Professor Stewart's Hoard of Mathematical Treasures (31 page)
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Authors: Ian Stewart
Tags: #Mathematics, #General
BOOK: Professor Stewart's Hoard of Mathematical Treasures
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A popular British journalist wrote, in his regular column, something to the effect ‘How come they can be so specific about the date, but they don’t know the year?’ Now, to be fair, it was a humorous column, and it’s quite an amusing question. But it has a serious answer.
Enlighten the journalist. [Hint: What is a year, astronomically speaking?]
Answer and discussion on page 315
What Are the Odds?Mathophila takes a pack of cards, and places the four aces on the table, face down. So two (spades, clubs) are black; the other two (hearts, diamonds) are red.
Shuffle, place face down, pick two.
‘Innumeratus?’
‘Yes?’
‘If you pick two of these cards at random, what is the probability that they have different colours?’
‘Ummm . . . ’
‘Well, either the colours are the same, or they’re not, right?’
‘Yes.’
‘And there’s the same number of cards of each colour.’
‘Yes.’
‘So the chances of your two cards being the same, or different, must be equal - so both are equal to
. Right?’
. Right?’‘Ummm . . . ’
Is Mathophila right?
Answer on page 316
A Potted History of Mathematics| c.23,000 BC | Ishango bone records the prime numbers between 10 and 20. Apparently. |
| c.1900 BC | Babylonian clay tablet Plimpton 322 lists what may be Pythagorean triples. Other tablets record movements of the planets and how to solve quadratic equations. |
| c.420 BC | Discovery of incommensurables (irrational numbers in geometric guise) by Hippasus of Metapontum.* |
| c.400 BC | Babylonians invent symbol for zero. |
| c.360 BC | Eudoxus develops a rigorous theory of incommensurables. |
| c.300 BC | Euclid’s Elements makes proof central to mathematics, and classifies the five regular solids. |
| c.250 BC | Archimedes calculates the volume of a sphere, and other neat stuff. |
| c.36 BC | Mayans reinvent symbol for zero. |
| c.250 | Diophantus writes his Arithmetica - how to solve equations in whole and rational numbers. Uses symbols for unknown quantities. |
| c.400 | Symbol for zero re-reinvented in India. Third time lucky. |
| 594 | Earliest evidence of positional notation in arithmetic. |
| c.830 | Muhammad ibn Musa al-Khwarizmi’s al-Jabr w’al-Muqabala manipulates algebraic concepts as abstract entities, not just placeholders for numbers, and gives us the word ‘algebra’. Doesn’t use symbols, however. |
| * Hippasus was a member of the Pythagorean cult, and it is said that he announced this theorem while he and some fellow cultists were crossing the Mediterranean in a boat. Since Pythagoreans believed that everything in the universe is reducible to whole numbers, the others were less than overjoyed, and he was expelled. From the boat, according to some versions. | |
| 876 | First undisputed use of a symbol for zero in positional base-10 notation. |
| 1202 | Leonardo’s Liber Abbaci introduces the Fibonacci numbers through a problem about the progeny of rabbits. Also promotes Arabic numerals and discusses applications of mathematics to currency trading. |
| 1500-1550 | Renaissance Italian mathematicians solve cubic and quartic equations. |
| 1585 | Simon Stevin introduces the decimal point. |
| 1589 | Galileo Galilei discovers mathematical patterns in falling bodies. |
| 1605 | Johannes Kepler shows that the orbit of Mars is an ellipse. |
| 1614 | John Napier invents logarithms. |
| 1637 | René Descartes invents coordinate geometry. |
| c.1680 | Gottfried Wilhelm Leibniz and Isaac Newton invent calculus and argue about who did it first. |
| 1684 | Newton sends Edmund Halley a derivation of elliptical orbits from the inverse square law of gravity. |
| 1718 | Abraham De Moivre writes first textbook on probability theory. |
| 1726-1783 | Leonhard Euler standardises notation such as e, i, π, systematises most known mathematics, and invents a huge amount of new mathematics. |
| 1788 | Joseph-Louis Lagrange’s Méchanique Analytique places mechanics on an analytic basis, getting rid of pictures. |
| 1796 | Carl Friedrich Gauss discovers how to construct a regular 17-gon. |
| 1799-1825 | Pierre Simon de Laplace’s five-volume epic Mécanique Céleste formulates the basic mathematics of the solar system. |
| 1801 | Gauss’s Disquisitiones Arithmeticae provides a basis for number theory. |
| 1821-1828 | Augustin-Louis Cauchy introduces complex analysis. |
| 1824-1832 | Niels Henrik Abel and Évariste Galois prove that the quintic equation is not soluble using radicals; Galois paves the way for modern abstract algebra. |
| 1829 | Nikolai Ivanovich Lobachevsky introduces non-Euclidean geometry, followed shortly by János Bolyai. |
| 1837 | William Rowan Hamilton defines complex numbers formally. |
| 1843 | Hamilton formulates mechanics and optics in terms of the Hamiltonian. |
| 1844 | Hermann Grassmann develops multidimensional geometry. |
| 1848 | Arthur Cayley and James Joseph Sylvester invent matrix notation. Cayley predicts that it will never have any practical uses. |
| 1851 | Posthumous publication of Bernard Bolzano’s Paradoxien des Unendlichen which tackles the mathematics of infinity. |
| 1854 | Georg Bernhard Riemann introduces manifolds—curved spaces of many dimensions - paving the way for Einstein’s general relativity. |
| 1858 | Augustus Möbius invents his band. |
| 1859 | Karl Weierstrass makes analysis rigorous with epsilon-delta definitions. |
| 1872 | Richard Dedekind proves that √2 × √3 = √6 - the first time this has been done rigorously - by developing the logical foundations of real numbers. |
| 1872 | Felix Klein’s Erlangen programme represents geometries as the invariants of transformation groups. |
| c.1873 | Sophus Lie starts working on Lie groups, and the mathematics of symmetry makes a huge leap forward. |
| 1874 | Georg Cantor introduces set theory and transfinite numbers. |
| 1885-1930 | Italian school of algebraic geometry flourishes. |
| 1886 | Henri Poincaré stumbles across hints of chaos theory and revives the use of pictures. |
| 1888 | Wilhelm Killing classifies the simple Lie algebras. |
| 1889 | Giuseppe Peano states his axioms for the natural numbers. |
| 1895 | Poincaré establishes basic ideas of algebraic topology. |
| 1900 | David Hilbert presents his 23 problems at the International Congress of Mathematicians. |
| 1902 | Henri Lebesgue invents measure theory and the Lebesgue integral in his PhD thesis. |
| 1904 | Helge von Koch invents the snowflake curve, which is continuous but not differentiable, simplifying an earlier example found by Karl Weierstrass and anticipating fractal geometry. |
| 1910 | Bertrand Russell and Alfred North Whitehead prove that 1 + 1 = 2 on page 379 of volume 1 of Principia Mathematica, and formalise the whole of mathematics using symbolic logic. |
| 1931 | Kurt Gödel’s theorems demonstrate the limitations of formal mathematics. |
| 1933 | Andrei Kolmogorov states axioms for probability. |
| c.1950 | Modern abstract mathematics starts to take off. After that it gets complicated. |
Let ε < 0.
If you don’t understand this one, see the note in the Answers section, page 317. If you do understand it and don’t find it funny, congratulations.
Global Warming SwindleMathematical models are central to the study of global warming, because they help us understand how the Earth’s atmosphere would behave with different levels of incoming radiation from the Sun, different levels of greenhouse gases such as carbon dioxide (CO
2
) and methane, and whatever else might go into the model. I’ll ignore the effect of methane - basically, it just makes everything worse. Climate change is a very complex topic, and this is just a quick look at one common misunderstanding.
2
) and methane, and whatever else might go into the model. I’ll ignore the effect of methane - basically, it just makes everything worse. Climate change is a very complex topic, and this is just a quick look at one common misunderstanding.
Nearly all scientists working on climate are now convinced that human activities have increased the amount of CO
2
in the atmosphere, and that this increase has caused temperatures to rise. A few still disagree, and in March 2007 Channel 4 Television broadcast a documentary, The Great Global Warming Swindle, about these dissident opinions. One of the more puzzling pieces of evidence put forward in this programme was the observed long-term relation between temperature and CO
2
. Former presidential candidate Al Gore, who has been very active trying to persuade the public that climate change is real, was shown delivering a lecture in front of a huge display of how temperature and CO
2
have changed in the past. These figures can be deduced from natural records such as ice cores.
2
in the atmosphere, and that this increase has caused temperatures to rise. A few still disagree, and in March 2007 Channel 4 Television broadcast a documentary, The Great Global Warming Swindle, about these dissident opinions. One of the more puzzling pieces of evidence put forward in this programme was the observed long-term relation between temperature and CO
2
. Former presidential candidate Al Gore, who has been very active trying to persuade the public that climate change is real, was shown delivering a lecture in front of a huge display of how temperature and CO
2
have changed in the past. These figures can be deduced from natural records such as ice cores.
Historical records of temperature and CO
2
, based on: J. R. Petit and others, ‘Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica’, Nature, vol. 399, pp. 429-436 (1999).
2
, based on: J. R. Petit and others, ‘Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica’, Nature, vol. 399, pp. 429-436 (1999).
The two curves go up and down almost together, a convincing association. But the programme pointed out that the temperature increases start and end before the CO
2
ones do, especially if you look closely at the most recent data. Clearly it is rising temperature that causes CO
2
to increase, not the other way round. This argument seems quite convincing, and the programme placed a lot of emphasis on it.
2
ones do, especially if you look closely at the most recent data. Clearly it is rising temperature that causes CO
2
to increase, not the other way round. This argument seems quite convincing, and the programme placed a lot of emphasis on it.
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