Education

"The first condition which is necessary in order that phenomena may admit of mathematical laws, susceptible of being discovered... is, that their different quantities should admit of being expressed by fixed numbers."

"Having thus exhibited the essential object and the principal composition of mathematical science, as well as its general relations with... natural philosophy, we have now to pass to... examination of the great sciences of which it is composed."

"John Stuart Mill, A System of Logic (1843) p. 369 of the 1846 edition."

"The want of a comprehensive map of the wide region of mathematical science—a bird's-eye view of its leading features, and of the true bearings and relations of all its parts—is felt by every thoughtful student. He is like the visitor to a great city, who gets no just idea of its extent and situation till he has seen it from some commanding eminence. To have a panoramic view of the whole district—presenting at one glance all the parts in due co-ordination, and the darkest nooks clearly shown—is invaluable to either traveller or student. It is this which has been most perfectly accomplished for mathematical science by the author whose work is here presented."

"[T]he whole of organic physics, and probably also the most complicated parts of inorganic physics, are necessarily inaccessible, by their nature, to our mathematical analysis, by reason of the extreme numerical variability of the corresponding phenomena."

"Passages which are obscure at the first reading will brighten up at the second; and as ...[the student's] studies cover a larger portion of... Mathematics, he will see more and more clearly their relations to one another, and to those which he is next to take up."

"The most complicated problem... of the modification produced in the motions of two bodies tending towards each other by virtue of their gravitation, by the influence of a third body acting on both of them in the same manner, is much less complex than the most simple terrestrial problem. And, nevertheless, even it presents difficulties so great that we yet possess only approximate solutions..."

"If... our solar system had been composed of a greater number of planets concentrated into a less space, and nearly equal in mass; if their orbits had presented very different inclinations, and considerable eccentricities; if these bodies had been of a more complicated form, such as very eccentric ellipsoids... supposing the same law of gravitation to exist, we should not yet have succeeded in subjecting the... celestial phenomena to our mathematical analysis, and probably we should not even have been able to disentangle the present principal law."

"[O]ur conceptions having been so generalized and simplified that a single analytical question, abstractly resolved, contains the implicit solution of a great number of diverse physical questions..."

"Mathematical analysis is, then, the true rational basis of the entire system of our actual knowledge. It constitutes the first and the most perfect of all the fundamental sciences. The ideas with which it occupies itself are the most universal, the most abstract, and the most simple which it is possible for us to conceive."

"[T]he human mind must necessarily acquire by these means a greater facility in perceiving relations between phenomena which at first appeared entirely distinct from one another."

"We are now able to define mathematical science... by assigning... as its object the indirect measurement of magnitudes, and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them."

"This enunciation, instead of giving the idea of only an art, as do... the ordinary definitions, characterizes... a true science, and shows it... to be composed of an immense chain of intellectual operations..."

"According[ly]... the spirit of mathematics consists in... regarding all the quantities which any phenomenon can present, as connected and interwoven..."

"Could we... without the aid of analysis, perceive the least resemblance between the determination of the direction of a curve at each of its points and that of the velocity acquired by a body at every instant of its variable motion? and yet these questions, however different they may be, compose but one in the eyes of the geometer."

"Astronomical Facts. It is by such calculations that man has been able to ascertain, not only the distances from the planets to the earth, and, consequently, from each other, but their actual magnitude, their true figure... their respective masses, their mean densities, the principal circumstances of the fall of heavy bodies on the surface of each of them, &c."

"The distance being once determined, the knowledge of it will frequently be sufficient for obtaining new quantities, which will become the subject of new mathematical questions. Thus, when we know at what distance any object is situated... its apparent diameter will... permit us to determine indirectly its real dimensions, however inaccessible it may be, and, by... analogous investigations, its surface... volume... weight, and a number of other properties... which seemed forbidden to us."

"By the power of mathematical theories, all these different results, and many others... have required no other direct measurements than... a very small number of straight lines, suitably chosen, and of a greater number of angles."

"[T]he knowledge of the desired distance, instead of being obtained directly, will be the result of a mathematical calculation, which will consist in deducing it from the observed elements by means of the relation which connects it with them."

"[C]alculation will become successively... more complicated, if the parts... supposed... known cannot themselves be determined (as is most frequently the case) except in an indirect manner, by the aid of new auxiliary systems, the number of which... becomes... considerable."

"[I]f we did not fear to multiply calculations unnecessarily... the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles."

"Mathematical science... pushes to the highest possible degree the same kind of researches which are pursued, in degrees more or less inferior, by every real science..."

"The high relative perfection of mathematical analysis... is not due, as some have thought, to the nature of the signs [mathematical notation] which are employed as instruments of reasoning, eminently concise and general... [A]ll great analytical ideas have been formed without the algebraic signs having been of any essential aid, except for working them out after the mind had conceived them."

"[O]n many occasions the... mind is obliged to establish a long series of intermediates between the system of unknown magnitudes which are the final objects of its researches, and the system of magnitudes susceptible of direct measurement, by whose means we... determine the first... which at first... appear to have no connexion."

"[T]his indirect determination of magnitudes may be indirect in very different degrees."

"Falling Bodies. ...The mind ...perceives that the two quantities which it presents— ...the height from which a body has fallen, and the time of its fall—are necessarily connected ...[I]n the language of geometers, that they are "functions" of each other. The phenomenon... gives rise then to a mathematical question... in substituting for the direct measurement of one... when it is impossible, the measurement of the other. ...[T]hus ...we may determine indirectly the depth of a precipice, by merely measuring the time that a heavy body would occupy in falling ...On other occasions it is the height ...will be easy to ascertain, while the time of the fall could not be observed directly; then the same phenomenon would give rise to the inverse question ..."

"The Object of Mathematics. Measuring Magnitudes. According to this definition... the science of mathematics—vast and profound as it is... instead of being an immense concatenation of prolonged mental labours... [of] our intellectual activity, would seem to consist of a simple series of mechanical processes for obtaining directly the ratios of the quantities to be measured to those by which we wish to measure... by... operations... similar... to the superposition of lines, as practiced by the carpenter with his rule."

"General Method. The general method... and evidently the only one conceivable, to ascertain magnitudes which do not admit of a direct measurement, consists in connecting them with others which are susceptible of being determined immediately, and by means of which we succeed in discovering the first through the relations which subsist between the two. Such is the precise object of mathematical science viewed as a whole."

"In this example the mathematical question is very simple... when we do not pay attention to the variation in the intensity of gravity, or the resistance of the fluid which the body passes through... But, to extend the question, we have only to consider the same phenomenon in its greatest generality..."

"[T]he high perfection to which solar astronomy has been able to elevate itself... is... essentially due to... all the particular, and... accidental facilities presented by the peculiarly favourable constitution of our planetary system. The planets... are quite few in number, and their masses... very unequal, and much less than that of the sun; they are... very distant from one another; they have forms almost spherical; their orbits are nearly circular, and only slightly inclined to each other, and so on. It results from all these circumstances that the perturbations are generally inconsiderable, and that... it is usually sufficient to take into the account, in connexion with the action of the sun... the influence of only one other planet..."

"[B]eing able to pass over the line from one end of it to the other, in order to apply the unit of measurement to its whole length... excludes... the greater part of the distances which interest us... all the distances between the celestial bodies, or from any one of them to the earth; and... even the greater number of terrestrial distances... so frequently inaccessible."

"Inaccessible Distances. ...[T]o determine a distance which is not susceptible of direct measurement; it will be ...conceived as making part of a figure, or ...system of lines, chosen ...such ...that all its other parts may be observed directly; thus, in the case ...most simple, and to which all ...others may be ...reduced, the proposed distance will be considered as belonging to a triangle, in which we can determine directly either another side and two angles, or two sides and one angle."

"This science, although nearer perfection than any other, is really little advanced as yet, so that this object is rarely attained in a manner completely satisfactory."

"It is this fact which makes necessary the formation of mathematical science... for the human mind has been compelled to renounce, in almost all cases, the direct measurement of magnitudes, and to seek to determine them indirectly, and it is thus... led to the creation of mathematics."

"The error of this definition consists in presenting as direct an object which is almost always, on the contrary, very indirect."

"The superior perfection of the science of the calculus is due principally to the extreme simplicity of the ideas which it considers, by whatever signs they may be expressed; so that there is not the least hope, by any artifice of scientific language, of perfecting to the same degree theories which refer to more complex subjects, and which are necessarily condemned by their nature to a greater or less logical inferiority."

"Every precise idea of fixed numbers is truly out of place in the phenomena of living bodies... when we attach any importance to the exact relations of the values assigned."

"The education of women, in particular their being awakened to new values, their being trained for new professions, their being awakened to their rights—all this is anathema; it is held to be injurious to them, in fact it is declared to be the way to disrupting society and undermining Islam."

"In these modern days there is a greater impetus towards higher education on the European lines, and the trend of opinion is strong towards women getting this higher education. Of course, there are some people in India who do not want it, but those who do want it carried the day. It is a strange fact that Oxford and Cambridge are closed to women today, so are Harvard and Yale; but Calcutta University opened its doors to women more than twenty years ago."

"The parents who send their daughters to college are the enemies of their daughters, not their friends... There is no doubt, that a collegiate girl becomes extremely free, purdahless, immodest and shameless. This is the general consequence of English education and college atmosphere... A girl who loses modesty loses everything... Modesty and faith—they are inseparable companions; when either of them is taken away, the other too goes away."

"Youth interest in civic engagement is soaring among the generation that the global volunteering nonprofit Points of Light says was already the most active in history. More than half (53%) of Generation Z individuals said they wanted to get more involved in their communities post-COVID, which was higher than any other generation, according to a 2020 Points of Light survey. “If there is something that is harming us directly, we should be the ones to take charge,’’ said Isaiah Llamas, a recent high school graduate who helped facilitate a spring youth leadership session in Albuquerque. The New Mexico meeting was one of six around the nation co-hosted and funded by America’s Promise Alliance, a national network of groups working to improve conditions for young people... Young people – and adults who support them – say they’re trying to use the pandemic as an opportunity to organize, connect and plan for a better future. “Our generation is more aware and takes the time to understand each other and advocate for diversity, and not division,” said Deyona Burton, senior class president at Robert E. Lee High School in Jacksonville, Florida, and founder of SPEAR (Showing Political Engagement and Responsibility), a youth-led social and political action group."

"Any U. S. politics, no matter how coalitional its compass, that identifies itself in terms of sexual orientation only (e.g., queer nation or lesbian and gay studies for example) will be a white-centered and dominated politics, since only white people in this society can afford to see their race as unmarked, as an irrelevant category of analysis."

"While disability theory has compellingly pointed to the ways in which the construction and production of (inaccessible) space renders disabled bodies abnormal or aberrant, Indigenous peoples informed by Indigenous epistemology, have consistently intimated that the disablement of space via settler colonial practices of land appropriation and destruction in pursuit of profit, concomitantly yields the disablement of Indigenous bodies and worldviews that are intricately woven together with space. These issues strike us as particularly urgent given that much of what is recognized as disability studies scholarship is produced by non-Indigenous people within settler-colonial states."

"Ordinary people began to approach the Dar al-Ulum very early on for rulings on all sorts of matters. Soon enough the demand became so considerable that it could not be handled on an ad hoc basis. In 1892 a separate department was set up for issuing fatwas. By now literally a few lakh matters have been settled by the institution’s fatwas. Initially the fatwa would be issued, and that would be the end of the matter. No copy of the fatwa would be kept, no record would remain. Eventually, copies began to be kept. For decades these were stored merely by the date on which the fatwa had been issued. On a visit to the institution soon after Independence, Maulana Azad, then the country’s education minister and one of the most important figures in Pandit Nehru’s government, himself commended the work which the institution had been doing in this field—it is a great religious service, he said, by which the difficulties of the people are being removed. He urged that a collection of them be published. Grouping the fatwas by subject, weeding out the repetitions, and selecting the ones that settled the more general principles of law on the matter took many years of painstaking effort. It was in 1962 that the Dar al-Ulum began publishing the fatwas in volumes organized around subjects."

"I am familiar with the traditions of this institution... If we examine the millennia-old gurukul tradition, we will find that most of those who became immortal [through their great actions] in history received their sanskars from the gurukul system. This tradition had so much samarthya (capability) because it did not just provide book knowledge, livelihood skills or merely train people to acquire degrees. This tradition taught human beings to become humane. This institution cultivates in men the capability to rise from being mere men to becoming divine (nar se narayan). This institution has cultivated an atmosphere that inculcates the sanskar of rising above aham (self-hood) towards vyam (ourness), whereby people are transformed from being self-centred to being inherently society oriented, and inclucate sanskars (values) of collectivity to widen people’s perspective towards life. This great tradition teaches students to honour their gurus; it cultivates shraddha towards sanskriti (culture) and the desire to dedicate one’s life to doing good, whereby there is constant inspiration to sacrifice all one has for achieving excellence. This institution carries out a nirantar (never ending) yagya for crafting such a lifestyle."

"The Dar al-Ulum is of course well known. Started in 1866, it is often referred to as the Al-Azhar of India. From its beginning it was profoundly anti-West, it was anti-modern. Accordingly, many persons associated with it exerted themselves to undermine the British. That opposition was an aspect of its commitment to orthodoxy."

"Lauding this commitment to orthodoxy as one of the hallmarks of the Dar al-Ulum, a Government of India publication, Centres of Islamic Learning in India, says : 'One of the main objects of the Darul Ulum was to provide the Indian Muslims with a direct access to the original sources of Islamic Learning, produce learned men with missionary zeal to work among the Muslim masses to create a truly religious awakening towards classical Islam, ridding the prevalent one in India of innovation and unorthodox practices, observances and beliefs that have crept into it and to impart instruction in classical religion. The Darul Ulum has achieved this aim to a great extent, having been undoubtedly the greatest source of orthodox Islam in India, fighting, on the one hand, religious innovation (bid’at) and, on the other, cultural and religious apostasy under Western or local influences. It has succeeded in instilling in its alumni the spirit of classical Islamic ideology which has been its motto. As a matter of fact, Deoband has established itself as a school of religious thought—a large number of religious madrasahs were founded on its lines throughout the country by those who graduated from it, thus bringing classic religious instruction to large sections of Muslim masses. Some of these schools and colleges have in their right become renowned centres of learning...' That praise for re-establishing orthodoxy in Islam, for purging it of bid’at, a condemnatory word for heretical ‘innovation’, for purging it of ‘religious apostasy’ which the study says had crept into it ‘under Western or local influences’, that approbation is from a publication of our secular government! But at the moment I am on the institution’s fatwas."

"The school Where round their Guru, in a grave half-moon, The Sâkya children sang their s through, And learned the greater and the lesser gods."

"In the abzu, the great crown of g, where sanctuaries are apportioned [...] -- when Enki, the great princely farmer of the awe-inspiring temple, the carpenter of Eridug, the master of purification rites, the lord of the great en priest's precinct, occupies E-engur, and when he builds up the abzu of Eridug; when he takes counsel in Hal-an-kug, when he splits with an axe the house of boxwood; when the sage's hair is allowed to hang loose, when he opens the house of learning, when he stands in the street of the door of learning; when he finishes the great dining-hall of cedar, when he grasps the date-palm mace, when he strikes the priestly garment with that mace, then he utters seven [words] to Nisaba, the supreme nursemaid: "O Nisaba, good woman, fair woman, woman born in the mountains! Nisaba, may you be the butter in the cattle-pen, may you be the cream in the sheepfold, may you be keeper of the seal in the treasury, may you be a good steward in the palace, may you be a heaper up of grain among the grain piles and in the grain stores!" Because the Prince Enki cherished Nisaba, O father Enki, it is sweet to praise you!"

"Hoe, do not start getting so mightily angry! Do not be so mightily scornful! Is not Nisaba the Hoe's inspector? Is not Nisaba its overseer? The scribe will register your work."