Analysis, Synthesis, the Infinite, and Numbers Chapter 15 Getting Our Bearings • Where are we in history? • We’ve finished the middle ages, and have embarked on the Early Modern period (about 1500 – 1800 or so). • In fact, we’ve finished the 1500’s as well. Not Europe Place Dates Names Mathematics China Ancient &Medieval The Nine Chapters, Square and cube roots, Sea Island indeterminate equations, Mathematical systems of linear Manual, others equations and linear congruences, polynomial equations India Ancient &Medieval Aryabhata, Square and cube roots, Bramagupta, indeterminate equations, others linear congruences, combinatorics. Islamic Empire Medieval Al Khwarizmi, Algebra, solutions of Omar Khayyam, al cubics by conics, Euclid’s Tusi, others 5th postulate, Hindu‐ Arabic numeration, Trigonometry Europe Place Dates Names Mathematics Europe Medieval Leonardo of Pisa, bar Islamic Methods introduced to Hiyya, ibn Ezra, ben Europe Gerson Europe Renaissance / Pacioli, Cardano, Solution of the cubic, Early Modern Targaglia, Ferrari, del systematic algebra & theory of Ferro, Bombelli, Viète, solving equations, Stevin, Regiomontanus, trigonometry, decimal Copernicus, Kepler, fractions, logarithms, Brahe, Napier, others heliocentric astronomy. Getting Our Bearings • Where are we in history? • We’ve finished the middle ages, and have embarked on the Early Modern period (about 1500 – 1800 or so). • In fact, we’ve finished the 1500’s as well. • We are now entering the 1600’s. Next up Place Dates Names Mathematics Europe 1600’s Fermat, Descartes, Analytic geometry, theory of equations, Newton, Liebniz, area, normals, tangents, max/min, Barrow, Pascal, systematic calculus Oughtred, Harriot, Wallis Europe 1700’s Bernoulli, Further development of the calculus: Bernoulli, differential equations, brachistochrone Bernoulli, …. problem, tautochrone problem, catenary Maclaurin, problem, calculus of variations, multi‐ L’Hopital, dimensional calculus, transcendental Euler functions (logs, exp, trig), complex numbers, multi‐dimensional calculus, partial differential equations, calculus texts, the foundations of calculus, theory of equations, number theory. Galileo and Cavalieri • For us, the major mathematical interest in his work is his treatment of the infinite, and we’ll discuss that later when we talk about Cantor. • We will also discuss Cavalieri’s work when we talk about the development of calculus prior to Newton and Leibniz. Pierre de Fermat • 1601‐1665 • Trained as a lawyer at the University of Toulouse. • Appointed as a judge in Toulouse in 1638. • Married and had five children. Pierre de Fermat • Considered math his hobby, and never really published any of his works. • Like many others, he was interested in “restoring” lost works of ancient Greek mathematics. • His work is known mainly because of his correspondences with other notable mathematicians, such as Mersenne (the “walking scientific journal of France.”) Pierre de Fermat • Made contributions to – Number theory – Analytic Geometry – Probability – Analysis (calculus) • Often didn’t provide proofs. Didn’t like to “polish” his work. This annoyed some of his contemporaries. He was often correct, but not always. Pierre de Fermat • His correspondence with Pascal signals the birth of modern probability theory, and the first major developments in this area since Cardano. • His best work was in number theory, inspired by his reading of Diophantus’ Arithmetica. We’ll read about how Euler filled in a great number of the proofs which Fermat overlooked. Pierre de Fermat • Of course, the most notorious of Fermat’s non‐proofs was of the statement that the equation has no non‐zero integer solutions for x, y, and z when . (He had already proved the case for n=3) • Known as “Fermat’s Last Theorem” • Fermat wrote, in the margins of his copy of Arithmetica, “I have discovered a truly remarkable proof which this margin is too small to contain.” • Mathematicians searched for a proof ever since this note was discovered and published by his son. “Fermat’s Last Theorem” • It is pretty well accepted that Fermat was in error about having a proof. I believe the direct quote from historian of mathematics Victor Katz was “Liar, Liar, Pants on Fire.”* • In 1993 English mathematician Andrew Wiles claimed to have a proof, although there were found to be some gaps in it. By 1994 he had closed these gaps and his proof is now accepted as correct. But it is a proof that relies on hundreds of pages and over three centuries of previous work from other mathematicians, and is nothing like anything Fermat could have envisioned. *Just kidding. He really cited 2 Nephi 9:34 René Descartes • 1596 (Tours, France) – 1650 (Stolkholm, Sweden) • Educated at the Jesuit college of La Flèche in Anjou from 1604 until 1612, studying classics, logic and traditional Aristotelian philosophy. • While in the school his health was poor and he was granted permission to remain in bed until 11 o'clock in the morning, a custom he maintained until the year of his death. René Descartes • “Philosophy affords the • Began to see mathesis means of discoursing universalis as the with an appearance of underpinnings of all truth on all matters, scientific explanation. and commands the • He devoted some study admiration of the more to this in order to later simple.” tackle the other sciences. René Descartes • Earned a law degree (which he largely ignored), travelled, joined the army. • Eventually (about 1628) he settled down in Holland. • Wrote a book Le Monde on Natural History (science/physics) but decided not to publish it after hearing what happened to Galileo. René Descartes • In 1637 published Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences. – ‘The Method’ of arriving at truth. – ‘‘I think, therefore I am.’’ • Three appendices to this work were La Dioptrique, Les Météores, and La Géométrie. • La Géométrie is what mainly concerns us. René Descartes • In La Géométrie Descartes developed his analytic geometry as an example of how his method could arrive at certain knowledge. • La Dioptrique contained a theory of light as instantaneous movement through an elastic medium. We experience light like we experience resistance that travels from the point of action to our hand through a walking stick. • Based on his earlier, unpublished work (Le Monde), and his vortex theory. René Descartes • He assumed that the universe is filled with matter which, due to some initial motion, had settled down into a system of “vortices” which carry the sun, the stars, the planets and comets in their paths. • Despite the problems with the vortex theory it was championed in France for nearly one hundred years even after Newton showed it was impossible as a dynamical system. René Descartes • In Le Dioptrique and Les Meteores, “only tried to convince people that my method is better than the ordinary one. I have proved this in my geometry, for in the beginning I have solved a question which, according to Pappus, could not have been solved by any of the ancient geometers.” ‐‐ Letter to Mersenne. • He misread Pappus, by the way. It had been solved. René Descartes • In 1641 he published a work called Meditationes, in which he explained at some length his views on philosophy as sketched out in the Discourse. • In 1644 he issued the Principia Philosophiae, the greater part of which was devoted to physical science, especially the laws of motion and the theory of vortices. René Descartes • In 1647 he received a pension from the French court in honor of his discoveries. • In 1649 he went to Sweden on the invitation of the Queen in order that she might learn some mathematics from him. She liked to draw her tangent lines at about 5 a.m. and so for the first time, Descartes had to give up his habit of lying in bed until almost noon. • He died a few months later of inflammation of the lungs / pneumonia, brought on by the climate of Sweden and having to arise that early. The Birth of Analytic Geometry • Born in the 1630’s of two fathers, Pierre de Fermat and René Descartes. • Although their approaches were somewhat different, they both attacked classic problems from ancient Greece, particularly the locus problems of Apollonius. • Both were searching, in a sense, for the “lost” analysis of the ancient Greeks. (Pappus’ work was lost, and The Method had not been found.) One of Fermat’s Problems • If, from any number of given points, straight lines are drawn to a point, and if the sum of the squares of the lines is equal to a given area, the point lies on a circle of a given position. (Apollonius, via Pappus) • Notice the generality here –perfectly set up for variables and hence for algebra, but the Greeks didn’t have that as an analytic tool. Fermat’s Solution • Used lengths GH, GL, and GK B P D • Also used GA, HB, LD, and KC C • Discovered A and O • Thus, his analysis allowed him toG H M L K find the center O of the circle. Fermat’s Methods B E Locus AB Variable line segment DE A D C Fixed straight line AC If, when solving a geometry problem algebraically, you end up with an equation in two unknowns, the resulting solution is a locus, whose points are determined by the motion of one endpoint of a variable line segment, the other endpoint of which moves along a fixed straight line. Fermat’s Methods • Note that Fermat used one axis, which the “variable line segment” moved along at a fixed angle and varied in length to define the locus of points. • The fixed angle didn’t have to be 90 degrees. • Notice also that this describes a “mechanical curve,” one that is drawn by the endpoint of a moving segment. Fermat’s Methods • Fermat proved that whenever the relationship among segment lengths gave rise to second degree equations in two variables, the loci defined were conics.