Courage is not a quality one normally associates with mathematicians. Yet it should apply to people who work in their attics in secret for seven years without cease on a problem that has eluded the greatest mathematical minds since first proposed in 1637.

There's something in the Western psyche, especially the Western liberal, pluralistic psyche, that finds it impossible to believe that radical ideologues mean what they say. We didn't believe communists or fascists, and in the early days we found it impossible to believe the Khomeinists. Yet these are people who do believe what they say. Attempts to moderate their behavior invariably fail; these are exercises in our heads. We need a policy of strong opposition to the fundamentalists and of strong support for those Muslims who stand up to them.

Donald Trump, the man who defied every political rule and prevailed to win his party’s nomination, last week took on perhaps the most sacred political rule of all: Never attack a Gold Star family. Not just because it alienates a vital constituency but because it reveals a shocking absence of elementary decency and of natural empathy for the most profound of human sorrows — parental grief. Why did Trump do it? It wasn’t a mistake. It was a revelation. It’s that he can’t help himself. His governing rule in life is to strike back when attacked, disrespected or even slighted. To understand Trump, you have to grasp the General Theory: He judges every action, every pronouncement, every person by a single criterion — whether or not it/he is “nice” to Trump.

In 1980, Reagan had to do just one thing: pass the threshold test for acceptability. He won that election because he did, especially in the debate with Jimmy Carter in which Reagan showed himself to be genial, self-assured and, above all, nonthreatening. You may not like all his policies, but you could safely entrust the nation to him. Trump badly needs to pass that threshold. If character is destiny, he won’t.

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I'm not a global warming believer. I'm not a global warming denier. I'm a global warming agnostic who believes instinctively that it can't be very good to pump lots of CO2 into the atmosphere but is equally convinced that those who presume to know exactly where that leads are talking through their hats.

If you’re very very rich, you can buy your Senate seat by spending as much of your money as you want. Meanwhile, your poor plebian opponent is running around groveling for the small contributions allowed by law. Hence the Corzines and the Kohls, who parachute into Congress seemingly out of nowhere. Having given this leg up to the rich, we should resist packing our legislatures with yet more privileged parachutists, the well-born. True, the Brits did it that way for centuries, but with characteristic honesty. They established a house of Parliament exclusively for high-born twits and ensconced them there for life. There they chatter away in supreme irrelevance deep into their dotage. Problem is that the U.S. Senate retains House of Commons powers even as it develops a House of Lords membership.

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Torture is an impermissible evil. Except under two circumstances. The first is the ticking time bomb. An innocent's life is at stake. The bad guy you have captured possesses information that could save this life. He refuses to divulge. In such a case, the choice is easy.

Communism always had sympathetic Westerners who amounted to a fifth column, including communist parties and intellectuals. Its appeal made it a double threat, internal and external. In the old days, communist parties were on the verge of getting parliamentary majorities in some European countries. That threat doesn't exist with Islam. The fundamentalists have nothing like that demographic or political import, even in Europe. The Muslim population in America is not much fundamentalist, nor radicalized. Rather, it accepts American religious pluralism and lives, like other religions, in a quite harmonious and pluralistic way. Of all the ideologies remaining in the world in the debris of the collapsed Soviet empire, fundamentalist Islam is the only one that, at least in our lifetime, appears to pose a serious problem to the West. It's the only expressly anti-Western ideology of any importance in the world and it means to destroy the Western position, Western institutions, Western culture, wherever it can. This occurred in Iran, and will happen again in Algeria. Should fundamentalists take power in Egypt, there will be profound geopolitical consequences. A region very important to us will be destabilized, with many problems resulting. Once that happens, we'll be asking ourselves why we weren't worrying about this years ago.

I’m not against a global pact to reduce CO2. Indeed, I favor it. But in the absence of one — and there is no chance of getting one in the foreseeable future — there is no point in America committing economic suicide to no effect on climate change, the reversing of which, after all, is the alleged point of the exercise. For a president to propose this with such aggressive certainty is incomprehensible. It is the starkest of examples of belief that is impervious to evidence. And the word for that is faith, not science.

Obama offered Muslims a careful admonition about women's rights, noting how denying women education impoverishes a country — balanced, of course, with this: "Issues of women's equality are by no means simply an issue for Islam." Example? "The struggle for women's equality continues in many aspects of American life." Well, yes. On the one hand, there certainly is some American university where the women's softball team has received insufficient Title IX funds — while, on the other hand, Saudi women showing ankle are beaten in the street, Afghan school girls have acid thrown in their faces, and Iranian women are publicly stoned to death for adultery. (Gays as well — but then again we have Prop 8.) We all have our shortcomings, our national foibles. Who's to judge?