Foundations of Computing and Communication Lecture 3 The Birth of the Modern Era Based on The Foundations of Computing and the Information Technology Age, Chapter 2 Lecture overheads c John Thornton 2
Lecture Objective To gain a basic understanding of the evolution of science, computation and mathematics from the fall of Rome in 76 CE to the scientific revolution of the 7th and 8th centuries. Overview of Topics The Hindu-Arabic Number System The Renaissance The Science of Aristotle The Scientific Revolution Pre-Mechanised Computation Mathematical Breakthroughs
After the Fall of Rome Last emperor deposed 76 CE: Gibbon fixes date of the fall in the Decline and Fall of the Roman Empire, although Rome continued as a centre of power Medieval or Middle Ages: 76-3: Defined by the fall of Constantinople to the Turks in 3, period from 76- generally considered as the Dark Ages, Europe in decline and disorder Non-European cultures flourish: Birth of Islam and continued development of Hindu culture in India 2
The Hindu-Arabic Number System Decimal positional system: modern decimal system developed in India, spread via Islam to Spain and Europe around 976 CE First true zero: zero used as padding in Babylon; Hindu zero has mathematical meaning: x x = and x = Negative number notation: Hindu mathematicians also introduced negative numbers again with arithmetic meaning: 7 = 2 Slow acceptance: not until mid-3th century did system make serious inroads on Roman numerals, thanks to translations of original works by Al Khowarizmi (hence algorism and algorithm). 3
The Islamic Legacy Alexandria: After death of Alexander (323 BCE) became centre of learning for, years, site of first university, great library destroyed in 62 CE by Islamic conquerors. Baghdad: Islam s House of Wisdom, became new centre of learning for next years preserving and extending Greek and Roman culture (founded by Caliph al Mansūr in 763 CE, sacked by Monguls in 28 CE). Scholars of Islam: Al Khowarizmi developed science of equations (algebra), Omar Khayyam went on to investigate cubic equations, Nasir-eddin developed trigonometry separate from astronomy
The th Century Renaissance Feudalism in Western Europe: Roman era ends: breakdown of commerce, decline of cities, shift of power to landowners, rise of Roman Catholic Church, scholarship confined mostly to monasteries Italy prospers: Cities of Northern Italy begin to prosper, trade with Islam and the East in goods and ideas; also invention of printing press, discovery of the Americas, sense of new possibilities Rediscovery and growth: fall of Constantinople, arrival of Byzantine scholars, Greek and Roman culture reclaimed, infusion of Islamic ideas breakup of static world view, Neo- Platonism, new realism in art, European universities expand, mathematics starts to develop again, general solution of cubic equations
The Science of Aristotle Scholastics: Aristotle dominated medieval thinking on science, scholastics added many commentaries but few new ideas Mechanics: Aristotle believed natural state of a body is rest, continuous force is needed for movement, no resistance = infinite speed, hence vacuum is impossible Projectiles: so why don t projectiles immediately fall to the ground after launching? Because they almost create a vacuum behind themselves causing the air in front to rush round and fill it; objects fall to earth because there is more air pushing down above than below, and they accelerate because they are joyful at returning home 6
Greek Astronomy Celestial spheres: Aristotle and Ptolemy proposed the earth as the centre of the universe with planets, sun and stars suspended on celestial spheres in a fifth element (ether) and moved by unseen Intelligences Epicycles: the unusual and retrograde movements of the planets were explained by epicycles: 7
Copernicus (3) Heliocentric model: Copernicus proposed that all heavenly bodies including earth revolve around the sun, in perfect circles Criticisms: predictions from Copernicus no better than updated Ptolemy; movement of the earth defied common sense and observation (objects fall perpendicular to the earth); theology placed the earth at the centre; the Greeks had already thought of this and rejected it for the same reasons (Aristarchus 3-23 BCE) Main difficulty: went against Aristotle s mechanics: how could the heavy earth move when everything falls towards it and nothing is pushing it? 8
The 7th Century Scientific Revolution New outlook: rejection of received authority, only accept what can be demonstrated or disproved by experience and experiment empirical science and induction of Francis Bacon (62) Mathematical science: Renewed interest in Plato and Pythagoras, belief that the true order of nature is best expressed mathematically, exemplified by Kepler s elliptical planetary orbits - it has to work because it is so perfect (69)! 9
Galileo New mechanics: Galileo broke away from Aristotlean tradition and developed our modern idea of inertia, i.e. a body will keep moving after a force is applied to it (632) Thought experiments: Galileo pioneered the method of thought experiments, removing complicating factors, theorising about underlying causes, arriving at general laws, then deducing actual behaviour Mathematisation of nature: Like Kepler, Galileo found that his ideas of motion were best expressed as mathematical formulae
Newton Gravity: Newton to synthesised the work of Copernicus, Kepler and Galileo into one grand and simple scheme: the basic laws of motion and the inverse square law of gravity (687) Earth and heavens united: the same simple law of gravity explained objects falling on earth and the movement of the planets, all made of tiny particles of matter The clockwork universe: no longer were unseen Intelligences needed to explain movement, the universe was seen one lawful whole, designed and set in motion by the perfect architect, God
Pre-Mechanised Computation Napier s bones: with the progress of mathematical science came the need for fast and accurate computation; an early aid to such calculation were Napier s bones; these reduced multiplication to addition by expressing the times tables on various strips or bones (67) 3 7 2 7 2 2 3 3 9 6 7 9 9 2 2 6 8 2 6 8 7 7 2 2 3 6 8 2 9 6 3 2 2 3 3 2 3 6 7 8 9 2
Logarithms Napier s most famous discovery was the use of logarithms to transform multiplication and division into addition and subtraction using the properties of geometric and arithmetic series (6): geometric 2 8 6 32 6 28 26 2 2 arithmetic 2 3 6 7 8 9 Looking at the arithmetic series, we can see 2 + = 6, but if we read off the terms corresponding to 2, and 6 from the geometric series, we obtain, 6 and 6, which gives 6 = 6 Napier s work led to the development of slide rules and log tables still widely used up to the 97 s 3
Logarithms We now understand logarithms as algebraic properties of numbers raised to powers. For instance, in the previous example, the geometric series represents the result of raising two to the power of the corresponding arithmetic series value, i.e. 2 =, 2 = 2, 2 2 =, 2 3 = 8, 2 = 6, etc. Logarithms therefore exploit the well known principle of multiplying powers by adding indices, e.g. 2 3 2 3 = 2 3+3 = 2 6, with the base of the logarithm being the number which is raised to the power (in this case base 2)
Mathematical Breakthroughs Descartes: René Descartes is famous as a philosopher for attempting to doubt everything not self-evidently certain and concluding Cogito, ergo sum (image courtesy of the Science Museum, London). Analytic geometry: Descartes also famous for starting analytic geometry i.e. using algebra to express geometric forms within a system of perpendicular axes mathematisation of space, and for developing the modern exponential notation (x 3, x, etc)
Infinitesimal Calculus Integration: Archimedes developed the method of exhaustion to calculate π, Cavalieri introduced techniques in Renaissance Europe to find areas and volumes by closer and closer approximation (63) later formalised as integration Limits: calculus uses idea of a limit, limits and infinitesimal distances already intuited by the Greeks (Zeno s paradox), in modern terms: lim n (.)n = 6
Infinitesimal Calculus Differentiation: used to find the slope of a curve by finding the slope between two points as the distance between the points approaches zero; Newton also used differentiation to find rates of change of motion. Unified algebraic method: Leibniz (673, 68) and Newton (66, 7) independently saw that differentiation and integration are two related processes, one being the reverse of the other, and both were able to unify these fields into what is now the general method of calculus. Theory of limits: calculus not given mathematical rigour until theory of limits developed by Cauchy in 82 7
A New Model of the Universe Scientific method: marriage of mathematical theorising (hypothesising) and experimental testing (falsification) The Age of Reason: reason not superstition, humanity in charge of its own destiny, ideals of humanitarianism and progress, end of the cyclical time of the Middle Ages Objectivity and Materialism: objective world described by science seen as more real than subjective world of the individual, emphasis on the measurable quantities not qualities, science builds an abstraction of a material universe explained by mathematical laws constraining idealised particles 8
Lecture Exercises Show, using diagrams, how Napier s bones could be used to multiply 2 by 6. Explain how the knowledge of the series 2 =, 2 = 2, 2 2 =, 2 3 = 8, 2 = 6,... can be used to transform the multiplication of 32 6 into an addition. Given the logarithmic table values of the numbers x and y are a x and a y respectively, and that antilog(a x ) = x and antilog(a y ) = y then what does x y equal in terms of a x and a y? Describe how the theory and practice of science after the scientific revolution differed from the earlier natural philosophy of Aristotle and his followers. 9