The arrow paradox is also often resolved by taking the ‘limit’ point of view, or, more precisely, calculus, which is what limits were invented for. In calculus, a moving object can have an instantaneous speed that is not zero, even though it has a fixed location at that instant. Making logical sense of this took a few centuries, and boils down to taking the limit of the average speed over shorter and shorter intervals of time. Again, some philosophers feel that this is not an acceptable approach.
I think there’s another interesting mathematical point buried in this one. Physically, there is a definite difference between an arrow that is moving and one that is not, even if they are both in the same place at some instant. The difference can’t be seen in an instantaneous ‘snapshot’, but nevertheless it is physically real (whatever that means). Anyone who does classical mechanics knows what the difference is: a moving body has momentum (mass times velocity). A snapshot tells you the position of the body, but not its momentum. These are independent variables: in principle, a body can have any position and any momentum.
While position is directly observable (see where the body is), momentum is not. The only way we know to observe it is to measure the velocity, which involves at least two positions, at closely spaced intervals of time. Momentum is a ‘hidden variable’, whose value must be inferred indirectly. Since 1833, the most popular formulation of mechanics has been the one proposed by Sir William Rowan Hamilton, which explicitly works with these two kinds of variables, position and momentum. So the difference between a moving arrow and a fixed one is that the moving one has momentum, whereas the fixed one does not. How can you tell the difference? Not by taking a snapshot. You have to wait and see what happens next. The main thing missing from this approach, philosophically, is any description of what momentum is, physically. And that’s probably a lot harder than anything that worried Zeno.
What of the stadium? One answer is that Zeno was hopelessly confused, and that his conclusion ‘half the time is equal to double the time’ does not follow from his set-up. But there is an interpretation that puts all four paradoxes in a more interesting light. The suggestion is that Zeno was trying to understand the nature of space and time.
The most obvious models of space are either that it is discrete, with isolated points placed at (say) integer positions 0, 1, 2, 3, and so on; or it is continuous, and points correspond to real numbers, which can be subdivided as finely as we wish. The same goes for time.
Possible structures for space and time.
Altogether, these choices give four distinct combinations for the structure of space and time. And these relate fairly convincingly to the four paradoxes, like this:
Paradox Space Time
Achilles and the tortoise Continuous Continuous
Dichotomy Discrete Continuous
Arrow Continuous Discrete
Stadium Discrete Discrete
Possibly Zeno was trying to show that each combination suffers from logical problems.
• The first requires infinitely many things to happen over a finite period of time.
• The second means that space cannot be subdivided indefinitely, while time can. So consider an object traversing the shortest possible unit of space, in some non-zero time t. At time 0, it is in one location; at time t, it is in the closest different location. So, where is it at time t? It should be halfway between, but in this discrete version of space, there is no point in between.
• If space is continuous and time discrete, then the same thing happens with time and space interchanged. The arrow manages to move from a fixed location at one instant to a different fixed location at the next. It could go in between, but there isn’t a time in between for it to get there.
• What of the stadium? Now both space and time are discrete. So imagine Zeno’s two rows of identical bodies passing each other. To clarify the problem, let’s add a third row of bodies, which doesn’t move, and compare each moving row with that. Assume that relative to the fixed row, they move as rapidly as possible: that is, each moves through the smallest possible unit of space in the smallest possible unit of time.
Successive positions of the rows of identical bodies.
You’ll notice that I’ve made two of the bodies black: this is for reference. At the first instant, the black bodies are one unit of space apart, with the top one on the left. At the next instant, they are one unit of space apart, with the top one on the right: they have swapped positions.
At which instant were they level with each other?
They weren’t. Because we are working with the smallest possible interval of time, what the pictures show is everything that happens. There is no ‘halfway time’ for the two black bodies to get level with each other. This problem is not insurmountable - we can just accept that a moving body makes this kind of ‘jump’, for instance. And maybe the whole neat and tidy classification of the paradoxes into four possibilities is misleading, and Zeno’s intentions were quite different.
Pieces of Five
‘Here’s a fine challenge for ye, me hearties!’ yelled Roger Redbeard, the pirate captain, who liked to keep his crew’s minds alert. If only to check they still had them.
He held up four coins, identical gold pieces-of-eight.
‘Now, me lads - what I wants ye to do is to place these four gold coins so that they be equidistant.’
Seeing the baffled looks on their faces, he explained. ‘What I means, lads, is that the shortest distance between any two coins’as to be the same as that between any other two coins.’
To his considerable surprise, the bosun immediately realised that it was no good ‘working in the plane’, and the solution required three dimensions of space. He quickly found an answer: place three of the coins touching each other in a triangle, and sit the fourth on top. All coins are touching, so all distances between them are zero, hence equal.
How to do it with four coins.
Redbeard, dismayed, thought for a moment. ‘So, ye think ye be smart? Try doin’ it with five coins, then. Make them all equidistant from each other!’
Eventually the bosun found an answer, but it wasn’t easy. What was it?
Answer on page 313
Pi in the Sky
It is not widely known that you can work out the value of π by observing the stars. Moreover, the reasoning behind this feat is not based on astronomy, but on number theory - and it works, not because of a pattern in the stars, but because there isn’t one.
Suppose you pick two non-zero whole numbers at random, less than or equal to some upper limit. The probability should be uniform - that is, each number should have the same chance of being chosen. For instance, the upper limit might be a million, and the numbers you get might be 14,775 and 303,254, say, each with probability one in a million. Now ask: Do those two numbers have a common factor (greater than 1) - or not? In this case they don’t. In general, number-theorists have proved that the proportion of pairs with no common factor tends to 6/π2 as the upper limit becomes arbitrarily large. This remarkable result is one of many properties of π that appear to have no connection to circles. It is exact, not an approximation, and it can (with some clever tricks) be deduced from the formula
In 1995, Robert Matthews wrote a short letter to the scientific journal Nature, pointing out that this theorem in number theory can be used to extract a reasonably accurate value of π from the stars in the night sky - on the assumption that the positions of the stars are random. His idea was to work out the angular distances between lots of stars (that is, the angle between the lines joining those stars to the observer’s eye) and then to transform those distances into large integers. (The actual formula he used was to take the cosine of the angle, add 1, and multiply by half a million.) If you ignore anything after the decimal point, and exclude zero, you get a list of positive integers between 1 and a million. Pick pairs at random, and let the proportion with no highest common factor be p. Then p is approximately 6/π2, so π is approximately .
Matthews
did this for the 100 brightest stars in the sky, producing a list of 4,095 integers between 1 and a million. From these he derived a million pairs of randomly chosen numbers, and found that p = 0.613333. Thus π should be approximately 3.12772. This isn’t as good as the school approximation 22/7, but it is within 0.4 per cent of the correct value. Using more stars should improve it. Matthews ended his letter by saying that ‘Latter-day Pythagoreans may take encouragement from learning that a 99.6 per cent accurate value for π can be found among the stars over their heads.’
The Curious Incident of the Dog
In Sir Arthur Conan Doyle’s Sherlock Holmes story ‘Silver Blaze’, we find:
‘Is there any other point to which you would wish to draw my attention?’
‘To the curious incident of the dog in the night-time.’
‘The dog did nothing in the night-time.’
‘That was the curious incident,’ remarked Sherlock Holmes.
Here is a sequence:
1, 2, 4, 7, 8, 11, 14, 16, 17, 19, 22, 26, 28, 29, 41, 44
Having taken Holmes’s point on board: what is the next number in the sequence?
Answer on page 313
Mathematics Made Difficult
There’s a snag with all these ‘find the next number in the sequence’ puzzles - the answer need not be unique. Carl Linderholm put his finger on this problem in the often-hilarious spoof Mathematics Made Difficult, published in 1971 when the ‘new math(s)’ was in vogue. In it, he remarks: ‘Mathematicians always strive to confuse their audiences; where there is no confusion, there is no prestige.’ As an example, Linderholm defines the natural number system as a ‘universal pointed function’.
His take on ‘guess the next number’ puzzles is unusual but logical. For example, to find the next number after
8, 75, 3, 9
he tells you to write down ‘the only answer any sensible person would put there’. Which is - what? Ah, that’s the clever part. As a clue, here are some more puzzles of the same type:
• What comes after 1, 2, 3, 4, 5?
• What comes after 2, 4, 6, 8, 10?
• What comes after 1, 4, 9, 16, 25?
• What comes after 1, 2, 4, 8, 16?
• What comes after 2, 3, 5, 7, 11?
• What comes after 139, 21, 3, 444, 65?
Here are the answers we would obtain using Linderholm’s method:
• 19
• 19
• 19
• 19
• 19
• 19
What is the justification for this bizarre set of answers? It is Lagrange’s interpolation formula, which provides a polynomial p(x) such that p(1), p(2), . . . , p(n) is any specified sequence of length n, for any finite n. Some such p must fit the sequence
1, 2, 3, 4, 5, 19
so the choice of 19 is justified by the polynomial. The same goes for all the other examples. As Linderholm explains, this answer is far superior to
1, 2, 3, 4, 5, 6
because his procedure ‘is much the simpler, and is easier to use, and is obtained by a more general method’.
Why 19? Choose your favourite number and add 1. Why add 1? To ‘make it more difficult to determine your character defects by analysing your favourite number. No technique by which a person’s character may be found out from his secret number is known to the author, but of course someone may some day invent such a technique.’
In the spirit of Linderholm’s book, I really ought to show you Lagrange’s interpolation formula. So it’s on page 313.
A Weird Fact about Egyptian Fractions
Ron Graham has proved that any number greater than 77 can be expressed as a sum of distinct positive integers, whose reciprocals (1 divided by the appropriate integer) add up to 1. So this represents 1 as an Egyptian fraction (see page 76).
For example, let n = 425. Then
and 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 63 + 105 + 135 = 425.
On the other hand, Derrick Henry Lehmer showed that 77 cannot be written in this form. So here we have a special property of the number 77, in the context of Egyptian fractions.
A Four Colour Theorem
If I arrange three equal circles so that each touches the other two, then it’s obvious that I need three colours if I wish to colour each circle so that circles that touch have different colours. The picture shows three circles, each touching the other two, so they all need different colours.
Three colours required.
Four equal circles in the plane can’t all touch each other, but that doesn’t mean three colours always work: there are more complicated ways to arrange lots of coins, and some of those might need four colours. What is the smallest number of equal circles that can be arranged so that four colours are needed? Again, the rule is that if two circles touch, they must have different colours.
Answer on page 314
Serpent of Perpetual Darkness
In 2004, astronomers discovered asteroid 99942, and named it Apophis after the ancient Egyptian serpent who attacks the Sun-god Ra during his nightly passage through the eternal darkness of the Underworld.33 It was an appropriate name in some ways, because the astronomers also announced that there was a serious danger that the newly discovered asteroid might collide with the Earth on 13 April 2029 - or, if not, on 13 April 2036. The chance of a collision was initially estimated at 1 in 200, and peaked at 1 in 37, but is now thought to be highly unlikely.
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?’
‘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.
&
nbsp; * 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.
Ian Stewart Page 13