However, there is no exact, universally agreed, definition of the term " discrete mathematics.

However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics.

However, Valla added the translation of ancient Greek works on mathematics ( firstly by Archimedes ), newly discovered and translated.

However, his abstract mathematics was denounced as " Jewish ", useless, and " un-German " and he lost his position in 1935.

However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that " most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.

However, among academic mathematics, the majority of mathematical papers published in the United States are written by academics outside of mathematics departments.

However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything he proves is in fact intuitionistically true.

" However, proponents of more limited forms of constructive mathematics would not assert that ZF itself is a constructive system.

However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics, medicine and allied disciplines, philosophy, physics, and psychology.

However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.

However, Pythagoras believed that the mathematics of music should be based on the specific ratio of 3: 2 whereas Ptolemy merely believed that it should just generally involve tetrachords and octaves.

However, for many Muslims there is no distinction ; all forms of art, the natural world, mathematics and science are all creations of God and therefore are reflections of the same thing-that is, God's will expressed through His Creation.

However, some finitist philosophers of mathematics and constructivists object to the notion.

However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department.

However, this criticism is rejected by supporters of the social constructivist perspective because it misses the point that the very objects of mathematics are social constructs.

However, Daniel refused, because he wanted to study mathematics.

However, many prominent economists come from a background in mathematics, business, political science, law, sociology, or history.

However, he was unable to complete the curriculum's rigorous mathematics requirement and was asked to repeat his sophomore year.

However, as the mid dot was already in common use in the mathematics world to indicate multiplication, the SI rejected its use as the decimal mark.

However these researches are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to Reverse Mathematics.

However, it is actually a broad, interdisciplinary field, incorporating not only aspects of computer science, but often diverse fields such as archival science, cognitive science, commerce, communications, law, library science, museology, management, mathematics, philosophy, public policy, and the social sciences.

However, the Weil conjectures in their scope are more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics.

However, the link between the Riemann hypothesis and the Prime Number Theorem had been known before in Continental Europe, and Littlewood also wrote later in his book A mathematician ’ s miscellany that his actually only rediscovered result did not shed a bright light on the isolated state of British mathematics at the time.

However, a rotating frame can be treated as if it were an inertial frame so that Newton's laws can be used if so-called fictitious forces ( also known as inertial or pseudo-forces ) are included in the sum of external forces on an object.

However Newton's laws of motion apply only in the stationary frame and describe dynamics in terms of the absolute acceleration d < sup > 2 </ sup > r / dt < sup > 2 </ sup >.

However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law.

However, a century later, Young and Fresnel revived the wave theory of light when they pointed out that light could be a transverse wave rather than a longitudinal wave — the polarization of a transverse wave ( like Newton's " sides " of light ) could explain birefringence, and in the wake of a series of experiments on diffraction the particle model of Newton was finally abandoned.

However, they were primarily written for immediate and day-to-day use in Newton's ministry of Olney.

However, the theory had difficulties in other matters, and was soon overshadowed by Isaac Newton's corpuscular theory of light.

However, many still favored Isaac Newton's corpuscular theory of light, among them the great theoretician Siméon-Denis Poisson.

However, unlike Archimedes ' method, which was based on Euclidean geometry, Newton's and Leibniz's integral calculus did not have a rigorous foundation.

However, Newton's laws ( combined with universal gravitation and classical electrodynamics ) are inappropriate for use in certain circumstances, most notably at very small scales, very high speeds ( in special relativity, the Lorentz factor must be included in the expression for momentum along with rest mass and velocity ) or very strong gravitational fields.

However, Newton's original precession equations did not work and were revised considerably by Jean le Rond d ' Alembert and subsequent scientists.

However, John Newton's heroics are said to be a product of Parson Weems, who also fictionalized the story of George Washington and the cherry tree.

" However, by definition, the validity of Newton's law of cooling requires that the rate of heat loss from convection be a linear function of (" proportional to ") the temperature difference that drives heat transfer, and in convective cooling this is sometimes not the case.

However the tendency of hot bodies to get cold ( see Thermodynamics ) and by Le Chatelier's Principle — the statistical mechanics extension of Newton's Third Law — to oppose this tendency should be noted.

However, Newton's notation is no longer standard ; instead this would be written with a prime or using Leibniz's notation.

However, the method was developed independently of Newton's method, and predated the latter by over 3, 000 years.

However, Newton's method requires the evaluation of both and its derivative at every step, while the secant method only requires the evaluation of.

However, if we let r >> n, then we get our usual Newton's law.

However, his assertion that a connection existed between the growth of knowledge and the art of war, and that ballistics played a central part of physics and Newton's world, was viewed with keen interest.

However, in early 1967, an English actor, Simon Prebble, coined the name " Newton's cradle " ( now used generically ) for the wooden version manufactured by his company, Scientific Demonstrations Ltd. After some initial resistance from retailers, they were first sold by Harrods of London, thus creating the start of an enduring market for executive toys.

However, he critically noted that the Relativity Principle, Newton's action and reaction, the Conservation of Mass, and the Conservation of Energy are not fully established and are even threatened by some experiments.

However, experiments and observations show that Einstein's description accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets.

However Clairaut showed shortly afterwards ( 1749 – 50 ) that at least the major cause of the discrepancy lay not in the lunar theory based on Newton's laws, but in excessive approximations that he and others had relied on to evaluate it.