...

That I might drink, and leave the world unseen,
And with thee fade away into the forest dim

(Keats, Ode to a nightingale)

Fernando Pessoa, I.

Whether we write or speak or do but look
We are ever unapparent. What we are
Cannot be transfused into word or book,
Our soul from us is infinitely far.

However much we give our thoughts the will
To be our soul and gesture it abroad,
Our hearts are incommunicable still.
In what we show ourselves we are ignored.

The abyss from soul to soul cannot be bridged
By any skill of thought or trick of seeming.
Unto our very selves we are abridged

When we would utter to our thought our being.
We are our dreams of ourselves souls by gleams,
And each to each other dreams of others' dreams.

Courbet : «L'Origine du Monde»

E. Bronner, NY Times : «U.S. Withdraws Fulbright Grants to Gaza»

U.S. Withdraws Fulbright Grants to Gaza

Published: May 30, 2008

GAZA — The American State Department has withdrawn all Fulbright grants to Palestinian students in Gaza hoping to pursue advanced degrees at American institutions this fall because Israel has not granted them permission to leave.

Israel has isolated this coastal strip, which is run by the militant group Hamas. Given that policy, the United States Consulate in Jerusalem said the grant money had been “redirected” to students elsewhere out of concern that it would go to waste if the Palestinian students were forced to remain in Gaza.

A letter was sent by e-mail to the students on Thursday telling them of the cancellation. Abdulrahman Abdullah, 30, who had been hoping to study for an M.B.A. at one of several American universities on his Fulbright, was in shock when he read it.

“If we are talking about peace and mutual understanding, it means investing in people who will later contribute to Palestinian society,” he said. “I am against Hamas. Their acts and policies are wrong. Israel talks about a Palestinian state. But who will build that state if we can get no training?”

Some Israeli lawmakers, who held a hearing on the issue of student movement out of Gaza on Wednesday, expressed anger that their government was failing to promote educational and civil development in a future Palestine given the hundreds of students who had been offered grants by the United States and other Western governments.

“This could be interpreted as collective punishment,” complained Rabbi Michael Melchior, chairman of the Parliament’s education committee, during the hearing. “This policy is not in keeping with international standards or with the moral standards of Jews, who have been subjected to the deprivation of higher education in the past. Even in war, there are rules.” Rabbi Melchior is from the Meimad Party, allied with Labor.

The committee asked the government and military to reconsider the policy and get back to it within two weeks. But even if the policy is changed, the seven Fulbright grantees in Gaza are out of luck for this year. Their letters urged them to reapply next year.

Israel’s policy appears to be in flux. At the parliamentary hearing on Wednesday, a Defense Ministry official recalled that the cabinet had declared Gaza “hostile territory” and decided that the safety of Israeli soldiers and civilians at or near the border should be risked only to facilitate the movement out of Gaza for humanitarian concerns, like medical treatment. Higher education, he said, was not a humanitarian concern.

But when a query about the canceled Fulbrights was made to the prime minister’s office on Thursday, senior officials expressed surprise. They said they did, in fact, consider study abroad to be a humanitarian necessity and that when cases were appealed to them, they would facilitate them.

They suggested that American officials never brought the Fulbright cases to their attention. The State Department and American officials in Israel refused to discuss the matter. But the failure to persuade the Israelis may have stemmed from longstanding tensions between the consulate in Jerusalem, which handles Palestinian affairs, and the embassy in Tel Aviv, which manages relations with the Israeli government.

The study grants notwithstanding, the Israeli officials argued that the policy of isolating Gaza was working, that Palestinians here were starting to lose faith in Hamas’s ability to rule because of the hardships of life.

Since Hamas, a radical Islamist group that opposes Israel’s existence, carried out what amounted to a coup d’état in Gaza against the more secular Fatah party a year ago, hundreds of rockets and mortar shells have been launched from here at Israeli civilians, truck and car bombs have gone off and numerous attempts to kidnap Israeli soldiers have taken place.

While Hamas says the attacks are in response to Israeli military incursions into Gaza, it also says it will never recognize Israel.

“We are using the rockets to shake the conscience of the world about Israeli aggression,” argued Ahmed Yusef, political adviser to the Hamas foreign minister in an interview in his office here. “All our rockets are a reaction to Israeli aggression.”

The Israeli closing of Gaza has added markedly to the difficulty of daily life here, with long lines for cooking gas and a sense across the population of being under siege. Israel does send in about 70 truckloads per day of wheat, dairy products and medical equipment as well as some fuel, and it permits some medical cases out.

But Israel’s stated goal is to support moderates among the Palestinians so that Hamas will lose power, and even some security-conscious Israeli hard-liners say that the policy of barring students with grants abroad is counterproductive.

“We correctly complain that the Palestinian Authority is not building civil society, but when we don’t help build civil society this plays into the hands of Hamas,” said Natan Sharansky, a former government official. “The Fulbright is administered independently, and people are chosen for it due to their talents.”

The State Department Web site describes the Fulbright, the American government’s flagship program in international educational exchange, as “an integral part of U.S. foreign relations.” It adds, “the Fulbright Program creates a context to provide a better understanding of U.S. views and values, promotes more effective binational cooperation and nurtures open-minded, thoughtful leaders, both in the U.S. and abroad, who can work together to address common concerns.”

Sari Bashi, who directs Gisha, an Israeli organization devoted to monitoring and increasing the free movement of Palestinians, said, “The fact that the U.S. cannot even get taxpayer-funded Fulbright students out of Gaza demonstrates the injustice and short-sightedness of a closure policy that arbitrarily traps 1.5 million people, including hundreds of Palestinian students accepted to universities abroad.” She said that their education was good not just for Palestinian society, but for Israel as well.

Some Israelis disagree strongly.

“We are fighting the regime in Gaza that does its utmost to kill our citizens and destroy our schools and our colleges,” said Yuval Steinitz, a lawmaker from the opposition Likud Party. “So I don’t think we should allow students from Gaza to go anywhere. Gaza is under siege, and rightly so, and it is up to the Gazans to change the regime or its behavior.”

Hadeel Abukwaik, a 23-year-old engineering software instructor in Gaza, had hoped to do graduate work in the United States this fall on the Fulbright that she thought was hers. She had stayed in Gaza this past winter when its metal border fence was destroyed and tens of thousands of Gazans poured into Egypt, including her sister, because the agency administering the Fulbright told her she would get the grant only if she stayed put. She lives alone in Gaza where she was sent to study because the cost is low; her parents, Palestinian refugees, live in Dubai.

“I stayed to get my scholarship,” she said. “Now I am desperate.”

She, like her six colleagues, was in disbelief. Mr. Abdullah, who called the consulate in Jerusalem for further explanation after receiving his letter, said to the official on the other end, “I still cannot believe that the American administration is not able to convince the Israelis to let seven Palestinians out of Gaza.”

Israel's "Separation Wall" - separating Palestinians from their land... (http://zope.gush-shalom.org)

Boycott settlement-made products !!!
(You may find a complete list here)

L. Kaufman, NY Times : «A Superhighway to Bliss»

A Superhighway to Bliss

May 25, 2008

JILL BOLTE TAYLOR was a neuroscientist working at Harvard’s brain research center when she experienced nirvana.

Dr. Taylor says the right, creative lobe can be used to foster contentment.

But she did it by having a stroke.

On Dec. 10, 1996, Dr. Taylor, then 37, woke up in her apartment near Boston with a piercing pain behind her eye. A blood vessel in her brain had popped. Within minutes, her left lobe — the source of ego, analysis, judgment and context — began to fail her. Oddly, it felt great.

The incessant chatter that normally filled her mind disappeared. Her everyday worries — about a brother with schizophrenia and her high-powered job — untethered themselves from her and slid away.

Her perceptions changed, too. She could see that the atoms and molecules making up her body blended with the space around her; the whole world and the creatures in it were all part of the same magnificent field of shimmering energy.

“My perception of physical boundaries was no longer limited to where my skin met air,” she has written in her memoir, “My Stroke of Insight,” which was just published by Viking.

After experiencing intense pain, she said, her body disconnected from her mind. “I felt like a genie liberated from its bottle,” she wrote in her book. “The energy of my spirit seemed to flow like a great whale gliding through a sea of silent euphoria.”

While her spirit soared, her body struggled to live. She had a clot the size of a golf ball in her head, and without the use of her left hemisphere she lost basic analytical functions like her ability to speak, to understand numbers or letters, and even, at first, to recognize her mother. A friend took her to the hospital. Surgery and eight years of recovery followed.

Her desire to teach others about nirvana, Dr. Taylor said, strongly motivated her to squeeze her spirit back into her body and to get well.

This story is not typical of stroke victims. Left-brain injuries don’t necessarily lead to blissful enlightenment; people sometimes sink into a helplessly moody state: their emotions run riot. Dr. Taylor was also helped because her left hemisphere was not destroyed, and that probably explains how she was able to recover fully.

Today, she says, she is a new person, one who “can step into the consciousness of my right hemisphere” on command and be “one with all that is.”

To her it is not faith, but science. She brings a deep personal understanding to something she long studied: that the two lobes of the brain have very different personalities. Generally, the left brain gives us context, ego, time, logic. The right brain gives us creativity and empathy. For most English-speakers, the left brain, which processes language, is dominant. Dr. Taylor’s insight is that it doesn’t have to be so.

Her message, that people can choose to live a more peaceful, spiritual life by sidestepping their left brain, has resonated widely.

In February, Dr. Taylor spoke at the Technology, Entertainment, Design conference (known as TED), the annual forum for presenting innovative scientific ideas. The result was electric. After her 18-minute address was posted as a video on TED’s Web site, she become a mini-celebrity. More than two million viewers have watched her talk, and about 20,000 more a day continue to do so. An interview with her was also posted on Oprah Winfrey’s Web site, and she was chosen as one of Time magazine’s 100 most influential people in the world for 2008.

She also receives more than 100 e-mail messages a day from fans. Some are brain scientists, who are fascinated that one of their own has had a stroke and can now come back and translate the experience in terms they can use. Some are stroke victims or their caregivers who want to share their stories and thank her for her openness.

But many reaching out are spiritual seekers, particularly Buddhists and meditation practitioners, who say her experience confirms their belief that there is an attainable state of joy.

“People are so taken with it,” said Sharon Salzberg, a founder of the Insight Mediation Society in Barre, Mass. “I keep getting that video in e-mail. I must have 100 copies.”

She is excited by Dr. Taylor’s speech because it uses the language of science to describe an occurrence that is normally ethereal. Dr. Taylor shows the less mystically inclined, she said, that this experience of deep contentment “is part of the capacity of the human mind.”

Since the stroke, Dr. Taylor has moved to Bloomington, Ind., an hour from where she was raised in Terre Haute and where her mother, Gladys Gillman Taylor, who nursed her back to health, still lives.

Originally, Dr. Taylor became a brain scientist — she has a Ph.D. in life sciences with a specialty in neuroanatomy — because she has a mentally ill brother who suffers from delusions that he is in direct contact with Jesus. And for her old research lab at Harvard, she continues to speak on behalf of the mentally ill.

But otherwise, she has dialed back her once loaded work schedule. Her house is on a leafy cul-de-sac minutes from Indiana University, which she attended as an undergraduate and where she now teaches at the medical school.

Her foyer is painted a vibrant purple. She greets a stranger at the door with a warm hug. When she talks, her pale blue eyes make extended contact.

Never married, she lives with her dog and two cats. She unselfconsciously calls her mother, 82, her best friend.

She seems bemused but not at all put off by the hundreds who have reached out to her on a spiritual level. Religious ecstatics who claim to see angels have asked her to appear on their radio and television programs.

She has declined these offers. Although her father is an Episcopal minister and she was raised in his church, she cannot be counted among the traditionally faithful. “Religion is a story that the left brain tells the right brain,” she said.

Still, Dr. Taylor says, “nirvana exists right now.”

“There is no doubt that it is a beautiful state and that we can get there,” she said.

That belief has certainly sparked debate. On Web sites like evolvingbeings.com and in Eckhart Tolle discussion groups, people debate whether she is truly enlightened or just physically damaged and confused.

Even her own scientific brethren have wondered.

“When I saw her on the TED video, at first I thought, Oh my god, is she losing it,” said Dr. Francine M. Benes, director of the Harvard Brain Tissue Resource Center, where Dr. Taylor once worked.

Dr. Benes makes clear that she still thinks Dr. Taylor is an extraordinary and competent woman. “It is just that the mystical side was not apparent when she was at Harvard,” Dr. Benes said.

Dr. Taylor makes no excuses or apologies, or even explanations. She says instead that she continues to battle her left brain for the better. She gently offers tips on how it might be done.

“As the child of divorced parents and a mentally ill brother, I was angry,” she said. Now when she feels anger rising, she trumps it with a thought of a person or activity that brings her pleasure. No meditation necessary, she says, just the belief that the left brain can be tamed.

Her newfound connection to other living beings means that she is no longer interested in performing experiments on live rat brains, which she did as a researcher.

She is committed to making time for passions — physical and visual — that she believes exercise her right brain, including water-skiing, guitar playing and stained-glass making. A picture of one of her intricate stained-glass pieces — of a brain — graces the cover of her book.

Karen Armstrong, a religious historian who has written several popular books including one on the Buddha, says there are odd parallels between his story and Dr. Taylor’s.

“Like this lady, he was reluctant to return to this world,” she said. “He wanted to luxuriate in the sense of enlightenment.”

But, she said, “the dynamic of the religious required that he go out into the world and share his sense of compassion.”

And in the end, compassion is why Dr. Taylor says she wrote her memoir. She thinks there is much to be mined from her experience on how brain-trauma patients might best recover and, in fact, she hopes to open a center in Indiana to treat such patients based on those principles.

And then there is the question of world peace. No, Dr. Taylor doesn’t know how to attain that, but she does think the right hemisphere could help. Or as she told the TED conference:

“I believe that the more time we spend choosing to run the deep inner peace circuitry of our right hemispheres, the more peace we will project into the world, and the more peaceful our planet will be.”

It almost seems like science.

Egon Schiele, «Freundschaft»

«Sexo sin amor 2», Nuria Vela

D. Brooks , NY Times : «The Neural Buddhists»

The Neural Buddhists

May 13, 2008

In 1996, Tom Wolfe wrote a brilliant essay called “Sorry, but Your Soul Just Died,” in which he captured the militant materialism of some modern scientists.

To these self-confident researchers, the idea that the spirit might exist apart from the body is just ridiculous. Instead, everything arises from atoms. Genes shape temperament. Brain chemicals shape behavior. Assemblies of neurons create consciousness. Free will is an illusion. Human beings are “hard-wired” to do this or that. Religion is an accident.

In this materialist view, people perceive God’s existence because their brains have evolved to confabulate belief systems. You put a magnetic helmet around their heads and they will begin to think they are having a spiritual epiphany. If they suffer from temporal lobe epilepsy, they will show signs of hyperreligiosity, an overexcitement of the brain tissue that leads sufferers to believe they are conversing with God.

Wolfe understood the central assertion contained in this kind of thinking: Everything is material and “the soul is dead.” He anticipated the way the genetic and neuroscience revolutions would affect public debate. They would kick off another fundamental argument over whether God exists.

Lo and behold, over the past decade, a new group of assertive atheists has done battle with defenders of faith. The two sides have argued about whether it is reasonable to conceive of a soul that survives the death of the body and about whether understanding the brain explains away or merely adds to our appreciation of the entity that created it.

The atheism debate is a textbook example of how a scientific revolution can change public culture. Just as “The Origin of Species reshaped social thinking, just as Einstein’s theory of relativity affected art, so the revolution in neuroscience is having an effect on how people see the world.

And yet my guess is that the atheism debate is going to be a sideshow. The cognitive revolution is not going to end up undermining faith in God, it’s going to end up challenging faith in the Bible.

Over the past several years, the momentum has shifted away from hard-core materialism. The brain seems less like a cold machine. It does not operate like a computer. Instead, meaning, belief and consciousness seem to emerge mysteriously from idiosyncratic networks of neural firings. Those squishy things called emotions play a gigantic role in all forms of thinking. Love is vital to brain development.

Researchers now spend a lot of time trying to understand universal moral intuitions. Genes are not merely selfish, it appears. Instead, people seem to have deep instincts for fairness, empathy and attachment.

Scientists have more respect for elevated spiritual states. Andrew Newberg of the University of Pennsylvania has shown that transcendent experiences can actually be identified and measured in the brain (people experience a decrease in activity in the parietal lobe, which orients us in space). The mind seems to have the ability to transcend itself and merge with a larger presence that feels more real.

This new wave of research will not seep into the public realm in the form of militant atheism. Instead it will lead to what you might call neural Buddhism.

If you survey the literature (and I’d recommend books by Newberg, Daniel J. Siegel, Michael S. Gazzaniga, Jonathan Haidt, Antonio Damasio and Marc D. Hauser if you want to get up to speed), you can see that certain beliefs will spread into the wider discussion.

First, the self is not a fixed entity but a dynamic process of relationships. Second, underneath the patina of different religions, people around the world have common moral intuitions. Third, people are equipped to experience the sacred, to have moments of elevated experience when they transcend boundaries and overflow with love. Fourth, God can best be conceived as the nature one experiences at those moments, the unknowable total of all there is.

In their arguments with Christopher Hitchens and Richard Dawkins, the faithful have been defending the existence of God. That was the easy debate. The real challenge is going to come from people who feel the existence of the sacred, but who think that particular religions are just cultural artifacts built on top of universal human traits. It’s going to come from scientists whose beliefs overlap a bit with Buddhism.

In unexpected ways, science and mysticism are joining hands and reinforcing each other. That’s bound to lead to new movements that emphasize self-transcendence but put little stock in divine law or revelation. Orthodox believers are going to have to defend particular doctrines and particular biblical teachings. They’re going to have to defend the idea of a personal God, and explain why specific theologies are true guides for behavior day to day. I’m not qualified to take sides, believe me. I’m just trying to anticipate which way the debate is headed. We’re in the middle of a scientific revolution. It’s going to have big cultural effects.

Un frisson naît et monte de ma forme élancée

Le soir quand je suis nue et que je me contemple

Et sur ma hanche souple au fléchissement ample

Passent des voluptés en caresses glacées,

Pâle en la psyché blonde, impavide et dressée

Comme une déité superbe dans un temple,

Et t’évoquant, Vénus Solitaire, en Exemple

L’hermaphrodite amour naît en ma pensée.

De fleurs mortes au sein de coupes transparentes

Une senteur morbide émane exaspérante

Et mes sens font un pas d’ardeur inanimée.

Alors, doubles, je vois s’adorer mes prunelles

Et mes lèvres se joindre et se baiser, pâmées,

L’une à l’autre écrasant mes quadruples mamelles.

Roger de Nereys

havia um nome...

Pessoa, « Hora Absurda», 04/07/1913

HORA ABSURDA

O teu silêncio é uma nau com todas as velas pandas...
Brandas, as brisas brincam nas flâmulas, teu sorriso...
E o teu sorriso no teu silêncio é as escadas e as andas
Com que me finjo mais alto e ao pé de qualquer paraíso... Meu coração é uma ânfora que cai e que se parte...
O teu silêncio recolhe-o e guarda-o, partido, a um canto...
Minha idéia de ti é um cadáver que o mar traz à praia..., e entanto
Tu és a tela irreal em que erro em cor a minha arte... Abre todas as portas e que o vento varra a idéia
Que temos de que um fumo perfuma de ócio os salões...
Minha alma é uma caverna enchida p'la maré cheia,
E a minha idéia de te sonhar uma caravana de histriões... Chove ouro baço, mas não no lá-fora... É em mim... Sou a Hora,
E a Hora é de assombros e toda ela escombros dela...
Na minha atenção há uma viúva pobre que nunca chora...
No meu céu interior nunca houve uma única estrela... Hoje o céu é pesado como a idéia de nunca chegar a um porto...
A chuva miúda é vazia... A Hora sabe a ter sido...
Não haver qualquer cousa como leitos para as naus!... Absorto
Em se alhear de si, teu olhar é uma praga sem sentido... Todas as minhas horas são feitas de jaspe negro,
Minhas ânsias todas talhadas num mármore que não há,
Não é alegria nem dor esta dor com que me alegro,
E a minha bondade inversa não é nem boa nem má... Os feixas dos lictores abriram-se à beira dos caminhos...
Os pendões das vitórias medievais nem chegaram às cruzadas...
Puseram in-fólios úteis entre as pedras das barricadas...
E a erva cresceu nas vias férreas com viços daninhos... Ah, como esta hora é velha!... E todas as naus partiram!
Na praia só um cabo morto e uns restos de vela falam
Do Longe, das horas do Sul, de onde os nossos sonhos tiram
Aquela angústia de sonhar mais que até para si calam... O palácio está em ruínas... Dói ver no parque o abandono
da fonte sem repuxo... Ninguém ergue o olhar da estrada
E sente saudades de si ante aquele lugar-outono...
Esta paisagem é um manuscrito com a frase mais bela cortada... A doida partiu todos os candelabros glabros,
Sujou de humano o lago com cartas rasgadas, muitas...
E a minha alma é aquela luz que não mais haverá nos candelabros...
E que querem ao lago aziago minhas ânsias, brisas fortuitas?... Por que me aflijo e me enfermo?... Deitam-se nuas ao luar
Todas as ninfas... Vejo o sol e já tinham partido...
O teu silêncio que me embala é a idéia de naufragar,
E a idéia de a tua voz soar a lira dum Apolo fingido... Já não há caudas de pavões todas olhos nos jardins de outrora...
As próprias sombras estão mais tristes... Ainda
Há rastros de vestes de aias (parece) no chão, e ainda chora
Um como que eco de passos pela alameda que eis finda... Todos os casos fundiram-se na minha alma...
As relvas de todos os prados foram frescas sob meus pés frios...
Secou em teu olhar a idéia de te julgares calma,
E eu ver isso em ti é um porto sem navios... Ergueram-se a um tempo todos os remos... Pelo ouro das searas
Passou uma saudade de não serem o mar... Em frente
Ao meu trono de alheamento há gestos com pedras raras...
Minha alma é uma lâmpada que se apagou e ainda está quente... Ah, e o teu silêncio é um perfil de píncaro ao sol!
Todas as princesas sentiram o seio oprimido...
Da última janela do castelo só um girassol
Se vê, e o sonhar que há outros põe brumas no nosso sentido... Sermos, e não sermos mais!... Ó leões nascidos na jaula!...
Repique de sinos para além, no Outro Vale... Perto?...
Arde o colégio e uma criança ficou fechada na aula...
Por que não há de ser o Norte o Sul?... O que está descoberto?... E eu deliro... De repente pauso no que penso... Fito-te
E o teu silêncio é uma cegueira minha... Fito-te e sonho...
Há cousas rubras e cobras no modo como medito-te,
E a tua idéia sabe à lembrança de um sabor de medonho... Para que não ter por ti desprezo? Por que não perdê-lo?...
Ah, deixa que eu te ignore... O teu silêncio é um leque-
Um leque fechado, um leque que aberto seria tão belo, tão belo,
Mas mais belo é não o abrir, para que a Hora não peque... Gelaram todas as mãos cruzadas sobre todos os peitos...
Murcharam mais flores do que as que havia no jardim...
O meu amar-te é uma catedral de silêncios eleitos,
E os meus sonhos uma escada sem princípio mas com fim... Alguém vai entrar pela porta... Sente-se o ar sorrir...
Tecedeiras viúvas gozam as mortalhas de virgens que tecem...
Ah, oteu tédio é uma estátua de uma mulher que há de vir,
O perfume que os crisântemos teriam, se o tivessem... É preciso destruir o propósito de todas as pontes,
Vestir de alheamento as paisagens de todas as terras,
Endireitar à força a curva dos horizontes,
E gemer por ter de viver, como um ruído brusco de serras... Há tão pouca gente que ame as paisagens que não existem!...
Saber que continuará a haver o mesmo mundo amanhã - como nos desalegra !...
Que o meu ouvir o teu silêncio não seja nuvens que atristem
O teu sorriso, anjo exilado, e o teu tédio, auréola negra... Suave, como ter mãe e irmãs, a tarde rica desce...
Não chove já, e o vasto céu é um grande sorriso imperfeito...
A minha consciência de ter consciência de ti é uma prece,
E o meu saber-te a sorrir é uma flor murcha a meu peito... Ah, se fôssemos duas figuras num longínquo vitral!...
Ah, se fôssemos as duas cores de uma bandeira de glória!...
Estátua acéfala posta a um canto, poeirenta pia batismal,
Pendão de vencidos tendo escrito ao centro este lema - Vitória! O que é que me tortura?... Se até a tua face calma
Só me enche de tédios e de ópios de ócios medonhos...
Não sei... Eu sou um doido que estranha a sua própria alma...
Eu fui amado em efígie num país para além dos sonhos...

Borges, «El enamorado»

Lunas, marfiles, instrumentos, rosas,
lámparas y la línea de Durero,
las nueve cifras y el cambiante cero,
debo fingir que existen esas cosas.

Debo fingir que en el pasado fueron
Persépolis y Roma y que una arena
sutil midió la suerte de la almena
que los siglos de hierro deshicieron.

Debo fingir las armas y la pira
de la epopeya y los pesados mares
que roen de la tierra los pilares.
Debo fingir que hay otros. Es mentira.

Sólo tú eres. Tú, mi desventura
y mi ventura, inagotable y pura.

Hamlet, Boris Pasternak

Hamlet

The buzz subsides. I have come on stage.
Leaning in an open door
I try to detect from the echo
What the future has in store.

A thousand opera-glasses level
The dark, point-blank, at me.
Abba, Father, if it be possible
Let this cup pass from me.

I love your preordained design
And am ready to play this role.
But the play being acted is not mine.
For this once let me go.

But the order of the acts is planned,
The end of the road already revealed.
Alone among the Pharisees I stand.
Life is not a stroll across a field.

(Translated by Jon Stallworthy and Peter France)

Stunde des Grams, Trakl de novo

Stunde des Grams

Schwärzlich folgt im herbstlichen Garten der Schritt
Dem glänzenden Mond,
Sinkt an frierender Mauer die gewaltige Nacht.
O, die dornige Stunde des Grams.

Silbern flackert im dämmernden Zimmer der Leuchter des Einsamen,
Hinsterbend, da jener ein Dunkles denkt
Und das steinerne Haupt über Vergängliches neigt,

Trunken von Wein und nächtigem Wohllaut.
Immer folgt das Ohr
Der sanften Klage der Amsel im Haselgebüsch.

Dunkle Rosenkranzstunde. Wer bis du
Einsame Flöte,
Stirne, frierend über finstere Zeiten geneigt.

Melancholie, Georg Trakl

Melancholie

Die blaue Seele hat sich stumm verschlossen,
Ins offne Fenster sinkt der braune Wald,
Die Stille dunkler Tiere; im Grunde mahlt
Die Mühle, am Steg ruhn Wolken hingegossen,

Die goldnen Fremdlinge. Ein Zug von Rossen
Sprengt rot ins Dorf. Der Garten braun und kalt.
Die Aster friert, am Zaun so zart gemalt
Der Sonnenblume Gold schon fast zerflossen.

Der Dirnen Stimmen; Tau ist ausgegossen
Ins harte Gras und Sterne weiß und kalt.
Im teuren Schatten sieh den Tod gemalt,
Voll Tränen jedes Antlitz und verschlossen.

Rilke

Ich bin auf der Welt zu allein und doch nicht allein genug,
um jede Stunde zu weihn.
Ich bin auf der Welt zu gering und doch nicht klein genug,
um vor dir zu sein wie ein Ding,
dunkel und klug.
Ich will meinen Willen und will meinen Willen begleiten
die Wege zur Tat;
und will in stillen, irgendwie zögernden Zeiten,
wenn etwas naht,
unter den Wissenden sein
oder allein.
Ich will dich immer spiegeln in ganzer Gestalt
und will niemals blind sein oder zu alt,
um dein schweres schwankendes Bild zu halten.
Ich will mich entfalten.
Nirgends will ich gebogen bleiben,
denn dort bin ich gelogen, wo ich gebogen bin.
Und ich will meinen Sinn
wahr vor dir. Ich will mich beschreiben
wie ein Bild, das ich sah
lange und nah,
wie ein Wort, das ich begriff,
wie meinen täglichen Krug,
wie meiner Mutter Gesicht,
wie ein Schiff,
das mich trug
durch den tödlichsten Sturm.

Keats : «You say you love me»

1.

You say you love; but with a voice
Chaster than a nun's, who singeth
The soft Vespers to herself
While the chime-bell ringeth -
O love me truly!

2.
You say you love; but with a smile
Cold as sunrise in September,
As you were Saint Cupid's nun,
And kept his weeks of Ember.
O love me truly!

3.

You say you love - but then your lips
Coral tinted teach no blisses.
More than coral in the sea -
They never pout for kisses -
O love me truly!

4.

You say you love; but then your hand
No soft squeeze for squeeze returneth,
It is like a statue's dead -
While mine to passion burneth -
O love me truly!

5.

O breathe a word or two of fire!
Smile, as if those words should burn be,
Squeeze as lovers should - O kiss
And in thy heart inurn me!
O love me truly!

Thurston : On proof and progress in mathematics

ON PROOF AND PROGRESS IN MATHEMATICS
WILLIAM P. THURSTON
This essay on the nature of proof and progress in mathematics was stimulated
by the article of Jaffe and Quinn, “Theoretical Mathematics: Toward a cultural
synthesis of mathematics and theoretical physics”. Their article raises interesting
issues that mathematicians should pay more attention to, but it also perpetuates
some widely held beliefs and attitudes that need to be questioned and examined.
The article had one paragraph portraying some of my work in a way that diverges
from my experience, and it also diverges from the observations of people in the field
whom I’ve discussed it with as a reality check.
After some reflection, it seemed to me that what Jaffe and Quinn wrote was
an example of the phenomenon that people see what they are tuned to see. Their
portrayal of my work resulted from projecting the sociology of mathematics onto a
one-dimensional scale (speculation versus rigor) that ignores many basic phenomena.
Responses to the Jaffe-Quinn article have been invited from a number of mathematicians,
and I expect it to receive plenty of specific analysis and criticism from
others. Therefore, I will concentrate in this essay on the positive rather than on
the contranegative. I will describe my view of the process of mathematics, referring
only occasionally to Jaffe and Quinn by way of comparison.
In attempting to peel back layers of assumptions, it is important to try to begin
with the right questions:
1. What is it that mathematicians accomplish?
There are many issues buried in this question, which I have tried to phrase in a
way that does not presuppose the nature of the answer.
It would not be good to start, for example, with the question
How do mathematicians prove theorems?
This question introduces an interesting topic, but to start with it would be to
project two hidden assumptions:
(1) that there is uniform, objective and firmly established theory and practice
of mathematical proof, and
(2) that progress made by mathematicians consists of proving theorems.
It is worthwhile to examine these hypotheses, rather than to accept them as obvious
and proceed from there.
The question is not even
How do mathematicians make progress in mathematics?
Rather, as a more explicit (and leading) form of the question, I prefer
How do mathematicians advance human understanding of mathematics?
This question brings to the fore something that is fundamental and pervasive:
that what we are doing is finding ways for people to understand and think about
mathematics.
The rapid advance of computers has helped dramatize this point, because computers
and people are very different. For instance, when Appel and Haken completed
a proof of the 4-color map theorem using a massive automatic computation,
it evoked much controversy. I interpret the controversy as having little to do with
doubt people had as to the veracity of the theorem or the correctness of the proof.
Rather, it reflected a continuing desire for human understanding of a proof, in
addition to knowledge that the theorem is true.
On a more everyday level, it is common for people first starting to grapple with
computers to make large-scale computations of things they might have done on a
smaller scale by hand. They might print out a table of the first 10,000 primes,
only to find that their printout isn’t something they really wanted after all. They
discover by this kind of experience that what they really want is usually not some
collection of “answers”—what they want is understanding.
It may sound almost circular to say that what mathematicians are accomplishing
is to advance human understanding of mathematics. I will not try to resolve this by
discussing what mathematics is, because it would take us far afield. Mathematicians
generally feel that they know what mathematics is, but find it difficult to give a good
direct definition. It is interesting to try. For me, “the theory of formal patterns”
has come the closest, but to discuss this would be a whole essay in itself.
Could the difficulty in giving a good direct definition of mathematics be an
essential one, indicating that mathematics has an essential recursive quality? Along
these lines we might say that mathematics is the smallest subject satisfying the
following:
• Mathematics includes the natural numbers and plane and solid geometry.
• Mathematics is that which mathematicians study.
• Mathematicians are those humans who advance human understanding of
mathematics.
In other words, as mathematics advances, we incorporate it into our thinking. As
our thinking becomes more sophisticated, we generate new mathematical concepts
and new mathematical structures: the subject matter of mathematics changes to
reflect how we think.
If what we are doing is constructing better ways of thinking, then psychological
and social dimensions are essential to a good model for mathematical progress.
These dimensions are absent from the popular model. In caricature, the popular
model holds that
D. mathematicians start from a few basic mathematical structures and a collection
of axioms “given” about these structures, that
T. there are various important questions to be answered about these structures
that can be stated as formal mathematical propositions, and
P. the task of the mathematician is to seek a deductive pathway from the
axioms to the propositions or to their denials.
We might call this the definition-theorem-proof (DTP) model of mathematics.
A clear difficulty with the DTP model is that it doesn’t explain the source of
the questions. Jaffe and Quinn discuss speculation (which they inappropriately label
“theoretical mathematics”) as an important additional ingredient. Speculation
consists of making conjectures, raising questions, and making intelligent guesses
and heuristic arguments about what is probably true.
Jaffe and Quinn’s DSTP model still fails to address some basic issues. We are not
trying to meet some abstract production quota of definitions, theorems and proofs.
The measure of our success is whether what we do enables people to understand
and think more clearly and effectively about mathematics.
Therefore, we need to ask ourselves:
2. How do people understand mathematics?
This is a very hard question. Understanding is an individual and internal matter
that is hard to be fully aware of, hard to understand and often hard to communicate.
We can only touch on it lightly here.
People have very different ways of understanding particular pieces of mathematics.
To illustrate this, it is best to take an example that practicing mathematicians
understand in multiple ways, but that we see our students struggling with. The
derivative of a function fits well. The derivative can be thought of as:
(1) Infinitesimal: the ratio of the infinitesimal change in the value of a function
to the infinitesimal change in a function.
(2) Symbolic: the derivative of xn is nxn−1, the derivative of sin(x) is cos(x),
the derivative of f ◦ g is f′ ◦ g ∗ g′, etc.
(3) Logical: f′(x) = d if and only if for every ǫ there is a δ such that when
0 < |
x| < δ,

f(x +
x) − f(x)

x
− d

< δ.
(4) Geometric: the derivative is the slope of a line tangent to the graph of the
function, if the graph has a tangent.
(5) Rate: the instantaneous speed of f(t), when t is time.
(6) Approximation: The derivative of a function is the best linear approximation
to the function near a point.
(7) Microscopic: The derivative of a function is the limit of what you get by
looking at it under a microscope of higher and higher power.
This is a list of different ways of thinking about or conceiving of the derivative,
rather than a list of different logical definitions. Unless great efforts are made to
maintain the tone and flavor of the original human insights, the differences start
to evaporate as soon as the mental concepts are translated into precise, formal and
explicit definitions.
I can remember absorbing each of these concepts as something new and interesting,
and spending a good deal of mental time and effort digesting and practicing
with each, reconciling it with the others. I also remember coming back to revisit
these different concepts later with added meaning and understanding.
The list continues; there is no reason for it ever to stop. A sample entry further
down the list may help illustrate this. We may think we know all there is to say
about a certain subject, but new insights are around the corner. Furthermore, one
person’s clear mental image is another person’s intimidation:
37. The derivative of a real-valued function f in a domain D is the Lagrangian
section of the cotangent bundle T ∗(D) that gives the connection form for
the unique flat connection on the trivial R-bundle D × R for which the
graph of f is parallel.
These differences are not just a curiosity. Human thinking and understanding
do not work on a single track, like a computer with a single central processing unit.
Our brains and minds seem to be organized into a variety of separate, powerful
facilities. These facilities work together loosely, “talking” to each other at high
levels rather than at low levels of organization.
Here are some major divisions that are important for mathematical thinking:
(1) Human language. We have powerful special-purpose facilities for speaking
and understanding human language, which also tie in to reading and writing.
Our linguistic facility is an important tool for thinking, not just for
communication. A crude example is the quadratic formula which people
may remember as a little chant, “ex equals minus bee plus or minus the
square root of bee squared minus four ay see all over two ay.” The mathematical
language of symbols is closely tied to our human language facility.
The fragment of mathematical symbolese available to most calculus students
has only one verb, “=”. That’s why students use it when they’re in
need of a verb. Almost anyone who has taught calculus in the U.S. has
seen students instinctively write “x3 = 3x2” and the like.
(2) Vision, spatial sense, kinesthetic (motion) sense. People have very powerful
facilities for taking in information visually or kinesthetically, and thinking
with their spatial sense. On the other hand, they do not have a very
good built-in facility for inverse vision, that is, turning an internal spatial
understanding back into a two-dimensional image. Consequently, mathematicians
usually have fewer and poorer figures in their papers and books
than in their heads.
An interesting phenomenon in spatial thinking is that scale makes a big
difference. We can think about little objects in our hands, or we can think
of bigger human-sized structures that we scan, or we can think of spatial
structures that encompass us and that we move around in. We tend to
think more effectively with spatial imagery on a larger scale: it’s as if our
brains take larger things more seriously and can devote more resources to
them.
(3) Logic and deduction. We have some built-in ways of reasoning and putting
things together associated with how we make logical deductions: cause and
effect (related to implication), contradiction or negation, etc.
Mathematicians apparently don’t generally rely on the formal rules of deduction
as they are thinking. Rather, they hold a fair bit of logical structure
of a proof in their heads, breaking proofs into intermediate results so that
they don’t have to hold too much logic at once. In fact, it is common
for excellent mathematicians not even to know the standard formal usage
of quantifiers (for all and there exists), yet all mathematicians certainly
perform the reasoning that they encode.
It’s interesting that although “or”, “and” and “implies” have identical formal
usage, we think of “or” and “and” as conjunctions and “implies” as a
verb.
(4) Intuition, association, metaphor. People have amazing facilities for sensing
something without knowing where it comes from (intuition); for sensing
that some phenomenon or situation or object is like something else (association);
and for building and testing connections and comparisons, holding
two things in mind at the same time (metaphor). These facilities are quite
important for mathematics. Personally, I put a lot of effort into “listening”
to my intuitions and associations, and building them into metaphors and
connections. This involves a kind of simultaneous quieting and focusing of
my mind. Words, logic, and detailed pictures rattling around can inhibit
intuitions and associations.
(5) Stimulus-response. This is often emphasized in schools; for instance, if
you see 3927 x 253, you write one number above the other and draw a
line underneath, etc. This is also important for research mathematics:
seeing a diagram of a knot, I might write down a presentation for the
fundamental group of its complement by a procedure that is similar in feel
to the multiplication algorithm.
(6) Process and time. We have a facility for thinking about processes or sequences
of actions that can often be used to good effect in mathematical
reasoning. One way to think of a function is as an action, a process, that
takes the domain to the range. This is particularly valuable when composing
functions. Another use of this facility is in remembering proofs:
people often remember a proof as a process consisting of several steps. In
topology, the notion of a homotopy is most often thought of as a process
taking time. Mathematically, time is no different from one more spatial
dimension, but since humans interact with it in a quite different way, it is
psychologically very different.
3. How is mathematical understanding communicated?
The transfer of understanding from one person to another is not automatic. It
is hard and tricky. Therefore, to analyze human understanding of mathematics, it
is important to consider who understands what, and when.
Mathematicians have developed habits of communication that are often dysfunctional.
Organizers of colloquium talks everywhere exhort speakers to explain things
in elementary terms. Nonetheless, most of the audience at an average colloquium
talk gets little of value from it. Perhaps they are lost within the first 5 minutes, yet
sit silently through the remaining 55 minutes. Or perhaps they quickly lose interest
because the speaker plunges into technical details without presenting any reason to
investigate them. At the end of the talk, the few mathematicians who are close to
the field of the speaker ask a question or two to avoid embarrassment.
This pattern is similar to what often holds in classrooms, where we go through the
motions of saying for the record what we think the students “ought” to learn, while
the students are trying to grapple with the more fundamental issues of learning
our language and guessing at our mental models. Books compensate by giving
samples of how to solve every type of homework problem. Professors compensate
by giving homework and tests that are much easier than the material “covered” in
the course, and then grading the homework and tests on a scale that requires little
understanding. We assume that the problem is with the students rather than with
communication: that the students either just don’t have what it takes, or else just
don’t care.
Outsiders are amazed at this phenomenon, but within the mathematical community,
we dismiss it with shrugs.
Much of the difficulty has to do with the language and culture of mathematics,
which is divided into subfields. Basic concepts used every day within one subfield are
often foreign to another subfield. Mathematicians give up on trying to understand
the basic concepts even from neighboring subfields, unless they were clued in as
graduate students.
In contrast, communication works very well within the subfields of mathematics.
Within a subfield, people develop a body of common knowledge and known
techniques. By informal contact, people learn to understand and copy each other’s
ways of thinking, so that ideas can be explained clearly and easily.
Mathematical knowledge can be transmitted amazingly fast within a subfield.
When a significant theorem is proved, it often (but not always) happens that the
solution can be communicated in a matter of minutes from one person to another
within the subfield. The same proof would be communicated and generally understood
in an hour talk to members of the subfield. It would be the subject of a 15-
or 20-page paper, which could be read and understood in a few hours or perhaps
days by members of the subfield.
Why is there such a big expansion from the informal discussion to the talk
to the paper? One-on-one, people use wide channels of communication that go
far beyond formal mathematical language. They use gestures, they draw pictures
and diagrams, they make sound effects and use body language. Communication
is more likely to be two-way, so that people can concentrate on what needs the
most attention. With these channels of communication, they are in a much better
position to convey what’s going on, not just in their logical and linguistic facilities,
but in their other mental facilities as well.
In talks, people are more inhibited and more formal. Mathematical audiences are
often not very good at asking the questions that are on most people’s minds, and
speakers often have an unrealistic preset outline that inhibits them from addressing
questions even when they are asked.
In papers, people are still more formal. Writers translate their ideas into symbols
and logic, and readers try to translate back.
Why is there such a discrepancy between communication within a subfield and
communication outside of subfields, not to mention communication outside mathematics?
Mathematics in some sense has a common language: a language of symbols, technical
definitions, computations, and logic. This language efficiently conveys some,
but not all, modes of mathematical thinking. Mathematicians learn to translate
certain things almost unconsciously from one mental mode to the other, so that
some statements quickly become clear. Different mathematicians study papers in
different ways, but when I read a mathematical paper in a field in which I’m conversant,
I concentrate on the thoughts that are between the lines. I might look over
several paragraphs or strings of equations and think to myself “Oh yeah, they’re
putting in enough rigamarole to carry such-and-such idea.” When the idea is clear,
the formal setup is usually unnecessary and redundant—I often feel that I could
write it out myself more easily than figuring out what the authors actually wrote.
It’s like a new toaster that comes with a 16-page manual. If you already understand
toasters and if the toaster looks like previous toasters you’ve encountered,
you might just plug it in and see if it works, rather than first reading all the details
in the manual.
People familiar with ways of doing things in a subfield recognize various patterns
of statements or formulas as idioms or circumlocution for certain concepts or mental
images. But to people not already familiar with what’s going on the same patterns
are not very illuminating; they are often even misleading. The language is not alive
except to those who use it.
I’d like to make an important remark here: there are some mathematicians who
are conversant with the ways of thinking in more than one subfield, sometimes
in quite a number of subfields. Some mathematicians learn the jargon of several
subfields as graduate students, some people are just quick at picking up foreign
mathematical language and culture, and some people are in mathematical centers
where they are exposed to many subfields. People who are comfortable in more than
one subfield can often have a very positive influence, serving as bridges, and helping
different groups of mathematicians learn from each other. But people knowledgeable
in multiple fields can also have a negative effect, by intimidating others, and
by helping to validate and maintain the whole system of generally poor communication.
For example, one effect often takes place during colloquium talks, where
one or two widely knowledgeable people sitting in the front row may serve as the
speaker’s mental guide to the audience.
There is another effect caused by the big differences between how we think
about mathematics and how we write it. A group of mathematicians interacting
with each other can keep a collection of mathematical ideas alive for a period of
years, even though the recorded version of their mathematical work differs from
their actual thinking, having much greater emphasis on language, symbols, logic
and formalism. But as new batches of mathematicians learn about the subject they
tend to interpret what they read and hear more literally, so that the more easily
recorded and communicated formalism and machinery tend to gradually take over
from other modes of thinking.
There are two counters to this trend, so that mathematics does not become
entirely mired down in formalism. First, younger generations of mathematicians
are continually discovering and rediscovering insights on their own, thus reinjecting
diverse modes of human thought into mathematics.
Second, mathematicians sometimes invent names and hit on unifying definitions
that replace technical circumlocutions and give good handles for insights. Names
like “group” to replace “a system of substitutions satisfying . . . ”, and “manifold”
to replace
We can’t give coordinates to parametrize all the solutions to our
equations simultaneously, but in the neighborhood of any particular
solution we can introduce coordinates
(f1(u1, u2, u3), f2(u1, u2, u3), f3(u1, u2, u3), f4(u1, u2, u3),
f5(u1, u2, u3))
where at least one of the ten determinants
. . .[ten 3 x 3 determinants of matrices of partial derivatives]. . .
is not zero
may or may not have represented advances in insight among experts, but they
greatly facilitate the communication of insights.
We mathematicians need to put far greater effort into communicating mathematical
ideas. To accomplish this, we need to pay much more attention to communicating
not just our definitions, theorems, and proofs, but also our ways of
thinking. We need to appreciate the value of different ways of thinking about the
same mathematical structure.
We need to focus far more energy on understanding and explaining the basic
mental infrastructure of mathematics—with consequently less energy on the most
recent results. This entails developing mathematical language that is effective for
the radical purpose of conveying ideas to people who don’t already know them.
Part of this communication is through proofs.
4. What is a proof?
When I started as a graduate student at Berkeley, I had trouble imagining how
I could “prove” a new and interesting mathematical theorem. I didn’t really understand
what a “proof” was.
By going to seminars, reading papers, and talking to other graduate students,
I gradually began to catch on. Within any field, there are certain theorems and
certain techniques that are generally known and generally accepted. When you
write a paper, you refer to these without proof. You look at other papers in the
field, and you see what facts they quote without proof, and what they cite in their
bibliography. You learn from other people some idea of the proofs. Then you’re
free to quote the same theorem and cite the same citations. You don’t necessarily
have to read the full papers or books that are in your bibliography. Many of the
things that are generally known are things for which there may be no known written
source. As long as people in the field are comfortable that the idea works, it doesn’t
need to have a formal written source.
At first I was highly suspicious of this process. I would doubt whether a certain
idea was really established. But I found that I could ask people, and they could
produce explanations and proofs, or else refer me to other people or to written
sources that would give explanations and proofs. There were published theorems
that were generally known to be false, or where the proofs were generally known
to be incomplete. Mathematical knowledge and understanding were embedded in
the minds and in the social fabric of the community of people thinking about a
particular topic. This knowledge was supported by written documents, but the
written documents were not really primary.
I think this pattern varies quite a bit from field to field. I was interested in
geometric areas of mathematics, where it is often pretty hard to have a document
that reflects well the way people actually think. In more algebraic or symbolic
fields, this is not necessarily so, and I have the impression that in some areas
documents are much closer to carrying the life of the field. But in any field, there
is a strong social standard of validity and truth. Andrew Wiles’s proof of Fermat’s
Last Theorem is a good illustration of this, in a field which is very algebraic. The
experts quickly came to believe that his proof was basically correct on the basis
of high-level ideas, long before details could be checked. This proof will receive a
great deal of scrutiny and checking compared to most mathematical proofs; but no
matter how the process of verification plays out, it helps illustrate how mathematics
evolves by rather organic psychological and social processes.
When people are doing mathematics, the flow of ideas and the social standard of
validity is much more reliable than formal documents. People are usually not very
good in checking formal correctness of proofs, but they are quite good at detecting
potential weaknesses or flaws in proofs.
To avoid misinterpretation, I’d like to emphasize two things I am not saying.
First, I am not advocating any weakening of our community standard of proof; I
am trying to describe how the process really works. Careful proofs that will stand
up to scrutiny are very important. I think the process of proof on the whole works
pretty well in the mathematical community. The kind of change I would advocate
is that mathematicians take more care with their proofs, making them really clear
and as simple as possible so that if any weakness is present it will be easy to
detect. Second, I am not criticizing the mathematical study of formal proofs, nor
am I criticizing people who put energy into making mathematical arguments more
explicit and more formal. These are both useful activities that shed new insights
on mathematics.
I have spent a fair amount of effort during periods of my career exploring mathematical
questions by computer. In view of that experience, I was astonished to see
the statement of Jaffe and Quinn that mathematics is extremely slow and arduous,
and that it is arguably the most disciplined of all human activities. The standard
of correctness and completeness necessary to get a computer program to work at
all is a couple of orders of magnitude higher than the mathematical community’s
standard of valid proofs. Nonetheless, large computer programs, even when they
have been very carefully written and very carefully tested, always seem to have
bugs.
I think that mathematics is one of the most intellectually gratifying of human
activities. Because we have a high standard for clear and convincing thinking and
because we place a high value on listening to and trying to understand each other,
we don’t engage in interminable arguments and endless redoing of our mathematics.
We are prepared to be convinced by others. Intellectually, mathematics moves very
quickly. Entire mathematical landscapes change and change again in amazing ways
during a single career.
When one considers how hard it is to write a computer programeven approaching
the intellectual scope of a good mathematical paper, and how much greater time and
effort have to be put into it to make it “almost” formally correct, it is preposterous
to claim that mathematics as we practice it is anywhere near formally correct.
Mathematics as we practice it is much more formally complete and precise than
other sciences, but it is much less formally complete and precise for its content
than computer programs. The difference has to do not just with the amount of
effort: the kind of effort is qualitatively different. In large computer programs,
a tremendous proportion of effort must be spent on myriad compatibility issues:
making sure that all definitions are consistent, developing “good” data structures
that have useful but not cumbersome generality, deciding on the “right” generality
for functions, etc. The proportion of energy spent on the working part of a large
program, as distinguished from the bookkeeping part, is surprisingly small. Because
of compatibility issues that almost inevitably escalate out of hand because
the “right” definitions change as generality and functionality are added, computer
programs usually need to be rewritten frequently, often from scratch.
A very similar kind of effort would have to go into mathematics to make it
formally correct and complete. It is not that formal correctness is prohibitively
difficult on a small scale—it’s that there are many possible choices of formalization
on small scales that translate to huge numbers of interdependent choices in the large.
It is quite hard to make these choices compatible; to do so would certainly entail
going back and rewriting from scratch all old mathematical papers whose results we
depend on. It is also quite hard to come up with good technical choices for formal
definitions that will be valid in the variety of ways that mathematicians want to
use them and that will anticipate future extensions of mathematics. If we were to
continue to cooperate, much of our time would be spent with international standards
commissions to establish uniform definitions and resolve huge controversies.
Mathematicians can and do fill in gaps, correct errors, and supply more detail
and more careful scholarship when they are called on or motivated to do so. Our
system is quite good at producing reliable theorems that can be solidly backed up.
It’s just that the reliability does not primarily come from mathematicians formally
checking formal arguments; it comes from mathematicians thinking carefully and
critically about mathematical ideas.
On the most fundamental level, the foundations of mathematics are much shakier
than the mathematics that we do. Most mathematicians adhere to foundational
principles that are known to be polite fictions. For example, it is a theorem that
there does not exist any way to ever actually construct or even define a well-ordering
of the real numbers. There is considerable evidence (but no proof) that we can get
away with these polite fictions without being caught out, but that doesn’t make
them right. Set theorists construct many alternate and mutually contradictory
“mathematical universes” such that if one is consistent, the others are too. This
leaves very little confidence that one or the other is the right choice or the natural
choice. Goedel’s incompleteness theorem implies that there can be no formal system
that is consistent, yet powerful enough to serve as a basis for all of the mathematics
that we do.
In contrast to humans, computers are good at performing formal processes.
There are people working hard on the project of actually formalizing parts of mathematics
by computer, with actual formally correct formal deductions. I think this
is a very big but very worthwhile project, and I am confident that we will learn
a lot from it. The process will help simplify and clarify mathematics. In not too
many years, I expect that we will have interactive computer programs that can
help people compile significant chunks of formally complete and correct mathematics
(based on a few perhaps shaky but at least explicit assumptions), and that they
will become part of the standard mathematician’s working environment.
However, we should recognize that the humanly understandable and humanly
checkable proofs that we actually do are what is most important to us, and that
they are quite different from formal proofs. For the present, formal proofs are
out of reach and mostly irrelevant: we have good human processes for checking
mathematical validity.
5. What motivates people to do mathematics?
There is a real joy in doing mathematics, in learning ways of thinking that explain
and organize and simplify. One can feel this joy discovering new mathematics,
rediscovering old mathematics, learning a way of thinking from a person or text,
or finding a new way to explain or to view an old mathematical structure.
This inner motivation might lead us to think that we do mathematics solely
for its own sake. That’s not true: the social setting is extremely important. We
are inspired by other people, we seek appreciation by other people, and we like to
help other people solve their mathematical problems. What we enjoy changes in
response to other people. Social interaction occurs through face-to-face meetings.
It also occurs through written and electronic correspondence, preprints, and journal
articles. One effect of this highly social system of mathematics is the tendency
of mathematicians to follow fads. For the purpose of producing new mathematical
theorems this is probably not very efficient: we’d seem to be better off having mathematicians
cover the intellectual field much more evenly. But most mathematicians
don’t like to be lonely, and they have trouble staying excited about a subject, even
if they are personally making progress, unless they have colleagues who share their
excitement.
In addition to our inner motivation and our informal social motivation for doing
mathematics, we are driven by considerations of economics and status. Mathematicians,
like other academics, do a lot of judging and being judged. Starting with
grades, and continuing through letters of recommendation, hiring decisions, promotion
decisions, referees reports, invitations to speak, prizes, . . . we are involved
in many ratings, in a fiercely competitive system.
Jaffe and Quinn analyze the motivation to do mathematics in terms of a common
currency that many mathematicians believe in: credit for theorems.
I think that our strong communal emphasis on theorem-credits has a negative
effect on mathematical progress. If what we are accomplishing is advancing human
understanding of mathematics, then we would be much better off recognizing and
valuing a far broader range of activity. The people who see the way to proving theorems
are doing it in the context of a mathematical community; they are not doing
it on their own. They depend on understanding of mathematics that they glean
from other mathematicians. Once a theorem has been proven, the mathematical
community depends on the social network to distribute the ideas to people who
might use them further—the print medium is far too obscure and cumbersome.
Even if one takes the narrow view that what we are producing is theorems, the
team is important. Soccer can serve as a metaphor. There might only be one or two
goals during a soccer game, made by one or two persons. That does not mean that
the efforts of all the others are wasted. We do not judge players on a soccer team
only by whether they personally make a goal; we judge the team by its function as
a team.
In mathematics, it often happens that a group of mathematicians advances with
a certain collection of ideas. There are theorems in the path of these advances that
will almost inevitably be proven by one person or another. Sometimes the group
of mathematicians can even anticipate what these theorems are likely to be. It is
much harder to predict who will actually prove the theorem, although there are
usually a few “point people” who are more likely to score. However, they are in
a position to prove those theorems because of the collective efforts of the team.
The team has a further function, in absorbing and making use of the theorems
once they are proven. Even if one person could prove all the theorems in the path
single-handedly, they are wasted if nobody else learns them.
There is an interesting phenomenon concerning the “point” people. It regularly
happens that someone who was in the middle of a pack proves a theorem that
receives wide recognition as being significant. Their status in the community—
their pecking order—rises immediately and dramatically. When this happens, they
usually become much more productive as a center of ideas and a source of theorems.
Why? First, there is a large increase in self-esteem, and an accompanying increase
in productivity. Second, when their status increases, people are more in the center
of the network of ideas—others take them more seriously. Finally and perhaps
most importantly, a mathematical breakthrough usually represents a new way of
thinking, and effective ways of thinking can usually be applied in more than one
situation.
This phenomenon convinces me that the entire mathematical community would
become much more productive if we open our eyes to the real values in what we are
doing. Jaffe and Quinn propose a system of recognized roles divided into “speculation”
and “proving”. Such a division only perpetuates the myth that our progress
is measured in units of standard theorems deduced. This is a bit like the fallacy of
the person who makes a printout of the first 10,000 primes. What we are producing
is human understanding. We have many different ways to understand and many
different processes that contribute to our understanding. We will be more satisfied,
more productive and happier if we recognize and focus on this.
6. Some personal experiences
Since this essay grew out of reflection on the misfit between my experiences
and the description of Jaffe and Quinn’s, I will discuss two personal experiences,
including the one they alluded to.
I feel some awkwardness in this, because I do have regrets about aspects of my
career: if I were to do things over again with the benefit of my present insights
about myself and about the process of mathematics, there is a lot that I would
hope to do differently. I hope that by describing these experiences rather openly as
I remember and understand them, I can help others understand the process better
and learn in advance.
First I will discuss briefly the theory of foliations, which was my first subject,
starting when I was a graduate student. (It doesn’t matter here whether you know
what foliations are.)
At that time, foliations had become a big center of attention among geometric
topologists, dynamical systems people, and differential geometers. I fairly rapidly
proved some dramatic theorems. I proved a classification theorem for foliations,
giving a necessary and sufficient condition for a manifold to admit a foliation. I
proved a number of other significant theorems. I wrote respectable papers and
published at least the most important theorems. It was hard to find the time to
write to keep up with what I could prove, and I built up a backlog.
An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation
of the field started to take place. I heard from a number of mathematicians
that they were giving or receiving advice not to go into foliations—they were saying
that Thurston was cleaning it out. People told me (not as a complaint, but
as a compliment) that I was killing the field. Graduate students stopped studying
foliations, and fairly soon, I turned to other interests as well.
I do not think that the evacuation occurred because the territory was intellectually
exhausted—there were (and still are) many interesting questions that remain
and that are probably approachable. Since those years, there have been interesting
developments carried out by the few people who stayed in the field or who entered
the field, and there have also been important developments in neighboring areas
that I think would have been much accelerated had mathematicians continued to
pursue foliation theory vigorously.
Today, I think there are few mathematicians who understand anything approaching
the state of the art of foliations as it lived at that time, although there are some
parts of the theory of foliations, including developments since that time, that are
still thriving.
I believe that two ecological effects were much more important in putting a
damper on the subject than any exhaustion of intellectual resources that occurred.
First, the results I proved (as well as some important results of other people)
were documented in a conventional, formidable mathematician’s style. They depended
heavily on readers who shared certain background and certain insights.
The theory of foliations was a young, opportunistic subfield, and the background
was not standardized. I did not hesitate to draw on any of the mathematics I
had learned from others. The papers I wrote did not (and could not) spend much
time explaining the background culture. They documented top-level reasoning and
conclusions that I often had achieved after much reflection and effort. I also threw
out prize cryptic tidbits of insight, such as “the Godbillon-Vey invariant measures
the helical wobble of a foliation”, that remained mysterious to most mathematicans
who read them. This created a high entry barrier: I think many graduate students
and mathematicians were discouraged that it was hard to learn and understand the
proofs of key theorems.
Second is the issue of what is in it for other people in the subfield. When I
started working on foliations, I had the conception that what people wanted was
to know the answers. I thought that what they sought was a collection of powerful
proven theorems that might be applied to answer further mathematical questions.
But that’s only one part of the story. More than the knowledge, people want
personal understanding. And in our credit-driven system, they also want and need
theorem-credits.
I’ll skip ahead a few years, to the subject that Jaffe and Quinn alluded to, when
I began studying 3-dimensional manifolds and their relationship to hyperbolic geometry.
(Again, it matters little if you know what this is about.) I gradually
built up over a number of years a certain intuition for hyperbolic three-manifolds,
with a repertoire of constructions, examples and proofs. (This process actually
started when I was an undergraduate, and was strongly bolstered by applications
to foliations.) After a while, I conjectured or speculated that all three-manifolds
have a certain geometric structure; this conjecture eventually became known as
the geometrization conjecture. About two or three years later, I proved the geometrization
theorem for Haken manifolds. It was a hard theorem, and I spent
a tremendous amount of effort thinking about it. When I completed the proof, I
spent a lot more effort checking the proof, searching for difficulties and testing it
against independent information.
I’d like to spell out more what I mean when I say I proved this theorem. It meant
that I had a clear and complete flow of ideas, including details, that withstood a
great deal of scrutiny by myself and by others. Mathematicians have many different
styles of thought. My style is not one of making broad sweeping but careless
generalities, which are merely hints or inspirations: I make clear mental models,
and I think things through. My proofs have turned out to be quite reliable. I have
not had trouble backing up claims or producing details for things I have proven.
I am good in detecting flaws in my own reasoning as well as in the reasoning of
others.
However, there is sometimes a huge expansion factor in translating from the
encoding in my own thinking to something that can be conveyed to someone else.
My mathematical education was rather independent and idiosyncratic, where for a
number of years I learned things on my own, developing personal mental models
for how to think about mathematics. This has often been a big advantage for me in
thinking about mathematics, because it’s easy to pick up later the standard mental
models shared by groups of mathematicians. This means that some concepts that
I use freely and naturally in my personal thinking are foreign to most mathematicians
I talk to. My personal mental models and structures are similar in character
to the kinds of models groups of mathematicians share—but they are often different
models. At the time of the formulation of the geometrization conjecture, my
understanding of hyperbolic geometry was a good example. A random continuing
example is an understanding of finite topological spaces, an oddball topic that can
lend good insight to a variety of questions but that is generally not worth developing
in any one case because there are standard circumlocutions that avoid it.
Neither the geometrization conjecture nor its proof for Haken manifolds was in
the path of any group of mathematicians at the time—it went against the trends
in topology for the preceding 30 years, and it took people by surprise. To most
topologists at the time, hyperbolic geometry was an arcane side branch of mathematics,
although there were other groups of mathematicians such as differential
geometers who did understand it from certain points of view. It took topologists
a while just to understand what the geometrization conjecture meant, what it was
good for, and why it was relevant.
At the same time, I started writing notes on the geometry and topology of 3-
manifolds, in conjunction with the graduate course I was teaching. I distributed
them to a few people, and before long many others from around the world were
writing for copies. The mailing list grew to about 1200 people to whom I was
sending notes every couple of months. I tried to communicate my real thoughts
in these notes. People ran many seminars based on my notes, and I got lots of
feedback. Overwhelmingly, the feedback ran something like “Your notes are really
inspiring and beautiful, but I have to tell you that we spent 3 weeks in our seminar
working out the details of §n.n. More explanation would sure help.”
I also gave many presentations to groups of mathematicians about the ideas of
studying 3-manifolds from the point of view of geometry, and about the proof of the
geometrization conjecture for Haken manifolds. At the beginning, this subject was
foreign to almost everyone. It was hard to communicate—the infrastructure was in
my head, not in the mathematical community. There were several mathematical
theories that fed into the cluster of ideas: three-manifold topology, Kleinian groups,
dynamical systems, geometric topology, discrete subgroups of Lie groups, foliations,
Teichm¨uller spaces, pseudo-Anosov diffeomorphisms, geometric group theory, as
well as hyperbolic geometry.
We held an AMS summer workshop at Bowdoin in 1980, where many mathematicans
in the subfields of low-dimensional topology, dynamical systems and Kleinian
groups came.
It was an interesting experience exchanging cultures. It became dramatically
clear how much proofs depend on the audience. We prove things in a social context
and address them to a certain audience. Parts of this proof I could communicate in
two minutes to the topologists, but the analysts would need an hour lecture before
they would begin to understand it. Similarly, there were some things that could be
said in two minutes to the analysts that would take an hour before the topologists
would begin to get it. And there were many other parts of the proof which should
take two minutes in the abstract, but that none of the audience at the time had
the mental infrastructure to get in less than an hour.
At that time, there was practically no infrastructure and practically no context
for this theorem, so the expansion from how an idea was keyed in my head to what
I had to say to get it across, not to mention how much energy the audience had to
devote to understand it, was very dramatic.
In reaction to my experience with foliations and in response to social pressures, I
concentrated most of my attention on developing and presenting the infrastructure
in what I wrote and in what I talked to people about. I explained the details to the
few people who were “up” for it. I wrote some papers giving the substantive parts
of the proof of the geometrization theorem for Haken manifolds—for these papers,
I got almost no feedback. Similarly, few people actually worked through the harder
and deeper sections of my notes until much later.
The result has been that now quite a number of mathematicians have what was
dramatically lacking in the beginning: a working understanding of the concepts
and the infrastructure that are natural for this subject. There has been and there
continues to be a great deal of thriving mathematical activity. By concentrating
on building the infrastructure and explaining and publishing definitions and ways
of thinking but being slow in stating or in publishing proofs of all the “theorems”
I knew how to prove, I left room for many other people to pick up credit. There
has been room for people to discover and publish other proofs of the geometriza16
tion theorem. These proofs helped develop mathematical concepts which are quite
interesting in themselves, and lead to further mathematics.
What mathematicians most wanted and needed from me was to learn my ways
of thinking, and not in fact to learn my proof of the geometrization conjecture
for Haken manifolds. It is unlikely that the proof of the general geometrization
conjecture will consist of pushing the same proof further.
A further issue is that people sometimes need or want an accepted and validated
result not in order to learn it, but so that they can quote it and rely on it.
Mathematicians were actually very quick to accept my proof, and to start quoting
it and using it based on what documentation there was, based on their experience
and belief in me, and based on acceptance by opinions of experts with whom I spent
a lot of time communicating the proof. The theorem now is documented, through
published sources authored by me and by others, so most people feel secure in
quoting it; people in the field certainly have not challenged me about its validity,
or expressed to me a need for details that are not available.
Not all proofs have an identical role in the logical scaffolding we are building
for mathematics. This particular proof probably has only temporary logical value,
although it has a high motivational value in helping support a certain vision for the
structure of 3-manifolds. The full geometrization conjecture is still a conjecture.
It has been proven for many cases, and is supported by a great deal of computer
evidence as well, but it has not been proven in generality. I am convinced that the
general proof will be discovered; I hope before too many more years. At that point,
proofs of special cases are likely to become obsolete.
Meanwhile, people who want to use the geometric technology are better off to
start off with the assumption “Let M3 be a manifold that admits a geometric
decomposition,” since this is more general than “Let M3 be a Haken manifold.”
People who don’t want to use the technology or who are suspicious of it can avoid
it. Even when a theorem about Haken manifolds can be proven using geometric
techniques, there is a high value in finding purely topological techniques to prove
it.
In this episode (which still continues) I think I have managed to avoid the two
worst possible outcomes: either for me not to let on that I discovered what I
discovered and proved what I proved, keeping it to myself (perhaps with the hope
of proving the Poincar´e conjecture), or for me to present an unassailable and hardto-
learn theory with no practitioners to keep it alive and to make it grow.
I can easily name regrets about my career. I have not published as much as
I should. There are a number of mathematical projects in addition to the geometrization
theorem for Haken manifolds that I have not delivered well or at all
to the mathematical public. When I concentrated more on developing the infrastructure
rather than the top-level theorems in the geometric theory of 3-manifolds,
I became somewhat disengaged as the subject continued to evolve; and I have not
actively or effectively promoted the field or the careers of the excellent people in it.
(But some degree of disengagement seems to me an almost inevitable by-product
of the mentoring of graduate students and others: in order to really turn genuine
research directions over to others, it’s necessary to really let go and stop oneself
from thinking about them very hard.)
On the other hand, I have been busy and productive, in many different activities.
Our system does not create extra time for people like me to spend on writing and
research; instead, it inundates us with many requests and opportunities for extra
work, and my gut reaction has been to say ‘yes’ to many of these requests and
opportunities. I have put a lot of effort into non-credit-producing activities that I
value just as I value proving theorems: mathematical politics, revision of my notes
into a book with a high standard of communication, exploration of computing in
mathematics, mathematical education, development of new forms for communication
of mathematics through the Geometry Center (such as our first experiment,
the “Not Knot” video), directing MSRI, etc.
I think that what I have done has not maximized my “credits”. I have been in
a position not to feel a strong need to compete for more credits. Indeed, I began
to feel strong challenges from other things besides proving new theorems.
I do think that my actions have done well in stimulating mathematics.