According to this report, Louis De Branges claims to have proved the Riemann Hypothesis. If correct, it’s very significant – much more so than the proof of Fermat’s Last Theorem by Wiles.

It is also, I think, the last of the big and well-known unsolved problems in mathematics, and its nice to see the search ending in success. Some of the other big problems have been closed, rather than solved. The classic problems of the Greeks such as squaring the circle were shown to be insoluble in the 19th century, and the Hilbert program of formalisation was shown by Godel to be infeasible. And the four-colour problem (not a really important problem, but a big one because it was easily described, interesting and very tough) was dealt with by a brute-force computer enumeration.

Almost instant update Commenter Eric points to Mathworld which says “Much ado about nothing”. On the other hand, the same page reports a proof of the infinitude of twin primes which has been an open question for a long time, though not a problem in the same league as those mentioned above.

• soma hernandez

Following up my three-way classification of paradoxes,[1] I want to argue that paradoxes involving infinity are always of type-3, that is, the result of ill-posed problems or inappropriate ways of taking limits. (Much the same position is defended in the comments thread by Bill Carone). In fact, I’d argue for the following general principle, applicable to all models relevant to human decisionmakers.

Whenever a result, true for all finite n, is strictly[2] reversed for the infinite case, the problem in question has been posed incorrectly

To defend this, I rely on the premise that we are finite creatures in a finite universe. If a mathematical representation of a decision problem involves an infinite set, such as the integers or the real line, it is only because this is more convenient than employing finite, but very large bounds, such as those derived from the number of particles in the universe. Any property that depends inherently on infinite sets and limits, such as the continuity of a function, can never be verified or falsified by empirical data. Since we are finite, any result that is true for all finite n is true for us.
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Suppose you have encountered Zeno’s Achilles paradoxfor the first time. Zeno offers a rigorous (looking) proof that, having once given the tortoise a head start, Achilles can never overtake it. Would you regard this as[1]

# A startling new discovery in athletics;

# A demonstration of the transcendent capacity of the human spirit – although the laws of logic forbid it, Achilles does in fact catch and overtake the tortoise; or

# A warning about how not to take limits?
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Via Juan at Philosophy617 (who doesn’t think much of the proffered solutions, and probably won’t like this one) I came back to this version of the two-envelope problem put forward by Brian over at Crooked Timber last year.

In this case, once you observe that Brian’s angel is giving you faulty theology, it’s easy to show that you should reject his[1] mathematics, and his offer. At the end of the problem, the angel says “It’s purgatory,” says the angel, “take all the time you want.” But the whole point of Purgatory is that it’s finite – you purge off your sins one at a time until they’re all paid off. Since we now have a finite problem, the solution is straightforward.
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Is Costello’s budget good or bad for Australian democracy? I’ve been critical of the exceptional reliance on lumpsum payments, which certainly looks like bribery. But this is a presentational trick Australians ought to be able to see through (if not, the adage about getting the government you deserve applies). Leaving this trick aside the Budget is clearly based on the premises that:

* what people need is money in their pockets rather than better publicly-funded health and education

* we need to redistribute income towards households with children and away from those without

* at least for households without children, tax cuts for ‘middle-income earners’ (that is, everyone with incomes over $50 000) are more important than for those with incomes less than $50 000[1].

Labor will presumably accept the second of these premises, and merely argue that it could do a better job of implementation. But that leaves plenty of room for disagreement on the first and third points. So, if Labor takes up the challenge, we might for once be faced with a real choice, something we haven’t really had for many years[2]. I’ll post on what some elements of that strategy should be in the near future.

fn1. As the scare quotes around ‘middle-income earners’ indicate, only around 20 per cent of all income-earners make this much money. And to get the full benefit of the tax cuts, you’ll need an income of at least 70 000 in 2005.

fn2. Neither Beazley in 1998 and 2001, nor Howard in 1996 nor Peacock in 1990 offered positive alternative to the government. Hewson certainly put up a radical strategy in 1993, but his radicalism consisted largely in being more Keatingesque than Keating.

In response to the exposure of widespread torture of Coalition prisoners in Iraq and elsewhere, it’s inevitable that the “ticking bomb” problem should be raised. As Harry Clarke asks in the comments to this thread

‘You hold a terrorist who knows the location of a defusable bomb which, if exploded, will kill x million people. Do you have the right to torture him/her to find the bomb?’

Instead of offering an answer to this question, I’m going to look at a question that follows immediately, but doesn’t seem to have been asked. Suppose that you have used torture to extract information from a prisoner in the belief (correct or not) that doing so was justified by a “ticking bomb” situation. What should you do next?

My answer is that you should turn yourself in, and plead guilty to the relevant criminal charges. I think this answer can be defended from a wide variety of perspectives, but I’ll take an intuitive one first. If the situation is grave enough to warrant resort to torture, it’s certainly grave enough to justify losing your job and going to jail.

In consequentialist terms, it’s desirable in general that laws against torture should be obeyed. Since few people will want to follow your example (particularly if they can’t plead a ticking bomb in mitigation) your action in such a case will undermine the law less than if you committed torture and got away with it. Other theories will, I think, give the same answer.

Turning from individual ethics to law and public policy what this means is that laws against torture should be enforced in all cases. A plea in mitigation might be considered in cases like the one described above – an urgent and immediate danger, followed by a voluntary confession. In any case where a confession is not made, no claims about mitigating circumstances should be admitted.

Since, to my knowledge, no torturer has ever made an immediate and voluntary confession, the practical impact is that the ticking bomb scenario should be disregarded in any consideration of the legal and political response to torture.

I’ve been meaning to post more on the situation on Iraq, but the complexity of the issues, and the rapid alternation of good, bad and ambiguous news makes things difficult. I’ve decided the best way to deal with things is to start with the big picture, and work down. At this stage, I don’t see any reason to change the prediction I made last November that the most plausible stable outcome in Iraq is, in fact if not in name, a two-state solution, with an Islamist Shiite majority government for Iraq as a whole, and the Kurds maintaining effective autonomy in the areas they already control. The recently-announced constitution contains various measures that are supposed to constrain this, but it’s already clear that they will be ineffectual. The Shiite leaders, most notably Sistani, have already stated that a document drawn up by an unelected council can’t constrain a democratically elected government. In any case, as long as they don’t challenge the regional autonomy of the Kurds, the Shiites will have all the votes they need to make the changes they want, particularly an enhancement of the role of Islam and the removal of requirements for power-sharing at the national level.

A second problem, pointed out by Alan at Southerly Buster is the attempt in Article 59 to maintain US military control until a permanent constitution is ratified, and a government elected pursuant to that constitution. This is the same kind of thinking that brought us the proposed regional caucuses. Faced with an elected government demanding the repudiation of this article, what are the Americans going to do? The article will fail in its intended purpose, but may cause a lot of trouble in the meantime. I predicted such an attempt, and its failure when I looked at the situation some months ago , and I see no reason to change this view now.

Despite these difficulties, and the still-real risk that the situation will collapse into civil war once it becomes clear that the historically-dominant Sunnis have been demoted to, at best, a subaltern role, I think the odds are in favour of a reasonably stable outcome.

Does this mean the war was justified? To answer this question, we need to ask two others: Compared to what? and Justified for whom?
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Howard’s sudden announcement of a proposal that the Commonwealth should take over the hospital system may be the result of desperation for a “Big Idea”, as some say. Still, since I have long advocated this idea myself (see the AFR article below), I’m happy to endorse it. The mixture of state and commonwealth funding for health is a recipe for cost-shifting and administrative duplication. If the Commonwealth took over health completely, and somehow managed to hand the GST properly to the states, it would also largely resolve the problem of vertical fiscal imbalance.

While we’ re in the business of tidying up spheres of responsibilities, I’d suggest getting the states out of the university and TAFE sectors (although they get almost no state funding, most universities operate under state acts of Parliament) and the Commonwealth out of school funding.

Update: It appears that I overstated the definiteness of this proposal. Here’s a semi-denial from Howard. In the current quasi-campaign environment, it’s increasingly hard to tell what’s policy and what’s not.
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