A businessman can slash his workforce in half, produce about the same as before, and be considered a big success; an economy that does the same plunges into depression, and ends up not being able to sell its goods.

So according to Krugman, it’s better for you and your spouse to earn $40,000 each than for one of you to earn $80,000 while the other stays home with the kids. I wonder how many two-earner families would agree with him.

Perhaps Krugman hasn’t noticed that the US economy, over the past century or so, has managed to cut its per capita labor input roughly in half without plunging into a 100-year-depression. There are even some people who think we’re better off these days partly because we work 35 hours a week instead of 70.

Apparently business isn’t the only profession where success is not always accompanied by wisdom.

Share/Save

A mere two days after I lavished praise on Alex Tabarrok’s new book, which (among many other things) makes an eloquent case for patent reform, the U.S. Patent Office has proved that nobody’s listening by issuing patent #8,082,523 to Apple, Incorporated for a “portable electronic device with graphical user interface supporting application switching”. The abstract, in its entirety, reads as follows:

A portable electronic device displays, on a touch screen display, a user interface for a phone application during a phone call. In response to detecting activation of a menu icon or menu button, the UI for the phone application is replaced with a menu of application icons, while maintaining the phone call. In response to detecting a finger gesture on a non-telephone service application icon, displaying a user interface for the non-telephone service application while continuing to maintain the phone call, the UI for the non-telephone service application including a switch application icon that is not displayed in the UI when there is no ongoing phone call. In response to detecting a finger gesture on the switch application icon, replacing display of the UI for the non-telephone service application with a respective UI for the phone application while continuing to maintain the phone call.

In other words, anything you’d recognize as a smartphone seems to be covered.

Would anyone care to applaud this broad assertion of intelletual property rights? Would anyone care to jump in on the other side? Have at it!

Share/Save

Welcome to the earth. Congratulations on being born in the 21st century, where the odds are excellent that you’ll live a richer, more prosperous and more fulfilling life than almost any of the 100 billion or so who preceded you — and paved the way for your prosperity with their investments and their inventions. Would that there had been more of them.

As you go through life, you will almost assuredly contribute to the world’s stock of ideas, diversity and love in ways your parents never contemplated — which is why the rest of us are a little sad that you might be their last child.

There’s certainly such a thing as a population that’s too large. Nobody disputes that. The interesting question is: Given the incentives faced by parents, it the population size we actually get too large or too small? And there are good reasons to think it’s too small.

In fact, population growth is a lot like pollution in reverse. Polluters don’t care about the damage they impose on strangers, so they pollute too much. Parents and potential parents don’t care about they joy and prosperity their chidren bring to strangers, so they reproduce too little.

When you’re older, maybe you’ll want to understand these arguments a little better. If so, you might want to read Chapter 3 of my book More Sex is Safer Sex. Or, if reading has gone out of fashion by then, you might want to watch the first video on this page. You’ll have to sit through eleven minutes of talk about sex before you get to the part about population, but I’m guessing sex will still be an interesting topic in twenty years.

Or maybe you’ll have no interest at all in this issue. But I’m guessing you’ll have some interest in something and that both friends and strangers will have reasons to be glad you did. when Baby Eight Billion comes along, the world is likely to be an even richer place, thanks partly to you.

Share/Save

There are, after all, multiple proofs that Peano Arithmetic (that is, the fragment of arithmetic described by the Peano axioms) is consistent. Among those, the simplest and most convincing (to the overwhelming majority of mathematicians) is this: The axioms of Peano Arithmetic, and therefore the theorems of Peano Arithmetic, are all true statements about the natural numbers — and a set of true statements cannot contradict itself.

Ed Nelson rejects that argument because (exempting himself from that overwhelming majority) he doesn’t believe in the set of natural numbers — or perhaps even in individual numbers when those numbers are very large. (How do you know that 8¹⁰⁰⁰⁰ exists? Have you ever counted to it?)

Needless to say, this announcement — and the announcement of a forthcoming book providing details — generated more than a flurry of excitement on the math blogs — including one of my very favorite blogs, the n-Category Cafe. After Fields Medalist Terry Tao raised a specific technical objection to Nelson’s argument, Nelson showed up in the comments section to defend himself — and then Tao showed up to expand on his objections. Nelson responded, Tao re-responded, and then Nelson posted:

You are quite right, and my original response was wrong. Thank you for spotting my error.

I withdraw my claim.

Just to be clear, here: That’s Ed Nelson cheerfully acknowledging that the book-length argument he’s been painstakingly constructing for (probably) years, and which was intended to shake the mathematical world to its foundations, doesn’t work. This says so many good things about the culture of mathematics, and so many good things about the Internet, and so many good things about the way they interact (see here and here for more examples), and it says those things so eloquently, that I see no further need for comment.

(On the other hand, if you’re hungry for additional comments, the philosopher Catarina Dutilh Novaes provides some good ones here.)

The Internet’s impact on mathematics is a huge huge thing. Not quite as huge as an inconsistency in Peano Arithmetic, but huge enough to count as a marvel.

Share/Save

If you’re interested in mathematics at the research level, MathOverflow is a place to learn something new and fascinating every single day. (If you are not doing mathematics at a research level, feel free to read but please don’t feel free to join the fray; questions at anything below about a second-year graduate level should be directed to MathStackExchange, another massively collaborative site aimed, roughly, at the college level — which reminds me that it’s not just mathematical research, but also mathematical education, that is being revolutionized before our eyes.)

It has been absolutely fascinating to watch MathOverflow and its sister sites develop. The mathematical content is awesome, but so is the sociological phenomenon — problems solved in hours instead of months via virtual brainstorming among some of the smartest and most knowledgable people in the world.

When MathOverflow first came on line, I thought it would be a superfluous addition to the many electronic resources already available. I couldn’t have been more wrong. I now suspect that like all other toddlers, it will be awesome at age five in ways that are only dimly imaginable at age two. If you like mathematics, these are very good times to live in.

Share/Save

Each generation wishes it could go back fifty years and shake some sense into those people who were so bound by unnecessary customs, and so blind to the options they could have chosen and the changes that loomed on the horizon. As I said on Tuesday, this was Edith Wharton’s theme when she wrote in 1920 about the 1870’s, and it’s the theme of Mad Men, written in 2010 about the 1960s.

I invited you on Tuesday to speculate about which of our own quirks will trigger this sort of bittersweet nostalgic frustration among our descendants fifty years from now. There were some great responses in the comments.

Here are some predictions of my own that I think are least plausible — some moreso than others, but I’ll throw them out in no particular order.

1. Future generations will look back with bemusement on a time when airline passengers couldn’t pay extra for a flight that’s guaranteed first place in the runway queue, and more generally on our odd reluctance to embrace prices. They’ll be unable to imagine why we thought it was better to let people die of liver disease than to pay organ donors, or why the “net neutrality” cult had a problem with Internet content providers being able to purchase resources to serve their customers better.
2. Future generations are likely to be appalled by the moral blindness of either their pro-life or pro-choice ancestors, though I’m not sure which. Like slavery, this issue will eventually be settled, whereupon the losing position (whichever it is) will start to seem not just wrong but unthinkable.
3. Future generations might look back tenderly on the naivete that led us to believe we (meaning, say, middle-class Americans) could go on much longer leading lives mostly untouched by violence.
4. Choose a random movie made before, say 1990, and the odds are good that all of the plot complications could have been resolved in the first five minutes if only somebody had a cellphone. Choose a random movie made today and the odds are good that all of the plot complications could be resolved in the first five minutes if only the characters were polyamorous. (Of course, then there would be a whole new set of complications.) As incomes, lifespans, and the quality of communication continue to improve, I expect that our societal fixation on monogamy will wither, and our grandchildren will look back in wonder at their ancestors’ blindness to the lifestyle options they could have chosen.
5. The moral circle will continue to expand. As we look back in horror on our ancestors’ harsh treatment of slaves or of Native Americans, our descendants will look back in horror on our treatment of immigrants and our reluctance to trade with foreigners. Slogans like “Buy American” will strike our grandchildrens’ ears the way “Buy White” would strike ours.
6. Our treatment of animals might seem almost as horrific as our treatment of foreigners.
7. Our descendants might well wonder why so few of us chose to be cryonically preserved (or to put this another way, why so many of us chose to die), and wish they could come back and shake some sense — and some life — into us.

Which of these strike you as likely, and which don’t? And do continue to add your own.

Share/Save

As Edith Wharton viewed the 1870s, and as Mad Men views the 1960s, so the fiction of the mid-to-late 21st century will probably view us. Which of our quaint but tragically foolish ways do you think it will emphasize?

Share/Save

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.

Share/Save

Here is what my iPhone does: It listens to the radio and tells me the name of the artist, song and album. It scans bar codes and tells me where to get the same item cheaper. It gives me step by step directions to anyplace I want to go. It points me to the nearest public bathroom. It recommends restaurants, based on cuisine, price, and proximity. It plays any music I want it to play, and it recommends new music based on what it’s learned about my preferences. It shows me a photograph of the entire earth and lets me slowly (or quickly) zoom in on my (or your) front porch. It takes pictures. It takes videos. It lets me edit those pictures and videos. It photographs 360 degree panoramas. It plays movies. It plays TV shows. It displays pretty much any book, newspaper or magazine I want to read. It reminds me where I parked my car. It lets me draw rough sketches of diagrams with my fingers and makes them look professional. It allows me to accept credit cards. It takes dictation. It checks the stock market or the weather with the push of a button. It reminds me of my appointments. It lets me browse the Web. It shows me my email. It locates and summons nearby taxicabs. It turns itself into a carpenter’s level. It turns itself into a flashlight. It makes phone calls. It makes video calls. And, oh yes — it has a calculator.

Now who would have been more astonished? Aristotle confronted with Martin Gardner’s calculator, or the Martin Gardner of 1975 confronted with my iPhone? I’m going to say it’s a close call.

Share/Save

First: For the past year or so, there has been a remarkable website called Math Overflow where research mathematicians gather to swap ideas, to ask for help when they get stuck, and to offer help when they can. Frequent contributors include the Fields Medalists (a Fields Medal is roughly the mathematical equivalent of a Nobel Prize) Terry Tao, Tim Gowers, Bill Thurston and Richard Borcherds. Others who have popped up from time to time include Vaughan Jones (yet another Fields Medalist), John Tate, whose thesis reshaped modern number theory, and Peter Shor, the pioneering figure in quantum computation. And every day, one runs across dozens of other folks who nearly any top math department would be proud to have (and in many cases are proud to have) on their faculties. If you already know a lot of math, you can get a hell of an advanced education browsing this site.

Second: For the past nine years or so, the Norwegian government has presented an annual Abel Prize for lifetime mathematical accomplishment. The winnners have been well chosen without exception — several are past Fields Medalists. The Abel prize differs from the Fields Medal in that you get it when you’re old (the Fields Medal must be won before age 40) and that it comes with about a million bucks (similar to the Nobel Prize). The Fields Medal has a longer and therefore more glorious history, but it’s clear that the Abel Prize is becoming similarly prestigious.

Third: This year’s Abel Prize was awarded to John Milnor for his dazzling career in topology. As part of the ceremony, Tim Gowers, the Fields Medalist and frequent Math Overflow participant, was called on to give an exposition of Milnor’s work, which you can watch here or read here. Gowers’s talk was to be given immediately following the announcement that Milnor had won, and therefore had to be prepared in great secrecy. (Milnor himself was not notified until one hour before the public announcement.)

Fourth: Grasping the essence of great work is not always easy, even for a Fields Medalist, especially when the work is as dramatic as Milnor’s. (And especially when, as in this case, the expositor and the prizewinner work in very different areas of mathematics.) Nobody I know of would want to write an exposition of someone else’s pathbreaking work without consulting a few colleagues to make sure they hadn’t missed the point. Gowers, working in top secrecy, didn’t have this luxury. But he did have Math Overflow.

So fifth: Gowers decided to post some questions to Math Overflow intended to check and expand his understanding of what Milnor accomplished and how. But he had to be very very careful that nobody figured out what he was up to. So instead of asking directly about Milnor’s work, he cleverly formulated his questions in a way that disguised the direct connection to Milnor — feigning, for example, a particular interest in four-dimensional topology, whereas Milnor’s most spectacular discoveries concern phenomena in seven dimensions and higher. The questions Gowers posed were just enough related to Milnor’s work to elicit some insight — but just enough unrelated to throw people off the scent. It worked! Read Gowers’s blog for the full story.

Share/Save