While the roots of formalised logic go back to Aristotele, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics.

While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both " an abomination " and " a cholera bacillus of mathematics ".

While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured.

While Babylonian number theory — or what survives of Babylonian mathematics that can be called thus — consists of this single, striking fragment, Babylonian algebra ( in the secondary-school sense of " algebra ") was exceptionally well developed.

While Greek astronomy — thanks to Alexander's conquests — probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition ; in particular, there is no evidence that Euclid's Elements reached India before the 18th century.

While OBE implementations often incorporate a host of many progressive pedagogical models and ideas, such as reform mathematics, block scheduling, project-based learning and whole language reading, OBE in itself does not specify or require any particular style of teaching or learning.

While working as an ambulance driver Wilson attended New York University, studying engineering and mathematics.

While the school specialized in the humanities, Banach and his best friend Witold Wiłkosz ( also a future mathematician ) spent most of their time working on mathematics problems during breaks and after school.

While studying mathematics as an undergraduate student at Harvard College, he began to write comic songs to entertain his friends, including " Fight Fiercely, Harvard " ( 1945 ).

While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft

While PM avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.

While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric linguistic variables are often used to facilitate the expression of rules and facts.

While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.

While his original writings demonstrate that he had a sincere passion for the seven liberal arts ( grammar, rhetoric, logic, mathematics, geometry, music, and astronomy ), his work in Quaestiones naturales illustrated a more encompassing dedication to subjects such as physics, the natural sciences, and possibly even metaphysics.

While it is often assumed that those with this intelligence naturally excel in mathematics, chess, computer programming and other logical or numerical activities, a more accurate definition places less emphasis on traditional mathematical ability and more on reasoning capabilities, recognizing abstract patterns, scientific thinking and investigation and the ability to perform complex calculations.

While some constructs, such as mathematics, are considered to be very reliable, all human constructs are subject to humanity's assumptions of cause and effect.

While the practice of mathematics previously developed in other civilizations, the special interests for its theoretical and foundational aspects really started with Ancient Greeks.

While this remains the case, the " string " must be thought of either as real but untestable, or simply as an illusion or artifact of either mathematics or cognition.

While the priests taught him mathematics and Latin, his mother completed his education.

While the Mizar proof checker remains proprietary, the Mizar Mathematical Library – the sizable body of formalized mathematics that it verified – is licensed open-source.

While learning these skills, he also intensely studied mathematics, architecture, perspective and map drawing.

While mathematicians usually study them for their own sake, by doing so results may be obtained that find application outside the realm of mathematics.

While he showed an early aptitude for mathematics, teaching himself calculus while a teenager, he left school after the 9th grade to work on his parents ' farm.

While in Geneva, he devoted himself to the study of mathematics under Georges-Louis Le Sage, and acquired from Switzerland an intense love of liberty.

While Aristotle censors literature only for the young, Plato would banish all poets from his ideal state.

While S.K. did not like Dylan Thomas, I liked his poems very much, but I made the mistake of telling Dylan Thomas so, whereupon he said to me, `` I suppose you think you know all about me ''.

While all citizens share in blame for lax municipal ethics the Wagner regime has seen serious problems in the schools, law enforcement and fiscal policies.

While there are many different possibilities for the timing of casework intervention, the experiments recently reported from a variety of traditional settings all point up the importance of an immediate response to the client's initial need for help.

While other conditions might be even more effective in bringing about a change from immobility to mobility in Kohnstamm reactivity, it is our hypothesis that all such conditions would have as a common factor the capacity to induce an attitude in the subject which enabled him to divorce himself temporarily from feelings of responsibility for his behavior.

While the Cold War raged it was easy to blame it all on Yalta.

While we, as American artists, believe deeply in the universal character of all intellectual activity, we would be less than honest with you, or ourselves, if we failed to state a specific attitude toward our own society as well as the international community as a whole.

While man shares this procreative function with all his predecessors in the evolutionary process, he is the only animal with a true non-instinctive and conscious creative ability.

While a Senator, Kennedy had unsuccessfully pushed a bill to preserve the Belasco Theater, as well as the Dolley Madison and the Benjamin Taylor houses, all scheduled for razing.

While not entirely synonymous with Anatolia, the term Asia Minor, derived from the Latin Asia Minores, refers to Asia inside the Roman Empire, versus Asia Magna, all of Asia beyond the borders.

While not all Southerners saw themselves as fighting to preserve slavery, most of the officers and over a third of the rank and file in Lee's army had close family ties to slavery.

While most everyone has an experience with anxiety at some point in their lives, as it is a common reaction to real or perceived threats of all kinds, most do not develop long-term problems with anxiety.

While the term's etymology might suggest that antisemitism is directed against all Semitic peoples, the term was coined in the late 19th century in Germany as a more scientific-sounding term for Judenhass (" Jew-hatred "),

While it would be rash to assert positively that no existing gems were the work of Gnostics, there is no valid reason for attributing any or all of them to such an origin.

While at Jerusalem, Absalom built support for himself among the populace by promising justice for all " if only I were appointed judge in the land ", and by showing humility by kissing those who approached him rather than accepting supplication.

While his predecessors Thales and Anaximander proposed that the arche, the underlying material of the world, were water and the ambiguous substance apeiron, respectively, Anaximenes asserted that air was this primary substance of which all other things are made.

While all modern variants of the acre contain 4, 840 square yards, there are alternative definitions of a yard, so the exact size of an acre depends on which yard it is based on.

" While his one-time acolytes The Rolling Stones and Cream made the front pages of music magazines all over the world, Korner was relegated to the role of ' elder statesman '.

While the NFL retained its old name and logo, it nevertheless claims the rights to all AFL products and trademarks, including the name and the eagle logo.

While the Ottomans held their own in the conflict they ultimately lost with Ochakov falling in 1788 to the Russians ( all of its inhabitants being massacred.

While all prokaryotes reproduce asexually ( without the formation and fusion of gametes ), mechanisms for lateral gene transfer such as conjugation, transformation and transduction are sometimes likened to sexual reproduction.

" While Sakharov strongly disagreed with Teller over nuclear testing in the atmosphere and Strategic Defense Initiative, he believed that American academics had been unfair to Teller ’ s resolve to get the H-bomb for the United States since " all steps by the Americans of a temporary or permanent rejection of developing thermonuclear weapons would have been seen either as a clever feint, or as the manifestation of stupidity.

While all real fluids are compressible, a flow problem is often considered incompressible if the density changes in the problem have a small effect on the outputs of interest.

While not all these conclusions have been accepted, it has had an enduring influence on archaeoastronomy, especially in Europe.

While this does bear some similarity with the classic posture of cowboys firing their twin revolvers from the hip, in games this posture is almost never reflected, with almost all game characters firing twin guns at shoulder level, straight-armed.