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

Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity

By Richard Taylor 

In modular arithmetic, one thinks of the whole numbers arranged around a circle, like the hours on a clock, instead of along an infinite straight line. Here we have seven “hours” on our clock—arithmetic modulo 7. To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count 3 clockwise 5 times, again ending up at 1.

Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs.

For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity. But in one of those strange pieces of serendipity which often characterize the advance of human knowledge, in the last half century modular arithmetic has found important applications in the “real world.” Today, the theory of modular arithmetic (e.g., Reed-Solomon error correcting codes) is the basis for the way DVDs store or satellites transmit large amounts of data without corrupting it. Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic. You can visualize the usual arithmetic as operating on points strung out along the “number line.”

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'An Artificially Created Universe': The Electronic Computer Project at IAS

By George Dyson 

In this 1953 diagnostic photograph from the maintenance logs of the IAS Electronic Computer Project (ECP), a 32-by-32 array of charged spots––serving as working memory, not display––is visible on the face of a Williams cathode-ray memory tube. Starting in late 1945, John von Neumann, Professor in the School of Mathematics, and a group of engineers worked at the Institute to design, build, and program an electronic digital computer.

I am thinking about something much more important than bombs. I am thinking about computers.––John von Neumann, 1946 

 

There are two kinds of creation myths: those where life arises out of the mud, and those where life falls from the sky. In this creation myth, computers arose from the mud, and code fell from the sky.
 
In late 1945, at the Institute for Advanced Study in Princeton, New Jersey, Hungarian-American mathematician John von Neumann gathered a small group of engineers to begin designing, building, and programming an electronic digital computer, with five kilobytes of storage, whose attention could be switched in 24 microseconds from one memory location to the next. The entire digital universe can be traced directly to this 32-by-32-by-40-bit nucleus: less memory than is allocated to displaying a single icon on a computer screen today.
 
Von Neumann’s project was the physical realization of Alan Turing’s Universal Machine, a theoretical construct invented in 1936. It was not the first computer. It was not even the second or third computer. It was, however, among the first compu­ters to make full use of a high-speed random-access storage matrix, and became the machine whose coding was most widely replicated and whose logical architecture was most widely reproduced. The stored-program computer, as conceived by Alan Turing and delivered by John von Neumann, broke the distinction between numbers that mean things and numbers that do things. Our universe would never be the same. 
 
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Extrasolar Planets and the New Astronomy

By Aristotle Socrates 

Figure 2: Orbits of the Earth, Venus, and Mercury superposed with that of HD 80606b (magenta). Not only is its orbit extreme in comparison with those of our inner-solar system, but its mass is extreme as well in that HD 80606b is a gas giant planet, like Jupiter.

The desire to discover distant, rare, and strange objects dominated twentieth-century astronomy, for which increasingly larger and more sensitive telescopes were constructed. 

The act of carrying out this objective has brought enormous—and somewhat unbelievable—rewards: We now accept that we orbit a thermonuclear furnace, the Sun, whose physical properties are quite common, so common that there are nearly 100 billion Sun-like stars within our galaxy, the Milky Way. It was discovered that the Milky Way was not, in fact, the entire Universe; the observable Universe is of order many billions of light years across (that’s big), and there are of order 100 billion galaxies like our own floating around within it. In the center of these galaxies there happen to be super-massive black holes whose masses can be up to 10 billion times the mass of the Sun. When these enormous black holes are built up by in-falling gas, they are called “quasars,” and produce the equivalent of 100 trillion Suns worth of light within a volume comparable to our solar system. The greater the separation between any two galaxies or quasars, the greater the rate at which they move apart or, in other words, the Universe is expanding. Perhaps even more surprising, the Universe is primarily made up of stuff that we can neither see nor feel, i.e., dark energy and dark matter. The strategy of building bigger and more sensitive telescopes, meanwhile, has produced a growing number of “smaller” results that continue to employ regiments of astronomers: gamma-ray bursts, pulsars, X-ray emitting binary stars, clusters of galaxies, cosmic microwave background radiation, and the list goes on.
 
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The Idea of Wartime

By Mary L. Dudziak 

When Mary Dudziak was the Ginny and Robert Loughlin Member in the School of Social Science in 2007-08, she intended to explore the history of war's impact on American law and politics. Instead, she found herself puzzling over ideas about time, which resulted in the book War-Time: An Idea, Its History, Its Consequences (Oxford University Press, 2012).

Does war have a time? The idea of “wartime” is regularly invoked by scholars and policymakers, but the temporal element in warfare is rarely directly examined. I came to the Institute in 2007–08 intent on exploring the history of war’s impact on American law and politics, but assumptions about wartime were so prevalent in the literature that first I found myself puzzling over ideas about time. Ultimately, this resulted in a book, War ·Time: An Idea, Its History, Its Consequences (Oxford University Press, 2012).

The idea that time matters to warfare appears in Thomas Hobbes’s Leviathan: “War consisteth not in battle only, or the act of fighting, but in a tract of time, wherein the will to contend by battle is sufficiently known; and therefore the notion of time is to be considered in the nature of war.” Time’s importance calls for critical inquiry, but time is often treated as if it were a natural phenomenon with an essential nature, shaping human action and thought. Yet our ideas about time are a product of social life, Émile Durkheim and others have argued. Time is of course not produced by clocks, which simply represent an understanding of time. Instead, ideas about time are generated by human beings working in specific historical and cultural contexts. Just as clock time is based on a set of ideas produced not by clocks but by the people who use them, wartime is also a set of ideas derived from social life, not from anything inevitable about war itself.

Yet war seems to structure time, as does the clock. Stephen Kern argues that World War I displaced a multiplicity of “private times,” and imposed “homogenous time,” through an “imposing coordination of all activity according to a single public time.” During World War I, soldiers synchronized their watches before heading into combat. In Eric J. Leed’s description of trench warfare, war instead disrupted time’s usual order. Battle became an extended present, as considerations of past and future were suspended by the violence of the moment. “The roar­ing chaos of the barrage effected a kind of hypnotic condition that shattered any rational pattern of cause and effect,” so that time had no sequence. And so one meaning of “wartime” is the idea that battle suspends time itself.

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Unpacking the Bachelor Pad

By Jessica Ellen Sewell 

Image from Pillow Talk (1959). Split screen heightens the contrast between the feminine apartment of Jan Morrow (Doris Day) and Brad Allen (Rock Hudson)’s bachelor pad. (© Universal Studios)

The mid-1950s saw the invention of a new, highly mythologized housing type, the bachelor pad, articulated most fully in the pages of Playboy and in films. The bachelor pad is an apartment for a single professional man, organized for entertaining and pleasure, and displaying tasteful consumption. The bachelor pad was culturally salient at this particular historical moment because it linked a culture increasingly focused on consumption and what sociologists and cultural commentators in the late 1950s argued was a “crisis in masculinity.” The bachelor pad provided a compelling fantasy of individual consumption and economic and sexual power to counter that crisis, but at the same time, helped to produce the masculinity crisis by problematizing straight male domesticity.

As described in Playboy, the pad “is, or should be, the outward reflection of his [the bachelor’s] inner self—a comfortable, livable, and yet exciting expression of the person he is and the life he leads.”1 It is precisely this inner self that was seen to be in crisis in the late 1950s: men’s sense of themselves as individuals had been stripped away, a state that was blamed partly on the conformity of corporate America and partly on women.

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