Articles from the Institute Letter

Additional articles from new and past issues of the Institute Letter will continue to be posted over time and as they become available.

by Vladimir Voevodsky

Diagram by Voevodsky
This three-dimensional diagram is an example of the kind of ­”formulas” that Voevodsky would have to use to support his arguments about 2-theories.

Professor Voevodsky’s Personal Mission to Develop Computer Proof Verification to Avoid Mathematical Mistakes

In January 1984, Alexander Grothendieck submitted to the French National Centre for Scientific Research his proposal “Esquisse d’un Programme.” Soon copies of this text started circulating among mathematicians. A few months later, as a first-year undergraduate at Moscow University, I was given a copy of it by George Shabat, my first scientific adviser. After learning some French with the sole purpose of being able to read this text, I started to work on some of the ideas outlined there.

In 1988 or 1989, I met Michael Kapranov who was equally fascinated by the perspectives of developing mathematics of new “higher-dimensional” objects inspired by the theory of categories and 2-categories.
The first paper that we published together was called “∞-Groupoids as a Model for a Homotopy Category.” In it, we claimed to provide a rigorous mathematical formulation and a proof of Grothendieck’s idea connecting two classes of mathematical objects: ∞-groupoids and homotopy types.

Later we decided that we could apply similar ideas to another top mathematical problem of that time: to construct motivic cohomology, conjectured to exist in a 1987 paper by Alexander Beilinson, Robert MacPherson (now Professor in the School of Mathematics), and Vadim Schechtman.

In the summer of 1990, Kapranov arranged for me to be accepted to graduate school at Harvard without applying. After a few months, while he was at Cornell and I was at Harvard, our mathematical paths diverged. I concentrated my efforts on motivic cohomology and later on motivic homotopy theory. My notes dated March 29, 1991, start with the question “What is a homotopy theory for algebraic varieties or schemes?”

The field of motivic cohomology was considered at that time to be highly speculative and lacking firm foundation. The groundbreaking 1986 paper “Algebraic Cycles and Higher K-theory” by Spencer Bloch was soon after publication found by Andrei Suslin to contain a mistake in the proof of Lemma 1.1. The proof could not be fixed, and almost all of the claims of the paper were left unsubstantiated.

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by Gabrielle Benette Jackson

Cartoon: cogito ergo sum consciousness

How Do We Understand Consciousness Without Becoming Complicit in That Understanding?

What is consciousness? “It is being awake,” “being responsive,” “acting,” “being aware,” “being self-aware,” “paying attention,” “perceiving,” “feeling emotions,” “feeling feelings,” “having thoughts,” “thinking about thoughts,” “it is like this!”

Who is conscious? “We humans, surely!” Well, maybe not all the time. “Animals!” Debatable. “Computers?” No—at least, not yet. “Other machines?” Only in fiction. “Plants?” Absolutely not, right?

Nearly twenty-five years ago, we lived through “the project of the decade of the brain,” a governmental initiative set forth by President George H. W. Bush.1 Presidential Proclamation 6158 begins, “The human brain, a three-pound mass of interwoven nerve cells that controls our activity, is one of the most magnificent—and mysterious—wonders of creation. The seat of human intelligence, interpreter of senses, and controller of movement, this incredible organ continues to intrigue scientists and laymen alike. Over the years, our understanding of the brain—how it works, what goes wrong when it is injured or diseased—has increased dramatically. However, we still have much more to learn.” And it concludes, “Now, Therefore, I, George Bush, President of the United States of America, do hereby proclaim the decade beginning January 1, 1990, as the Decade of the Brain. I call upon all public officials and the people of the United States to observe that decade with appropriate programs, ceremonies, and activities.”

What the former President did not say—what is perhaps understood by his readers—is that the brain is quite different from other body parts that have come under scientific investigation. We might be grateful to receive a donated kidney, or to have an artificial heart. But unlike every other body part, without my brain, there may be no I. Our sense of self, of awareness, of life—are profoundly connected to a working brain. A philosopher once said, “in a brain transplant, one wants to be the donor not the recipient.” Indeed, saying that the brain is the seat of mentality is like saying that the sun is a source of light.
 

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by Yvonne Chiu

A wounded German soldier lighting a cigarette for a wounded British soldier at a British field hospital during the Battle of Épehy, near the end of the First World War (1918)
A wounded German soldier lighting a cigarette for a wounded British soldier at a British field hospital during the Battle of Épehy, near the end of the First World War (1918) (Photo: Lt. Thomas K. Aitken, British Army photographer/Imperial War Museums)

‘Live and Let Live’ as a Representative Element of War

Images that convey the essence of war are more likely to resemble the frenzied, merciless, mutual slaughter between the Aegeans and the Trojans as told in The Iliad, the rapes depicted in Goya’s The Disasters of War, the torture portrayed in The Battle of Algiers, or the indiscriminate napalm bombing in Vietnam dramatized in Apocalypse Now. It is commonly believed—and for good reason—that morality and civilization are inevitably forgotten in war, as participants become desperate to survive, get caught up in the bloodlust, or lose touch with their humanity. There is truth to that, so it might be surprising to think of banning hollow point bullets (Hague Convention, 1899) or regulating prisoner-of-war treatment (from the 1648 Peace of Westphalia through the 1949 Geneva Conventions) as simultaneously capturing an essential element of warfare, but in fact they represent a significant component of war, which is cooperation between enemies.

Some of the more amazing stories of cooperation in warfare come from the trenches of World War I. During the Christmas truces in 1914, and to a lesser extent in 1915, not only did 100,000 British and German soldiers in WWI unofficially stop fighting, but in some places in Belgium, German soldiers who decorated their trenches with candles and trees and sang carols were met with British soldiers singing in kind; eventually, the two sides mingled in No Man’s Land, exchanging gifts, food, and souvenirs, and even engaging in short, casual football games.

In addition to ad hoc cooperation on a shared holy day, opposing trenches spontaneously developed a longer-lived system of timed shellings to allow the other side to anticipate and avoid their impact. While trench warfare was a large part of the WWI experience, it is not particularly interesting militarily. Rather, it is noteworthy for what fighting did not happen. This “live and let live” system has been recounted in marvelous detail by Tony Ashworth (Trench Warfare 1914–1918). That reciprocal exchange—of minimization of injury and death—took different forms during the war: truces lasted anywhere from a few minutes to several months; some were explicit agreements between fraternizing soldiers in close quarters, while others were indirect (due to legal sanctions), over long distances, and involving large numbers of people. There were numerous reports of people walking openly above trenches; unrestricted movement in and out of the trenches; Germans frying sausages and photos of Brits frying bacon in the trenches, despite the fact that smoke from the fires would have attracted gunfire on active fronts; and descriptions of “quiet” fronts, where there were no ammunition shortages. In some trenches, people hunted and retrieved small game, harvested vegetables, kept milking cows for fresh milk, and had pianos and books.

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Prospects in Theoretical Physics at the Institute for Advanced Study (Photo: Alexandra Altman)

The Institute’s thirteenth annual Prospects in Theoretical Physics (PiTP) summer program for graduate students and postdoctoral scholars, which focused on string theory, was truly extraordinary in that it overlapped with Strings 2014. This is one of the field’s most important gatherings, which the Institute hosted with Princeton University, convening international experts and researchers to discuss string theory and its most recent developments. Six hundred attendees gathered for Strings 2014, which made it one of the largest Strings conferences since their inception in 1995.

Strings 2014 talks, which covered topics from B-mode cosmology and the theory of inflation to quantum entanglement, the amplituhedron, and the fate of spacetime, may be viewed at https://physics.princeton.edu/strings2014/Talk_titles.shtml. The program for PiTP and videos of its string theory talks may be viewed at https://pitp2014.ias.edu/schedule.html.

As part of the PiTP program, the Institute showed a screening of Particle Fever, a new film that follows six scientists, including the Institute’s Nima Arkani-Hamed, during the launch of the Large Hadron Collider and fortutiously captures the discovery of the Higgs particle. Peter Higgs, who predicted the existence of the particle fifty years ago, gave one of his first seminars on the topic at the Institute in 1966.

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Exploring the Arguments of Independence

The following text is excerpted from Our Declaration: A Reading of the ­Declaration of Independence in Defense of Equality (Liveright Publishing Corporation, 2014) by Danielle Allen, UPS Foundation Professor in the School of Social Science.

The Declaration of Independence matters because it helps us see that we cannot have freedom without equality. It is out of an egalitarian commitment that a people grows—a people that is capable of protecting us all collectively, and each of us individually, from domination. If the Declaration can stake a claim to freedom, it is only because it is so clear-eyed about the fact that the people’s strength resides in its equality.

The Declaration also conveys another lesson of paramount importance. It is this: language is one of the most potent resources each of us has for achieving our own political empowerment. The men who wrote the Declaration of Independence grasped the power of words. This reveals itself in the laborious processes by which they brought the Declaration, and their revolution, into being. It shows itself forcefully, of course, in the text’s own eloquence.

When we think about how to achieve political equality, we have to attend to things like voting rights and the right to hold office. We have to foster economic opportunity and understand when excessive material inequality undermines broad democratic political participation. But we also have to cultivate the capacity of citizens to use language effectively enough to influence the choices we make together.
 

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