MATH 4400, History of Mathematics

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1 MATH 4400, History of Mathematics Lecture 4: Emergence of modern science Professor: Peter Gibson October 6, 2015

2 Overview and historical context The early 16th century brought great change to the world. For the first time since the Ice Age, the Americas became reconnected to Afro-Eurasia. This had devastating consequences for the pre-existing cultures of the Americas, bringing to an end the great Aztec and Incan empires. Diseases such as smallpox devastated indigenous North and South Americans. New foods and products flowed from the Americas to the rest of the world. Precious metals were extracted for world trade. New colonies were established. P. Gibson (YorkU) Math / 1

3 The world of knowledge also changed fundamentally. The end of the 16th century saw major breakthroughs in the applications of mathematics to an understanding of the physical world and of the heavens. Galileo, Kepler and Descartes all left a lasting mark, both for their particular discoveries and perhaps even more importantly for their new methods. P. Gibson (YorkU) Math / 1

4 About Kepler Kepler lived from in the Holy Roman Empire (present day Germany, Czech Republic and Austria) P. Gibson (YorkU) Math / 1

5 A scene from Ulm... P. Gibson (YorkU) Math / 1

6 Among his works: Mysterium cosmographicum Dioptrice Harmonice Mundi Tabulae Rudolphinae P. Gibson (YorkU) Math / 1

7 An important picture from Kepler s first great work: A beautiful, mathematical theory of the planets P. Gibson (YorkU) Math / 1

8 From Harmonice mundi: P. Gibson (YorkU) Math / 1

9 Kepler s work reflects the mathematics of antiquity (Pythagorean harmony, Platonic solids). But he was also a bridge to modernity, in particular with respect to precise measurement of the stars in order to test his theories. Kepler s work on polytopes endures. His work on ellipses was a crucial step (as will be seen in later lectures). P. Gibson (YorkU) Math / 1

10 Galileo Galileo lived from 1564 to 1642 in the Duchy of Florence (now Italy) His work marks the transition to fully modern science P. Gibson (YorkU) Math / 1

11 Galileo s works include: Siderius Nuncius (1610) Discorso del flusso e reflusso del mare (1616) Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze (1638) We shall look closer at (an English translation of) the latter work. P. Gibson (YorkU) Math / 1

12 Galileo studied statics P. Gibson (YorkU) Math / 1

13 He studied motion, based on physical experiment In this he broke firmly with academic tradition. P. Gibson (YorkU) Math / 1

14 Galileo made extensive use of the language of the universe But mathematics was not always sufficiently advanced. P. Gibson (YorkU) Math / 1

15 Two New Sciences Some basic considerations... How does the author address the reader? What is the essential content of the section on uniform motion? What is natural acceleration? How does Galileo reason about comparative magnitudes? Is his reasoning rigorous? Are the results correct? P. Gibson (YorkU) Math / 1

16 What is the basic importance of Galileo s work on motion?...this and other facts, not few in number or less worth knowing, I have succeeded in proving; and what I consider more important, there have been opened up to this vast and most excellent science, of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore it s remote corners. P. Gibson (YorkU) Math / 1

17 From the modern perspective, Galileo s results on motion, serve as the basis for an analysis of frames of reference. (This requires some illustration...) P. Gibson (YorkU) Math / 1

18 Descartes Rene Descartes lived from P. Gibson (YorkU) Math / 1

19 A brief timeline Jesuit college of La Fleche in Anjou 1616 Law degree at University of Poitiers 1618 studies with Isaac Beekman in Holland 1619 enlists in the Bavarian army travels widely in Europe 1628 moves to Holland 1649 moves to Stockholm P. Gibson (YorkU) Math / 1

20 Important works... (1641) Meditationes de prima philosophia (1637) Discours de la methode pour bien conduire sa raison et chercher la verite dans les sciences La Dioptrique Les Meteores La Geometrie P. Gibson (YorkU) Math / 1

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23 Descartes is often credited with the introduction of analytic geometry, but this is not clear... P. Gibson (YorkU) Math / 1

Tycho Brahe and Johannes Kepler

Tycho Brahe and Johannes Kepler The Music of the Spheres 1 Tycho Brahe 1546-1601 Motivated by astronomy's predictive powers. Saw and reported the Nova of 1572. Considered poor observational data to be