The New York Times, December 5, 2011, by Alva Noe  —  What is art? What does art reveal about human nature? The trend these days is to approach such questions in the key of neuroscience.

“Neuroaesthetics” is a term that has been coined to refer to the project of studying art using the methods of neuroscience. It would be fair to say that neuroaesthetics has become a hot field. It is not unusual for leading scientists and distinguished theorists of art to collaborate on papers that find their way into top scientific journals.

Semir Zeki, a neuroscientist at University College London, likes to say that art is governed by the laws of the brain. It is brains, he says, that see art and it is brains that make art. Champions of the new brain-based approach to art sometimes think of themselves as fighting a battle with scholars in the humanities who may lack the courage (in the words of the art historian John Onians) to acknowledge the ways in which biology constrains cultural activity. Strikingly, it hasn’t been much of a battle. Students of culture, like so many of us, seem all too glad to join in the general enthusiasm for neural approaches to just about everything.

 

 

Leif Parsons

 

What is striking about neuroaesthetics is not so much the fact that it has failed to produce interesting or surprising results about art, but rather the fact that no one — not the scientists, and not the artists and art historians — seem to have minded, or even noticed. What stands in the way of success in this new field is, first, the fact that neuroscience has yet to frame anything like an adequate biological or “naturalistic” account of human experience — of thought, perception, or consciousness.

The idea that a person is a functioning assembly of brain cells and associated molecules is not something neuroscience has discovered. It is, rather, something it takes for granted. You are your brain. Francis Crick once called this “the astonishing hypothesis,” because, as he claimed, it is so remote from the way most people alive today think about themselves. But what is really astonishing about this supposedly astonishing hypothesis is how astonishing it is not! The idea that there is a thing inside us that thinks and feels — and that we are that thing — is an old one. Descartes thought that the thinking thing inside had to be immaterial; he couldn’t conceive how flesh could perform the job. Scientists today suppose that it is the brain that is the thing inside us that thinks and feels. But the basic idea is the same. And this is not an idle point. However surprising it may seem, the fact is we don’t actually have a better understanding how the brain might produce consciousness than Descartes did of how the immaterial soul would accomplish this feat; after all, at the present time we lack even the rudimentary outlines of a neural theory of consciousness.

 

Leif Parsons

 

What we do know is that a healthy brain is necessary for normal mental life, and indeed, for any life at all. But of course much else is necessary for mental life. We need roughly normal bodies and a roughly normal environment. We also need the presence and availability of other people if we are to have anything like the sorts of lives that we know and value. So we really ought to say that it is the normally embodied, environmentally- and socially-situated human animal that thinks, feels, decides and is conscious. But once we say this, it would be simpler, and more accurate, to allow that it is people, not their brains, who think and feel and decide. It is people, not their brains, that make and enjoy art. You are not your brain, you are a living human being.

We need finally to break with the dogma that you are something inside of you — whether we think of this as the brain or an immaterial soul — and we need finally take seriously the possibility that the conscious mind is achieved by persons and other animals thanks to their dynamic exchange with the world around them (a dynamic exchange that no doubt depends on the brain, among other things). Importantly, to break with the Cartesian dogmas of contemporary neuroscience would not be to cave in and give up on a commitment to understanding ourselves as natural. It would be rather to rethink what a biologically adequate conception of our nature would be.

But there is a second obstacle to progress in neuroaesthetics. Neural approaches to art have not yet been able to find a way to bring art into focus in the laboratory. As mentioned, theorists in this field like to say that art is constrained by the laws of the brain. But in practice what this is usually taken to come down to is the humble fact that the brain constrains the experience of art because it constrains all experience. Visual artists, for example, don’t work with ultraviolet light, as Zeki reminds us, because we can’t see ultraviolet light. They do work with shape and form and color because we can see them.

 

Leif Parsons

 

Now it is doubtless correct that visual artists confine themselves to materials and effects that are, well, visible. And likewise, it seems right that our perception of works of art, like our perception of anything, depends on the nature of our perceptual capacities, capacities which, in their turn, are constrained by the brain.

But there is a problem with this: An account of how the brain constrains our ability to perceive has no greater claim to being an account of our ability to perceive art than it has to being an account of how we perceive sports, or how we perceive the man across from us on the subway. In works about neuroaesthetics, art is discussed in the prefaces and touted on the book jackets, but never really manages to show up in the body of the works themselves!

Some of us might wonder whether the relevant question is how we perceive works of art, anyway. What we ought to be asking is: Why do we value some works as art? Why do they move us? Why does art matter?  And here again, the closest neural scientists or psychologists come to saying anything about this kind of aesthetic evaluation is to say something about preference. But the class of things we like, or that we prefer as compared to other things, is much wider than the class of things we value as art. And the sorts of reasons we have for valuing one art work over another are not the same kind of reasons we would give for liking one person more than another, or one flavor more than another. And it is no help to appeal to beauty here. Beauty is both too wide and too narrow. Not all art works are beautiful (or pleasing for that matter, even if many are), and not everything we find beautiful (a person, say, or a sunset) is a work of art.

Again we find not that neuroaesthetics takes aim at our target and misses, but that it fails even to bring the target into focus.

Yet it’s early. Neuroaesthetics, like the neuroscience of consciousness itself, is still in its infancy. Is there any reason to doubt that progress will be made? Is there any principled reason to be skeptical that there can be a valuable study of art making use of the methods and tools of neuroscience? I think the answer to these questions must be yes, but not because there is no value in bringing art and empirical science into contact, and not because art does not reflect our human biology.

To begin to see this, consider: engagement with a work of art is a bit like engagement with another person in conversation; and a work of art itself can be usefully compared with a humorous gesture or a joke. Just as getting a joke requires sensitivity to a whole background context, to presuppositions and intended as well as unintended meanings, so “getting” a work of art requires an attunement to problems, questions, attitudes and expectations; it requires an engagement with the context in which the work of art has work to do. We might say that works of art pose questions and encountering a work of art meaningfully requires understanding the relevant questions and getting why they matter, or maybe even, why they don’t matter, or don’t matter any more, or why they would matter in one context but not another. In short, the work of art, whatever its local subject matter or specific concerns ― God, life, death, politics, the beautiful, art itself, perceptual consciousness ― and whatever its medium, is doing something like philosophical work.

One consequence of this is that it may belong to the very nature of art, as it belongs to the nature of philosophy, that there can be nothing like a settled, once-and-for-all account of what art is, just as there can be no all-purpose account of what happens when people communicate or when they laugh together. Art, even for those who make it and love it, is always a question, a problem for itself. What is art? The question must arise, but it allows no definitive answer.

For these reasons, neuroscience, which looks at events in the brains of individual people and can do no more than describe and analyze them, may just be the wrong kind of empirical science for understanding art.

Far from its being the case that we can apply neuroscience as an intellectual ready-made to understand art, it may be that art, by disclosing the ways in which human experience in general is something we enact together, in exchange, may provide new resources for shaping a more plausible, more empirically rigorous, account of our human nature.

 

 

UX and your brain – By uxtraordinary Alex O’Neal
An image for a 2008 presentation on UX and the brain.

Neuroscience research is mine, based on academic courses and professional journals. I hand-drew the brain from multiple anatomical resources, then scanned and PhotoShopped for the rest of the effects. Lassigue D’mato is the handwriting font.

 


 

Alva Noë is a philosopher at CUNY’s Graduate Center. He is the author of “Out of Our Heads: Why You Are Not Your Brain and Other Lessons From The Biology of Consciousness.” He is now writing a book on art and human nature. Noë writes a weekly column for NPR’s science and culture blog.

Shiho Fukada for The New York Times

The world took notice of China‘s technological prowess in late 2010, when a Chinese supercomputer, the Tianhe-1A, became the world’s fastest.

 

 

Shiho Fukada for The New York Times

 

 


A lab worker with the Tianhe-1A, a Chinese supercomputer, at the
National Supercomputer Center in Tianjin, China.

 

 

Shiho Fukada for The New York Times

A monitor used to oversee the Tianhe-1A supercomputer

 

 

The New York Times, December 5, 2011, by David Barboza and John Markoff, BEIJING — In an otherwise nondescript conference room, Wu Jianping stands before a giant wall of frosted glass. He toggles a switch and the glass becomes transparent, looking down on an imposing network operations center full of large computer displays. They show maps of China and the world, pinpointing China’s IPv6 links, the next generation of the Internet.

 

China already has almost twice the number of Internet users as in the United States, and Dr. Wu, a computer scientist and director of the Chinese Educational and Research Network, points out that his nation is moving more quickly than any other in the world to deploy the new protocol.

IPv6 — Internet Protocol version 6 — offers advanced security and privacy options, but more important, many more I.P. addresses, whose supply on the present Internet (IPv4) is almost exhausted.

“China must move to IPv6,” Dr. Wu said. “In the U.S., some people don’t believe it’s urgent, but we believe it’s urgent.”

If the future of the Internet is already in China, is the future of computing there as well?

Many experts in the United States say it could very well be. Because of the ready availability of low-cost labor, China has already become the world’s dominant maker of computers and consumer electronics products. Now, these experts say, its booming economy and growing technological infrastructure may thrust it to the forefront of the next generation of computing.

For China, the quest to develop advanced computing centers is not simply a matter of national pride. It is an attempt to lay the groundwork for innovative Chinese companies and to reshape the technological landscape by doing something more than assembling the world’s desktop PCs.

Never mind that there may be no Chinese Steve Jobs, said Clyde V. Prestowitz Jr., president of the Economic Strategy Institute.

“There are different kinds of innovation,” he said. “We tend to equate innovation with companies that start from garages based on brainstorms.

“There is another kind of innovation that results in constant improvement that we are not good at — and they are.”

The view is not universal. Still, other experts say it would be a mistake to underestimate China’s capacity for rapid progress.

“When I went to China for the first time in 1978, I saw workers stringing together computer memories with sewing needles,” said Patrick J. McGovern, the founder of the International Data Group, an early investor in Tencent Holdings, one of the most successful Chinese Internet companies. “Now innovation is accelerating, and in the future, patents on smartphones and tablets will be originated by the Chinese people.”

A New Kind of Challenge

Going back six decades — to Eniac, considered to be the first electronic computer — the United States has set both the pace and the path of modern computing and communication. From mainframes to iPhones, from the Arpanet to WiFi, innovation has been as American as Norman Rockwell.

And for more than a generation, the hub of innovation has been Silicon Valley, a multicultural melting pot that has supported the singular amalgam of computer-hacker ethos and entrepreneurial aggressiveness that made it the envy of the world.

Probably the most serious challenge to the Valley’s dominance came in the late 1980s from Japan, which seemed on the brink of taking command of the semiconductor and computer industries until its economy foundered.

Today, China poses a very different kind of challenge. While Japan’s economy has long been driven by exports, China will soon have the world’s largest domestic market for both Internet commerce and computing.

The world took notice of Chinese technological prowess in late 2010, when a Chinese supercomputer, the Tianhe-1A, briefly became the world’s fastest. Though it was made from American processors and was soon surpassed by a Japanese machine, it was still indisputable evidence that the Chinese had achieved world-class computing designs.

Then, this October, another Chinese supercomputer, the Sunway Bluelight MPP, broke the petaflop barrier — a quadrillion calculations per second — putting it among the world’s 20 fastest computers.

This machine proved even more surprising in the West. Not only was it based on a Chinese-made microprocessor, but it also achieved a significant advance in low-power operation. That might indicate the Chinese now have a significant lead in “performance per watt” — a measure of energy-efficient computing that will prove crucial to reaching the next generation of so-called exascale supercomputers, which are computers that will be a thousand times faster than the world’s fastest today, and which are scheduled to arrive by the end of this decade.

“This is what Chinese companies need to do,” said Hu Weiwu, a professor at the Chinese Academy of Sciences who is the chief designer of another Chinese family of microprocessor chips. “We can send a spaceship to space. We can design high-performance computers.”

American officials agree, saying the Chinese government’s investment in supercomputing is paying off.

“The overall point of all of this is that the Chinese understand the importance of high-performance computing,” said Donna Crawford, the associate director of computation at the Lawrence Livermore National Laboratory. “They are executing on the plan as a key enabler for their whole society.”

Obstacles to Dominance

Last year, in an interview that would have been seen as extraordinary if the remarks had been made by a United States president, Prime Minister Wen Jiabao committed China to creating an “Internet of things.”

Connecting homes and smart power grids has been a driving principle behind the next-generation Internet in the United States. And it goes hand in hand with “ubiquitous computing,” the idea that computing power transforms everyday devices like smartphones and digital music players.

But China’s efforts at dominance are hardly without obstacles. The country has fallen far short on a decade-long commitment to build the world’s leading semiconductor industry, and it still imports a vast majority of microchips for the products it assembles. Its best chip factories are two to three generations behind world leaders like Intel, in the United States, and T.S.M.C., in Taiwan.

China’s great weakness may prove to be too much government control. Chinese innovation may also be limited by the relative lack of intellectual property protection, discouraging entrepreneurs from breaking new ground.

The 13th International Conference on Ubiquitous Computing, in Beijing in September, left American technology experts underwhelmed.

“There was nothing that really leaped out at me,” said John Seeley Brown, who directed the Xerox Palo Alto Research Center in the 1980s, when the concept was invented.

In contrast, he said, he observed true innovation at companies like Foxconn, the Taiwanese manufacturing firm with extensive operations in mainland China that has done much of Apple’s assembly work.

“There is the deep embedding of the research and design culture driving a place like Foxconn to do things that they would never have done on their own,” he said. “We’ve now combined the best thinkers in the U.S. sitting side by side with the people who are best at manufacturing in the world.”

Still, other investors have glimpsed similar possibilities in China itself. Ruby Lu, a Beijing-based partner in DCM, a venture capital firm with investments in China, described a semiconductor firm begun with lead designers in San Diego and support engineers in Shanghai, working at one-sixth the cost.

“It was the Shanghai team who came up with a groundbreaking solution to an existing problem,” she recalled in an e-mail. What scares competitors is that China has begun producing waves of amazing hardware engineers and software programmers, winning international competitions and beginning to dominate the best engineering programs in the United States. The University of California, Berkeley, is about to announce a deal to create an engineering campus in Shanghai, raising fears about transferring technology from one of the best American engineering schools.

Much has been made of computer science “returnees,” most notably Andrew Chi-Chih Yao, who left Princeton to create an institute at Tsingtao University in Beijing that has already made breakthroughs in game theory and computer security.

“Overnight there is lots of activity coming from Beijing,” said Christos Papadimitriou, a computer scientist at Berkeley.

And there is little question that the structure of Chinese industry is becoming more innovation-oriented. This summer Dieter Ernst, senior fellow at the East-West Center, testified before a Congressional commission that the Chinese had overtaken South Korea and Europe in total patents and were catching up with the United States and Japan.

Moreover, China is now the world’s second-largest venture capital market, growing to $7.6 billion from just $2.2 billion in 2005, while the American venture capital market has remained largely stagnant, according to Rebecca A. Fannin, author of the new book “Startup Asia” (Wiley).

Researchers at the Stanford Program on Regions of Innovation and Entrepreneurship recently mapped the investment activity of 769 firms investing in 2,203 Chinese companies and found patterns reminiscent of Silicon Valley.

“You have lead individuals and firms that have established beachheads of success and power,” said Marguerite Gong Hancock, a professor at the Stanford Graduate School of Business, who directed the study. “They are the same firms that were successful in Silicon Valley that have transplanted their expertise to China.”

The similarities to Silicon Valley can be eerie. “All the symptoms of a bubble are here,” said Anne Stevenson-Yang, co-director of J Capital Research, a technology investment research firm in Beijing. “It’s an unsettling instability.”

At the same time, there is a consensus that China’s entrepreneurs have a workaholic culture that is unmatched anywhere in the world.

“What I found in doing five startups in China is the culture makes Silicon Valley look laid-back and slow,” said Tom Melcher, an entrepreneur who left California a decade ago to move to Beijing. “In Beijing, if you want to find a chief executive at 7:30 p.m. on a Friday, it’s guaranteed you will find him at the office.”

Not every China specialist buys such comparisons.

“When we look at China through the lens of American decline, we see the Chinese ascendancy, we see the modern skylines and the fastest computers and the new airports, and we see an invincible force building,” said Orville Schell, director of the Center on U.S.-China Relations at the Asia Society. “Through Chinese eyes it looks tremendously uncertain and provisional. They are not filled with self-confidence.”

But Mr. McGovern, the investor and venture capitalist, thinks that may be an advantage. In the United States, he says, he is often approached by overconfident entrepreneurs who walk off in a huff when he is less than impressed by their ideas.

The Chinese entrepreneurs he deals with are different.

“They will come to me and show me their language-translation software that will convert Mandarin to English and back again,” he said. “Then I will tell them that I don’t think there is a market because it is too difficult to protect the intellectual property.”

Rather than send them away, however, he might ask them if they would be interested in working on a different idea that his firm has been considering.

“They will respond, ‘Can I get rich?’ When I tell them that I think that there is a good chance, they say, ‘O.K., I’ll do it!’ ”

 

David Barboza reported from Beijing and John Markoff from San Francisco.

 

More info on the Tianhe – 1A

 

In October 2010, Tianhe-1A, an upgraded supercomputer, was unveiled at HPC 2010 China. It is now equipped with 14,336 Xeon X5670 processors and 7,168 Nvidia Tesla M2050 general purpose GPUs. 2,048 NUDT FT1000 heterogeneous processors are also installed in the system, but their computing power was not counted into the machine’s official Linpack statistics as of October 2010. Tianhe-1A has a theoretical peak performance of 4.701 petaflops. NVIDIA suggests that it would have taken “50,000 CPUs and twice as much floor space to deliver the same performance using CPUs alone.” The current heterogeneous system consumes 4.04 megawatts compared to over 12 megawatts had it been built only with CPUs.

The Tianhe-1A system is composed of 112 computer cabinets, 12 storage cabinets, 6 communications cabinets, and 8 I/O cabinets. Each computer cabinet is composed of four frames, with each frame containing eight blades, plus a 16-port switching board. Each blade is composed of two computer nodes, with each computer node containing two Xeon X5670 6-core processors and one Nvidia M2050 GPU processor. The system has 3584 total blades containing 7168 GPUs, and 14,336 CPUs, managed by the SLURM job scheduler. The total disk storage of the systems is 2 Petabytes implemented as a Lustre clustered file system, and the total memory size of the system is 262 Terabytes.

Another significant reason for the increased performance of the upgraded Tianhe-1A system is the Chinese-designed NUDT custom designed proprietary high-speed interconnect called Arch that runs at 160 Gbps, twice the bandwidth of InfiniBand.

The supercomputer is installed at the National Supercomputing Center, Tianjin, and is used to carry out computations for petroleum exploration and aircraft design. It is an “open access” computer, meaning it provides services for other countries. The supercomputer will be available to international clients.

The computer cost $88 million to build. Approximately $20 million is spent annually for electricity and operating expenses. Approximately 200 workers are employed in its operation.

Tianhe-IA was ranked as the world’s fastest supercomputer in the TOP500 list until July 2011 when the K computer overtook it.

 

 

The K Super Computer

A cabinet of RIKEN’s K computer prototype, manufactured by Fujitsu

 

A cabinet of RIKEN’s K computer prototype, manufactured by Fujitsu

Active Operational June 2011
Sponsors MEXT, Japan
Operators Fujitsu
Location RIKEN Advanced Institute for Computational Science
Architecture 88,128 SPARC64 VIIIfx processors, Tofu interconnect, Linux-based enhanced operating system
Speed 10.51 petaflops (Rmax)
Ranking TOP500: 1, November 2011

 

 

Performance

On 20 June 2011, the TOP500 Project Committee announced that K scored a LINPACK record with a performance of 8.162 petaflops, making it the fastest supercomputer in the world at the time; it achieved this performance with a computing efficiency ratio of 93.0%. The previous record holder was the Chinese National University of Defense Technology‘s Tianhe-1A, which performed at 2.507 petaflops. The TOP500 list is revised semiannually, and the rankings change frequently, indicating the speed at which computing power is increasing. In November 2011, RIKEN reported that K had become the first supercomputer to exceed 10 petaflops, achieving a LINPACK performance of 10.51 quadrillion computations per second with a computing efficiency ratio of 93.2%. K received top-ranking in all four performance benchmarks at the 2011 HPC Challenge Awards.

Node architecture

As of the November 2011 TOP500 list, the K computer uses 88,128 2.0GHz 8-core SPARC64 VIIIfx processors packed in 864 cabinets, for a total of 705,024 cores, manufactured by Fujitsu with 45 nm CMOS technology. Each cabinet contains 96 computing nodes, in addition to 6 I/O nodes. Each computing node contains a single processor and 16 GB of memory. The computer’s water cooling system is designed to minimize failure rate and power consumption.

Network

The K computer uses a proprietary six-dimensional torus network interconnect called Tofu, and a Tofu-optimized Message Passing Interface based on the open-source Open MPI library. Users can create application programs adapted to either a one-, two-, or three-dimensional torus network.

Filesystem

The system adopts a two-level local/global file system with parallel/distributed functions, and provides users with an automatic staging function for moving files between global and local file systems. Fujitsu developed an optimized parallel file system based on Lustre, called Fujitsu Exabyte File System, scalable to several hundred petabytes.

Power consumption

While the K computer reports the highest total power consumption of any TOP500 supercomputer (9.89 MW – the equivalent of almost 10,000 suburban homes), the computer is relatively efficient, achieving 824.6 GFlop/kWatt. This is 29.8% more efficient than China’s NUDT TH MPP (ranked #2 – 2011/06), and 225.8% more efficient than Oak Ridge’s Jaguar-Cray XT5-HE (ranked #3 – 2011/06). However, K’s efficiency rating still falls far short of the 2097.2 GFlops/kWatt supercomputer record set by IBM’s NNSA/SC Blue Gene/Q Prototype 2, which is currently the world’s 109th-fastest supercomputer. For comparison, the average power consumption of a TOP 10 system is 4.3 MW, and the average efficiency is 463.7 GFlop/kW. According to TOP500 compiler Jack Dongarra, professor of electrical engineering and computer science at the University of Tennessee, the K computer’s performance equals “one million linked desktop computers”. The computer’s annual running costs are estimated at US$10 million.

 

The New York Times, December 5, 2011

 

Steven Strogatz on the Elements of Math

 

About the Column
Steven Strogatz, an award-winning professor, takes readers from the basics to the baffling in a 15-part series on mathematics. Beginning with a column on why numbers are helpful, he goes on to investigate topics including negative numbers, calculus and group theory, finishing with the mysteries of infinity.

 

 

Columnist Biography
Steven Strogatz is a professor of applied mathematics at Cornell University. In 2007 he received the Communications Award, a lifetime achievement award for the communication of mathematics to the general public. He previously taught at the Massachusetts Institute of Technology, where he received the E.M. Baker Award, an institute-wide teaching prize selected solely by students. “Chaos,” his series of 24 lectures on chaos theory, was filmed and produced in 2008 by The Teaching Company. He is the author, most recently, of “The Calculus of Friendship,” the story of his 30-year correspondence with his high school calculus teacher.

 

Click on the 15 links below, one for each Lesson

 

Columns

1)  From Fish to Infinity

http://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/

 

A debut column on math features an introduction to numbers, from upsides (they’re efficient) to down (they’re ethereal).

January 31, 2010

2)  Rock Groups

http://opinionator.blogs.nytimes.com/2010/02/07/rock-groups/

 

Treating numbers concretely — think rocks, for instance — can make calculations less baffling.

February 7, 2010

 

3)  The Enemy of My Enemy

http://opinionator.blogs.nytimes.com/2010/02/14/the-enemy-of-my-enemy/

 

The disturbing concept of subtraction, and how we deal with the fact that negative numbers are so . . . negative.

February 14, 2010

4)  Division and Its Discontents

http://opinionator.blogs.nytimes.com/2010/02/21/division-and-its-discontents/

 

This week, division — where many students hit the mathematical wall — is made less confusing.

February 21, 2010

5)  The Joy of X

http://opinionator.blogs.nytimes.com/2010/02/28/the-joy-of-x/

 

The series moves on to high school math, specifically algebra and formulas.

February 28, 2010

6)  Finding Your Roots

http://opinionator.blogs.nytimes.com/2010/02/28/the-joy-of-x/

 

Complex numbers, a hybrid of the imaginary and the real, are the pinnacle of number systems.

March 7, 2010

7)  Square Dancing

http://opinionator.blogs.nytimes.com/2010/03/14/square-dancing/

 

Geometry, intuition and the long road from Pythagoras to Einstein.

March 14, 2010

8)  Think Globally

http://opinionator.blogs.nytimes.com/2010/03/21/think-globally/

 

Differential geometry can show us the shortest route between two points.

March 21, 2010

9)  Power Tools

http://opinionator.blogs.nytimes.com/2010/03/28/power-tools/

 

In math, the function of functions is to transform.

March 28, 2010

10)  Take It to the Limit

http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit/

 

Archimedes recognized the power of the infinite, and in the process laid the groundwork for calculus.

April 4, 2010

11)  Change We Can Believe In

http://opinionator.blogs.nytimes.com/2010/04/11/change-we-can-believe-in/?gwh=165F2C0F38954A75E41B10709D877F2A

 

Differential calculus can show you the best path from A to B, and Michael Jordan’s dunks can help explain why that is.

April 11, 2010

12)  It Slices, It Dices

http://opinionator.blogs.nytimes.com/2010/04/18/it-slices-it-dices/?gwh=91569133B43406C7E9608B0EC5651643

 

The integral, perhaps mathematics’ most graceful sign, is a foundation of calculus.

April 18, 2010

13)  Chances Are

http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/?gwh=167D84955788A5BC15C01FAD377E75B5

 

The improbable thrills of probability theory.

April 25, 2010

14)  Group Think

http://opinionator.blogs.nytimes.com/2010/05/02/group-think/?gwh=8744624BDC289CD2676B2816E3239FB6

 

Group theory, one of the most versatile parts of math, bridges the arts and sciences.

May 2, 2010

15)  The Hilbert Hotel

http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/?gwh=0B6F74FBF151705E8F5432C1F2EB611D

 

An exploration of infinity as this math series, not being infinite, comes to an end.

 

 

 

 

 

http://topics.nytimes.com/top/opinion/series/steven_strogatz_on_the_elements_of_math/index.html