Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.

Although Xenakis could well have composed this music by hand, the intensity of the calculations needed to transform probabilistic mathematics into musical notation was best left to the number-crunching power of the computer.

Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

" Although deliberately framed as if it were a law of nature or of mathematics, its purpose has always been rhetorical and pedagogical: I wanted folks who glibly compared someone else to Hitler or to Nazis to think a bit harder about the Holocaust ," Godwin has written.

Although he enjoyed the civil engineering curriculum at the academy and did well in mathematics, his poor drawing skills hampered his engineering studies, and he finished 29th in that discipline.

Although Kurt had first excelled in languages, he later became more interested in history and mathematics.

Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy.

Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled

Although her father refused to grant this wish of joining a convent, he agreed to let her live from that time on in an almost conventual semi-retirement, avoiding all interactions with society and devoting herself entirely to the study of mathematics.

Although Escher did not have mathematical training — his understanding of mathematics was largely visual and intuitive — Escher's work had a strong mathematical component, and more than a few of the worlds which he drew are built around impossible objects such as the Necker cube and the Penrose triangle.

Although at odds with the scientific establishment for most of his life, Heaviside changed the face of mathematics and science for years to come.

Although Descartes did not pursue the subject, he preceded Leibniz in envisioning a more general science of algebra or " universal mathematics ," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.

Although Dirac did not at first fully appreciate what his own equation was telling him, his resolute faith in the logic of mathematics as a means to physical reasoning, his explanation of spin as a consequence of the union of quantum mechanics and relativity, and the eventual discovery of the positron, represents one of the great triumphs of theoretical physics, fully on a par with the work of Newton, Maxwell, and Einstein before him.

Although he worked in the subjects of ballistics, physics and astronomy, his major contributions are to the mathematics of the second half of the 18th century.

Although most mathematicians and physicists ( and many philosophers ) would accept the statement " mathematics is a language ", linguists believe that the implications of such a statement must be considered.

Although Marshall took economics to a more mathematically rigorous level, he did not want mathematics to overshadow economics and thus make economics irrelevant to the layman.

Although he graduated from high school, Tange's poor results in mathematics and physics meant that he had to pass entrance exams to qualify for admission to the prestigious universities.

Sir Michael Atiyah said of Witten, " Although he is definitely a physicist, his command of mathematics is rivaled by few mathematicians ... Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems ... he has made a profound impact on contemporary mathematics.

Although Ackermann did not choose a university career and rather continued as a high school teacher, he was continually engaged in research and published many contributions to the foundations of mathematics until the end of his life.

Although the concepts of a law or principle in nature is borderline to philosophy, and presents the depth to which mathematics can describe nature, scientific laws are considered from a scientific perspective and follow the scientific method ; they " serve their purpose " rather than " questioning reality " ( philosophical ) or " statements of logical absolution " ( mathematical ).

Although it was dark as usual I could see that the hall had only recently contained a great many people.

Although New Orleans was not to learn of it for a spell, she also was a sadist, a nymphomaniac and unobtrusively mad -- the perpetrator of some of the worst crimes against humanity ever committed on American soil.

Although Faulkner was the heir in his own family to this tradition, he did not have Stark Young's inclination to romanticize and sentimentalize the planter society.

Although his tender nights were not the ones I dreamed of, nor was it for yachts, sports cars, tall drinks, and swimming pools, nor yet for money or what money buys that I burned, I too was burning and watching myself burn.

Although after much trouble he did manage to get it back, he discovered there was no trade to be had.

Although he did not attend any celebrated schools or universities, he was a master of Greek and Hebrew and could read the Bible in the original.

Although the fort was evacuated in the face of the force of Cornwallis, Morgan and his men did have a chance to take another swing at the redcoats.

Although because of the important achievements of nineteenth century scholars in the field of textual criticism the advance is not so striking as it was in the case of archaeology and place-names, the editorial principles laid down by Stevenson in his great edition of Asser and in his Crawford Charters were a distinct improvement upon those of his predecessors and remain unimproved upon today.

Although Mr. Brown was not himself its inventor ( it was a French idea ), it is typical that his intuition first conceived the importance of mass producing this basic tool for general use.

Although it was a mere fragment of the victim's remains, it was enough.

Although there were only four fluids tested, it was apparent that there were two distinct types.

Although the tape was run for over 1 hr., a steady state was not reached, and it was concluded that the reason for this was that the back pressure of the manometer was built up from the material fed from between the blocks and this was available at a very slow rate.

-- Although there was some variation in results which must be attributed either to trace impurities or to variation in wall effects, the photochemical exchange in the gas phase was sufficiently reproducible so that it seemed meaningful to compare the reaction rates in different series of reaction tubes for the purpose of obtaining information on the effect of chlorine concentration and of carbon tetrachloride concentration on the reaction rate.

Although the Taylor Scale was designed as a group testing device, in this study it was individually administered by psychologically trained workers who established rapport and assisted the children in reading the items.

Although the armored vest fitted the upper part of his body snugly, he felt no security.

Although no drugs act exclusively on the hypothalamus or a part of it, there is sufficient specificity to distinguish drugs which shift the hypothalamic balance to the sympathetic side from those which produce a parasympathetic dominance.

Although they " were expecting to see activity in the brain's reward centers ", based on the idea that " people perform altruistic acts because they feel good about it ", what they found was that " another part of the brain was also involved, and it was quite sensitive to the difference between doing something for personal gain and doing it for someone else's gain ".

Although discomforts are made part of certain Jewish conceptions of the afterlife, the concept of " eternal damnation ", so prevalent in other religions, is no tenet of the Jewish afterlife.

Although there is no reference to reincarnation in the Talmud or any prior writings, according to rabbis such as Rabbi Avraham Arieh Trugman, reincarnation is recognized as being part and parcel of Jewish tradition.

Although punishments are made part of certain Christian conceptions of the afterlife, the prevalent concept of " eternal damnation " is a tenet of the Christian afterlife.

Although details of the negotiations are lacking, Gruffydd ap Llywelyn swore loyalty to King Edward, but the oath may not have had any obligations on Gruffydd's part to Edward.

Although it does not match the German exactly, Wheen's title has justly become part of the English language and is retained here with gratitude.

Although the duty of Abbreviators was originally to make abstracts and abridgments of the apostolic letters, diplomas, etc., using the legal abbreviations, clauses, and formularies, in course of time, as their office grew in importance they delegated that part of their office to their substitute and confined themselves to overseeing the proper expedition of the apostolic letters.

Although AVMs are often associated with the brain and spinal cord, they can develop in any part of the body.

Although technically a part of the fourth university, it largely functions autonomously.

Although the use of aircraft has for the most part always been used as a supplement to land or naval engagements, since their first major military use in World War I aircraft have increasingly taken on larger roles in warfare.

Although details remain disputed, the vast majority of recent studies agree with Martin Noth's thesis, published in 1943, that the book of Samuel was composed as part of the Deuteronomistic history, the books of Deuteronomy, Joshua, Judges, Samuel and Kings.

Although players usually use their sticks to move the ball around, they may use any part of their bodies other than their heads, hands or arms and may use their skates in a limited manner.

Although his name does not appear in any other part of the Jewish Bible, Rabbinic tradition holds Habakkuk to be the Shunammite woman's son, who was restored to life by Elisha in 2 Kings 4: 16.

Although Douglas was always second billed under Lancaster in these films, their roles were usually more or less the same size with the exceptions of I Walk Alone, in which Douglas played a villain, and in Seven Days in May, where Douglas ' part was larger than Lancaster's but not as dramatic.

Although he initially declined the part of Dr. David Banner in The Incredible Hulk because of its comic book origins, on reading Kenneth Johnson's script for the pilot episode, he was persuaded to change his mind ( and agreed to remain involved with the series for as long as Johnson was to be involved ).

Although they can sometimes be found on InterCity routes in the continental part of the country.

Although the " self-important rhetoric and whines of persecution " on the part of cyberpunk fans were irritating at worst and humorous at best, Brin declares that the " rebels did shake things up.

Although Louisbourg was captured by New Englanders with British naval assistance in 1745 and by the British again in 1758, Île Royale remained formally part of colonial France until it was ceded to Britain under the Treaty of Paris in 1763.

Although female roles were performed by castrati in some of the papal states, this was increasingly rare ; by 1680, they had supplanted " normal " male voices in lead roles, and retained their position as primo uomo for about a hundred years ; an Italian opera not featuring at least one renowned castrato in a lead part would be doomed to fail.

Although the Apocrypha was part of the 1611 edition of the KJV, the LDS Church does not currently use the Apocrypha as part of its canon.

Although not part of the Cursus Honorum, upon completing a term as either Praetor or Consul, an officer was required to serve a term as Propraetor and Proconsul, respectively, in one of Rome's many provinces.

Although, in these early years, engines came in a number of configurations, typically Q < sub > H </ sub > was supplied by a boiler, wherein water was boiled over a furnace ; Q < sub > C </ sub > was typically a stream of cold flowing water in the form of a condenser located on a separate part of the engine.