What's the Use? Page 2
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On a more pragmatic level, it can be argued that mathematics has several features that help to make it unreasonably effective in Wigner’s sense. One is, I agree, its many links with natural science, which transfer to the human world as transformative technology. Many of the great mathematical innovations have indeed arisen from scientific enquiries. Others are rooted in human concerns. Numbers arose from basic accountancy (how many sheep have I got?). Geometry means ‘earth-measurement’, and was closely related to the taxation of land and in ancient Egypt to the construction of pyramids. Trigonometry emerged from astronomy, navigation, and map-making.
However, that alone isn’t an adequate explanation. Many other great mathematical innovations have not arisen from scientific enquiry or specific human issues. Prime numbers, complex numbers, abstract algebra, topology – the primary motivation for these discoveries/inventions was human curiosity and a sense of pattern. This is a second reason why mathematics is so effective: mathematicians use it to seek patterns and to tease out underlying structure. They search for beauty, not of form but of logic. When Newton wanted to understand the motion of the planets, the solution came when he thought like a mathematician and looked for deeper patterns beneath the raw astronomical data. Then he came up with his law of gravity.5 Many of the greatest mathematical ideas had no real-world motivation at all. Pierre de Fermat, a lawyer who did mathematics for fun in the seventeenth century, made fundamental discoveries in number theory: deep patterns in the behaviour of ordinary whole numbers. It took three centuries for his work in that area to acquire practical applications, but right now the commercial transactions that drive the Internet wouldn’t be possible without it.
Another feature of mathematics that’s become increasingly evident since the late 1800s is generality. Different mathematical structures have many common features. The rules of elementary algebra are the same as those of arithmetic. Different kinds of geometry (Euclidean, projective, non-Euclidean, even topology) are all closely related to each other. This hidden unity can be made explicit by working, from the start, with general structures that obey specified rules. Understand the generalities, and all the special examples become obvious. This saves a lot of effort, which would otherwise be wasted doing essentially the same thing many times in slightly different language. It has one downside, however: it tends to make the subject more abstract. Instead of talking about familiar things such as numbers, the generalities must refer to anything obeying the same rules as numbers, with names like ‘Noetherian ring’, ‘tensor category’, or ‘topological vector space’. When this kind of abstraction is carried to extremes, it can become difficult to understand what the generalities are, let alone how to make use of them. Yet they’re so useful that the human world would no longer function without them. You want Netflix? Someone has to do the maths. It’s not magic; it just feels like it.
A fourth feature of mathematics, highly relevant to this discussion, is its portability. This is a consequence of its generality, and it’s why abstraction is necessary. Irrespective of the problem that motivated it, a mathematical concept or method possesses a level of generality that often makes it applicable to quite different problems. Any problem that can be recast in the appropriate framework then becomes fair game. The simplest and most effective way to create portable mathematics is to design portability in from the start, by making the generalities explicit.
For the last two thousand years, mathematics has taken its inspiration from three main sources: the workings of nature, the workings of humanity, and the internal pattern-seeking tendencies of the human mind. These three pillars support the entire subject. The miracle is that despite its multifarious motivations, mathematics is all one thing. Every branch of the subject, whatever its origins and aims, has become tightly bound to every other branch – and the bonds are becoming ever stronger and ever more entangled.
This points to a fifth reason why mathematics is so effective, and in such unexpected ways: its unity. And alongside this goes a sixth, for which I’ll provide ample evidence as we proceed: its diversity.
Reality, beauty, generality, portability, unity, diversity. Which, together, lead to utility.
It’s as simple as that.
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How Politicians Pick Their Voters
Ankh-Morpork had dallied with many forms of government and had ended up with that form of democracy known as One Man, One Vote. The Patrician was the Man; he had the Vote.
Terry Pratchett, Mort
The ancient Greeks gave the world many things – poetry, drama, sculpture, philosophy, logic. They also gave us geometry and democracy, which have turned out to be more closely linked than anyone might have expected, least of all the Greeks. To be sure, the political system of ancient Athens was a very limited form of democracy; only free men could vote, not women or slaves. Even so, in an era dominated by hereditary rulers, dictators, and tyrants, Athenian democracy was a distinct advance. As was Greek geometry, which, in the hands of Euclid of Alexandria, emphasised the importance of making your basic assumptions clear and precise, and deriving everything else from them in a logical and systematic fashion.
How on earth can mathematics be applied to politics? Politics is about human relationships, agreements, and obligations, whereas mathematics is about cold, abstract logic. In political circles, rhetoric trumps logic, and the inhuman calculations of mathematics seem far removed from political bickering. But democratic politics is carried out according to rules, and rules have consequences that aren’t always foreseen when they’re first drawn up. Euclid’s pioneering work in geometry, collected in his famous Elements, set a standard for deducing consequences from rules. In fact, that’s not a bad definition of mathematics as a whole. At any rate, after a mere 2,500 years, mathematics is beginning to infiltrate the political world.
One of the curious features of democracy is that politicians who claim to be devoted to the idea that decisions should be made by ‘The People’ repeatedly go out of their way to ensure that this doesn’t happen. This tendency goes right back to the first democracy in ancient Greece, where the right to vote was given only to adult male Athenians, about one third of the adult population. From the moment the idea of electing leaders and selecting policy by popular vote was conceived, so was the even more attractive idea of subverting the entire process, by controlling who voted and how effective their votes are. This is easy, even when every voter gets one vote, because the effectiveness of a vote depends on the context in which it’s cast, and you can rig the context. As journalism professor Wayne Dawkins delicately put it, this amounts to politicians picking their voters instead of voters picking their politicians.6
That’s where mathematics comes in. Not in the cut-and-thrust of political debate, but in the structure of the debating rules and the context in which they apply. Mathematical analysis cuts both ways. It can reveal new, cunning methods for rigging votes. It can also shine a spotlight on such practices, providing clear evidence of that kind of subversion, which can sometimes be used to prevent it happening.
Mathematics also tells us that any democratic system must involve elements of compromise. You can’t have everything you want, however desirable that might be, because the list of desirable attributes is self-contradictory.
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On 26 March 1812 the Boston Gazette gave the world a new word: gerrymander. Originally spelt Gerry-mander, it’s what Lewis Carroll later called a portmanteau word, created by combining two standard words. ‘Mander’ was the tail end of ‘salamander’, and ‘Gerry’ was the tail end of Elbridge Gerry, governor of Massachusetts. We don’t know for sure who first put the two tails together, but on circumstantial grounds, historians tend to opt for one of the newspaper’s editors, Nathan Hale, Benjamin Russell, or John Russell. Incidentally, ‘Gerry’ was pronounced with a hard G, like ‘Gary’, but ‘gerrymander’ has a soft G, like ‘Jerry’.
What was Elbridge Gerry doing that got him combined with a lizard-like creature, which, in medieval folklore, was reputed to dwell in fire?
Rigging an election.
More precisely, Gerry was responsible for a bill that redrew the district boundaries in Massachusetts for elections to the state senate. Districting, as it’s called, naturally leads to boundaries; it is, and has long been, common in most democracies. The overt reason is practicality: it’s awkward to take decisions if the entire nation gets to vote on every proposal. (Switzerland comes close: up to four times a year the Federal Council chooses proposals for citizens to vote on, essentially a series of referendums. On the other hand, women didn’t get the vote there until 1971, and one canton held out until 1991.) The time-honoured solution is for voters to elect a much smaller number of representatives, and let the representatives make the decisions. One of the fairer methods is proportional representation: the number of representatives of a given political party is proportional to the number of votes that that party receives. More commonly, the population is divided into districts, and each district elects a number of representatives, roughly proportional to the number of electors in that district.
For example, in American presidential elections, each state votes for a specific number of ‘electors’ – members of the Electoral College. Each elector has one vote, and who becomes President is decided by a simple majority of these votes. It’s a system that originated when the only way to get a message from the American hinterland to the centres of power was to carry a letter on horseback or in a horse-drawn coach. Long-distance rail and the telegraph came later. In those days, totalling up the votes of huge numbers of individuals was too slow.7 But this system also ceded control to the elite members of the Electoral College. In British parliamentary elections, the country is divided into (mainly geographical) constituencies,
each of which elects one Member of Parliament (MP). Then the party (or combination of parties in a coalition) with the most MPs forms the government, and chooses one of its MPs to be Prime Minister, by a variety of methods. The Prime Minister has considerable powers, and in many ways is more like a President.
There’s also a covert reason for funnelling democratic decisions through a small number of gatekeepers: it’s easier to rig the vote. All such systems have innate flaws, which often lead to strange results, and on occasion they can be exploited to flout the Will of the People. In several recent US presidential elections, the total number of votes cast by the People for the candidate who lost was greater than the number of votes for the candidate who won. Agreed, the current method for choosing a President doesn’t depend on the popular vote, but with modern communications the only reason not to change to a fairer system is that a lot of powerful people prefer it the way it is.
The underlying problem here is ‘wasted votes’. In each state, a candidate needs half the total plus one vote (or half a vote if the total is odd) to win; any extra votes beyond that threshold make no difference to what happens at the Electoral College stage. Thus, in the 2016 presidential election, Donald Trump received 304 votes in the Electoral College compared to 227 for Hillary Clinton, but Clinton’s popular vote exceeded Trump’s by 2·87 million. Trump thereby became the fifth US President to be elected while losing the popular vote.
The Gerry-mander, thought to have been drawn in 1812 by Elkanah Tisdale.
The boundaries of American states are effectively immutable, so this is not a districting issue. In other elections, the boundaries of districts can be redrawn, usually by the party in power, and a more insidious flaw appears. Namely, that party can draw the boundaries to ensure that unusually large numbers of votes for the opposing party are wasted. Cue Elbridge Gerry and the senate vote. When Massachusetts voters saw the map of electoral districts, most of them looked entirely normal. One didn’t. It combined twelve counties from the west and north of the state into a singe, sprawling, meandering region. To the political cartoonist responsible for the drawing that shortly appeared in the Boston Gazette – probably the painter, designer, and engraver Elkanah Tisdale – this district closely resembled a salamander.
Gerry belonged to the Democratic-Republican Party, which was in competition with the Federalists. In the 1812 election the Federalists won the House and governorship in the state, which put Gerry out of office. However, his redistricting of the state senate worked a treat, and it was held comfortably by the Democratic-Republicans.
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The mathematics of gerrymandering begins by looking at how people do it. There are two main tactics, packing and cracking. Packing spreads your own vote as evenly as possible, with a small but decisive majority, in as many districts as possible, and cedes the rest to the enemy. Sorry, opposition. Cracking breaks up the opposition’s votes so that they lose as many districts as possible. Proportional representation, in which the number of representatives is proportional to each party’s total votes (or as close to that as possible given the numbers) avoids these tricks, and is fairer. Unsurprisingly, the US constitution makes proportional representation illegal, because as the law stands, districts must have only one representative. In 2011 the UK held a referendum on another alternative, the single transferable vote: the people voted against this change. There’s never been a referendum on proportional representation in the UK.
Here’s how packing and cracking work, in an artificial example with very simple geography and voting distributions.
The state of Jerimandia is contested by two political parties, the Lights and the Darks. There are fifty regions, to be split into five districts. In recent elections, Light has a majority in twenty of them, all in the north, while Dark has a majority in the thirty southern regions (top left). The Light administration, which just scraped in at the previous vote, has redistricted the state by packing more of its voters into three of the districts (top right) so that it wins three and Dark gets only two. Dark subsequently challenges this redistricting in court, on the grounds that the shapes of the districts are obviously gerrymandered, and manages to gain control of redistricting for the next election, when it uses cracking (bottom left) to ensure that it will win all five districts.
Carving up Jerimandia. Top left: Fifty regions to be cut into five districts of ten each. Voters are known to prefer the Light or Dark party according to the shading. Top right: Packing gives Light three districts and Dark only two. Bottom left: Cracking gives Dark all five districts. Bottom right: This arrangement would give proportional representation.
If the districts must be composed of ten of the small square regions, the best that Light can achieve by packing is three regions out of the five. They need to win six out of ten regions to win a district, and they control twenty regions; that gives them three sixes plus a two, which is wasted. The best that Dark can achieve by cracking is all five. Proportional representation would give Light two districts and Dark three, like the bottom right picture. (In practice, proportional representation is not achieved by districting.)
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Countries ruled by dictators, or what amount to dictators, commonly run elections to prove to the world how democratic they are. These elections are generally rigged, and even if legal challenges are permitted, they never succeed because the courts are rigged too. In other countries, it’s not only possible to challenge any particular instance of redistricting, but there’s a chance of winning, because the court’s judgement is mostly independent of the governing party. Except for appointments of judges on a partisan basis, of course.
In such cases the main problem facing the judges isn’t political. It’s to find objective ways to assess whether gerrymandering has occurred. For every ‘expert’ who eyeballs the map and declares a gerrymander, you can always find another who comes to the opposite conclusion. More objective methods than opinions and verbal arguments are needed.
This is a clear opportunity for mathematics. Formulas or algorithms can quantify whether district boundaries are reasonable and fair, or artificial and biased, in some clearly defined sense. The design of these formulas or algorithms is not of itself an objective process, of course, but once they’re agreed upon (partly a political process) everyone concerned knows what they are, and their results can be verified independently. This provides the court with a logical basis for its decision.
Having understood the underhand methods that politicians can use to implement partisan redistricting, you can invent mathematical quantities or rules to detect them. No such rule can be perfect – in fact, there’s a proof that this is impossible, which I’ll come to once we have the background to appreciate what it tells us. There are five types of approach in current use:
• Detect strangely shaped districts.
• Detect imbalances in the proportion of seats to votes.
• Quantify how many wasted votes a given division creates, and compare that to what has been legally decided to be acceptable.
• Consider all possible electoral maps, estimate the likely outcome in terms of seats based on existing voter data, and see if the proposed map is a statistical outlier.
• Set up protocols that guarantee the eventual decision is fair, is seen to be fair, and is agreed to be fair by both parties.
The fifth approach is the most surprising, and the surprise is that it can actually be done. Let’s take them in turn, saving the surprise till last.
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First, weird shapes.
As long ago as 1787 James Madison wrote in The Federalist Papers that ‘the natural limit of a democracy is that distance from the central point which will just permit the most remote citizens to assemble as often as their public functions demand’. Taken literally, he was proposing that districts should be roughly circular, and not so large that travel times from the periphery to the centre become unreasonable.