Cohen Goes to Work

Paul Cohen is not famous. And he has nothing to do. He got his PhD from University of Chicago under Zygmund–the godfather of American analysis school, but has done nothing good in the area. He would’ve been stuck at Rochester if his high school and grad school pal Stein didn’t help him get short stints at MIT and IAS. Now his Stanford professorship is again in peril.

“I have nothing to do and I’m not famous…” Basking in California sunshine in the Quad, he had a revelation: he has to do something that makes him famous, and do it in 5 years before he’s too old for the Fields Medal.

During lunch, he casually asks his mates: “How can I get famous?”

“Shoot the President.”

“I’m not in CIA yet.”

“Break Dick Feynman’s safe.”

“He hasn’t got Nobel yet.”

“Solve any Hilbert’s Problem.”

“They haven’t been solved yet?”

That pesky Dilberto von Hilberto made a list of things he couldn’t figure out on his own, so his name will always be tagged along. Nobody cares if you solved something really important yet not in any of those “problems of the century” lists, but as soon as your hiccup sounds like “Hilbert 9” your picture is on the front page.

One day Cohen sneaks out of a snoring seminar to the library and found a list of Hilbert Problems. He flips to a page and sees No. 8, Riemann Hypothesis. “That will make me even more famous after I become famous. I’ll leave it for icing on the cake.”

A few more random page turns give him solved problems (“they are too trivial!”) and unsolvable problems (“they are too vague!”). “Is there anything not crappy?!” Going back to the Table of Contents, he sees a page that is almost blank.

Almost, except a single line:

No. 1. Continuum Hypothesis

Cantor Goes to Sleep

Georg Cantor wakes up in cold sweat all over his body. He cannot recall to which cardinality he had counted when he fell asleep.

It started a few days ago when he realized how to count infinity. In his definition, anyway. If he can establish an exactly one-to-one correspondence between the elements of two arbitrary sets, even if they have infinite number of elements, it means the two sets have the same “size”–cardinality as he calls it. In this sense, the set of all natural numbers is as large as the set of all integers, and–surprise!–the set of all rational numbers too.

He started to conjure up an infinite set with a larger cardinality, and he found one by brute force: the power set, which is the combination of all subsets of a set, including itself and the empty set. And it took him two sleepless nights to prove that power set has more elements than the original set.

He would have had a nice sleep had he stopped right there. But how could he stop? The power set has its own power set with an even larger cardinality. And it can go on and on and on, ad infinitum, just like Hugh Everett’s many worlds.

Is there an end? If there is, what is it?

That was what startled Cantor from his sleep, dreaming about larger and larger cardinality. Too scared to recall the dream, he goes back to the first power set of counting numbers. It doesn’t take too long to realize that power set is exactly the set of all real numbers, since any real number can be represented by a subset of counting numbers.

But soon a question looms: is there anything in between the set of counting numbers and real numbers? The real numbers is a continuum on the number axle in which the counting numbers take infinitely small space, so it seems likely that there’s a set whose cardinality is between the two.

“I’m too tired to solve it, so let’s call it a hypothesis for now.”

Cantor knew his mentor Weierstrasse would like these results, but suspected Kronecker the Red-Knecker would throw him into an asylum over such crazy ideas. “God made the integers; all else is the work of man”, so proclaimed Prof. Knecker, sitting firmly at the top of University of Berlin’s math department. “Knecker is right that the cardinality of an infinite set is the work of man. One Man, that is me.” Cantor’s conscious faded with a grin.

Gödel Goes to the Door

Knock, knock. “Who is it?”

Kurt Gödel asks even though he knows who is at the door. But he doesn’t actually know the person, and in this crazy world, it doesn’t hurt to be a little paranoid.

“Mr. Gödel? This is Paul Cohen from Stanford. The secretary said she had called you about my paper…” The door squeaks open a small slit.

“What do you want?”

“People told me you’re the only one who can put the final word on the Continuum Hypothesis.”

“What do you have?”

“You proved the hypothesis can’t be disproved within Zermelo–Fraenkel set theory, and I think I proved the other half: the hypothesis can’t be proved within ZF. So it’s a certain piece of uncertainty.”

Gödel grabs the envelope. “Quite forceful. Are you certain about the uncertainty?” He shuts the door without waiting for an answer.

Three days later, Mrs. Gödel answers the phone.

“Hello, Adele, this is Magritte from the Institute. A fella from Stanford sent a paper to Mr. Gödel a few days ago, and we’d like to know what he thinks of it. You see, we can’t accommodate the guy any longer if the paper is no good.”

“Hi Magritte. You haven’t seen Kurt yet? He went to the Institute this morning. I’m not sure if he likes it or not. He looked quite distressed for the last few days, and I heard him saying ‘I can’t let Albert know it’. He even forgot to ask me to taste his food once.”

Gödel hasn’t been to Princeton for years, so he stays in the only familiar common room and grabs any passer-by:

“Where is the famous Paul Cohen? He has done something that makes the rest of my life meaningless!”

Everyone Goes Away

Paul Cohen (1934 – 2007) won the 1966 Fields Medal for resolving the Continuum Hypothesis in 1963 with the forcing method that he invented. He died from a rare lung disease, after spending much of his late years in solving the Riemann Hypothesis with no success.

Kurt Gödel (1906 – 1978) created the two Incompleteness Theorems in 1931 and resolved part of the Continuum Hypothesis in 1940. He died from starvation when his wife was hospitalized for an extended period of time and could not taste his food for him.

Georg Cantor (1845 – 1918) created set theory and proposed the Continuum Hypothesis around 1880. His career was gravely compromised by strong opposition from renowned mathematicians like Kronecker and Poincaré. He was hospitalized in mental institutions many times, and died in one.

The basic story came from my colleague LJ, who is Cohen’s academic grand-nephew as LJ comes from the Elias Stein line. It can’t be all bogus, according to a 1st-person account of Cohen’s IAS trip within a super tedious autobiography page (search for Cohen). No wonder why most math is so hard: most mathematicians can’t write! Cohen’s 1-page memorial website provides some corroboration as well.

Mayor Bloomberg asked, and my answer is: in the office. There can’t be a less common answer than that, but everything else is uncommon.

ConEd, the utility company for New York City, came to our building in the morning and reset the meter, so we were asked to turn off everything, and the power went out for less than one minute. With our CEO on vacation, the rest of us chatted happily with sunlight seeping through the blinds. So when the power went off at about 4:10pm, we had no other thought than that it’s another meter reset. Our office manager went out to check with the building owner, and she came back quickly to tell us that the whole block was out of power. Soon everyone found out that no one’s cell phone can connect a call.

Of course the first thing that came across my mind was that it’s 9/11 all over again. The Empire State Building must have been hit. I ran down to the street. There were not too much more people on the street, and no one was panicking. I walked half a block and saw the tip of the Empire State Building through the forest of buildings under a bright blue sky. No smoke, no fire, no siren. It’s just a blackout.

And it’s a big one! There was a blackout in 1965 that affected 25 million people in Northeast US. In 1977, 9 million people in New York City lost power. This time it’s reported that 50 million people throughout southern Canada (Toronto and Ottawa), Michigan (Detroit), Ohio (Cleveland), Pennsylvania, Connecticut, New Jersey, and New York may have an answer to the Bloomberg Question.

We soon learned about the blackout from a battery-powered radio, the same good old one that brought us news on 9/11. Fortunately our cordless phone base station has batteries in it, and the whole phone system must have an independent power supply, so we were able to make phone calls as usual. I called Ding Qi, and learned that her building was operating on backup power and she still got Internet on her laptop. The cell phones were actually working, only that the networks were too overwhelmed.

As we learned from 9/11, we need to get out of Manhattan as quickly as possible. There’s a ferry station right beneath Ding Qi’s building, and she would take the ferry across the river to Hoboken. We four Chinese engineers at BrainMedia would go to Port Authority Bus Terminal. It was about 5pm when we started walking. The streets were jammed with no traffic light. Lots more people were on the streets, and more every minute. People were popping out of the ground through evacuation wells from stranded subway trains. I couldn’t help but have Bob Marley singing “Exodus/movement of the people” in my head.

We got to the bus terminal in less than an hour, not much longer than it usually takes. It was closed. We continued to the entrance of Lincoln Tunnel two blocks further away, wondering if we could walk to the other side. It’s also not far from the midtown ferry stations, where we could use the ferry to get across the river (later we learned from other people who went to the ferry that it’s a huge mess and the waiting was 3-4 hours). Turned out that it’s not allowed to walk in the tunnel, but we saw cars and buses moving, slowly but indeed going through the tunnel. Most drivers were kind enough to give people a ride, and we got on a New Jersey Transit bus going to northern New Jersey and upstate New York. I asked the driver to drop us off before getting onto New Jersey Turnpike, where it’s just a couple of miles from our home. When I saw the light at the New Jersey side of the tunnel, I knew it’s nothing like 9/11.

We got off the bus, slide down a few meters on a slope from the highway to a fenced area, and found ourselves on the street of Secaucus through a door secured only by a bungee cord. There happened to be a taxi company close by, and 5 minutes later we arrived at the basement of our building. It’s surreal that everything was normal in our building because of the powerful backup generators. Next month I’ll feel much more at ease when I pay the maintenance fee.

Meanwhile Ding Qi was waiting to get on a ferry with at least one thousand people. It’s grueling under the sun, but it’s orderly and she got to Hoboken in 1.5 hours. We drove to the ferry station to picked her up, and had a nice big dinner with our colleagues celebrating the easy escape.

When I turned off the light before sleep, I still forgot to remember that it’s quite a privilege to be able to turn off the light by yourself.



昨天灾后第一天进城上班,与Penn Station依旧川流的人群一起浮上地面,发现在门口等出租车的长队不见了,因为任何车辆不准靠近Penn Station的大门。每个街口都有警察,每个电话亭上都贴满了寻找亲人的绝望。经常经过的一个消防站里,死于废墟中的消防队牧师在相片里微笑着,透过天堂般的烛光。从第六大道上向南望,第一次看到了尖顶的世界金融中心一号楼(美国运通大厦),才发现其实它比周围的楼已经都高了。灰烟散尽以后,估计从帝国大厦顶上就可以直接看到自由女神像了。

今天到了公司才知道8:48时全美国静默了一分钟。那时我一定刚刚从火车上下来,在自动扶梯上闭着眼睛醒觉;等我上到地面,街道繁忙如常。Canal Street以上的纽约到底并没有什么不同,出租车依然鸣叫,Broadway两边批发店的韩国老板们依然抽着烟搬着箱子。百万城市居民加百万进城的上班族,受到直接影响的几万人最多只是一个统计百分点。纽约市长朱利阿尼早就开始呼吁人们恢复正常生活,而正常的生活和华尔街一旦重新启动,恐怕要再倒一座楼才能轰然停止。忘却的力量是残酷的,可我们只好倚仗那力量来摆脱电视里无数次地重播各个角度的撞击和坍塌,以及一家家的人们一次次的缅怀。说实话我无法想象他们如何面对着摄像机平静地讲述着他们接到的最后一次电话,但愿那也是一种忘却的手段。





大约9点,我看到一则Yahoo Alert说飞机撞到了世贸中心。我们公司的秘书接到她姐姐的电话,说电视上在直播。我还以为是小事故,但也试图给丁琦的手机打电话,总也不通。


大约9点半我和两个中国同事上街看“热闹”。我们沿着第五大道往南走,街上全是人。快到华盛顿广场的时候,一辆大卡车挡住了我们的视线。忽然周围的人开始大叫大嚷地往前跑。等卡车开走了,WTC二楼(南楼)已经只剩一股浓烟了。我心里开始发虚,但是想着要去找丁琦所以继续向downtown走。与此同时丁琦已经走开大约10个街区,听到了一声响回头正好看到WTC二楼坍塌的过程。她继续向 uptown走,幸运地打到了一辆出租车到了我们公司。

快10点时我们走到了大约Houston街与Canal街交界处,大概离WTC有不到20个街区的地方。路边有一家酒吧敞开着所有的门,里面放着电视直播。我们走得累了,我也想再打个电话试试,就进了酒吧。等电话的人有五六个,因为手机根本打不出去(我根本就把手机落在家里了)。排队的时候我看着电视,忽然一楼开始坍塌。我一心以为是在重播刚才二楼坍塌,播音员也在说“is this live??!!”(这是实况吗)。我跑到街上看,一楼已经没了,一片极浓的烟从地面迅速卷起,立刻把整个downtown 地区淹没了。我跑回酒吧里打电话,丁琦的手机还是不通。我又给公司打电话,我们秘书听到是我立刻大叫“your wife is here!!!”我跟丁琦说了几句话就往回走了。

中午和下午我们一直在公司上网看新闻。几乎所有连接曼哈顿与周围地方的桥与隧道都封锁了。我从广播里听到有些火车通了就准备走,后来一察发现只是新泽西部分的线路。我同事郭林峰的夫人在家里看电视看到有许多人从曼哈顿最北端的 George Washington桥上走到新泽西去,于是我们决定离开公司。


大约快7点知道是虚惊一场,开始通车。公共汽车走得不慢,但排在外圈的人开始等不及,只要有车经过就会上前请求搭车。许多车上就挤满了人,也有不少司机把门锁住把窗摇起绝尘而去。我们终于等上了一辆公共汽车,慢慢地开过了桥。同时步行通道也开放了,有不少人还是象以色列人出埃及似的排队走过桥。汽车过了桥就把人都放下来了。许多人换上其他的汽车走了,其他人和我们一样在陌生的Fort Lee的路上等人来接。

这时大约快8点,天已经黑了。郭林峰家离Fort Lee大约有30英里,但是与曼哈顿相连的所有高速公路都封闭了,他夫人只能从城里的道路慢慢开过来,在Fort Lee的街上找到我们已经10点多了。他们把我们送回家,大概是11点。



早上7点多有人打电话来问候。之后电话不断,我们也就起来继续看电视打电话。我的老板晚上打电话来说明天我自己决定去不去上班。新闻里报道帝国大厦和Penn Station都因假炸弹警报而被疏散。明天还是不去的好。丁琦的公司总机没有人接。

A friend of mine who’s been in America for a couple of years told me a funny experience. Once he was visiting NYC with a few fellow Chinese students. All of them were in Manhattan for the first time, so normally they felt frighteningly lost, especially in the subway. Whenever in a subway car, they always tried to find a corner and sardined themselves on the bench no matter how vacant the car was. My friend said that in those moments he felt completely identified with those “min2 gong1” in Beijing. I guess I would’ve acted the same under similar circumstances. But after 5 months of commute and a dozen visits around town, now I feel quite safe and unlostable above and under the streets of Manhattan. It’s said that every street, every park, and every subway train in New York can be dangerous at anytime except for a commuter.

I would’ve been happier if the streets that I zigzag through everyday were 20 blocks more uptown–the real Midtown where all offices, restaurants, shops, and theatres are. The regions I tread upon is actually called Chelsea and Gramercy. Yes there’s the Empire State Building to the north and the Flat Iron Building to the south, but what lie in-between are mostly, and strangely, hundreds of small wholesale stores that sell everything from stuffed animals to Chinese medicine. Many owners are Chinese or Korean. I just can’t imagine how they survive by selling Made-in-China anonymously-branded items that you may not even find in Kmart. Maybe that’s exactly New York–everyone can survive and anything can happen.

There is one place in my commute route that I really love: the Madison Square Park. It’s one of the numerous angular-shaped parks created by Broadway cutting across the otherwise perfectly checker-boarded midtown streets. People know the Madison Square Garden much better from the Knicks and Rangers, the arena used to be right beside the park then it relocated a couple times and finally settled down right above Penn Station. Every day I start from the Garden and usually pass through the Park–sometimes deliberately. It’s literally an oasis in the heart of the city, and I’ve been witnessing its reincarnation for months. Fountains are cleaned, lawns replanted, and sculptures erected. Now there’re always all kinds of people in the park, some for a break, some for a book, some for the whole summer. Dogs bark merrily in the playground, or gaze curiously at the squirrels running around and up and down. Pigeons sleep with heads tucked under wings when I walk by in the morning, and lurk on the branches in the setting sun when I return. I feel refreshed by the scent of the grass and elevated by the facade of the Flat Iron Building every time in the park. Why? Well…

“I don’t have any reasons
I’ve left them all behind.
I’m in a New York state of mind.”
(Billy Joel, New York State of Mind, from his 1976 album “Turnstiles”)

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