematics appears to become a microcosm for the most general and central issues in philosophy—issues in epistemology, metaphysics, and philosophy of language—and the study of those parts of mathematics to which philosophers... most often attend (logic, set theory, aritmetic) seems designed to test the merits of large philosophical views about the existence of abstract entities of the tenability of a certain picture of human knowledge. ...[A]re [there] not other tasks ...that arise either from the current practice of mathematics or the history of the subject... the kinds of issues that occupy those who study the other branches of human knowledge... as: How does mathematical knowledge grow? What is mathematical progress? What makes some mathematical ideas (or theories) better than others? What is mathematical explanation?"

"Philosophers and logicians have been so busy trying to provide mathematics with a "foundation" in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don't think mathematics is unclear; I don't think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs "foundations." The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. The systems are doubtless interesting as intelIectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won't, but one can always hope) that the various systems of mathematicaI philosophy, without exception, need not be taken seriously."

"Philosophy of mathematics has been slow to draw the analogy from the Kuhnian sea-change in philosophy of science, but during the last decade, a growing number of younger philosophers of mathematics have turned their attention to the history of mathematics and tried to make use of it in their investigations. The most exciting of these concern how mathematical discovery takes place, how new discoveries are structured and integrated into existing knowledge, and what light these processes shed on the existence and applicability of mathematical objects."

"According to the dominant view, the reflection on mathematics is the task of a specialized discipline, the philosophy of mathematics, starting with Frege, characterized by its own problems and methods, and in a sense “the easiest part of philosophy”. In this view, the philosophy of mathematics “is a specialized area of philosophy... Many of the questions that arise within it... occur within the philosophy of mathematics in an especially pure, or especially simplified, form”. ...[However,] like applied mathematics, pure mathematics draws its concepts from experience, observation, scientific theories and even economics. The questions considered by the reflection on mathematics have, therefore, all the impurity and complexity of which philosophical problems are capable."

"We are still in the aftermath of the great foundationist controversies of the early twentieth century. Formalism, and , each left its trace in the form of a certain mathematical research program that ultimately made its own contribution to the corpus of mathematics... As philosophical programs, as attempts to establish a secure foundation for mathematical knowledge, all have run their course and petered out or dried up. Yet there remains, as a residue, an unstated consensus that the philosophy of mathematics is research on the foundations of mathematics. If I find [that] uninteresting or irrelevant, I conclude that I'm simply not interested in philosophy (thereby depriving myself of any chance of confronting my own uncertainties about the meaning, nature, purpose or significance of mathematical research)."

"If one excludes the philosophy of science from the ambit of a study of its history, then one is obliged to do history with the default philosophy of science. In our case this means that one must then accept the present-day Western philosophy of mathematics, not only as a privileged philosophy, but as the only possible philosophy of mathematics."

"The doctrine that mathematical knowledge is a priori—mathematical apriorism...—has been articulated in many different ways... To name only the most prominent defenders... since the seventeenth century, Descartes, Locke, Berkeley, Kant, Frege, Hilbert, Brouwer, and Carnap... Most of the disputes... conducted in our century represent internal differences... among apriorists. ...I shall offer a picture of mathematical knowledge which rejects mathematical apriorism."

"The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!"

"Now Gödel's proof, Russell's original paradox, all these things, all stem from one common root which is inherent in all symbolic languages, including the language we use. ...the problem which dogs all formal systems, the problem of self-reference; that is, the language can be used to refer to sentences in the language. Indeed, between 1900 and 1910 Russell tried to forbid this, to say you cannot do mathematics if you can do that, and so he invented the theory of types. Of course, no sooner had he invented it than it turned up you could not do mathematics at all if you obeyed the theory of types. So then he had to put in an , which allows a certain amount of self-reference. And by this time everyone was pretty bored."

"In the summer of 1914 I attended Frege's course, Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions, and proofs, even in the works of the more prominent mathematicians. ...Unfortunately, his admonitions go unheeded even today."

"On the subject of demonstrations, it is to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically: as is particularly noticed by Bhaskara himself, towards the close of his algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which involve a factum of two unknown quantities."

"Proofs have gaps and are... inherently incomplete and sometimes wrong. ...There is another reason ...Humans err. ...and others do not necessarily notice our mistakes. ...This suggests an important reason why "more elementary" proofs are better... The more elementary... the easier it is to check, and the more reliable its verification."

"The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the states that every polynomial of degree n with complex coefficients has exactly n complex roots. D'Alembert published a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong. Gauss published his first proof... in 1799, but this, too, had gaps. Gauss's subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof... was correct. Why?"

"Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored..."

"Comparatively few of the propositions and proofs in the Elements are [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions."

"Social pressure often hides mistakes in proofs. In a seminar lecture... interrupt the speaker... to ask for more explanation... [often] the response will be that it is "obvious" or "clear" or "follows easily..." Occasionally... a look... conveys the message that the questioner is an idiot. That's why most mathematicians sit quietly... understanding very little... and applauding politely... One of the joys of Gel'fand's seminar... he would constantly interrupt... to ask questions and give elementary examples... [T]he audience would actually learn some mathematics."

"M. Poincaré finds the answer to these questions in the so-called 'mathematical induction' which proceeds from the particular to the more general, but at the same time does so by steps of the highest degree of certitude. In this process he sees the creative force of mathematics, which leads to real proofs and not mere sterile verifications. ...No arithmetic could be built up, rejecting the axiom of mathematical induction, as the non-Euclidean geometries have been built up, rejecting the postulate of Euclid."

"An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it."

"Under the same assumptions made in the Chord papers, the [SIGCOMM] version of the protocol is not correct, and not one of the properties claimed invariant in [PODC] is actually invariantly true of it."

"There are... masterpieces of... exposition... Two examples... are Weil's Number Theory for Beginners... and Artin's '. Mathematics can be done scrupulously."

"Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different... specifically Greek character... in the Pythagorean school and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to a state of perfection, beauty and exactness, which we admire in the elements of Euclid. ...the mathematical method of proof served as a prototype for Plato's dialectics and for Aristotle's logic."