Mathematicians

Descartes is a famous and well noted European mathematician who lived hundreds of years ago. He pioneered many of the complex math procedures, equations, and mathematical formulas that we still use today.

Descartes is the namesake of the standard "Cartesian" or rectangular coordinate system.

He also is famous for saying "I think, therefore I am."

Descartes' theory provided the basis for the calculus of Newton and Leibniz, by applying infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics. This appears even more astounding considering that the work was just intended as an example to his Discours de la méthode pour bien conduire sa raison, et chercher la verité dans les sciences (Discourse on the Method to Rightly Conduct the Reason and Search for the Truth in Sciences, better known under the shortened title, "Discours de la méthode").

Descartes' rule of signs is also a commonly used method in modern mathematics to determine possible quantities of positive and negative zeros of a function.

Descartes created analytic geometry, and discovered the law of conservation of momentum. He outlined his views on the universe in his Principles of Philosophy.

Descartes also made contributions to the field of optics. He showed by using geometric construction and the law of refraction (also known as Descartes' law) that the angular radius of a rainbow is 42 degrees (i.e. the angle subtended at the eye by the edge of the rainbow and the ray passing from the sun through the rainbow's centre is 42°). He also independently discovered the law of reflection, and his essay on optics was the first published mention of this law.

One of Descartes most enduring legacies was his development of Cartesian geometry, the algebraic system taught in schools today. He also created exponential notation, indicated by numbers written in what is now referred to as superscript.

This information was gathered from Wikipedia, see attached link.

When assuming the task of profiling Socrates utilizing these four criteria, one must take as requirement the "Principle of Textual Fidelity" and balance it with the "Principle of Interpretive Plausibility". With the body of work on Socrates being replete with secondhand sources, satisfying these two principles can be tricky.

I. Theory of Value: What knowledge and skills are worthwhile learning? What are the goals of education?

Socrates believed that there were different kinds of knowledge, important and trivial. He acknowledges that most of us know many "trivial" things. He states that the craftsman possesses important knowledge, the practice of his craft, but this is important only to himself, the craftsman. But this is not the important knowledge that Socrates is referring to. Through his method of powerfully questioning his students, he seeks to guide them to discover the subject matter rather than simply telling them what they need to know. The goals of education are to know what you can; and, even more importantly, to know what you do not know.

II. Theory of Knowledge: What is knowledge? How is it different from belief? What is a mistake? A lie?

Socrates makes the claim there are two very different sorts of knowledge. One is ordinary knowledge. This is of very specific (and ordinary) information. He claims that to have such knowledge does not give the possessor of said knowledge any expertise or wisdom worth mentioning.

Socrates devotes much thought to the concept of belief, through the use of logic. He spars with students early in his career and later with his accusers, at his trial, on the nature of his belief regarding the gods. To define belief, according to Socrates, was to use naturalistic explanations for phenomena traditionally explained in terms of Divine Agency.

III.Theory of Human Nature: What is a human being? How does it differ from other species? What are the limits of human potential?

The being in human is an inner-self. This inner-self is divine, cannot die, and will dwell forever with the gods. Only human beings can distinguish virtue, which is knowledge, from ignorance, which is the root of moral evil.

The human being is so constituted that he "can" know the good. And, knowing it, he can follow it, for no one who truly knows the good would deliberately choose to follow the evil. This is a typically Greek notion, and is attractive to all rationalists

From experience, it can be known that intellectually the human potential is infinitesimal. The mind of man is constantly reaching out for more and more knowledge, just as his will is desirous of more and more love. The search for knowledge varies with the individual, but the race of man has always carried on the quest in accordance with its nature and for the practical and speculative value that knowledge brings with it

IV. Theory of Learning: What is learning? How are skills and knowledge acquired?

Learning is the seeking of truth in matters, and it occurs when after questioning and interpreting the wisdom and knowledge of others, one comes to recognize their own ignorance. Skills and knowledge are acquired by: (1) interpreting the statements of others; (2) testing or examining the knowledge or wisdom of those reputed (by themselves or others) to be wise; (3) showing those who are not wise their ignorance; (4 ) learning from those who are wise; (5) examining oneself; (6) exhorting others to philosophy; (7) examining the lives of others; (8) attaining moral knowledge.

V. Theory of Transmission: Who is to teach? By what methods? What will be the curriculum be?

Socrates does not believe that any one person or any one school of thought is authoratative or has the wisdom to teach "things." Socrates repeatedly disavows his own knowledge and his own methods. However, this appears to be a technique for engaging others and empowering the conversator to openly dialogue.

Be that as it may, Socrates is widely regarded as one of the great teachers of all time. The Socratic Method is one in which a teacher, by asking leading questions, guides students to discovery. It was a dialectical method that employs critical inquiry to undermine the plausibility of widely-held doctrine.

Socrates devoted himself to a free-wheeling discussion with the aristocratic young citizens of Athens, insistently questioning their unwarranted confidence in the truth of popular opinions, even though he often offered them no clear alternative teaching.

VI. Theory of Society: What is society? What institutions are involved in the education process?

To the class of Athenians that Socrates was born into, society existed to provide the best life for the individual. The Athenians of Socrates' day assumed just as their ancestors had assumed that the best life one could have, required the acquisition of what was called virtue, or excellence. A truly good person succeeded in doing great things for the city, strictly obeyed its law, honored parents and ancestors, scrupulously paid homage to the gods by strictly obeying the conventions governing prayer and sacrifice.

Athens' political system was a radical, participating democracy in which every Athenian male citizen could-and was expected to-vote, hold office, and serve on the very powerful Athenian juries.

Societies are invariably formed for a particular purpose. Individuals are not self-sufficient, no one working alone can acquire all the genuine necessities of life. Separations of functions and specialization of labor are key. Society is composed of distinct classes (clothiers, farmers, builders, etc.). In addition, there are those that manage society and settle disputes. In Plato's Republic, he uses the fictional character Socrates as spokesman for explaining the fundamental principles for the conduct of human life.

Education took place in magnificent buildings such as the Parthenon and Hephaisteion, which adorn the Acropolis and the Agora, the large open area at the front of the Acropolis that consisted of the Athenian market place and Public Square.

It has been said that Bertrand Russell, a prominent 20th century philosopher (particularly for his work in analytic philosophy and logic) made this highly interesting list of commandments for intellectual independence.

1. Do not feel certain of anything.
2. Do not feel it's worthwhile to hold on to a belief by concealing contrary evidence, for that evidence will surely come to light.
3. Never discourage thinking (in a philosophical sense.)
4. Overcome opposition via argument, not force. A victory based upon force is unreal and illusionary.
5. Have no respect for the authority of others, for there will always be counter-authorities.
6. Don't use force to suppress opinions which you think are dangerous, for if you do they will surely suppress you.
7. Don't fear being different and eccentric, for every mainstream idea was at one time eccentric and different.
8. Find pleasure in intelligent, not passive, agreement.
9. Be scrupulously truthful, even when the truth is inconvenient, for it is always more inconvenient when you try to conceal truth.
10. Don't feel envious of the happiness of those living in a fool's paradise, for only a fool would see it as true happiness.

Al-Khwarizmi (Mohammad ebne Mūsā Khwārazmī محمد بن موسی خوارزمی) was a Persian mathematician, astronomer, astrologer and geographer. He was born around 780 in Khwārizm, then part of the Persian Empire (now Khiva, Uzbekistan) and died around 850. He worked most of his life as a scholar in the House of Wisdom in Baghdad.

His Algebra was the first book on the systematic solution of linear and quadratic equations. Consequently he is considered by many to be the father of algebra, a title some scholars assign to Diophantus. Latin translations of his Arithmetic, on the Indian numerals, introduced the decimal positional number system to the Western world in the twelfth century. He revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology.

His contributions not only made a great impact on mathematics, but on language as well. The word algebra is derived from al-jabr, one of the two operations used to solve quadratic equations, as described in his book.

This answer was taken from a Wikipedia page that is linked below.