Ian Stewart
Page 14
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.
The Shortest Mathematical Joke Ever
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 Swindle
Mathematical 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 (CO2) 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 CO2 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 CO2. 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 CO2 have changed in the past. These figures can be deduced from natural records such as ice cores.
Historical records of temperature and CO2, 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 CO2 ones do, especially if you look closely at the most recent data. Clearly it is rising temperature that causes CO2 to increase, not the other way round. This argument seems quite convincing, and the programme placed a lot of emphasis on it.
Temperature always changes first (schematic picture for illustrative purposes).
Climate science relies heavily on mathematical models of the physical processes that influence the climate, so this is a mathematical issue as well as a scientific one. The best data available to date indicate that this effect is real, with the CO2 peaks and troughs appearing about 100 years after those of temperature. So does this relationship prove that rising temperatures cause rising CO2, rather than the other way round? And what, if anything, does all this have to do with global warming?
Let’s put our brains in gear first. The graphs are well known to climatologists, and indeed are a big part of the evidence that human production of CO2 is causing temperatures to rise. If those graphs really do prove that CO2 is not responsible for rising temperatures, the climatologists are likely to have noticed. Yes, it could all be a big conspiracy, but governments worldwide would be much happier if climate change turned out to be a delusion, and they’re the ones paying for the research. If there’s a conspiracy, it’s far more likely to be one that tries to suppress evidence of climate change. So it seems likely that the climatologists have worked out why this delay occurs, and have concluded that it does not show that CO2 plays no significant role in climate change. And indeed they have: it takes 30 seconds on the internet to find the explanation.
What happens at times A, B, C, D and E?
So why does that 100-year delay happen? The full story is complicated, but the broad outlines aren’t hard to grasp if we think about the schematic picture, where the issues are easier to follow. The key facts are these:
• There is a natural cycle of changes in temperature caused by systematic changes in the Earth’s orbit, the tilt of its axis, and the direction in which the axis points.
• Rises in temperature do indeed cause CO2 levels to rise, with a time delay of tens or hundreds of years for nature to respond to the temperature change.
First, observe that most of the time, temperature and CO2 levels rise together (between times B and C), or fall together (between times D and E). This shows temperature and CO2 are linked, but it doesn’t tell us which is cause and which is effect. In fact, each causes the other.
What is actually going on here, according to the vast majority of climatologists, is roughly this. At time A the natural cycle causes temperatures to start rising, though not by much. By time B, a century or so later, the effect on CO2 becomes apparent. This rise feeds back into the temperature, which responds far more quickly to CO2 levels than CO2 levels do to temperature. So the temperature rises. Now temperature and CO2 reinforce each other though positive feedback, and both climb together (times B to C). At time C, the external temperature cycle and other factors cause temperatures to begin to fall. The CO2 levels don’t show much effect until time D, but
as soon as they do, the fall in CO2 reinforces the fall in temperature, and both drop together. This continues until time E, when the whole process repeats itself.
Next question: what does this have to do with global warming?
Not a lot.
What we’ve been discussing is a natural free-running cycle, without human intervention. The terms ‘global warming’ and ‘climate change’ do not refer to increasing temperatures or changes in climate as such. They refer, very specifically, to deviations from the natural cycle.
The term ‘global warming’ was used first by scientists who understood that point and also understood that what was being discussed was medium-term average global temperatures, not short-term local ones. It was widely misunderstood, because some parts of the globe may cool for a time, while others get warmer. So the term ‘climate change’ started to be used in the hope of avoiding confusion. But that phrase doesn’t just mean ‘the climate is changing’: that happens during the natural cycle. It means ‘the climate is changing in a way that the natural cycle does not explain’.
In the natural cycle, as we have seen, temperature influences CO2 and CO2 influences temperature. When the atmosphere is ‘forced’ by a changing cycle of solar radiation, both quantities respond. The issue of ‘global warming’ is: what do we expect to happen to that cycle if humans cause large quantities of CO2 to enter the atmosphere? Mathematically, this amounts to giving a big kick to CO2, and seeing what the system does. And the answer is: the temperature promptly goes up too, because it responds fairly rapidly to changes in CO2.
So the graphs, with that puzzling time delay, show what a free-running atmospheric system does when it is forced by variations in incoming radiation. ‘Global warming’ isn’t about that at all. It’s about what this free-running system will do when you give it a kick. We know that human activity has raised CO2 levels significantly over the past 50 years or so; in fact, they are now distinctly higher than anything found earlier in the ice core record. Look at the right-hand end of the graph of CO2 on page 164. The proportions of various isotopes of carbon (different forms of the carbon atom with different atomic weight) show that this rise is mainly the result of human activity - and the unprecedented level of CO2 in modern times confirms that.
To test the hypothesis that this rise in CO2 has led to global warming, the mathematical kick that we give to any model of the atmosphere also has to be a rise in CO2. So we are asking what effect this rise in CO2 causes - in that context.
To check what happens, and to make it clear that this really is mathematics, I set up a simple system of model equations for how temperature T and carbon dioxide levels C change over time. It’s not ‘realistic’, but it has the basic features we are discussing, and illustrates the key point. It looks like this:
Here temperature is forced periodically (the sin t term) which models the changing heat coming from the Sun. Moreover, any change in C produces a proportionate change in T (the 0.25C term), and any change in T produces a proportionate change in C (the 0.1T term). So my model is set up so that higher temperatures cause more CO2, and more CO2 causes higher temperatures, just like the real world. Since 0.25 is bigger than 0.1, temperature responds faster to changes in CO2 than CO2 does to changes in temperature. Finally, I subtract 0.01T2 and 0.01C2 to mimic the cut-off effects known to occur.
I now solve these equations on my computer, and see what I get. Here are three pictures of how T (black curve) and C (grey curve) change over time. I have plotted 4y - 60 rather than y to move the two curves close enough together to see the relationship.
• When the system is free-running, both T and C fluctuate periodically, and C lags behind T. This is the paradoxical time delay, which according to the TV programme means that rising CO2 does not cause rising temperatures. But, in our model, rising CO2 does cause rising temperatures, thanks to the 0.25C term in the first equation, yet we still see that time delay. The time delay is a consequence of non-linear effects in the model, not delays in what affects what.
How temperature (black line) and CO2 (grey line) vary over time. Note that CO2 lags behind temperature.
• When I give C a sudden brief increase at time 25, both T and C react. However, C still appears to lag behind T, and T doesn’t seem to change much.
The effect of a sudden increase in CO2 (grey line).
• However, if I graph the changes in T and C between the two runs of the equations, then I see that T starts to increase as soon as C does. So a change in C does cause an immediate change in T. What’s interesting here is how temperature continues to increase while the spike in CO2 is dying down. Non-linear dynamics can be counter-intuitive, which is why we have to use mathematics rather than naive verbal arguments.
Differences in CO2 and temperature between the two runs show that temperature rises immediately CO2 does.
So the issue of ‘global warming’ or ‘climate change’ is not what causes what in the free-running system, where both rising temperature and rising CO2 cause each other. Climate scientists don’t dispute that and have known about it for a long time. The issue is: what happens when we know that one of these quantities has suddenly been changed by human activity? That much-trumpeted time delay is irrelevant to this question - in fact, it is misleading. The resulting temperature change begins immediately, and rises.
For further information, take a look at:
en.wikipedia.org/wiki/Climate_change
en.wikipedia.org/wiki/Global_warming
And you may find it informative to look at what happened after that Channel 4 broadcast, at:
en.wikipedia.org/wiki/The_Great_Global_Warming_Swindle
Name the Cards
‘Ladies and gentlemen,’ the Great Whodunni announced, ‘My assistant Grumpelina will ask a member of the audience to place three cards in a row on the table, while I am blindfolded. I will then ask her to provide some limited information, after which I will name those cards.’
The cards were chosen and placed in a row. Grumpelina then recited a strange list of statements:
‘To the right of a King there’s a Queen or two.
‘To the left of a Queen there’s a Queen or two.
‘To the left of a Heart there’s a Spade or two.
‘To the right of a Spade there’s a Spade or two.’
Instantly, Whodunni named the three cards.
What were they? [Note that ‘two’ here means two cards, not a card with two spots.]
Answer on page 317
What Is Point Nine Recurring?
The first place most of us encounter mathematical infinity is when we study decimals. Not only do exotic numbers like π ‘go on for ever’ - so do more prosaic ones. Probably the first example We get to see is the fraction . In decimals, this becomes 0.333333 . . . , and the only way to make the decimal exactly equal to is to let it continue for ever.
The same problem arises for any fraction p/q where q is not just a lot of 2’s and 5’s multiplied together (which in particular includes all powers of 10). But unlike π, the decimal form of a fraction repeats the same pattern of digits over and over again, perhaps after some initial digits that don’t fit that pattern. For instance, = 2.3714285714285714285 . . . , repeating the 714285 indefinitely. These are called recurring decimals, and the part that repeats is usually marked with a dot, or dots at each end if it involves several digits:
All this sounds reasonable, but the number 0.999999 . . . , or 0.9, often causes trouble. On the one hand, it is obviously equal to 3 times 0.3, which is 3 × , which is 1. On the other hand, 1 in decimals is 1.000000 . . . , which doesn’t look the same.
It seems to be widely believed that 0.9 is slightly less than 1. The reason for thinking that is presumably that whenever you stop, say at 0.9999999999, the resulting number differs from 1. The difference isn’t very big - here it’s 0.0000000001 - but it’s not zero. But, of course, the point is that you shouldn’t stop. So that argument doesn’t hold water. Nevertheless, many people get a sneaky feeling
that 0.9 still ought to be less than 1. How much less? Well, by a number that is smaller than anything looking like 0.000 . . . 01, no matter how many 0’s there are.
A friend of mine, who worked in mathematics education, used to ask people how big 0.3 is, and then how big 0.9 is. Everyone was happy that the first decimal is exactly , but on being told to multiply by 3, they became nervous. One said: ‘That’s sneaky! At first I thought that point three recurring is exactly one-third, but now I see it must be slightly less than one-third!’
We get confused about this point because it’s a subtle feature of infinite series, and though we all do decimals, we don’t do infinite series at school. To see the connection, observe that
This series converges, that is, it has a well-defined sum, and the rules of algebra apply. So we can use a standard trick. If the sum is s, then
so 9s = 9, and s = 1.
There are lots of other calculations like this. They all tell us that 0.9 = 1.
So what about that number that is smaller than anything looking like 0.000 . . . 01, no matter how many 0’s there are? Is it an ‘infinitesimal’ - whatever that may mean?
Not in the real number system, no. There the only such number is 0. Why? Any (small) non-zero number has a decimal representation with a lot of 0’s, but eventually some digit must be non-zero - otherwise the number is 0.000 . . . , which is 0. As soon as we reach that position, we see that the number is greater than or equal to 0.000 . . . 01 with the appropriate number of 0’s. So it doesn’t satisfy the definition. In short: the difference between 1 and 0.9 is 0, so they are equal.