O1 History of Mathematics Lecture XV Probability, geometry, and number theory. Monday 28th November 2016 (Week 8)

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1 O1 History of Mathematics Lecture XV Probability, geometry, and number theory Monday 28th November 2016 (Week 8)

2 Summary Early probability theory Probability theory in the 18th century Euclid s Elements Geometry A little number theory

3 Probability

Probability: the beginnings Girolamo Cardano (1501 1576), Liber de ludo aleae [A book of the game of dice] (unpublished until 1663).

4 Probability: the beginnings Girolamo Cardano ( ), Liber de ludo aleae [A book of the game of dice] (unpublished until 1663). Written to show how gambling can be considered a just activity; contains: probabilities for sums on two and three dice simple calculations related to card games a version of the multiplication rule for probabilities advice on protection against cheating

5 Some recommended books On Cardano s probabilistic work: Øystein Ore, Cardano, the gambling scholar, Princeton University Press, 1953 On Cardano himself: The book of my life (NYRB: an unvarnished and often outrageous account of his character and conduct ) On combinatorics more generally: Robin Wilson and John J. Watkins (eds.), Combinatorics: ancient and modern, Oxford University Press, 2013

Early probability theory Correspondence between Fermat and Pascal, 1654

divided if a game is interrupted? Led to.

arithmetic triangle] (1655): properties of Pascal s triangle Christiaan

6 Early probability theory Correspondence between Fermat and Pascal, 1654 (Fermat s side of the correspondence published 1679): how should stakes be divided if a game is interrupted? Led to... Blaise Pascal, Traité du triangle arithmétique [Treatise on the arithmetic triangle] (1655): properties of Pascal s triangle Christiaan Huygens, De ratiociniis in ludo aleae [On reasoning in a game of dice] (1657): concept of expectation interrupted games chances in dice

Further probability Jacob Bernoulli (1654 1705), Ars conjectandi [The art of conjecture] (1713) in four parts: (i) reprint (with commentaries) of Huygens De

7 Further probability Jacob Bernoulli ( ), Ars conjectandi [The art of conjecture] (1713) in four parts: (i) reprint (with commentaries) of Huygens De ratiociniis in ludo aleae (ii) first printed treatise on combinations (iii) applications to games of chance (iv) applications to civil, moral, and economic affairs (incomplete)

8 Some of Jacob Bernoulli s concerns But, who from among the mortals will be able to determine, for example, the number of diseases, that is, the same number of cases which at each age invade the innumerable parts of the human body and can bring about our death; and how much more easily one disease (for example, the plague) can kill a man than another one (for example, rabies; or, the rabies than fever), so that we would be able to conjecture about the future state of life or death? And who will count the innumerable cases of changes to which the air is subjected each day so as to form a conjecture about its state in a month, to say nothing about a year? [Translation by Oscar Sheynin, Berlin, 2005]

9 Jacob Bernoulli and the Law of Large Numbers (1) To make clear [...] by illustration, I suppose that without your knowledge three thousand white pebbles and two thousand black ones are hidden in an urn, and that, to determine [the ratio of] their numbers by experiment, you draw one pebble after another (but each time returning the drawn pebble before extracting the next one so that their number in the urn will not decrease), and note how many times is a white pebble drawn, and how many times a black one. It is required to know whether you are able to do it so many times that it will become ten, a hundred, a thousand, etc., times more probable (i.e., become at last morally certain) that the number of the white and the black pebbles which you extracted will be in the same ratio, of 3 to 2, as the number of pebbles themselves, [...] than in any other different ratio.

10 Jacob Bernoulli and the Law of Large Numbers (2) [...] it ought to be noted that the ratio between the numbers of cases which we desire to determine experimentally is accepted not as precise and strict [...] but that this ratio be accepted with a certain latitude, that is, contained between two limits [bounds] which could be taken as close as you like. Indeed, if in the example just provided concerning pebbles, we will assume two ratios, 301/200 and 299/200, or 3001/2000 and 2999/2000, etc., one of which is very near but greater, and the other one very near but smaller than 3/2, it will be shown that, [...] it can be made more probable that the ratio derived from many observations will be contained within these limits of 3/2 rather than outside. [Translation by Oscar Sheynin, Berlin, 2005]

Abraham de Moivre s contributions (1) Abraham de Moivre (1667 1754), Annuities upon lives (1725; reprinted 1731, 1743, 1752, 1756) The doctrine of chances: or, a method of calculating the probability

11 Abraham de Moivre s contributions (1) Abraham de Moivre ( ), Annuities upon lives (1725; reprinted 1731, 1743, 1752, 1756) The doctrine of chances: or, a method of calculating the probability of events in play (1718; second edition 1738; third edition 1756) Confidence intervals from the 1738 edition: What reasonable Conjectures may be derived from Experiments, or what are the Odds that after a certain number of Experiments have been made concerning the happening or failing of Events, the Accidents of Contingency will not afterwards vary from those of Observation beyond certain Limits?

12 Abraham de Moivre s contributions (2) 1730: a version of Stirling s approximation Used in 1738 to produce an approximation to the normal distribution and from that an answer to his question (See: Mathematics emerging, )

13 Bayes Theorem Rev. Thomas Bayes, An essay towards solving a problem in the doctrine of chances, Phil. Trans. Roy. Soc. London, 1763 (two years posthumous) Bayes Theorem on conditional probabilities (See: Mathematics emerging, )

14 Laplace and probability theory Pierre-Simon Laplace, Théorie analytique des probabilités [Analytic theory of probabilities] (1812). Contains: probability generating functions large number approximations probabilities of compound events applications to mortality and life expectancy (See: Mathematics emerging, )

15 Geometry

16 Euclid s Elements Euclid s Elements, in 13 books, compiled c. 250 BC. Books I V: Book VI: Books VII IX: Book X: Books XI XIII: definitions, postulates, plane geometry of lines and circles similarity, proportion number theory commensurability, irrational numbers, surds solid geometry ending with the classification of the regular polyhedra Canonical English edition by Sir Thomas L. Heath, 1908

17 Billingsley s Euclid, 1570 The Elements of Geometrie: Faithfully (now first) translated into the Englishe toung by H. Billingsley, London, 1570

18 Billingsley s Preface, pp. 1, 3

19 Euclid: the fifth postulate 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles Heath s edition, 1908, p. 155: Equivalent: given a straight line L and a point P not on L there is one and only one straight line through P that is parallel to L.

20 Disquiet about the fifth postulate Does the fifth postulate follow from the rest...? Many failed attempts at proof 100 AD 1900 AD: Ptolemy, Proclus, ibn al-haytham, Wallis, Saccheri, Lambert, Legendre,... (see, for example, Heath, pp ) So can we have a consistent geometry in which it fails?

Non-Euclidean geometries Gauss (perhaps, but tentative and

facsimile with English translation 2010) Non-Euclidean geometries

21 Non-Euclidean geometries Gauss (perhaps, but tentative and unpublished), János Bolyai, 1832 Nikolai Ivanovich Lobachevsky, 1829, 1837,..., 1855 (and his Pangeometry, published in French and Russian facsimile with English translation 2010) Non-Euclidean geometries overturned old ideas of mathematical certainty introduced new ideas about space helped drive the late 19th-century move towards axiomatisation

22 Number theory

23 Elementary number theory Euclid s Elements, Book VII, Proposition 2: Given two numbers [...], to find their greatest common measure Euclid s Algorithm Euclid s Elements, Book IX, Proposition 20: Prime numbers are more than any assigned multitude of prime numbers. (See: Mathematics emerging, ) Euclid s Elements, Book IX, Proposition 36: If as many numbers as we please beginning from an unit be set out in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. (See: Mathematics emerging, ) Note: A number is perfect if it equals the sum of its proper divisors (examples: 6, 28,...)

Perfect numbers Euclid s Theorem: if 2 n 1 is prime then 2 n 1 (2 n 1) is

Mersenne (1644): if p 257 and 2 p 1 is prime then p is one of 2, 3, 5, 7, 13,

Not quite right: 2 89 1, 2 107 1 are prime and 2 257 1 is composite.

24 Perfect numbers Euclid s Theorem: if 2 n 1 is prime then 2 n 1 (2 n 1) is perfect Fermat to Mersenne (1640): if 2 n 1 is prime then n must be prime Mersenne (1644): if p 257 and 2 p 1 is prime then p is one of 2, 3, 5, 7, 13, 17, 67 (a misprint for 61 perhaps?), 127, 257. Not quite right: , are prime and is composite. Euler: proof that all even perfect numbers are of Euclid s form (proved 1749, but published posthumously) (See: Mathematics emerging, )

25 Fermat and his theorems Fermat s Little Theorem: if a is any integer and p is prime then p divides a p a Studies of Pell s Equation x 2 Dy 2 = 1 Studies of diophantine problems leading to Fermat s Last Theorem Published nothing had to be exhorted to write his ideas down (See: Mathematics emerging, ; also: Simon Singh, Fermat s Last Theorem, Fourth Estate, 1998.)

26 Later number theory Number theory was taken up seriously only in the 18th century, first by Euler, then by Lagrange and Legendre Gauss s Disquisitiones arithmeticae (1801) became a key text for many years to come: modular arithmetic, quadratic forms, cyclotomy,... Some sources on the history of number theory: Leonard Eugene Dickson, History of the theory of numbers, 3 vols., Carnegie Institution of Washington, : I, II, III Øystein Ore, Number theory and its history, McGraw-Hill, 1948 John J. Watkins, Number theory: a historical approach, Princeton University Press, 2014

In Defense of Euclid. The Ancient Greek Theory of Numbers

In Defense of Euclid. The Ancient Greek Theory of Numbers In Defense of Euclid The Ancient Greek Theory of Numbers The Poetry of Euclid A unit is that by virtue of which each of the things that exist is called one.» The Elements, book VII, definition 1. Our Goal: