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Maths and Philosophy

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WHY MATHS?WHY MATHS?by the students and the teachers from:

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IntroductionMathematics and Philosophy have had historically close ties: many thinkers appear both in books than in Mathematics in Philosophy and often the subjects of investigation have boundless from one to another discipline. This study aims to highlight just a few of these links, we studied in these years of high school.

The origins: Greeks philosophersGreek Mathematics is mainly Geometry and Greek Philosophy often uses mathematical tools to affirm or refute theories. Logic and demonstrations as we know them today have their bases in the Hellenic peninsula, Southern Italy and Sicily. Greek philosophers and mathematicians had great fame in their society and the subtlety of their ideas still has influence on our modern way of thinking.Above the entrance to the Academy was inscribed the phrase "Let None But Geometers Enter Here.": this means they considered essential knowing Geometry and mainly the way of thinking students get from learning deductions and arguing in a precise way. This is something Maths and Philosophy have still in common today.

PythagorasOften described as the first pure mathematician, he is a very important figure in the development of Mathematics yet we know relatively little about his mathematical achievements as we have none of Pythagoras's writings: the society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.

Fortunately we do have details of Pythagoras's life from early biographies which use important original sources: there is a general agreement on the main events of his life but most of the dates are disputed.

Pythagoras he was born and spent his early years in Samos but traveled a lot with his father: it seems that he also visited Italy. Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer, for sure he heard about Thales and his pupil Anaximander who both lived on Miletus and probably he met them there when he was between 18 and 20 years old: by this time Thales was an old man and, although he created a strong impression on Pythagoras and contributed to his interest in Mathematics and Astronomy advising him to travel to Egypt to learn more of these subjects. The tradition says he attended some lectures from Anaximander and he got interest into Geometry and Cosmology.

A few years after the tyrant Polycrates seized control of the city of Samos, in about 535 BC, Pythagoras went to Egypt, on behalf of the tyrant himself, who was his friend.

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Many of Pythagoras's beliefs can be related to Egyptian customs: for instance the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt.

In 525 BC the king of Persia Cambyses II invaded Egypt: Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis, Egyptian resistance collapsed and Pythagoras was taken prisoner and taken to Babylon. In about 520 BC Pythagoras left Babylon and could return to Samos as both Polycrates and Cambyses died in 522 BC. We don't know how he obtained his freedom. After his return to Samos, Pythagoras made a journey to Crete where he studied the system of laws there. Back in Samos he founded a school which was called the semicircle, but the Samians were not very keen on this method and treated him in a rude and improper manner. So he left Samos and went to Croton, where he founded a philosophical and religious school that had many followers: Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. who lived permanently with the Society, had no personal possessions and were vegetarians and both men and women were permitted to become members of the Society. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians.

Pythagoras's Society at Croton, despite his desire to stay out of politics, was affected by political events. Pythagoras went to Delos in 513 b.C. to nurse his old teacher Pherekydes who was dying. He remained there for a few months until the death of his friend and teacher and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris and it seems probable that Pythagoras was involved in the dispute. Then in around 508 b.C. the Pythagorean Society at Croton was attacked by Cylon, a noble from Croton itself. Pythagoras escaped to Metapontium and probably he died there, perhaps committing suicide because of the attack on his Society. The Pythagorean Society expanded rapidly after 500 b.C., became political in nature and also split into a number of factions.

The beliefs that Pythagoras held were:

(1) that at its deepest level, reality is mathematical in nature,(2) that philosophy can be used for spiritual purification,(3) that the soul can rise to union with the divine,(4) that certain symbols have a mystical significance, and(5) that all brothers of the order should observe strict loyalty and secrecy.We don't know anything about Pythagoras's actual work: his school practiced secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers. Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions. They were not acting as a mathematics research group does in a modern university, there were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems.

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Rather Pythagoras was interested in the principles of Mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof. Pythagoras believed that all relations could be reduced to number relations and studied properties of numbers which would be familiar to mathematicians today, such as even, odd, triangular and perfect numbers. He noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments: he was a fine musician, playing the lyre, and he used music as a means to help those who were ill.

Today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it.

Here is a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.

(i) The sum of the angles of a triangle is equal to two right angles: the Pythagoreans knew the generalization which states that a polygon with n sides has sum of interior angles 2n

(ii) The theorem of Pythagoras: to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side and to say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square

(iii) Constructing figures of a given area and geometrical algebra: for example they geometrically solved equations such as a (a ─ x) = x2

(iv) The discovery of irrational numbers: it seems unlikely to have been due to Pythagoras himself, as it was against his philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number. He also knew about Golden Number and many properties of the regular pentagon (it was a sacred figure as the number 5 was sacred itself)

(v) The five regular solids (Plato learned about them and now they are known and Platonic solids)

(vi) In astronomy Pythagoras taught that the Earth was a sphere at the center of the Universe. He also recognized that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realize that Venus as an evening star was the same planet as Venus as a morning star.

Primarily, however, Pythagoras was a philosopher and he had many ideas sometimes original, sometimes learned during his staying in Egypt:

• the dependence of the dynamics of world structure on the interaction of contraries, or pairs of opposites

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• the viewing of the soul as a self-moving number experiencing a form of metempsychosis, or successive reincarnation in different species until its eventual purification

• the understanding that all existing objects were fundamentally composed of form and not of material substance.

ZenoZeno from Elea is famous for his paradoxes that stumped mathematicians for millennia and provided enough aggravation to lead to numerous discoveries in the attempt to solve them. He was born in the Greek colony of Elea in southern Italy around 495 b.C. and we know very little about him: he was a student of the philosopher Parmenides and accompanied his teacher on a trip to Athens in 449 b.C. where he met a young Socrates. On his return to Elea he became active in politics and was arrested and tortured for taking part in a plot against Nearchus, the city's tyrant. He was a philosopher and logician, not a mathematician, even if many mathematicians studied his ideas: he probably invented dialectic, a form of debate in which one arguer supports a premise while another one attempts to reduce the idea to nonsense, that is at the basis of the process of reductio ad absurdum, often used in Maths demonstrations. Parmenides believed that reality was one, immutable and unchanging: motion, change, time and plurality were all mere illusions.

The four most famous paradoxes are the Dichotomy, the Achilles, the Arrow, and the Stadium. For that time, they were not considered very important, but they were studied by XX century's mathematicians as Bertrand Russell and Lewis Carroll. Today, knowing the converging series and the theories on infinite sets, these paradoxes can be explained. However, even today the debate continues on the validity of both the paradoxes and the rationalizations.

The Dichotomy: Motion can't exist as before that the moving object can reach its destination, it must reach the midpoint of its course, but before it can reach the middle, it must reach the quarterpoint, but before it reaches the quarterpoint, it first must reach the eigthpoint, etc. Hence, motion can never start.The Achilles: Achilles while is running can never reach the tortoise ahead of him because he must first reach the point where the tortoise started, but when he arrives there, the tortoise has moved ahead, and Achilles must now run to the new position, which by the time he reaches the tortoise has moved ahead, etc. Hence the tortoise will always be ahead.The fundamental error in the theory is now obvious: Achilles and the tortoise are moving independently through space-time. Achilles' position is not dependent upon that of the tortoise: he overtakes the tortoise when his trajectory crosses that of the tortoise. The only problem was in calculating the precise point in time when that happened. The wrong Math was being used to calculate it so it could not give the correct answer. In fact, things were more complicated than Xeno was allowing for, and this shows the danger of relying on intuition to

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assess reality. The logic seemed sound because it was intuitive. Maths tells us intuition was wrong. The main flaw of Zeno’s paradox is that he uses the concept of “eternity”: if we record the story mathematically, the time taken for Achilles to run the footrace is (if it took him 10 seconds to run 100m): 10 + 1 + 0.1 + 0.01 + 0.001… = 11.111… So, the tortoise is only ahead of Achilles for less than 11.2 seconds (rounded). After 11.2 seconds pass, the time passed exceeds the sum of the infinite series and the paradox no longer applies. What Zeno didn't know was the infinite concept, that was finally systematized around the XVII century.The Arrow: According to Zeno, time is made up of instants (“nows”), which are the smallest measure and indivisible. Consider an arrow, apparently in motion, at any instant. First, Zeno assumes that it travels no distance during that instant —‘it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving. Zeno was so clever in formulating his arrow paradox that it has resisted convincing resolution into our days. The commonly accepted "resolution" is the "at-at" theory formulated by Bertrand Russell: he says that the motion of an arrow appearing at different positions at different times can be described by a continuous function where is the position at time and with a non-constant continuous function of the real variable, the arrow is thus changing position with increasing time and thus is moving. The Stadium: Half the time is equal to twice the time. The three rows here start at the first position. Row A stays stationary while rows B & C move at equal speeds in opposite directions. When they have reached the second position, each B has passed twice as many C's as A's. Thus it takes row B twice as long to pass row A as it does to pass row C. However, the time for rows B & C to reach the position of row A is the same. So half the time is equal to twice the time: if there is a smallest instant of time and if the farthest that a block can move in that instant is the length of one block, then if we move the set B to the right that length in the smallest instant and the set C to the left in that instant, then the net shift of the sets B and C is two blocks. Thus there must be a smaller instant of time when the relative shift is just one block.

PlatoPlato (427-347 B.C.) is considered to be one of the philosophers who contributed much in shaping western Philosophy. He was born from an aristocratic family and he had early interests were in poetry and politics: he learned philosophy from Socrates. As a consequence of the political execution of his teacher, he abandoned temporarily his political interests and left Athens. In his travels, he got in touch with the Pythagoreans from whom he gained the interest in Mathematics: when he finally returned to Athens about 385 B.C., he founded his Academy, that was considered one of the main centers of intellectual life at the time, where he was lecturing for the rest of his life, attracting many talented people in Greece. The Academy flourished until 529 when it was closed down by the Christian Emperor Justinian, who claimed it was a pagan establishment. The topics of study in the Academy included Philosophy, Mathematics and Law. One of his student, Eudoxus from Cindus, became one of the most able

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mathematicians at the time, gained fame from his work on the theory of proportions and developing the idea of the method of exhaustion. Plato’s contributions to Mathematics were focused on its foundations: he discussed the importance of examining the hypotheses of Maths, underlying the importance of making mathematical definitions clear and precise as these definitions are fundamental entities in this discipline. Another indirect contribution of Plato was the important role he played in encouraging and inspiring people to study mathematics: he proposed many problems and encouraged the students of the Academy to investigate. Why did Plato stress on the study of Maths? We can find the answer in the seventh book of his masterpiece, The Republic, where he stated some of his views on its importance: to him, the idea of good is the ultimate objective of Philosophy as “in the world of knowledge the idea of good appears last of all, and is seen only with an effort; and, when seen, is also inferred to be the universal author of all things beautiful and right.”. Arithmetic and Geometry have two important characteristics that make them valuable in comprehending the idea of good:

• these subjects lead the mind to reflect and hence enabling the mind to reach truth;

• the advanced parts of Arithmetic and Geometry have the power to draw the soul from becoming to beings as “arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number and repelling against the introduction of visible or tangible objects into the argument.”. Studying arithmetic as an amateur or like a merchant with a view to buying or selling will not help the soul to make this transition from becoming to being.

He also thought that people who are good in Mathematics will do well in any other field of knowledge and payed specifically attention to the irrational numbers, as for him Science is incomplete without them and that a comprehensive study of irrationals is necessary to build “a coherent and universal philosophy free of the difficulties that wrecked the Phythagorean system.”. He concentrated on the idea of "proof" and insisted on accurate definitions and clear hypotheses: this laid the foundations for Euclid's systematic approach to Maths.He didn't deny the important applications of Mathematics in people’s daily life, but, to him, the philosophical importance of this subject is more important and more rewarding as it may affect one’s understanding of his being, even if he didn't made any important discovery in this filed: he supplied many Mathematical problems in his writing, especially in the Meno, where Socrates asks to a slave boy to double the area of a square or asks to the audience if a given triangle can be inscribed in a given circle.

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In Mathematics Plato's name is linked to the “Platonic solids”, that he learned from Pythagorean school: in the Timaeus, there is a mathematical construction of the elements earth, fire, air, and water being represented by the cube, tetrahedron, octahedron, and icosahedron respectively. The fifth Platonic solid, the dodecahedron, is Plato's model for the whole universe: this was the first attempt Kepler used when he had to fit Brahe's data collection studying the Solar System structure.

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AristotleAristotle (Stagirus 384 a.C. - Euboea 322 a.C.), another giant of the Western culture whose influence has been immeasurably vast, set the Philosophy of Physics, Mathematics, and Science on a foundations that would carry it to modern times. He was teacher of Alexander the Great and this gave him many opportunities and an abundance of supplies. He founded a school called Lyceum where he established a library which aided in the production of many of his hundreds of books. He was a pupil of Plato but immersed himself in empirical studies and shifted from the theoretical Platonism to empiricism. He believed all peoples' concepts and all of their knowledge was ultimately based on perception. Aristotle's views on natural sciences represent the groundwork underlying many of his works and profoundly shaped medieval scholarship .

He viewed the sciences as being of three types:• theoretical (Maths, Physics, Logic and Metaphysics)• productive (the Arts)• practical (Ethics, Politics).

He contributed little to Mathematics however, his views on the nature of this subject and its relations to the physical world were highly influential. Whereas Plato believed that there was an independent, eternally existing world of ideas which constituted the reality of the universe and that mathematical concepts were part of this world, Aristotle favored concrete matter or substance.Aristotle regarded the notion of definition as a significant aspect of argument: he required that definitions reference to prior objects. For instance, the definition, 'A point is that which has no part', was unacceptable for him, as it does not assert or deny. Instead a hypothesis asserts one part of a contradiction, for instance that something is or is not. There are many views as to what Aristotle's hypotheses are:

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▪ existence claims▪ any true assumption within a science ▪ the stipulation of objects at the beginning of a typical proof in Greek

mathematics. For instance in AB is a line stipulated to exist a hypothesis is: ‘Let there be a line AB’.

He treated the basic principles of Mathematics, distinguishing between axioms (a statement worthy of acceptance and is needed prior to learning anything) and postulates (that need not be self-evident, but their truth must be sustained by the results derived from them): this distinction was later on used by Euclid who organized all Greek Mathematical (Geometrical) knowledge in the same way we have been using nowadays. Aristotle explored the relation of the point to the line dealing with the problem of the indecomposable and decomposable and made the distinction between potential infinity and actual infinity: he stated that only the former actually exists, in all regards.He is also credited with the invention of Logic, through the syllogism (the most famous is this one: “All men are mortal, Socrates is a man, Socrates is mortal”), the law of contradiction (two antithetical propositions cannot both be true at the same time and in the same sense. X cannot be non-X. A thing cannot be and not be simultaneously. And nothing that is true can be self-contradictory or inconsistent with any other truth → ¬( X ∧¬ X ) ) and the law of the excluded middle (there is no third alternative to truth or falsehood: in other words, for any statement X, either X or not-X must be true and the other must be false). His logic remained unchallenged until the century. Aristotle regarded logic as an independent subject that should precede Science and Maths.

In any society at every moment all the strong and developing forces work in relation to each other, sometimes unknowingly, sometimes quite obviously. As we look back into the past, where there are many chapters permanently erased from our view, we tend to see books and men and ideas as particular and individual, and it is easy to forget that intellectual fields cross all lines of human endeavor constantly. So the study of Greek medicine is necessary for an understanding of Greek drama and Aristotle's critique of it. Acknowledge of Greek history is required for understanding of Aristotle's Ethics, and as we learn more about the ancient world to know a great deal more about the nature and the influences of Greek Mathematics, which was perhaps the single greatest achievement of the Greek mind, and one which certainly influenced the general tenor of the society more thoroughly and in more ways than we us can yet fathom.

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Maths and Philosophy in XVI and XVII centuriesIn the late 16th and 17th centuries Mathematicians began to deep the question about the continua — straight lines, plane figures, solids – For instance one question is: “Is a line segment composed of an infinite number of indivisible points?” If so, and if these infinitesimals have zero width, how does the line segment come to have a positive length? And if they have nonzero widths, why isn’t the sum of their widths infinite?Already the ancient Greeks posed questions like these and Aristotle, trying to solve it, had argued that continua could not consist of indivisibles. but in these centuries philosophers try to demonstrate that thinking of them this way yielded insights not easily obtained from traditional Euclidean geometry.In Italy, Torricelli, Cavalieri and Galileo were approached Geometry in a new way involving infinitesimals. If classical Euclidean Geometry is conceived as a top-down approach with all theorems following by pure logic from a few self-evident axioms, the new approaches can be thought of as bottom-up, inspired by experience: a plane can be considered an infinite number of parallel lines and a solid an infinite number of parallel planes. But in some part of Europe philosophers could go on with their thinking while others couldn't as the Jesuits and the Roman Catholic associated chaos, confusion and paradoxes with infinitesimals and the motley array of proliferating Protestant sects. The philosophers we are going to analyze are involved in this revolution of thinking: two of them worked inside the Catholicism, one lived in the Protestant environment.

DescartesRené Descartes was born on March 31, 1596, in the French town called La Haye. He was entered into Jesuit College at the age of eight, where he studied for about eight years, then he spent several more years in Paris studying Mathematics with friends, such as Mersenne. He then received a law degree in 1616. At that point in time, a man that held that type of education either joined the army or the church: he chose to join the army of a nobleman in 1617, so a young man he traveled around Europe. While serving, Descartes came across a certain geometrical problem that had been posed as a challenge to the entire world to solve. Upon solving the problem in only a few hours, he had met a man named Isaac Beeckman, a Dutch scientist who became a very close friend of his. Since becoming aware of his mathematical abilities, the life of the army was unacceptable to Descartes. His health had not been good for all his life and the soldier life was not suitable for him. So, in 1621, Descartes resigned from the army and traveled for five years. During this period, he continued studying pure Maths, then, in 1626, he settled in Paris where he constructed optical instruments. In 1628 he devoted his life to seeking the truth about the science of nature and moved to Holland where he remained for twenty years, dedicating his time to Philosophy and Maths. During this time, Descartes wrote the book Meditations on First Philosophy, where he introduced the famous phrase "I think, therefore I am." meaning he wanted to find truth by the use of reason, taking complex ideas and breaking them down into simpler ones that were clear.Since he was young, he showed an interest in Physics, but, while he was writing his first book, he heard that Galileo had been arrested for claiming that the Earth rotates around the Sun, so he decided not to publish his book, and to concentrate on Mathematics instead. But his friends wanted him to publish his ideas, so in 1637 Descartes published a book called Discours de la méthod pour bien conduire sa raison et chercher la vérité dans les

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sciences, or Discourse on methods for conducting reason and seeking truth in the sciences in which he described the methods of studying Science. This book has had three appendices and one was about Optics (light and lenses), one was about weather, and one was about Mathematics, called La Géometrie and, despite its title, it is focused on the connections between Geometry and Algebra. In this book we can see many modern algebraic conventions: for example, Descartes used letters from the beginning of the alphabet for constants and known quantities, and letters from the end of the algebra for variables: this is the reason why we usually solve equations for x, and not some other symbol. Descartes believed that Mathematics was the only thing that is certain or true, so it could be used to model the complex ideas of the universe into simpler ideas that were true. La Géometrie, even it is an appendix of the main Philosophy book, is divided into several chapters: the first one explains connections between Algebra and Geometry, and gives both algebraic and geometric ways to solve equations, the second and third ones investigate more complicated subjects as conic sections and plane loci (curves at a fixed distance from different lines in a plane): we can find general equations for certain types of parabolas and hyperbolas and the classification of linear function (first degree) and curves (second and upper degrees) depending on their algebraic representation. Here he introduced his theory about determining a point in a plane by pairs of real numbers (ordered pairs), now known as Cartesian Plane. Although this had been done before by other mathematicians earlier in the history of Maths, his become a standard as he had gathered all the tools for coordinate graphing, using these reference lines to analyze the curves he studied. Like Apollonius, he usually drew his curves first, then drew reference lines to analyze them with but thought that negative numbers did not represent "real" physical quantities, so he ignored negative roots of equations, and he avoided measuring in more than one direction on a line whenever possible. Probably because of his not good health, he never got out of bed before 11 in the morning, and it is said (although the story is probably a myth) that Descartes came up with the idea for his coordinate system while lying in bed and watching a fly crawl on the ceiling of his room. Descartes had gathered all the tools for coordinate graphing. Because of this accomplishment, he is often given credit for inventing the coordinate plane, even though he never graphed an equation.La Géometrie was soon recognized as an important work of Maths, but it still took several years for its ideas to become well known. The information moved slowly for two reasons: Descartes wrote his book in French, so scholars who didn't know French could not read it and he discussed many difficult problems without giving examples or explaining simpler cases. Eventually, Descartes' friends and students published guides and commentaries for La Géometrie: many of these guides were in Latin, since almost all educated people knew Latin at that time and gave examples and explained simpler problems.In 1649, Descartes was invited by the Queen to Sweden as Court Mathematician. It is said that the Queen wanted to work on Maths at an early hour in the mornings. Thus, Descartes

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had to rise early to reach the palace. Due to the cold climate, he developed pneumonia after just a few months and died on February 11, 1650.

PascalBlaise Pascal (Clermont-Ferrand 1623 – 1662 Paris) was a French philosopher and scientist, one of the greatest and most influential mathematical writers of all time: he was also an expert in many fields, including various languages, and a well-versed religious philosopher. When he was only three years old Blaise's mother died and in 1632 the family left Clermont and settled in Paris. His father had unorthodox educational views and decided to teach his son himself, deciding that Blaise was not to study mathematics before the age of 15 and all mathematics texts were removed from their house. Blaise however, his curiosity raised by this, started to work on geometry himself at the age of 12. He discovered that the sum of the angles of a triangle are two right angles and, when his father found out, he surrendered and gave to Blaise a copy of Euclid's book “Elements”. When he was 14 he started to accompany his father to Mersenne's meetings: Mersenne belonged to the religious order of the Minims, and his cell in Paris was a frequent meeting place for many mathematicians, Gassendi, Desargues and others. When he was 16, Blaise presented a single piece of paper containing a number of projective geometry theorems, including his mystic hexagon to one of Mersenne's meetings. In December 1639 the Pascal family left Paris to live in Rouen where his father had been appointed as a tax collector for Upper Normandy. Shortly after moving to Rouen, Blaise published his first work, Essay on Conic Sections, in February 1640. In this six-pages paper he wrote the fundamental theorem that states that the three points of intersection of the opposite sides of a hexagon inscribed in a conic section lie on a straight line. A hexagon inscribed in a conic section essentially consists of six points (called “vertices”) anywhere on the conic section The the segments connecting consecutive vertices are called the "sides" of the hexagon. The straight line on which the three points of intersection lie is called the Pascal line. The hexagon itself is called a Pascal hexagon.Working three years (1642-45) Pascal invented the first digital calculator (called the Pascaline) to help his father with his work collecting taxes. Designing the device he had to face with three problems due to the design of the French currency at that time: there were 20 sols in a livre and 12 deniers in a sol. Pascal had to solve much harder technical problems to work with this division of the livre into 240 than he would have had if the division had been 100.The year 1646 was very significant for the young Pascal as his father injured his leg and had to recuperate in his house. He was looked after by two young brothers from a religious movement just outside Rouen. They had a profound effect on the young Pascal who became deeply religious. In this period Pascal began a series of experiments on atmospheric pressure: by 1647 he proved to his satisfaction that a vacuum

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existed. Descartes visited Pascal on 23 September: his visit only lasted two days and the two argued about the vacuum which Descartes did not believe in. Descartes wrote, rather cruelly, in a letter to Huygens after this visit that Pascal “...has too much vacuum in his head.”In August of 1648 Pascal observed that the pressure of the atmosphere decreases with height and deduced that a vacuum existed above the atmosphere. In October 1647 Pascal wrote New Experiments Concerning Vacuums which led to disputes with a number of scientists who, like Descartes, did not believe in a vacuum.

Pascal's father died in September 1651 and after he wrote to one of his sisters giving a deeply Christian meaning to death in general and his father's death in particular: these ideas here are at the basis of his later philosophical work Pensées.

In 1653 he published the Treatise on the Equilibrium of Liquids in which he explains the law of pressure that now we know as Pascal's law: “that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure variations (initial differences) remain the same“. In other words, if no other forces are acting on a fluid, the pressure will be the same throughout the fluid and the same in all directions In this period he also composed, but never published, another monographs that was discovered among his manuscripts after his death: the Treatise on the Weight of the Mass of Air. These two treatises represent a very important contributions to the Hydraulics and Hydrostatics. It is in recognition of his important work in the study of fluid mechanics that a standard unit of pressure is today known as the pascal (Pa), defined as a force equal to 1 Newton per square meter. Between 1648 and 1654 he worked on conic sections and produced important theorems in projective geometry. In The Generation of Conic Sections, he first part of a treatise on conics which Pascal never completed, he considered conics generated by central projection of a circle: the work is now lost but Leibniz and Tschirnhaus made notes from it and so now we can have a fairly complete picture of the work. Studying the results obtained by Tartaglia, the came out with the so called “Pascal's triangle” and wrote about this in Treatise on the Arithmetical Triangle (1653-1654) was the most important on this topic that lead Newton to his discovery of the general binomial theorem for fractional and negative powers.Pascal called the square containing each number in the array a cell. The numeral 1’s at the top of his triangle head perpendicular rows; those on the left side of the triangle head parallel rows. He called the third side of the triangle the base (diagonal) and the cells along any diagonal row “cells of the same base”. The first diagonal row (consisting of the number 1) was for him row 0, the second diagonal row (1, 1) was row 1; and so on. He constructed the triangle calculating each the number value of each cell is equal to the sum of its immediately preceding perpendicular and parallel cells (i.e.item 4 in row 7 in the base diagonal 120 = 36 + 84). Furthermore, the number value of each cell is also equal to the sum of all the cells of the preceding row (from the first cell to the cell immediately above the target cell). For example, 126 (the number value of cell 6 in row 5) = 1 + 4 + 10 + 20 + 35 + 56 (the sum of cells 1-6 of row 4).Pascal explained also in detail how the Triangle could be used to calculate combinations (n

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things taken r at a time indicated now with the symbol Cn ,r ): we would move perpendicularly down to the nth row and then move diagonally r cells (i.e. to calculate C5,4

we have to go perpendicularly down to row 5 and then move diagonally 4 cells and find that the number of combinations is 5). In the Summer of 1654 he exchanged some letters with Fermat where they laid the foundations for the theory of probability. They considered the dice problem (how many times one must throw a pair of dice before one expects a double six), already studied by Cardano, and the problem of points (how to divide the stakes if a game of dice is incomplete) also deepened by Cardano, Pacioli and Tartaglia Pascal and Fermat solved the problem of points for a two player game but did not develop powerful enough mathematical methods to solve it for three or more players. In a section of the Treatise, Pascal explained how to use the Triangle to solve the Problem of Points: “A needs 3 more points, B needs 5 more points and the game will end after seven more tries since at that juncture one of the players must reach ten points: count 3 + 5 rows on the Triangle; then sum the first 5 items. That sum divided by the sum of all items in the row is A’s portion of the stakes. Then sum the remaining 3 items in the row and divide that total by the sum of all the items in the row. That will be B’s portion.”From the Triangle:(1+7+21+35+35) ÷ (1+7+21+35+35+21+7+1) = 99/128 = A’s portion.(1+7+21) ÷ (1+7+21+35+35+21+7+1) = 29/128 = B’s portion.Expressed as a percentage, A receives 77.34375 percent of the stake; B receives 22.65625 percent of the stake.By October 1654 he almost lost his life in an accident: the horses pulling his carriage bolted, the carriage was left hanging over a bridge above the river Seine and he was rescued without any physical injury but he was much affected psychologically. Not long after he underwent another religious experience, on 23 November 1654, and he pledged his life to Christianity.

After this time Pascal made visits to the Jansenist monastery at Port-Royal des Champs about 30 km south west of Paris and between 1656 and early 1657 he published anonymous 18 writings on religious topics (Provincial Letters) being published during 1656 and early 1657 in defense of his friend Antoine Arnauld, a Jansenist who was an opponent of the Jesuits and who was on trial before the faculty of theology in Paris for his controversial religious works. In this period (late 1656 -1658) he also wrote his most famous work in Philosophy, Pensées, a collection of personal thoughts on human suffering and faith in God : this work contains 'Pascal's wager' which claims to prove that belief in God is rational with the following argument. He said that “If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.”: here he uses probabilistic and mathematical arguments but his main conclusion is that “...we are compelled to gamble...”.

His last work (1658) was on the cycloid, the curve traced by a point on the circumference of a rolling circle: he stayed up late the night unable to sleep for pain thinking about these mathematical problems. He applied Cavalieri's calculus of indivisibles to the problem of the

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area of any segment of the cycloid and the centre of gravity of any segment and could calculate the volume and surface area of the solid of revolution formed by rotating the cycloid about the x-axis. He published a challenge offering two prizes for solutions to these problems to many famous mathematicians, as Wren, Laloubère, Leibniz, Huygens, Wallis, Fermat and several others, many of whom communicated their discoveries to Pascal without entering the competition with him and could find the arc length, the length of the arch, of the cycloid. Pascal published his own solutions to his challenge problems in the Letters to Carcavi.

After that time on he took little interest in scientific subjects and spent his last years giving to the poors and going from church to church in Paris attending one religious service after another. He died at the age of 39 in intense pain after a malignant growth in his stomach spread to the brain.

Leibniz Gottfried Wilhelm Leibniz (Leipzig July 1, 1646 - Hanover November 14, 1716) was a German philosopher, mathematician, physicist and statesman who occupies a prominent place in the history of both Maths and Philosophy: noted for his independent invention of the differential and integral calculus, is one of the greatest and most influential metaphysicians, thinkers and logicians in history and also invented the “Leibniz wheel” and suggested important theories about force, energy and time. Leibniz also developed the binary number system, which is at the foundation of virtually all digital computers. Leibniz made major contributions to Physics and Technology anticipating notions that were developed much later in Biology, Medicine, Geology, Psychology, Linguistics: his writings (many of them never published) show major contributions to Politics, Ethics, Theology and History. Modern Physics, Maths, Engineering would be unthinkable without his contribution about the fundamental method of dealing with infinitesimal numbers. In Philosophy, Leibniz is mostly noted for his optimism: our Universe is the best possible one that a God could have created. Leibniz, along with Descartes and Baruch Spinoza, was one of the three great 17th century advocates of rationalism but his philosophy also looks back to the scholastic tradition, in which conclusions are produced by applying reason to “a priori” definitions rather than to empirical evidence.His father Friedrich was a professor of moral Philosophy at the University of Leipzig and died when he was only six, while his mother, Catharina Schmuck, was the daughter of a rich local lawyer who influenced his philosophical ideas. As a result, his early education was somewhat

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haphazard, but he became fluent in Latin and studied works of Greeks scholars so he was fluent at them by the age of 12 and he was ready for the university at the age of 15. He pursued the course in law in preparation for a political career and also studied Theology, Mathematics, and the new natural Philosophy of the Enlightenment, received his Bachelor’s Degree in Philosophy by 1662 and earning his Masters Degree by 1664. He also gained his Bachelors degree in Law by 1665: this shows the extent and diversity of his academic interests. After graduation, he applied for a doctorate in Law, but was refused due to his young age, so he presented his thesis to the University of Altdorf, where professors were so impressed that they awarded him the degree of Doctor of Laws (1667) and gave him a job of professorship: he declined the offer and accepted instead a position in the service of the elector of Mainz.In this period Louis XIV's aggressive activities were a serious threat to the German states so in 1670 Leibniz published a pamphlet proposing a defensive coalition of the northern European Protestant countries, recently weakened by the Thirty Years War. In the mean time, to give these principalities an economic recovery, he conceived a plan whereby Louis might gain Holland's valuable possessions in Asia by sort of a "holy war" against non-Christian Egypt: Leibniz was invited to Paris to present his plan that was not adopted at the end. Despite this, stayed 4 years in the French capital, visiting twice London (1673 and 1676) and this experience was crucial for his intellectual development.Before going to Paris, Leibniz developed a calculating machine based on the principles of the “Pascalina” but capable of performing much more complicated mathematical operations; his demonstrations of this machine before the Académie Royale des Sciences and the Royal Society of London aroused much interest and led to fruitful relations with members of these groups and to his election to membership in the Royal Society shortly after his first visit in London.In Paris he met the Dutch mathematician Christiaan Huygens who was very important as a stimulus to Leibniz's interest in Mathematics during the years of his residence there Leibniz developed both the integral and the differential calculus.

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In 1676 Leibniz transferred his services to the house of Brunswick and moved to Hanover, town which became his home and where he mainly lived for the rest of his life. He was sent on important diplomatic missions, with freedom to seek out leading scholars wherever he went and he received many honors, as well as a generous stipend, and had ample leisure for pursuing his own interests. He was in charge to write a history of Brunswick from earliest times and he could access not only to the resources of the ducal library but also to the historical repositories in Germany and Italy.He couldn't complete the history itself (at his death he had completed to the year 1005) but he added geological data to bear for the first time on historical interpretation and used original documents in a modern way. In 1672 he published a pamphlet where he proposed an alliance of all the European countries against Turkey, an example of a possible reunification of all Christians, adducing historical evidence of this unity in correspondence with the French prelate Jacques Bossuet: he didn't succeeded again in his attempts at mediation of differences.In 1678 he founded journal for the publication of scholarly papers named “Acta eruditorum”,that gained wide circulation in Europe: over the next 35 years, most of his own published writings appeared. He imprinted his name in the world of mechanical calculation, he was also the first to create a pinwheel calculator (step rekoner) explaining in detail how it worked in 1685 and developed the “Leibniz Wheel” that helped to improve the binary number system which is at the basis of the digital computers.

In 1700 he was elected a member of the French Académie Royale and, in the same year, the

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Akademie der Wissenschaften was founded at Berlin because his recommendation: following the patterns of the French Académie and the Royal Society of London, he drew up its statutes and was its first president, maintaining that position for the rest of his life. Through his influence similar academies were established at Dresden, St. Petersburg, and Vienna.Because of his disposition to moderation and tolerance and his position of leadership among European scholars, he had an active role as diplomat: the warmth and loyalty of many friends and supporters can be seen in his enormous correspondence with them. Between these people there were also women and he had a positive influence to develop the new "learned woman" idea.Leibniz' last years were overshadowed by a controversy with the powerful president of the Royal Society, Isaac Newton, who said he had invented calculus before Leibniz, without publishing it, and who made friends sign texts he wrote to support this claim: now we considered to have been a case of independent discovery by two highly gifted minds but at the time the quarrel was really hard for the exchanges of accusations that lasted for more than 10 years by partisans on one side and then the other. Newton was in a sense essentially practical: he invented tools then showed how these could be used to compute practical results about the physical world, but Leibniz had a broader and more philosophical view and saw calculus not just as a specific tool in itself, but as an example that should inspire efforts at other kinds of formalization and other kinds of universal tools. Finally, an investigating commission got some results and exonerated Newton and failed to remove the charge of plagiarism against Leibniz: the cutting off of free communication of ideas between the English scientists and those of the Continent was ironically to the detriment of the former: Leibniz's notation was more efficient than Newton's and facilitated the great strides in mathematical Physics made on the Continent during the next hundred years, in which the participation of English scientists was negligible. The whole procedure was a crushing offense for Leibniz, who had always been a proponent of free interchange among scholars and who was really sad when the Duke of Brunswick's refused to include him in his entourage when, in 1714, he became England's George I, as he considered him a controversial figure. Leibniz died at Hanover in 1716 and his popularity with his own countrymen had waned with his declining court favor: his only worthy eulogy was composed on the first anniversary of his death by the French academician Bernard de Fontenelle and it was read before the meeting of Leibniz's colleagues in Paris and recorded in their archives.

http://www-history.mcs.st-and.ac.uk/Biographies/Pythagoras.htmlhttp://www.iep.utm.edu/zeno-par/http://www.answers.com/topic/platonic-academy-1http://www.ucmp.berkeley.edu/history/aristotle.htmlhttp://www.math.wichita.edu/history/men/descartes.htmlhttp://www-history.mcs.st-and.ac.uk/Biographies/Pascal.htmlhttp://www.iep.utm.edu/pascal-b/http://www.famousscientists.org/gottfried-leibniz/http://www.dmoz.org/Society/Philosophy/Philosophy_of_Science/Mathematics/

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