History of Mathematics Course Outline PDF

Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Course Title: History of Mathematics Course Code: SEMA 30013 Course Credits: 3 units Course Description: The course presents the humanistic aspects of mathematics which provide the historical context and timeline that led to the present understanding and applications of the different branches of mathematics Topics included in this course are not very technical and rigid aspects of mathematics; rather they are early, interesting, and light developments of the field. They are intended to enrich the background of the students in the hope that the students find value and inspiration in the historical approach to mathematical concepts. Course Learning Outcomes: At the end of the course, the pre-service teachers should be able to: A. Demonstrate knowledge and understanding of the historical facts and landmarks that led to the development of the different branches and schools of thought in mathematics; B. Show critical and creat7ive thinking in analyzing popular problems involving foundational concepts in mathematics; and C. Manifest appreciation for mathematics as a dynamic field through sharing of personal experiences of enlightenment relative to the evolution of the different branches of mathematics. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Course Content Course Contents The Development of mathematics: ancient period Unit 1 Origins of Mathematics: Egypt and Babylonia Mathematics of Ancient Greece The Development of mathematics: a historical overview: Medieval Period Unit 2 Medieval Period and the Renaissance Euler, Fermat and Descartes The Development of mathematics: a historical overview: Modern Period Unit 3 Non-Euclidean Geometries Birth of set theory and problems in the foundations of mathematics The Nature of Mathematics Patterns and Relationships Unit 4 Mathematics, Science, and Technology Mathematical Inquiry Course Grading System Class Standing 70% Submitted Activities Portfolio/Output Final Examination 30% FINAL RATING 100% Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Unit 1 : The Development of mathematics: ancient period The history of mathematics is nearly as old as humanity itself. Since antiquity, mathematics has been fundamental to advances in science, engineering, and philosophy. It has evolved from simple counting, measurement, and calculation, and the systematic study of the shapes and motions of physical objects, through the application of abstraction, imagination, and logic, to the broad, complex, and often abstract discipline we know today. From the notched bones of early man to the mathematical advances brought about by settled agriculture in Mesopotamia and Egypt and the revolutionary developments of ancient Greece and its Hellenistic empire, the story of mathematics is a long and impressive one. The East carried on the baton, particularly China, India, and the medieval Islamic empire, before the focus of mathematical innovation moved back to Europe in the late Middle Ages and Renaissance. Then, a whole new series of revolutionary developments occurred in 17th Century and 18th Century Europe, setting the stage for the increasing complexity and abstraction of 19th Century mathematics, and finally the audacious and sometimes devastating discoveries of the 20th Century. Origins of Mathematics: Egypt and Babylonia Sumer (a region of Mesopotamia, modern-day Iraq) was the birthplace of writing, the wheel, agriculture, the arch, the plow, irrigation and many other innovations, and is often referred to as the Cradle of Civilization. The Sumerians developed the earliest known writing system – a pictographic writing system known as cuneiform script, using wedge-shaped characters inscribed on baked clay tablets – and this has meant that we actually have more knowledge of ancient Sumerian and Babylonian mathematics than of early Egyptian mathematics. Indeed, we even have what appear to school exercises in arithmetic and geometric problems. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde EGYPTIAN MATHEMATICS – NUMBERS & NUMERALS The early Egyptians settled along the fertile Nile valley as early as about 6000 BCE, and they began to record the patterns of lunar phases and the seasons, both for agricultural and religious reasons. The Pharaoh’s surveyors used measurements based on body parts (a palm was the width of the hand, a cubit the measurement from elbow to fingertips) to measure land and buildings very early in Egyptian history, and a decimal numeric system was developed based on our ten fingers. The oldest mathematical text from ancient Egypt discovered so far, though, is the Moscow Papyrus, which dates from the Egyptian Middle Kingdom around 2000 – 1800 BCE. Ancient Egyptian hieroglyphic numerals ANCIENT EGYPTIAN NUMBER SYSTEM It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BCE (and probably much early). Written numbers used a stroke for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands, as well as other hieroglyphic symbols for higher powers of ten up to a million. However, there was no concept of place value, so larger numbers were rather unwieldy (although a million required just one character, a million minus one required fifty-four characters). The Rhind Papyrus, dating from around 1650 BCE, is a kind of instruction manual in arithmetic and geometry, and it gives us explicit demonstrations of how multiplication and division was carried out at that time. It also contains evidence of other mathematical knowledge, including unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means, and how to solve first order linear equations as well as arithmetic and geometric series. The Berlin Papyrus, which dates from around 1300 BCE, shows that ancient Egyptians could solve second-order algebraic (quadratic) equations. Multiplication, for example, was achieved by a process of repeated doubling of the number to be multiplied on one side and of one on the other, essentially a kind of multiplication of binary factors similar to that used by modern computers (see the Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde example). These corresponding blocks of counters could then be used as a kind of multiplication reference table: first, the combination of powers of two which add up to the number to be multiplied by was isolated, and then the corresponding blocks of counters on the other side yielded the answer. This effectively made use of the concept of binary numbers, over 3,000 years before Leibniz introduced it into the west, and many more years before the development of the computer was to fully explore its potential. Ancient Egyptian method of multiplication Practical problems of trade and the market led to the development of a notation for fractions. The papyri which have come down to us demonstrate the use of unit fractions based on the symbol of the Eye of Horus, where each part of the eye represented a different fraction, each half of the previous one (i.e. half, quarter, eighth, sixteenth, thirty- second, sixty-fourth), so that the total was one-sixty-fourth short of a whole, the first known example of a geometric series. Unit fractions could also be used for simple division sums. For example, if they needed to divide 3 loaves among 5 people, they would first divide two of the loaves into thirds and the third loaf into fifths, then they would divide the left over third from the second loaf into five pieces. Thus, each person would receive one-third plus one-fifth plus one- fifteenth (which totals three-fifths, as we would expect). The Egyptians approximated the area of a circle by using shapes whose area they did know. They observed that the area of a circle of diameter 9 units, for example, was very close to the area of a square with sides of 8 units, so that the area of circles of other Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde diameters could be obtained by multiplying the diameter by 8⁄9 and then squaring it. This gives an effective approximation of π accurate to within less than one percent. Ancient Egyptian method of division The pyramids themselves are another indication of the sophistication of Egyptian mathematics. Setting aside claims that the pyramids are first known structures to observe the golden ratio of 1 : 1.618 (which may have occurred for purely aesthetic, and not mathematical, reasons), there is certainly evidence that they knew the formula for the volume of a pyramid – 1⁄3 times the height times the length times the width – as well as of a truncated or clipped pyramid. They were also aware, long before Pythagoras, of the rule that a triangle with sides 3, 4 and 5 units yields a perfect right angle, and Egyptian builders used ropes knotted at intervals of 3, 4 and 5 units in order to ensure exact right angles for their stonework (in fact, the 3-4-5 right triangle is often called “Egyptian”). As in Egypt, Sumerian mathematics initially developed largely as a response to bureaucratic needs when their civilization settled and developed agriculture (possibly as early as the 6th millennium BCE) for the measurement of plots of land, the taxation of Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde individuals, etc. In addition, the Sumerians and Babylonians needed to describe quite large numbers as they attempted to chart the course of the night sky and develop their sophisticated lunar calendar. They were perhaps the first people to assign symbols to groups of objects in an attempt to make the description of larger numbers easier. They moved from using separate tokens or symbols to represent sheaves of wheat, jars of oil, etc, to the more abstract use of a symbol for specific numbers of anything. Starting as early as the 4th millennium BCE, they began using a small clay cone to represent one, a clay ball for ten, and a large cone for sixty. Over the course of the third millennium, these objects were replaced by cuneiform equivalents so that numbers could be written with the same stylus that was being used for the words in the text. A rudimentary model of the abacus was probably in use in Sumeria from as early as 2700 – 2300 BCE. SUMERIAN & BABYLONIAN NUMBER SYSTEM: BASE 60 Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand the five fingers on the other hand. Unlike those of the Egyptians, Greeks and Romans, Babylonian numbers used a true place-value system, where digits written in the left column represented larger values, much as in the modern decimal system, although of course using base 60 not base 10. Thus, in the Babylonian system represented 3,600 plus 60 plus 1, or 3,661. Also, to represent the numbers 1 – 59 within each place value, two distinct symbols were used, a unit symbol ( ) and a ten symbol ( ) which were combined in a similar way to the familiar system of Roman numerals (e.g. 23 would be shown as ). Thus, represents 60 plus 23, or 83. However, the number 60 was represented by the same symbol as the number 1 and, because they lacked an equivalent of the decimal point, the actual place value of a symbol often had to be inferred from the context. Babylonian Numerals Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde It has been conjectured that Babylonian advances in mathematics were probably facilitated by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 – in fact, 60 is the smallest integer divisible by all integers from 1 to 6), and the continued modern-day usage of of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are all testaments to the ancient Babylonian system. It is for similar reasons that 12 (which has factors of 1, 2, 3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12 inches, 12 pence, 2 x 12 hours, etc). The Babylonians also developed another revolutionary mathematical concept, something else that the Egyptians, Greeks and Romans did not have, a circle character for zero, although its symbol was really still more of a placeholder than a number in its own right. Babylonian Clay tablets We have evidence of the development of a complex system of metrology in Sumer from about 3000 BCE, and multiplication and reciprocal (division) tables, tables of squares, square roots and cube roots, geometrical exercises and division problems from around 2600 BCE onwards. Later Babylonian tablets dating from about 1800 to 1600 BCE cover topics as varied as fractions, algebra, methods for solving linear, quadratic and even some cubic equations, and the calculation of regular reciprocal pairs (pairs of number which multiply together to give 60). One Babylonian tablet gives an approximation to √2 accurate to an astonishing five decimal places. Others list the squares of numbers up to 59, the cubes of numbers up to 32 as well as tables of compound interest. Yet another gives an estimate for π of 3 1⁄8 (3.125, a reasonable approximation of the real value of 3.1416). The idea of square numbers and quadratic equations (where the unknown quantity is multiplied by itself, e.g. x2) naturally arose in the context of the measurement of land, and Babylonian mathematical tablets give us the first ever evidence of the solution of quadratic equations. The Babylonian approach to solving them usually revolved around a kind of geometric game of slicing up and rearranging shapes, although the use of algebra and quadratic equations also appears. At least some of the examples we have appear to indicate problem-solving for its own sake rather than in order to resolve a concrete practical problem. The Babylonians used geometric shapes in their buildings and design and in dice for the leisure games which were so popular in their society, such as the ancient game of backgammon. Their geometry extended to the calculation of the areas of rectangles, triangles and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders (although not pyramids). Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Plimpton 322 clay tablet The famous and controversial Plimpton 322 clay tablet, believed to date from around 1800 BCE, suggests that the Babylonians may well have known the secret of right- angled triangles (that the square of the hypotenuse equals the sum of the square of the other two sides) many centuries before the Greek Pythagoras. The tablet appears to list 15 perfect Pythagorean triangles with whole number sides, although some claim that they were merely academic exercises, and not deliberate manifestations of Pythagorean triples. Babylonian Clay tablets from c. 2100 BCE showing a problem concerning the area of an irregular shape 1. By which BC two earlier nations had joined to form a single Egyptian nation under a single ruler? a. 38 b. 300 c. 3000 d. 30 2. Knowing which season was about to arrive was vital and the study of astronomy developed to provide calendar information? a. Summer b. winter c. spring d. None of the above 3. The Egyptians used________ as base number a. 10 b. 100 c. 56 d. 30 4. Why did Egyptians introduced a new symbol for 10? a. Because it was fun to invent new symbols b. Because there would be too many lines c. Because they were bored of the lines d. None of the above 5. What was the symbol for 10000 in Egyptian Mathematics? a. Single pencil b. Two fingers c. Single finger d. A tadpole Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde 6. How can we write 345? a. Three coil of rope and four piece of rope and five lines b. Two coil of rope and four piece of rope c. Three piece of rope and four coil of rope and five lines d. None of the above 7. Tell the similarities between Hindu Arabic form and Egyptian form. a. Base as ten b. The way they represent the 1 is just a line. Usually, when we write a 1 on a piece of paper its just a line. c. Ten as piece of rope d. There no similarities 8. The system of Egyptian Numerals was used In Egypt around________until the first millennium. a. 2,888 BC b. 2,888 AD c. 2,889 BC d. 2889 AC 9. How many characters are there in the Egyptian numerals? a. 9 b. 7 c. 6 d. 5 10. The Egyptian number system was written in____ a. hieroglyphics b. hierogly c. hair d. hierlogyph 11. Egyptians needed a symbol for zero. a. true b. false 12. Meaning of Hieroglyph? a. a character used in a system of pictorial writing, particularly that form used on ancient Egyptian monuments. b. name of the Egyptian civilization c. a character used in a system of fictional writing, particularly that form used on ancient Egyptian monuments. d. a character used in a system of pictorial writing, particularly that form used on ancient indian monuments. 13. What kind of mathematical operations were used in the Egyptian number system? a. papyrus, hieroglyphics. b. paper and hieroglyphics. c. Decimals d. all of the above 14. Egyptian Numerals was used In Egypt around_____ until the first millennium. a. 28888 BC b. 2887 AC c. 2888 AD d. 2888 BC Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde 15. The Egyptians were are the first civilisation that used a civilised system of numbers. a. True b. false c. both Mathematics of Ancient Greece As the Greek empire began to spread its sphere of influence into Asia Minor, Mesopotamia and beyond, the Greeks were smart enough to adopt and adapt useful elements from the societies they conquered. This was as true of their mathematics as anything else, and they adopted elements of mathematics from both the Babylonians and the Egyptians. But they soon started to make important contributions in their own right and, for the first time, we can acknowledge contributions by individuals. By the Hellenistic period, the Greeks had presided over one of the most dramatic and important revolutions in mathematical thought of all time. Ancient Greek Herodianic numerals Attic or Herodianic numerals The ancient Greek numeral system, known as Attic or Herodianic numerals, was fully developed by about 450 BCE, and in regular use possibly as early as the 7th Century BCE. It was a base 10 system similar to the earlier Egyptian one (and even more similar to the later Roman system), with symbols for 1, 5, 10, 50, 100, 500 and 1,000 repeated as many times needed to represent the desired number. Addition was done by totalling separately the symbols (1s, 10s, 100s, etc) in the numbers to be added, and multiplication was a laborious process based on successive doublings (division was based on the inverse of this process). Thales’ Intercept Theorem But most of Greek mathematics was based on geometry. Thales, one of the Seven Sages of Ancient Greece, who lived on the Ionian coast of Asian Minor in the first half of the 6th Century BCE, is usually considered to have been the first to lay down guidelines Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde for the abstract development of geometry, although what we know of his work (such as on similar and right triangles) now seems quite elementary. Thales established what has become known as Thales’ Theorem, whereby if a triangle is drawn within a circle with the long side as a diameter of the circle, then the opposite angle will always be a right angle (as well as some other related properties derived from this). He is also credited with another theorem, also known as Thales’ Theorem or the Intercept Theorem, about the ratios of the line segments that are created if two intersecting lines are intercepted by a pair of parallels (and, by extension, the ratios of the sides of similar triangles). To some extent, however, the legend of the 6th Century BCE mathematician Pythagoras of Samos has become synonymous with the birth of Greek mathematics. Indeed, he is believed to have coined both the words “philosophy” (“love of wisdom“) and “mathematics” (“that which is learned“). Pythagoras was perhaps the first to realize that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers. Pythagoras’ Theorem (or the Pythagorean Theorem) is one of the best known of all mathematical theorems. But he remains a controversial figure, as we will see, and Greek mathematics was by no means limited to one man. Thale’s Theorem https://www.youtube.com/watch?v=0PvSfDE6fKs Three geometrical problems Three geometrical problems in particular, often referred to as the Three Classical Problems, and all to be solved by purely geometric means using only a straight edge and a compass, date back to the early days of Greek geometry: “the squaring (or quadrature) of the circle”, “the doubling (or duplicating) of the cube” and “the trisection of an angle”. These intransigent problems were profoundly influential on future geometry and led to many fruitful discoveries, although their actual solutions (or, as it turned out, the proofs of their impossibility) had to wait until the 19th Century. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde The Three Classical Problems Hippocrates of Chios (not to be confused with the great Greek physician Hippocrates of Kos. A detailed biography here.) was one such Greek mathematician who applied himself to these problems during the 5th Century BCE (his contribution to the “squaring the circle” problem is known as the Lune of Hippocrates). His influential book “The Elements”, dating to around 440 BCE, was the first compilation of the elements of geometry, and his work was an important source for Euclid‘s later work. Squaring the Circle https://www.youtube.com/watch?v=TfH4DtZGP3I Doubling the Cube https://www.youtube.com/watch?v=4Ncc5A2xT78 Trisecting the Angle https://www.youtube.com/watch?v=5VxMUhxkBqA Zeno’s Paradox of Achilles and the Tortoise It was the Greeks who first grappled with the idea of infinity, such as described in the well-known paradoxes attributed to the philosopher Zeno of Elea in the 5th Century BCE. The most famous of his paradoxes is that of Achilles and the Tortoise, which describes a theoretical race between Achilles and a tortoise. Achilles gives the much slower tortoise a head start, but by the time Achilles reaches the tortoise’s starting point, the tortoise has already moved ahead. By the time Achilles reaches that point, the tortoise has moved on again, etc, etc, so that in principle the swift Achilles can never catch up with the slow tortoise. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Paradoxes such as this one and Zeno’s so-called Dichotomy Paradox are based on the infinite divisibility of space and time, and rest on the idea that a half plus a quarter plus an eighth plus a sixteenth, etc, etc, to infinity will never quite equal a whole. The paradox stems, however, from the false assumption that it is impossible to complete an infinite number of discrete dashes in a finite time, although it is extremely difficult to definitively prove the fallacy. The ancient Greek Aristotle was the first of many to try to disprove the paradoxes, particularly as he was a firm believer that infinity could only ever be potential and not real. Zeno’s Paradox of Achilles and the Tortoise Democritus, most famous for his prescient ideas about all matter being composed of tiny atoms, was also a pioneer of mathematics and geometry in the 5th – 4th Century BCE, and he produced works with titles like “On Numbers“, “On Geometrics“, “On Tangencies“, “On Mapping” and “On Irrationals“, although these works have not survived. We do know that he was among the first to observe that a cone (or pyramid) has one-third the volume of a cylinder (or prism) with the same base and height, and he is perhaps the first to have seriously considered the division of objects into an infinite number of cross-sections. However, it is certainly true that Pythagoras in particular greatly influenced those who came after him, including Plato, who established his famous Academy in Athens in 387 BCE, and his protégé Aristotle, whose work on logic was regarded as definitive for over two thousand years. Plato the mathematician is best known Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde for his description of the five Platonic solids, but the value of his work as a teacher and popularizer of mathematics can not be overstated. Plato’s student Eudoxus of Cnidus is usually credited with the first implementation of the “method of exhaustion” (later developed by Archimedes), an early method of integration by successive approximations which he used for the calculation of the volume of the pyramid and cone. He also developed a general theory of proportion, which was applicable to incommensurable (irrational) magnitudes that cannot be expressed as a ratio of two whole numbers, as well as to commensurable (rational) magnitudes, thus extending Pythagoras’ incomplete ideas. Perhaps the most important single contribution of the Greeks, though – and Pythagoras, Plato and Aristotle were all influential in this respect – was the idea of proof, and the deductive method of using logical steps to prove or disprove theorems from initial assumed axioms. Older cultures, like the Egyptians and the Babylonians, had relied on inductive reasoning, that is using repeated observations to establish rules of thumb. It is this concept of proof that give mathematics its power and ensures that proven theories are as true today as they were two thousand years ago, and which laid the foundations for the systematic approach to mathematics of Euclid and those who came after him. Zeno’s Paradox of Achilles and the Tortoise https://www.youtube.com/watch?v=NCtw5f6XPF4 Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Unit 2 :The Development of mathematics: a historical overview: Medieval Period During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated. Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as “How many angels can stand on the point of a needle?“ From the 4th to 12th Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such as Nicomachus and Euclid. All trade and calculation was made using the clumsy and inefficient Roman numeral system, and with an abacus based on Greek and Roman models. By the 12th Century, though, Europe, and particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Chester translated Al-Khwarizmi‘s important book on algebra into Latin in the 12th Century, and the complete text of Euclid‘s “Elements” was translated in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The great expansion of trade and commerce in general created a growing practical need for mathematics, and arithmetic entered much more into the lives of common people and was no longer limited to the academic realm. The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education. Europe’s first great medieval mathematician was the Italian Leonardo of Pisa, better known by his nickname Fibonacci. Although best known for the so-called Fibonacci Sequence of numbers, perhaps his most important contribution to European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the 13th Century, which soon made the Roman numeral system obsolete, and opened the way for great advances in European mathematics. An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was the Frenchman Nicole Oresme. He used a system of rectangular coordinates centuries before his countryman René Descartes popularized the idea, as well as perhaps the first time-speed-distance graph. Also, leading from his research into musicology, he was the first to use fractional exponents, and also worked on infinite series, being the first to prove that the harmonic series 1⁄1 + 1⁄2 + 1⁄3 + 1⁄4 + 1⁄5… is a divergent infinite series (i.e. not tending to a limit, other than infinity). The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century, his main contribution to mathematics being in the area of trigonometry. He helped separate trigonometry from astronomy, and it was largely through his efforts that trigonometry came to be considered an independent branch of mathematics. His book “De Triangulis“, in which he described much of the basic trigonometric knowledge which is now taught in high school and college, was the first great book on trigonometry to appear in print. Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician and astronomer, whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor. He also held some distinctly non-standard intuitive ideas about Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde the universe and the Earth’s position in it, and about the elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of Copernicus and Kepler. Renaissance - 16TH CENTURY MATHEMATICS The cultural, intellectual and artistic movement of the Renaissance, which saw a resurgence of learning based on classical sources, began in Italy around the 14th Century, and gradually spread across most of Europe over the next two centuries. Science and art were still very much interconnected and intermingled at this time, as exemplified by the work of artist/scientists such as Leonardo da Vinci, and it is no surprise that, just as in art, revolutionary work in the fields of philosophy and science was soon taking place. THE SUPERMAGIC SQUARE It is a tribute to the respect in which mathematics was held in Renaissance Europe that the famed German artist Albrecht Dürer included an order-4 magic square in his engraving “Melencolia I“. In fact, it is a so-called “super magic square” with many more lines of addition symmetry than a regular 4 x 4 magic square (see image at right). The year of the work, 1514, is shown in the two bottom central squares. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde An important figure in the late 15th and early 16th Centuries is an Italian Franciscan friar called Luca Pacioli, who published a book on arithmetic, geometry and book-keeping at the end of the 15th Century which became quite popular for the mathematical puzzles it contained. It also introduced symbols for plus and minus for the first time in a printed book (although this is also sometimes attributed to Giel Vander Hoecke, Johannes Widmann and others), symbols that were to become standard notation. Pacioli also investigated the Golden Ratio of 1 : 1.618… (see the section on Fibonacci) in his 1509 book “The Divine Proportion”, concluding that the number was a message from God and a source of secret knowledge about the inner beauty of things. The supermagic square shown in Albrecht Dürer’s engraving “Melencolia I” https://www.youtube.com/watch?v=-Tbd3dzlRnY Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde During the 16th and early 17th Century, the equals, multiplication, division, radical (root), decimal and inequality symbols were gradually introduced and standardized. The use of decimal fractions and decimal arithmetic is usually attributed to the Flemish mathematician Simon Stevin the late 16th Century, although the decimal point notation was not popularized until early in the 17th Century. Stevin was ahead of his time in enjoining that all types of numbers, whether fractions, negatives, real numbers or surds (such as √2) should be treated equally as numbers in their own right. In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competition that the unlikely figure of the young, self-taught Niccolò Fontana Tartaglia revealed to the world the formula for solving first one type, and later all types, of cubic equations (equations with terms including x3), an achievement hitherto considered impossible and which had stumped the best mathematicians of China, India and the Islamic world. Building on Tartaglia’s work, another young Italian, Lodovico Ferrari, soon devised a similar method to solve quartic equations (equations with terms including x4) and both solutions were published by Gerolamo Cardano. Despite a decade-long fight over the publication, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers (although it fell to another Bologna resident, Rafael Bombelli, to explain what imaginary numbers really were and how they could be used). Tartaglia went on to produce other important (although largely ignored) formulas and methods, and Cardano published perhaps the first systematic treatment of probability. With Hindu-Arabic numerals, standardized notation and the new language of algebra at their disposal, the stage was set for the European mathematical revolution of the 17th Century. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Basic mathematical notation, with dates of first use Euler, Fermat and Descartes Leonhard Euler – Swiss Mathematician Leonhard Euler was one of the giants of 18th Century mathematics. Like the Bernoulli’s, he was born in Basel, Switzerland, and he studied for a while under Johann Bernoulli at Basel University. But, partly due to the overwhelming dominance of the Bernoulli family in Swiss mathematics, and the difficulty of finding a good position and recognition in his hometown, he spent most of his academic life in Russia and Germany, especially in the burgeoning St. Petersburg of Peter the Great and Catherine the Great. Despite a long life and thirteen children, Euler had more than his fair share of tragedies and deaths, and even his blindness later in life did not slow his prodigious output – his collected works comprise nearly 900 books and, in the year 1775, he is said to have produced on average one mathematical paper every week – as he compensated for it with his mental calculation skills and photographic memory (for example, he could repeat the Aeneid of Virgil from beginning to end without Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde hesitation, and for every page in the edition he could indicate which line was the first and which the last). Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music. Mathematical notation created or popularized by Euler Much of the notation used by mathematicians today – including e, i, f(x), ∑, and the use of a, b and c as constants and x, y and z as unknowns – was either created, popularized or standardized by Euler. His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems. He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, eiπ = -1, sometimes known as Euler’s Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called “the most remarkable formula in mathematics”, “uncanny and sublime” and “filled with cosmic beauty”, among other descriptions. Another such discovery, often known simply as Euler’s Formula, is eix = cosx + isinx. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers. The discovery that initially sealed Euler’s reputation was announced in 1735 and concerned the calculation of infinite sums. It was called the Basel problem after the Bernoulli’s had tried and failed to solve it, and asked what was the precise sum of the of the reciprocals of the squares of all the natural numbers to infinity i.e. 1⁄12 + 1⁄22 + 1⁄32 + 1⁄42 … (a zeta function using a zeta constant of 2). Euler’s friend Daniel Bernoulli had estimated the sum to be about 13⁄5, but Euler’s superior method yielded the exact but rather unexpected result of π2⁄6. He also showed that the infinite series was equivalent to an infinite product of prime numbers, an identity which would later inspire Riemann’s investigation of complex zeta functions. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Mathematical notation created or popularized by Euler The Seven Bridges of Königsberg Problem Also in 1735, Euler solved an intransigent mathematical and logical problem, known as the Seven Bridges of Königsberg Problem, which had perplexed scholars for many years, and in doing so laid the foundations of graph theory and presaged the important mathematical idea of topology. The city of Königsberg in Prussia (modern-day Kaliningrad in Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem was to find a route through the city that would cross each bridge once and only once. In fact, Euler proved that the problem has no solution, but in doing so he made the important conceptual leap of pointing out that the choice of route within each landmass is irrelevant and the only important feature is the sequence of bridges crossed. This allowed him to reformulate the problem in abstract terms, replacing each land mass with an abstract node and each bridge with an abstract connection. This resulted in a mathematical structure called a “graph”, a pictorial representation made up of points (vertices) connected by non-intersecting curves (arcs), which may be distorted in any way without changing the graph itself. In this way, Euler was able to deduce that, because the four land masses in the original problem are touched by an odd number of bridges, the existence of a walk traversing each bridge once only inevitably leads to a contradiction. If Königsberg had had Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde one fewer bridges, on the other hand, with an even number of bridges leading to each piece of land, then a solution would have been possible. https://www.youtube.com/watch?v=2iovbcPwAro List of theorems and methods pioneered by Euler The list of theorems and methods pioneered by Euler is immense, and largely outside the scope of an entry-level study such as this, but mention could be made of just some of them: the demonstration of geometrical properties such as Euler’s Line and Euler’s Circle; the definition of the Euler Characteristic χ (chi) for the surfaces of polyhedra, whereby the number of vertices minus the number of edges plus the number of faces always equals 2 (see table at right); a new method for solving quartic equations; the Prime Number Theorem, which describes the asymptotic distribution of the prime numbers; proofs (and in some cases disproofs) of some of Fermat’s theorems and conjectures; Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde the discovery of over 60 amicable numbers (pairs of numbers for which the sum of the divisors of one number equals the other number), although some were actually incorrect; a method of calculating integrals with complex limits (foreshadowing the development of modern complex analysis); the calculus of variations, including its best-known result, the Euler-Lagrange equation; a proof of the infinitude of primes, using the divergence of the harmonic series; the integration of Leibniz‘s differential calculus with Newton‘s Method of Fluxions into a form of calculus we would recognize today, as well as the development of tools to make it easier to apply calculus to real physical problems; etc, etc. In 1766, Euler accepted an invitation from Catherine the Great to return to the St. Petersburg Academy, and spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy, including a fire in 1771 which cost him his home (and almost his life), and the loss in 1773 of his dear wife of 40 years, Katharina. He later married Katharina’s half-sister, Salome Abigail, and this marriage would last until his death from a brain hemorrhage in 1783. The Euler Characteristic Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Pierre De Fermat Mathematician Biography. Another Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Stimulated and inspired by the “Arithmetica” of the Hellenistic mathematician Diophantus, he went on to discover several new patterns in numbers which had defeated mathematicians for centuries, and throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory. Although he showed an early interest in mathematics, he went on study law at Orléans and received the title of councillor at the High Court of Judicature in Toulouse in 1631, which he held for the rest of his life. He was fluent in Latin, Greek, Italian and Spanish, and was praised for his written verse in several languages, and eagerly sought for advice on the emendation of Greek texts. Fermat’s mathematical work was communicated mainly in letters to friends, often with little or no proof of his theorems. Although he himself claimed to have proved all his arithmetic theorems, few records of his proofs have survived, and many mathematicians have doubted some of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat. The Two Square Theorem One example of his many theorems is the Two Square Theorem, which shows that any prime number which, when divided by 4, leaves a remainder of 1 (i.e. can be written in the form 4n + 1), can always be re-written as the sum of two square numbers (see image at right for examples). His so-called Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers a and p, where p is a prime number and not a factor of a, then a multiplied by itself p-1 times and then divided by p, will always leave a remainder of 1. In mathematical terms, this is written: ap-1 = 1(mod p). Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde For example, if a = 7 and p = 3, then 72 ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1. Fermat’s Theorem on Sums of Two Squares Fermat numbers Fermat identified a subset of numbers, now known as Fermat numbers, which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 22n + 1. The first five such numbers are: 21 + 1 = 3; 22 + 1 = 5; 24 + 1 = 17; 28 + 1 = 257; and 216 + 1 = 65,537. Interestingly, these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to to show the value of inductive proof in mathematics. Last Theorem Fermat’s pièce de résistance, though, was his famous Last Theorem, a conjecture left unproven at his death, and which puzzled mathematicians for over 350 years. The theorem, originally described in a scribbled note in the margin of his copy of Diophantus‘ “Arithmetica”, states that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two (i.e. squared). This seemingly simple conjecture has proved to be one of the world’s hardest mathematical problems to prove. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde There are clearly many solutions – indeed, an infinite number – when n = 2 (namely, all the Pythagorean triples), but no solution could be found for cubes or higher powers. Tantalizingly, Fermat himself claimed to have a proof, but wrote that “this margin is too small to contain it”. As far as we know from the papers which have come down to us, however, Fermat only managed to partially prove the theorem for the special case of n = 4, as did several other mathematicians who applied themselves to it (and indeed as had earlier mathematicians dating back to Fibonacci, albeit not with the same intent). Over the centuries, several mathematical and scientific academies offered substantial prizes for a proof of the theorem, and to some extent it single-handedly stimulated the development of algebraic number theory in the 19th and 20th Centuries. It was finally proved for ALL numbers only in 1995 (a proof usually attributed to British mathematician Andrew Wiles, although in reality it was a joint effort of several steps involving many mathematicians over several years). The final proof made use of complex modern mathematics, such as the modularity theorem for semi-stable elliptic curves, Galois representations and Ribet’s epsilon theorem, all of which were unavailable in Fermat’s time, so it seems clear that Fermat’s claim to have solved his last theorem was almost certainly an exaggeration (or at least a misunderstanding). Fermat’s Last Theorem In addition to his work in number theory, Fermat anticipated the development of calculus to some extent, and his work in this field was invaluable later to Newton and Leibniz. While investigating a technique for finding the centres of gravity Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation. Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series. Fermat’s correspondence with his friend Pascal also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values. RENé DESCARTES: Father of Modern Philosophy Biography – Who is Descartes René Descartes has been dubbed the “Father of Modern Philosophy“, but he was also one of the key figures in the Scientific Revolution of the 17th Century, and is sometimes considered the first of the modern school of mathematics. As a young man, he found employment for a time as a soldier (essentially as a mercenary in the pay of various forces, both Catholic and Protestant). But, after a series of dreams or visions, and after meeting the Dutch philosopher and scientist Isaac Beeckman, who sparked his interest in mathematics and the New Physics, he concluded that his real path in life was the pursuit of true wisdom and science. Back in France, the young Descartes soon came to the conclusion that the key to philosophy, with all its uncertainties and ambiguity, was to build it on the indisputable facts of mathematics. To pursue his rather heretical ideas further, though, he moved from the restrictions of Catholic France to the more liberal environment of the Netherlands, where he spent most of his adult life, and where he worked on his dream of merging algebra and geometry. In 1637, he published his ground-breaking philosophical and mathematical treatise “Discours de la méthode” (the “Discourse on Method”), and one of its appendices in particular, “La Géométrie”, is now considered a landmark in the history of mathematics. Following on from early movements towards the use of symbolic expressions in mathematics by Diophantus, Al-Khwarizmi and François Viète, “La Géométrie” introduced what has become known as the standard algebraic notation, using lowercase a, b and c for known quantities and x, y and z for unknown quantities. It was perhaps the first book to look like a modern mathematics textbook, full of a‘s and b‘s, x2‘s, etc. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Cartesian Coordinate System It was in “La Géométrie” that Descartes first proposed that each point in two dimensions can be described by two numbers on a plane, one giving the point’s horizontal location and the other the vertical location, which have come to be known as Cartesian coordinates. He used perpendicular lines (or axes), crossing at a point called the origin, to measure the horizontal (x) and vertical (y) locations, both positive and negative, thus effectively dividing the plane up into four quadrants. Any equation can be represented on the plane by plotting on it the solution set of the equation. For example, the simple equation y = x yields a straight line linking together the points (0,0), (1,1), (2,2), (3,3), etc. The equation y = 2x yields a straight line linking together the points (0,0), (1,2), (2,4), (3,6), etc. More complex equations involving x2, x3, etc, plot various types of curves on the plane. As a point moves along a curve, then, its coordinates change, but an equation can be written to describe the change in the value of the coordinates at any point in the figure. Using this novel approach, it soon became clear that an equation like x2 + y2 = 4, for example, describes a circle; y2 – 16x a curve called a parabola; x2⁄a2 + y2⁄b2 = 1 an ellipse; x2⁄a2 – y2⁄b2 = 1 a hyperbola; etc. Descartes’ ground-breaking work, usually referred to as analytic geometry or Cartesian geometry, had the effect of allowing the conversion of geometry into algebra (and vice versa). Thus, a pair of simultaneous equations could now be solved either algebraically or graphically (at the intersection of two lines). It allowed the development of Newton’s and Leibniz’s subsequent discoveries of calculus. It also unlocked the possibility of navigating geometries of higher dimensions, impossible to physically visualize – a concept which was to become central to modern technology and physics – thus transforming mathematics forever. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Cartesian Coordinates Rule of Signs Although analytic geometry was far and away Descartes’ most important contribution to mathematics, he also: developed a “rule of signs” technique for determining the number of positive or negative real roots of a polynomial; “invented” (or at least popularized) the superscript notation for showing powers or exponents (e.g. 24 to show 2 x 2 x 2 x 2); and re-discovered Thabit ibn Qurra’s general formula for amicable numbers, as well as the amicable pair 9,363,584 and 9,437,056 (which had also been discovered by another Islamic mathematician, Yazdi, almost a century earlier). For all his importance in the development of modern mathematics, though, Descartes is perhaps best known today as a philosopher who espoused rationalism and dualism. His philosophy consisted of a method of doubting everything, then rebuilding knowledge from the ground, and he is particularly known for the often-quoted statement “Cogito ergo sum”(“I think, therefore I am”). He also had an influential rôle in the development of modern physics, a rôle which has been, until quite recently, generally under-appreciated and under-investigated. He provided the first distinctly modern formulation of laws of nature and a conservation principle of motion, made numerous advances in optics and the study of the reflection and refraction of light, and constructed what would become the most popular theory of planetary motion of the late 17th Century. His commitment to the scientific method was met with strident opposition by the church officials of the day. His revolutionary ideas made him a centre of controversy in his day, and he died in 1650 far from home in Stockholm, Sweden. 13 years later, his works were placed on the Catholic Church’s “Index of Prohibited Books”. ACTIVITY 2: I. You meet the ideal mathematician at a party. Make up a possible conversation you might have with him. II. Multiple Choice: 1. Tracing all edges on a figure without picking up your pencil or repeating and starting and stopping at different spots a. Euler Circuit b. Euler Path 2. Tracing all edges on a figure without picking up your pencil and repeating and starting and stopping in the same spot a. Euler Circuit b. Euler Path 3. Circuits start and stop at a. Same vertex b. different vertices 4. Paths start and stop at Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde a. same vertex b. different vertices 5. Euler paths must touch a. all edges b. all vertices 6. Which of the following is false? a. Euler Paths exist when there are exactly two vertices of odd degree. b. Euler circuits exist when the degree of all vertices are even. c. A graph with more than two odd vertices will never have an Euler Path or Circuit. d. A graph with one odd vertex will have an Euler Path but not an Euler Circuit. Descartes’ Rule of Signs Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde Unit 3 : The Development of mathematics: a historical overview: Modern Period The modern period of mathematics was characterized by the comprehensive and systematic synthesis of mathematical knowledge. It is remarkable for its uncovering of deep structural phenomena, and the generalization, unification, and synthesis of all of mathematics. Modern mathematics can be said to have been born in the 1800s, and characterized by grappling with the challenges from the Classical period, as well with addditional disturbances that had been found and continued to be found with the theory of mathematics as then understood: the basis of the integral and differential calculus, the impossibility of a solution by radicals of polynomials of degree five or higher (which explains why the classical geometric problems had no solution), paradoxes in logical foundations (Russell, Burale Forte, etc.), shocking results about higher orders of infinity and Cantor’s theory of sets (the Continuum Hypothesis), the “monsters” of real analysis functions and measure theory (continuous but nowhere differentiable functions, etc.), and the shocking limitations of logic in Godel’s Incompleteness Theorems. What resulted was a rich development and re-working of mathematics: the Galois theory, that resolved as impossible the unsolved problems from classical geometry and also the unsolved problems from classical algebra and theory of equations; the careful definition of the concept of limit, the treatment of infinite series as a limit of partial sums, and the foundation of analysis on arithmetical terms, i.e. the construction of the real number system as equivalence classes of Cauchy sequences, thus effectively completing the number system and including the irrational numbers; the investigation of algebraic structure of integers, polynomials, number theory, of matrices, quaternions, and vectors, modern algebraic structures, and algebraic mathematics applied to geometry and the continuum; the resolution of the parallel postulate unsolved problem by the demonstration of logically valid non-Euclidean geometries; the establishment of a set theory able to handle the infinite and higher orders of infinity; the demonstration of the existence of transcendental numbers, and indeed their dominance among all numbers and the relative small infinity of the rationals and even the algebraic numbers, as well as the demonstration of the transcendence of and. Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde So modern mathematics is modern algebra, Galois theory of algebraic equations, modern number theory, analysis, set theory, complex variables, and Fourier analysis, etc.: much of the content of advanced undergraduate and graduate level mathematics. Modern mathematics can be said to have been from the mid 1800s to the early middle 1900s, with mathematicians such as Cauchy, Weierstrass, Riemann, Dedekind, Bolzano, Cantor, and Hilbert, all establishing the language and patterns of thinking characteristic of modern mathematics. Though Laplace, Poisson, Gauss, Fourier, and Lagrange contributed to the establishment of many modern areas of investigation, in the late 1700s and through the 1800s, and uncovered important parts of the structures of modern mathematics, the form of their work and the style of their exposition would now appear archaic, being, as it usually was, in the style of pre-modern mathematics. Modern mathematics, though more unified, abstract, and diverse than the pre-modern mathematics, is still not the mathematics of today. Though the deeper structures of mathematical fields were being uncovered, they were not yet reflected in a standardized approach to its various areas. This is the legacy that has characterized the post-modern period. Non-Euclidean Geometries Non-Euclidean geometry, literally any geometry that is not the same as Euclidean geometry. Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries (hyperbolic and spherical) that differ from but are very close to Euclidean geometry. Comparison of Euclidean, spherical, and hyperbolic geometries Given a line and a point not on the line, there exist(s) ____________ through the given point and parallel to the given line. a) exactly one line (Euclidean) b) no lines (spherical) c) infinitely many lines (hyperbolic) Euclid’s fifth postulate is ____________. a) true (Euclidean) b) false (spherical) c) false (hyperbolic) The sum of the interior angles of a triangle ______ 180 degrees. a) = (Euclidean) b) > (spherical) c) < (hyperbolic) Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde The non-Euclidean geometries developed along two different historical threads. The first thread started with the search to understand the movement of stars and planets in the apparently hemispherical sky. For example, Euclid (flourished c. 300 BCE) wrote about spherical geometry in his astronomical work Phaenomena. In addition to looking to the heavens, the ancients attempted to understand the shape of the Earth and to use this understanding to solve problems in navigation over long distances (and later for large- scale surveying). These activities are aspects of spherical geometry. The second thread started with the fifth (“parallel”) postulate in Euclid’s Elements: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles. For 2,000 years following Euclid, mathematicians attempted either to prove the postulate as a theorem (based on the other postulates) or to modify it in various ways. (See geometry: Non-Euclidean geometries.) These attempts culminated when the Russian Nikolay Lobachevsky (1829) and the Hungarian János Bolyai (1831) independently published a description of a geometry that, except for the parallel postulate, satisfied all of Euclid’s postulates and common notions. It is this geometry that is called hyperbolic geometry. Birth of set theory and problems in the foundations of mathematics Set theory is one of the greatest achievements of modern mathematics. Basically all mathematical concepts, methods, and results admit of representation within axiomatic set theory. Thus, set theory has served quite a unique role by systematizing modern mathematics, and approaching in a unified form all basic questions about admissible mathematical arguments—including the thorny question of existence principles. This entry covers in outline the convoluted process by which set theory came into being, covering roughly the years 1850 to 1930. In 1910, Hilbert wrote that set theory is that mathematical discipline which today occupies an outstanding role in our science, and radiates [ausströmt] its powerful influence into all branches of mathematics. [Hilbert 1910, 466; translation by entry author] Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde This already suggests that, in order to discuss the early history, it is necessary to distinguish two aspects of set theory: its role as a fundamental language and repository of the basic principles of modern mathematics; and its role as an independent branch of mathematics, classified (today) as a branch of mathematical logic. Both aspects are considered here. The first section examines the origins and emergence of set theoretic mathematics around 1870; this is followed by a discussion of the period of expansion and consolidation of the theory up to 1900. Section 3 provides a look at the critical period in the decades 1897 to 1918, and Section 4 deals with the time from Zermelo to Gödel (from theory to metatheory), with special attention to the often overlooked, but crucial, descriptive set theory. Emergence The concept of a set appears deceivingly simple, at least to the trained mathematician, and to such an extent that it becomes difficult to judge and appreciate correctly the contributions of the pioneers. What cost them much effort to produce, and took the mathematical community considerable time to accept, may seem to us rather self- explanatory or even trivial. Three historical misconceptions that are widespread in the literature should be noted at the outset: 1. It is not the case that actual infinity was universally rejected before Cantor. 2. Set-theoretic views did not arise exclusively from analysis, but emerged also in algebra, number theory, and geometry. 3. In fact, the rise of set-theoretic mathematics preceded Cantor’s crucial contributions. All of these points shall become clear in what follows. The notion of a collection is as old as counting, and logical ideas about classes have existed since at least the “tree of Porphyry” (3rd century CE). Thus it becomes difficult to sort out the origins of the concept of set. But sets are neither collections in the everyday sense of this word, nor “classes” in the sense of logicians before the mid-19th century. The key missing element is objecthood — a set is a mathematical object, to be operated upon just like any other object (the set NN is as much ‘a thing’ as number 3). To clarify this point, Russell employed the useful distinction between a class-as-many (this is the traditional idea) and a class-as-one (or set). Ernst Zermelo, a crucial figure in our story, said that the theory had historically been “created by Cantor and Dedekind” [Zermelo 1908, 262]. This suggests a good pragmatic criterion: one should start from authors who have significantly influenced the Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde conceptions of Cantor, Dedekind, and Zermelo. For the most part, this is the criterion adopted here. Nevertheless, as every rule calls for an exception, the case of Bolzano is important and instructive, even though Bolzano did not significantly influence later writers. In 19th century German-speaking areas, there were some intellectual tendencies that promoted the acceptance of the actual infinite (e.g., a revival of Leibniz’s thought). In spite of Gauss’s warning that the infinite can only be a manner of speaking, some minor figures and three major ones (Bolzano, Riemann, Dedekind) preceded Cantor in fully accepting the actual infinite in mathematics. Those three authors were active in promoting a set-theoretic formulation of mathematical ideas, with Dedekind’s contribution in a good number of classic writings (1871, 1872, 1876/77, 1888) being of central importance. Chronologically, Bernard Bolzano was the first, but he exerted almost no influence. The high quality of his work in logic and the foundations of mathematics is well known. A book entitled Paradoxien des Unendlichen was posthumously published in 1851. Here Bolzano argued in detail that a host of paradoxes surrounding infinity are logically harmless, and mounted a forceful defence of actual infinity. He proposed an interesting argument attempting to prove the existence of infinite sets, which bears comparison with Dedekind’s later argument (1888). Although he employed complicated distinctions of different kinds of sets or classes, Bolzano recognized clearly the possibility of putting two infinite sets in one-to-one correspondence, as one can easily do, e.g., with the intervals [0,5][0,5] and [0,12][0,12] by the function 5y=12x5y=12x. However, Bolzano resisted the conclusion that both sets are “equal with respect to the multiplicity of their parts” [1851, 30–31]. In all likelihood, traditional ideas of measurement were still too powerful in his way of thinking, and thus he missed the discovery of the concept of cardinality (however, one may consider Non-Cantorian ideas, on which see Mancosu 2009). The case of Bolzano suggests that a liberation from metric concepts (which came with the development of theories of projective geometry and especially of topology) was to have a crucial role in making possible the abstract viewpoint of set theory. Bernhard Riemann proposed visionary ideas about topology, and about basing all of mathematics on the notion of set or “manifold” in the sense of class (Mannigfaltigkeit), in his celebrated inaugural lecture “On the Hypotheses which lie at the Foundations of Geometry” (1854/1868a). Also characteristic of Riemann was a great emphasis on conceptual mathematics, particularly visible in his approach to complex analysis (which again went deep into topology). To give but the simplest example, Riemann was an enthusiastic follower of Dirichlet’s idea that a function has to be conceived as an arbitrary correspondence between numerical values, be it representable by a formula or not; this Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde meant leaving behind the times when a function was defined to be an analytic expression. Through this new style of mathematics, and through his vision of a new role for sets and a full program for developing topology, Riemann was a crucial influence on both Dedekind and Cantor (see Ferreirós 1999). The five-year period 1868–1872 saw a mushrooming of set-theoretic proposals in Germany, so much so that we could regard it as the birth of set-theoretic mathematics. Riemann’s geometry lecture, delivered in 1854, was published by Dedekind in 1868, jointly with Riemann’s paper on trigonometric series (1854/1868b, which presented the Riemann integral). The latter was the starting point for deep work in real analysis, commencing the study of “seriously” discontinuous functions. The young Georg Cantor entered into this area, which led him to the study of point-sets. In 1872 Cantor introduced an operation upon point sets (see below) and soon he was ruminating about the possibility to iterate that operation to infinity and beyond: it was the first glimpse of the transfinite realm. Meanwhile, another major development had been put forward by Richard Dedekind in 1871. In the context of his work on algebraic number theory, Dedekind introduced an essentially set-theoretic viewpoint, defining fields and ideals of algebraic numbers. These ideas were presented in a very mature form, making use of set operations and of structure-preserving mappings (see a relevant passage in Ferreirós 1999: 92–93; Cantor employed Dedekind’s terminology for the operations in his own work on set theory around 1880 [1999: 204]). Considering the ring of integers in a given field of algebraic numbers, Dedekind defined certain subsets called “ideals” and operated on these sets as new objects. This procedure was the key to his general approach to the topic. In other works, he dealt very clearly and precisely with equivalence relations, partition sets, homomorphisms, and automorphisms (on the history of equivalence relations, see Asghari 2018). Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. Several years later (in 1888), Dedekind would publish a presentation of the basic elements of set theory, making only a bit more explicit the operations on sets and mappings he had been using since 1871. The following year, Dedekind published a paper in which he provided an axiomatic analysis of the structure of the set RR of real numbers. He defined it as an ordered field that is also complete (in the sense that all Dedekind-cuts on RR correspond to an element in RR); completeness in that sense has the Archimedean axiom as a consequence. Cantor too provided a definition of RR in 1872, employing Cauchy sequences of rational numbers, which was an elegant simplification of the definition offered by Carl Weierstrass in his lectures. The form of completeness axiom that Weierstrass preferred was Bolzano’s principle that a sequence of nested closed intervals in RR (a sequence such that [am+1,bm+1]⊂[am,bm][am+1,bm+1]⊂[am,bm]) “contains” at least one real number (or, as we would say, has a non-empty intersection). Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde The Cantor and Dedekind definitions of the real numbers relied implicitly on set theory, and can be seen in retrospect to involve the assumption of a Power Set principle. Both took as given the set of rational numbers, and for the definition of RR they relied on a certain totality of infinite sets of rational numbers (either the totality of Cauchy sequences, or of all Dedekind cuts). With this, too, constructivistic criticism of set theory began to emerge, as Leopold Kronecker started to make objections to such infinitary procedures. Simultaneously, there began a study of the topology of RR, in particular in the work of Weierstrass, Dedekind, and Cantor. The set-theoretic approach was also exploited by several authors in the fields of real analysis and complex analysis (e.g., Hankel, du Bois-Reymond, H.J.S. Smith, U. Dini) and by Dedekind in joint work with Weber (1882), pioneering algebraic geometry. Cantor’s derived sets are of particular interest (for the context of this idea in real analysis, see e.g., Dauben 1979, Hallett 1984, Lavine 1994, Kanamori 1996, Ferreirós 1999). Cantor took as given the “conceptual sphere” of the real numbers, and he considered arbitrary subsets PP, which he called ‘point sets’. A real number rr is called a limit point of PP, when all neighbourhoods of rr contain points of PP. This can only happen if PP is infinite. With that concept, due to Weierstrass, Cantor went on to define the derived set P′P′ of PP, as the set of all the limit points of PP. In general P′P′ may be infinite and have its own limit points (see Cantor’s paper in Ewald [1996, vol. 2, 840ff], esp. p. 848). Thus one can iterate the operation and obtain further derived sets P′′P″, P′′′P‴… P(n)P(n) … It is easy to give examples of a set PP that will give rise to non-empty derived sets P(n)P(n) for all finite nn. (A rather trivial example is P=Q[0,1]P=Q[0,1], the set of rational numbers in the unit interval; in this case P′=[0,1]=P′′P′=[0,1]=P″.) Thus one can define P(∞)P(∞) as the intersection of all P(n)P(n) for finite nn. This was Cantor’s first encounter with transfinite iterations. Then, in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind (see Ewald 1996, vol. 2), Cantor asked the question whether the infinite sets NN of the natural numbers and RR of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set AA of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with NN). A few days later, Cantor was able to prove that the assumption that RR is denumerable leads to a contradiction. To this end, he employed the Bolzano- Weierstrass principle of completeness mentioned above. Thus he had shown that there are more elements in RR than in NN or QQ or AA, in the precise sense that the cardinality of RR is strictly greater than that of NN. Consolidation Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. But this viewpoint was contested, and its consolidation took a rather long time. Dedekind’s algebraic style only began to find followers in the 1890s; Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde David Hilbert was among them. The soil was better prepared for the modern theories of real functions: Italian, German, French and British mathematicians contributed during the 1880s. And the new foundational views were taken up by Peano and his followers, by Frege to some extent, by Hilbert in the 1890s, and later by Russell. Meanwhile, Cantor spent the years 1878 to 1885 publishing key works that helped turn set theory into an autonomous branch of mathematics. Let’s write A≡BA≡B in order to express that the two sets AA, BB can be put in one-to-one correspondence (have the same cardinality). After proving that the irrational numbers can be put in one-to-one correspondence with RR, and, surprisingly, that also Rn≡RRn≡R, Cantor conjectured in 1878 that any subset of RR would be either denumerable (≡N)(≡N) or ≡R≡R. This is the first and weakest form of the celebrated Continuum Hypothesis. During the following years, Cantor explored the world of point sets, introducing several important topological ideas (e.g., perfect set, closed set, isolated set), and arrived at results such as the Cantor- Bendixson theorem. A point set PP is closed iff its derived set P′⊆PP′⊆P, and perfect iff P=P′P=P′. The Cantor-Bendixson theorem then states that a closed point set can be decomposed into two subsets RR and SS, such that RR is denumerable and SS is perfect (indeed, SS is the aath derived set of PP, for a countable ordinal aa). Because of this, closed sets are said to have the perfect set property. Furthermore, Cantor was able to prove that perfect sets have the power of the continuum (1884). Both results implied that the Continuum Hypothesis is valid for all closed point sets. Many years later, in 1916, Pavel Aleksandrov and Felix Hausdorff were able to show that the broader class of Borel sets have the perfect set property too. His work on points sets led Cantor, in 1882, to conceive of the transfinite numbers (see Ferreirós 1999: 267ff). This was a turning point in his research, for from then onward he studied abstract set theory independently of more specific questions having to do with point sets and their topology (until the mid-1880s, these questions had been prominent in his agenda). Subsequently, Cantor focused on the transfinite cardinal and ordinal numbers, and on general order types, independently of the topological properties of RR. The transfinite ordinals were introduced as new numbers in an important mathematico-philosophical paper of 1883, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (notice that Cantor still uses Riemann’s term Mannigfaltigkeit or ‘manifold’ to denote sets). Cantor defined them by means of two “generating principles”: the first (1) yields the successor a+1a+1 for any given number aa, while the second (2) stipulates that there is a number bb which follows immediately after any given sequence of numbers without a last element. Thus, after all the finite numbers comes, by (2), the first transfinite number, ωω (read: omega); and this is followed by ω+1ω+1, ω+2ω+2, …, ω+ω=ω⋅2ω+ω=ω⋅2, …, ω⋅nω⋅n, ω⋅n+1ω⋅n+1, …, ω2ω2, ω2+1ω2+1, …, ωωωω, … Republic of the Philippines Polytechnic University of the Philippines COLLEGE OF EDUCATION DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATION By : Asst. Prof. Mavel Besmonte - Lagarde and so on and on. Whenever a sequence without last element appears, one can go on and, so to say, jump to a higher stage by (2). The introduction of these new numbers seemed like idle speculation to most of his contemporaries, but for Cantor they served two very important functions. To this end, he classified the transfinite ordinals as follows: the “first number class” consisted of the finite ordinals, the set NN of natural numbers; the “second number class” was formed by ω and all numbers following it (including ωωωω, and many more) that have only a denumerable set of predecessors. This crucial condition was suggested by the problem of proving the Cantor-Bendixson theorem (see Ferreirós 1995). On that basis, Cantor could establish the results that the cardinality of the “second number class” is greater than that of NN; and that no intermediate cardinality exists. Thus, if you write card(N)=ℵ0card(N)=ℵ0 (read: aleph zero), his theorems justified calling the cardinality of the “second number class” ℵ1ℵ1. After the second number class comes a “third number class” (all transfinite ordinals whose set of predecessors has cardinality ℵ1ℵ1); the cardinality of this new number class can be proved to be ℵ2ℵ2. And so on. The first function of the transfinite ordinals was, thus, to establish a well-defined scale of increasing transfinite cardinalities. (The aleph notation used above was introduced by Cantor only in 1895.) This made it possible to formulate much more precisely the problem of the continuum; Cantor’s conjecture became the hypothesis that card(R)=ℵ1card(R)=ℵ1. Furthermore, relying on the transfinite ordinals, Cantor was able to prove the Cantor-Bendixson theorem, rounding out the results on point sets that he had been elaborating during these crucial years. The Cantor- Bendixson theorem states: closed sets of RnRn (generalizable to Polish spaces) have the perfect set property, so that any closed set SS in RnRn can be written uniquely as the disjoint union of a perfect set PP and a countable set RR. Moreover, PP is SαSα for α countable ordinal. The study of the transfinite ordinals directed Cantor’s attention towards ordered sets, and in particular well-ordered sets. A set SS is well-ordered by a relation < iff < is a total order and every subset of SS has a least element in the

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