Well, I was pretty young in 1963, probably too young to think about such matters. I remember having little interest in the Beatles, but being being very aware that they were something very big. Everyone was aware of that. But unless I am mistaken, pretty much nobody realized that we were witnessing something really big and lasting. More generally, I doubt that anyone at the time had any inkling of the long-term significance of rock ‘n’ roll. We knew it was popular, but we had no idea it would change the world. I’m not sure that in 1963 anyone knew that it was possible for music to change the world.

This led to the more general question: How quickly are great cultural watersheds recognized for what they are? In the few areas I know something about, I think the answer is “usually pretty quickly”. I remember 1910 even less vividly than I remember 1963, but I am pretty sure that it wasn’t long between the appearance of The Love Song of J. Alfred Prufrock and the realization (at least among people who care about this sort of thing) that poetry had changed forever. In mathematics, at least in the past century (and I’m pretty sure for several centuries, or even millenia, before that), major paradigm shifts have generally been recognized very quickly. When a Serre or a Grothendieck upends the mathematical world, the mathematical world quickly knows it’s been upended.

On the other hand, it took people remarakably long to catch on to the significance of the Internet. I remember trying to tell people in 1992 that this Internet thing was going to be very big someday, and meeting a lot of blank stares. And even I, who was a very early adopter of email, Usenet, FTP and IRC, initially dismissed the World Wide Web as a passing fad.

So here’s the (extremely vague) question of the day: How often are cultural watersheds widely and quickly recognized, and what characterizes those that are and those that aren not? I’m not talking about fads here (so LOLcats don’t count); I’m talking about real lasting world-shaking changes. Feel free to interpret the question in any way you please, and have at it.

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Last week, Britain had a referendum to decide whether or not to replace its current “first past the post” electoral system with the alternative vote system (AV). During the campaign, the No to AV campaign claimed that changing to AV could cost £250 million, in part because voting machines would be introduced with it. Yes campaigner and member of parliament, Simon Hughes claimed that this was false and that the No campaigners knew it was. He asked the electoral commission to stop the No campaigners from lying.

Similar appeals are often made by other frustrated political disputants. But the idea that electioneering politicians should be allowed to say, and voters to hear, only what the electoral commission deems to be true and honestly believed is outrageous. It would make election outcomes depend on the judgement, not of the voters, but of the electoral commissioners.

The proposal is also unnecessary. As anyone who has argued with blowhards will know, there is an easy way of showing that someone does not really believe what he says. Challenge him to a wager. Demand that he put his money where his mouth is.

If the No campaigners really believe that changing to AV would cost £250 million, they will be willing to bet on it. By offering the wager, and having it declined, Mr Hughes would expose their insincerity. Equally, Mr Hughes’ failure to suggest the wager may tell us something about his own alleged certainty on the matter.

Politicians should generally be obliged to bet on the outcomes their various claims. This would discourage their lying which, incredible as it may sound, is even more widespread than people working with the standard definition of lying realise.

Lying is not a matter of saying something you do not really believe. This is because, on matters subject to doubt, we believe “both sides of the debate”. For example, I believe that Osama Bin Laden is dead. But I am not certain of it. Or, in other words, I believe, to a small degree, that Bin Laden is alive. So, if I said “Bin Laden Lives!”, though I would be lying, this would not be because I do not believe it; I do believe it a little. I would be lying because I believe it with less confidence than my assertion suggests.

Once you see lying as misrepresenting your degree of belief, it is clear that politicians lie incessantly. They pretend to a level of confidence that they cannot really feel. Forcing them to bet material sums of money on their claims would encourage them to reveal their true confidence.

For example, Tony Blair said that Saddam Hussein almost certainly possessed weapons of mass destruction (WMD) despite the available evidence making the proposition far from certain. If someone had bet him £100,000 at odds of 5:1 – surely attractive to someone of Mr Blair’s professed certainty – he would have lost £500,000 when the WMD failed to appear. Or, had he refused to wager at these odds, we would have known what to think of his certainty.

Or consider the politicians who bailed out Greek sovereign bond holders on the ground that this would prevent “contagion” and further European sovereign debt crises. This was an implausible idea. What odds do you think those who so enthusiastically peddled it would have taken on a €100,000 bet that there would be no more euro bailouts within two years? What odds would Mervin King, Governor of the Bank of England, take on a bet that inflation will be below 3% in a year? What odds would Donald Trump have taken a month ago on President Obama’s place of birth?

Of course, compulsory betting is an imperfect path to honesty. Much of the nonsense that politicians talk is not amenable to verification and so cannot be bet on. How will we know, for example, if Britain really has become “fair” under the influence of our government’s social mobility policies? And politicians may sometimes be willing to take the betting losses caused by their misrepresentations. Indeed, they could do deals that mean they do not have to. The holders of Greek government debt would surely be glad to cover the gambling losses of the European politicians who gave them taxpayers’ money.

Imperfect but not worthless. If a politician were a long-run loser in his compulsory betting, we would know that he was either a well funded liar or a fool. Knowing which may be required to make a moral judgement about him. But not to know whether we should take him seriously.

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Before I explain the problem, let’s talk a little about tic-tac-toe. As you probably figured out long ago, intelligent players of ordinary tic-tac-toe (on a 3 by 3 board) will invariably battle to a draw. But, as you probably also figured out, not every game ends in a draw, because not every player is intelligent.

On the other hand, if we blacken out the three squares on the main diagonal and don’t allow anyone to play there (so the game ends when the remaining six squares are filled, then every game is sure to end in a draw. There’s simply no way to get three in a row when you’re not allowed to play on the diagonal:

Okay. Now let’s play tic-tac-toe in three dimensions, with three ordinary boards stacked on top of each other (giving you a total of 27 places to place your X.) How many of those 27 squares would I have to blacken to insure that winning is flat-out impossible? The answer, it turns out, is 9—as long as you choose the right 9. And if we go to 4 dimensions? Now there are 81 squares, and if you want to prevent any possibility of winning, you’re going to have to blacken at least 29 of them. In 5 dimensions, you’ve got to blacken at least 93.

(See yesterday’s post if you’re puzzled about how to play tic-tac-toe in four dimensions).

The density Hales Jewett Theorem says that as you go to higher and higher dimensions, the number of squares you must black out to prevent a win gets arbitrarily close to 100% of the squares available. In some high enough dimension, you’ll have to black out at least 90% of the squares; in some higher dimension, you’ll have to black out 95%, and then 99% and 99.9999%. If you’re not sure why anybody would care about such a thing, take my word for it—there are many applications to other areas of mathematics.

(My statement of the theorem glosses over some minor technicalities; the actual theorem is slightly stronger than what I’ve quoted here.)

Now until a few months ago, the only known proof of the density Hales Jewett theorem was extremely difficult. But Tim Gowers, a Fields-Medal winning Cambridge mathematician, thought there ought to be an easier proof. So he did what everyone with an opinion about anything does nowadays; he posted his opinion on his blog. He also did what no mathematician had ever done before, and invited the entire world to collaborate with him in proving his opinion correct. Following an initial post asking “Is Massively Collaborative Mathematics Possible?”, he posted a description of the problem and invited his readers to have at it in comments.

Commenters leapt in. In response to a couple of dozen blog posts by Gowers and others, roughly spanning the calendar year 2009, commenters continued to build on each others’ ideas until they produced the (relatively) simple proof Gowers had been hoping for. Along the way, they accomplished a lot more—for example, we now know that in 5 and 6 dimensions, you’ve got to black out exactly 93 and 279 squares (again, glossing over some minor technicalities); these numbers were not known before the blogging project. For any single mathematician—or team of mathematicians—this would have been a singular accomplishment. It’s not clear it would ever have happened in a world without blogs.

Gowers believes this could be the beginning of a whole new way of doing mathematics, allowing hundreds or thousands of mathematicians to contribute to the solution of a single problem. Of course this raises all sorts of questions about rewards and incentives, many of which are addressed (but not, or course, settled) in Gowers’s “Massive Collaboration” post. Still, I have an inkling that this is a big freaking deal.

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Last year, the British government decided to lift the top rate of income tax from 41 to 52 percent. Last month, Lord Myners, the UK Secretary of State for Financial Services, said that the policy would raise not nearly as much revenue as had been expected. People are apparently making efforts to avoid paying it. A host of politicians and commentators responded that it was always a foolish idea, a purely “political” policy.

But how can a bad policy be good politics? What defect in the electoral system can explain this?

The most popular explanation these days is the malign influence of “special interests”. Perhaps there is something in this. But a more fundamental defect is always overlooked, presumably because it is mistaken for a virtue of modern democracies. The reason so many bad policies are good politics is that so many people vote: about 62 percent of adults at the last general election, both in Great Britain and in the United States. The best way to get more sensible policies would be to reduce the number of voters to less than 0.01 percent of the population.

To see why, consider a question that arises in banking. How many bankers should be involved in deciding whether to approve a loan application? The ideal number may vary with the complexity of the application. But the right answer is always, “very few.”

If a loan officer’s initial decision required sign-off by a majority of 100 other bankers, his own judgement would have little effect on the final outcome. So he would have little incentive to think hard about the application and the likelihood that the loan will be repaid. Since this would be equally true for each of the other 100 bankers, none would bother to think hard. Why struggle to make the right decision when your decision will have no effect?

This is the position of voters in a general election. Each individual’s vote makes no difference to the outcome. Even marginal districts are won with majorities of hundreds. If you had stayed home instead of voting, the same candidate would have been elected.

If each person’s vote makes no difference to the candidate elected, why do so many people vote? One answer, as the economist Geoffrey Brennan has argued, is that people enjoy it. The simple act of going to a polling booth and ticking a box is imagined to display democratic virtue. And, by ticking one box rather than others, people can feel themselves to be generous or pragmatic or progressive or something else they like to be.

Enjoying such feelings is easily worth the cost of taking two hours off work on a Tuesday every couple of years. But it is not worth the effort of learning anything about economics, jurisprudence, international relations or even the policies of the candidate you vote for. Research into voters’ knowledge shows a stunning degree of ignorance. Most voters would be as likely to vote for the best candidate if they entered the polling booth blindfolded.

In fact, blindfolds would increase most voters’ chance of making the best choice. Because, as Bryan Caplan shows in The Myth of the Rational Voter, ignorant voters do not make their mistakes randomly. They are biased towards particular errors; they tend to underestimate the benefits of trade and they believe that the prices of goods and labour are determined by corporate greed rather than by supply and demand, to take but two of many examples.

Hence the many foolish policies followed by democratic governments. And hence politicians’ sentimental and grandiose rhetoric. Modern politics is just as you should expect it to be when votes are cast by ignorant people taking advantage of a low-cost source of emotional gratification.

So what is the best way to improve modern politics? The answer is not to increase voter turnout. On the contrary, the number of voters should be drastically reduced so that each voter realizes that his vote will matter. Something like 12 voters per district should be about right. If you were one of these 12 voters then, like one of 12 jurors deciding if someone should be imprisoned, you would take a serious interest in the issues.

These 12 voters should be selected at random from the electorate. With 535 districts in Congress – 435 in the House and 100 in the Senate – there would be 6,420 voters nationally. A random selection would deliver a proportional representation of sexes, ages, races and income groups. This would improve on the current system, in which the voting population is skewed relative to the general population: the old vote more than the young, the rich vote more than the poor, and so on.

To safeguard against the possibility of abuse, these 6,420 voters would not know that they had been selected at random until the moment when the polling officers arrived at their house. They would then be spirited away to a place where they will spend a week locked away with the candidates, attending a series of speeches, debates and question-and-answer sessions before voting on the final day. All of these events should be filmed and broadcast, so that everyone could make sure that nothing dodgy was going on.

Some will complain that this system would disenfranchise most of the population. It would not, because every adult would be eligible for random selection. Of course, each of us would have a tiny chance of being selected. But, on the current system, it is equally improbable that any individual’s vote will make a difference to the election’s outcome. The difference with this “jury” system is that those whose votes make a difference would know who they are. And that would give them a reason to take the job seriously.

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Jamie Whyte is the author of Crimes Against Logic: Exposing the Bogus Arguments of Politicians, Priests, Journalists and Other Serial Offenders.

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