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Socrates (Greek: Σωκράτης c. 470 BC–399 BC[1]), was a Classical Greek philosopher. Considered one of the founders of Western philosophy,[1] he strongly influenced Plato, who was his student, and Aristotle, whom Plato taught. His work continues to form an important part of the study of philosophy.

Usually, renowned for his contribution to the field of ethics, Socrates also lends his name to the concepts of Socratic irony and the Socratic Method, or elenchus. The latter remains a commonly used tool in a wide range of discussions, and is a type of pedagogy in which a series of questions are asked not only to draw individual answers, but to encourage fundamental insight into the issue at hand. Socrates also made important and lasting contributions to the fields of epistemology and logic, and the influence of his ideas and approach, remains strong in providing a foundation for much western philosophy which followed.

Pythagoras of Samos (Greek: Πυθαγόρας; born between 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician[1] and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy.[2] Herodotus referred to him as "the most able philosopher among the Greeks".[3] His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and of benefit to humankind.[4]

He is best known for the Pythagorean theorem, which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. We do know that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus, Pythagoras once said that "number is the ruler of forms and ideas and the cause of gods and demons."

He was the first man to call himself a philosopher, or lover of wisdom,[5] and Pythagorean ideas exercised a marked influence on Plato. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors.

Pythagoras

Descartes 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician[1] and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy.[2] Herodotus referred to him as "the most able philosopher among the Greeks".[3] His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and of benefit to humankind.[4]

René Descartes (French IPA: [ʁəne de'kaʁt] Latin:Renatus Cartesius) (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. He has been dubbed the "Father of Modern Philosophy," and much of subsequent Western philosophy is a response to his writings, which continue to be studied closely. His influence in mathematics is also apparent, the Cartesian coordinate system that is used in plane geometry and algebra being named for him, and he was one of the key figures in the Scientific Revolution.

L’Hopital 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician[1] and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy.[2] Herodotus referred to him as "the most able philosopher among the Greeks".[3] His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and of benefit to humankind.[4]

In calculus, l'Hôpital's rule (also spelled l'Hospital) uses

derivatives to help compute limits with indeterminate forms.

Application (or repeated application) of the rule often converts

an indeterminate form to a determinate form, allowing easy

computation of the limit. The rule is named after the 17th-century

French mathematician Guillaume de l'Hôpital, who published

the rule in his book l'Analyse des Infiniment Petits pour

l'Intelligence des Lignes Courbes (literal translation:

Analysis of the infinitely small to understand curves) (1696),

the first book about differential calculus.

In simple cases, l'Hôpital's rule states that for functions f(x) and g(x), if:lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,

or:

\lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x)=\pm\infty,

then:lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}

where the prime (') denotes the derivative.

Among other requirements, for this rule to hold, the limit \lim_{x\to c}\frac{f'(x)}{g'(x)} must exist. Other requirements are detailed below, in the formal statement.

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.

Riemann made major contributions to real analysis. He defined the Riemann integral by means of Riemann sums, developed a theory of trigonometric series that are not Fourier series—a first step in generalized function theory—and studied the Riemann-Liouville differintegral.

He made some famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he introduced the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful heuristic rather than a rigorous method. Its justification took at least a generation. His work on monodromy and the hypergeometric function in the complex domain made a great impression, and established a basic way of working with functions by consideration only of their singularities.

Riemann

Laplace analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.

Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799-1825). This seminal work translated the geometric study of classical mechanics, used by Isaac Newton, to one based on calculus, opening up a broader range of problems.

He formulated Laplace's equation, and invented the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in applied mathematics, is also named after him.

Independently from Immanuel Kant, he formulated the nebular hypothesis of the origin of the solar system and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.

He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton or Newton of France, with a natural phenomenal mathematical faculty possessed by none of his contemporaries.[1]

He became a count of the First French Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.

Cauchy analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.

Augustin Louis Cauchy (August 21, 1789 – May 23, 1857; pronounced [o.gysˈtɛ̃ lwi koˈʃi]) was a French mathematician. He started the project of formulating and proving the theorems of calculus in a rigorous manner and was thus an early pioneer of analysis. He also gave several important theorems in complex analysis and initiated the study of permutation groups. A profound mathematician, Cauchy exercised by his perspicuous and rigorous methods a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.

Blaise Pascal (pronounced [bl analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.ɛz paskal]), (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. He was a child prodigy who was educated by his father. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the construction of mechanical calculators, the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli. Pascal also wrote in defense of the scientific method.

Pascal was a mathematician of the first order. He helped create two major new areas of research. He wrote a significant treatise on the subject of projective geometry at the age of sixteen, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science.

Following a mystical experience in late 1654, he abandoned his scientific work and devoted himself to philosophy and theology. His two most famous works date from this period: the Lettres provinciales and the Pensées. Pascal suffered from ill health throughout his life and died two months after his 39th birthday.

Pascal

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