To this date, although extensive computer tests have found no counter-examples to these conjectures, no proofs are known. Deshoulliers, Effinger, te Riele and Zinoviev published a proof of the odd Goldbach conjecture, but it assumed the generalized Riemann hypothesis (another pre-emiment unsolved conjecture of mathematics), and so it is not really a proof of the original conjecture. Collectively these conjectures are among the oldest and most prominent unsolved problems in mathematics.

A few weeks ago (1 Feb 2012), the well-known Australian Fields medalist mathematician Terence Tao posted an arXiv paper entitled “Every odd number greater than 1 is the sum of at most five primes.” The title tells it all. Although numerous weaker results have been published in this area, Tao’s result, if it survives peer review by highly qualified mathematicians, would clearly be the strongest and most satisfactory yet.

A brief summary of prior results (taken from MathWorld) runs as follows:

The conjecture that all odd numbers  are the sum of three odd primes is called the “weak” Goldbach conjecture. Vinogradov (1937ab, 1954) proved that every sufficiently large odd number is the sum of three primes. (Nagell 1951, p. 66; Guy 1994), and Estermann (1938) proved that almost all even numbers are the sums of two primes. Vinogradov’s original “sufficiently large”  was subsequently reduced to  by Chen and Wang (1989). Chen (1973, 1978) also showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes (Guy 1994, Courant and Robbins 1996).

Tao presents a short synopsis of his wonderful result on his blog. He mentions that his paper utilizes the Hardy-Littlewood circle method, one of the most frequently employed techniques of analytic number theory, due to famed British mathematicians G. H. Hardy and J. E. Littlewood in the 1920s, but broadly based on Hardy’s earlier work with Indian mathematician Srinivasa Ramanujan.

While Terry Tao’s manuscript includes much impressive analysis of his own, he carefully notes that he relies on the results of numerous other contemporary mathematicians (the bibliography includes 39 references). Among the key results he utilizes are those of Jean Bourgain, Jing Run Chen, Xavier Gourdon, Ming-Chit Liu, Hugh Montgomery, Harmut Siebert, Ivan Vinogradov and Tianze Wang. As Newton, in a moment of unusual candor,  once confessed, “If I have seen further it is by standing on ye sholders of Giants.”

Stanley Kubrick made the ultimate science-fiction movie and it is going to be very hard for somebody to come along and make a better movie.

From a scientific point of view, the film’s account of the discovery of evidence for extraterrestrial intelligence reflects the ever-growing fascination with fundamental scientific questions such as “Are there extraterrestrials?,” “Where are they?”, “What are they like?” — questions directly tied to Fermi’s paradox. In general, the film faithfully reflects humankind’s eternal fascination with the planets and stars and our future destiny in outer space.

The movie has also been influential in technology, and so it is worth taking score on which of its numerous predictions have been borne out in the 43 years since its release, and which have not:

1. Space travel. Perhaps the biggest disappointment since the movie was released has been our failure to seriously embark on space travel. We have an International Space Station orbiting the planet, but it is a far cry from the huge double-wheel structure depicted in the movie. We traveled to the moon shortly after the film’s release, but we have not returned there in four decades, even for brief exploration trips, much less to construct permanent colonies. Several U.S. attempts to initiate human trips to Mars have been scuttled, victims of crushing budget deficits. Space vehicles of the enormous scale depicted in “2001″ remain far-future fantasies.  NASA has had to bring back seventy-year old engineers to tell their “war stories” about the Apollo program, in order to preserve a rapidly disappearing corporate memory.
2. Computer technology. While the film does not provide details of the construction of the HAL-9000 onboard computer, it is fair to say that few, if anyone, at the time could have predicted the enormous increase in computer power that has been achieved in the interim, with Moore’s Law marching relentlessly forward for 45 years and running. It is worth pointing out that a physically large, centralized computer was required in the movie to monitor and control all onboard systems. Nowadays this task could be handled by a handful of $1000 computers with relative ease. And no one in 1968 dreamed that by 2001 (and much more so in 2012), many individual households would have multiple computers (counting PCs, tablets and smartphones), each more powerful and capacious than the world’s most powerful systems in 1968. Here as elsewhere, reality has imitated art. Apple’s “Siri” assistant (and Steven Hawking as well) sounds like HAL because Steve Jobs wanted it to.
3. Videophones. In “2001,” when Frank Bowman arrives at the space station, he places a videophone call to his daughter from a pay telephone booth. Nowadays, many of us carry in our pocket a smartphone that can place color video calls (e.g., using Microsoft’s Skype and Apple’s FaceTime) to anyone else with a similar smartphone, and millions of others use similar services on PCs. Along this line, the visual resolution of the videophone screen in the movie was not very good — present-day Internet video is typically much better. Finally, it is worth pointing out the laughable suggestion that Frank Bowman should be charged $1.75 for placing a 2-minute videophone call to his daughter. Nowadays no one pays per-minute charges for FaceTime or Skype video calls, provided one has a suitable high-speed Internet service.
4. Tablet computers. One prediction was right on the money: early prototypes of flat-screen display tablets appeared in roughly 2001. In the past few years they have exploded in popularity, and are now part of the daily routine of many millions of devoted users worldwide. It is amusing that in August 2011, in response to a patent infringement lawsuit by Apple Computers against Samsung, the latter argued that Apple’s iPad was too closely modeled on the tablets depicted in “2001.”
5. Computer chess. Computer-based chess-playing programs have steadily increased in power in the intervening years since “2001.” Finally, in 1998, IBM’s “Deep Blue” computer defeated Garry Kasparov, the reigning world champion chess player. Nowadays even PC-based chess programs can defeat most strong chess players. See the Math Drudge blog on games and our Huffington Post article for details.
6. Voice recognition. While voiceprint identification was available not long after the movie’s release, full-fledged, multi-person, reasonably reliable voice recognition technology has only recently come to full flower. Apple’s voice-activated “Siri” assistant, now available on the latest iPhones, is a harbinger of the future, and similar products are in development by other high-tech firms.
7. Artificial intelligence. At the time “2001″ was produced, researchers were confident that fully operational artificial intelligence systems would be available in just a few years. Alas, even now we do not have the equivalent of the HAL-9000 system aboard the movie’s spacecraft, which effortlessly exchanged information with the astronauts. But we are getting much closer, as evidenced by last year’s stunning victory of IBM’s Watson computer system in the American TV game show Jeopardy! Renewed interest in the Turing test speaks to the revived sense of progress in this area.

So how many more years will transpire before we truly return to space in the style of “2001″? Recently NASA, recognizing continuing problems with safety and technological obsolescence, terminated its Space Shuttle program, leaving the world’s largest economy without a viable space transport. NASA is placing its hopes on private ventures such as SpaceX, founded by PayPal entrepreneur Elon Musk, which is developing its Falcon 9 spacecraft to deliver cargoes (and later people) to the International Space Station. But NASA’s longer-term plans to send humans to Mars remain mired in budget cuts and high-level indecision. Even American Presidential candidate Newt Gingrich’s more modest suggestion to create a colony on the moon was met with derision. We hope that it was the messenger, not his message and long-term vision, that was being derided.

Meanwhile, China has announced an ambitious five-year plan to develop space technology. China’s initial moon plans include orbiters that will make soft lunar landings, survey the lunar landscape, and then return collect samples of the moon’s surface to earth for analysis. Ultimately China plans to place astronauts on the moon. One advantage of China’s space program is that most likely it will not be subject to the “fits and starts” and political infighting that have plagued the U.S. space program.

Looking further into the future, NASA and the Defense Advanced Projects Research Agency (DARPA) recently embarked on a 100-Year Starship Study, as a first step to chart out the future of space exploration. This includes plans for interstellar travel, energy generation and requisite medical and radiation-resistance technologies.

There is clearly a public appetite for human space discovery, independent of the public’s appreciation of spin-off technologies generated from the first space age. What is missing is a sales pitch — like the arms race but ideally less grim — that made possible the Apollo program. Time will tell when and how the 2001 vision will be realized, and if it will be human or robotic. In the meantime we all can dream.

This article also appeared in the Huffington Post.

In the past few years, modern genome sequencing and computer technology have placed enormous volumes of DNA data at the fingertips of researchers worldwide. The first complete human genome sequence was completed in 2000, after a ten-year effort that cost over USD$500 million. But genome sequencing technology is advancing very rapidly — human genomes can now be sequenced for roughly $100,000, and some groups are targeting a price as low as $1,000 [Pollack2008]. This same sequencing technology has enabled biologists to study the genomes of thousands of other biological species, including many common (and not-so-common) plants and animals. This has resulted in an enormous repository of data available for the study of evolution at the most basic level.

Amino acid data

One example of DNA-type data is the table below, which compares the 146-unit amino acid sequences of beta globin (a component of hemoglobin) among various species of animals. Amino acids are coded directly by triplets of DNA letters, and thus the study of amino acid sequences is very close to the study of DNA sequences themselves. Note that human beta globin is identical to that of chimpanzees, differs in only one location from that of gorillas, yet is increasingly distinct from that in red foxes, polar bears, horses, rats, chicken and salmon. Anyone can generate similar data using online tools and databases [Evolution2009]:

Mutations

The picture is the same if we consider the pattern of mutations between closely related species. One particularly interesting example that has recently been uncovered is the “GULO” gene, which is an essential part of the machinery that makes Vitamin C in most animals. Humans lack a functioning copy of this gene — our copy is highly mutated fragment, classified as a relic gene or pseudogene. Scurvy, that scourge of British sailors and Mormon pioneers crossing the plains, occurs in humans when they do not get enough Vitamin C. Interestingly, although the GULO pseudogene is highly mutated and utterly useless, humans and chimpanzees have almost identical copies of it — the human and chimp versions are 98% identical. Evidently a common ancestor of humans and chimps adopted a diet rich in fruits and vegetables, and thus a chance mutation that disabled Vitamin C production was no longer a fatal one and was passed on to posterity [Fairbanks2007, pg. 53-55; Coyne2009, pg. 67-69].

Transposons

Another recent development in this arena is the analysis of “transposons” or “jumping genes.” These are sections of DNA that have been randomly copied from one part of an organism’s genome to another. Most of the time, these inserted genes do no damage, because they “land” in relatively unimportant sections of DNA. But they do provide an excellent means to classify species into their phylogenetic (“family tree”) relationship. This is because it is exceedingly unlikely that the same random insertion of an entire gene would occur at the same spot in the genomes of two or more different organisms or species, unless, of course, each inherited this curious feature from a common ancestor, and it is also exceedingly unlikely that a group of species with “random” assortments of transposons could be organized into a family tree. Transposon data has been used, for instance, to classify a large number of vertebrate species into a “family tree,” with a result that is virtually identical to what biologists had earlier reckoned based only physical features and biological functions [Rogers2011, pg. 25-30].

Here is an example of how transposon data can be used to determine the phylogenetic relationships (i.e., “family tree”) of various primates including humans. The columns labeled ABCDE denote five blocks of transposons, and x and o respectively denote that the block is present or absent in the genome of the given species. It is clear from this data that our closest primate relatives are chimpanzees and bonobos [Rogers2011, pg. 89; Salem2003].

						Transposon blocks
			Species		A	B	C	D	E
        /---------	Human		o	x	x	x	x
       /----------	Bonobo		x	x	x	x	x
      / \---------	Chimp		x	x	x	x	x
     /------------	Gorilla		o	o	x	x	x
-----|------------	Orangutan	o	o	o	x	x
     \------------	Gibbon		o	o	o	o	o

Other areas of research

Another research arena that is exploding with activity is in analyzing DNA of groups of existing species, then employing advanced statistical methods (e.g., “maximum likelihood analysis”), running on powerful computer systems, to reconstruct the most likely family tree for a given set of organisms. Soon much of evolutionary history will be deducible purely from this type of automatic computer-based analysis. Already, significant results have been obtained in this area. In May 2010, a researcher announced, on the basis of a very carefully performed statistical analysis, that the hypothesis of a “universal common ancestor” (a conjecture, dating back to Charles Darwin, that all life arose from a single common ancestral organism) has been resoundingly confirmed. The author, Prof. Douglas L. Theobald of Brandeis University, found that the universal common ancestor hypothesis is at least 10²⁸⁶⁰ times more likely to have produced the modern-day protein sequences that we observe in living organisms, compared to the next most probable scenario that involves multiple original ancestors [Harmon2010; Theobald2010].

Researchers are also combining analyses of DNA sequences with paleontological (fossil) data, resulting in more precise determinations of various branches in the tree of life. For example, a study published in November 2010 that combined both paleontological and molecular data established that divergence of humans and chimpanzees very likely took place eight million years in the past instead of five to six million years, as generally believed until recently [SD2010d; Wilkinson2010].

Summary

The explosion of genome sequences and DNA data banks in recent years has provided an enormous storehouse of data for biologists. Analyses of these data have dramatically confirmed the central tenets of evolution, including the common ancestry of all biological organisms, all arranged convincingly in a phylogenetic family tree, in most cases exactly as had been previously reckoned based solely on similarities of physical forms and biological functions. As anthropologist Alan R. Rogers recently noted, “Phylogenetic pattern is everywhere in nature. It makes sense only if all living things evolved from a single ancestor.” [Rogers2011, pg. 31]. Similarly, genetist Daniel J. Fairbanks emphasizes that [Fairbanks2007, pg. 170]:

[The] obvious hierarchical arrangement of life, and the literally millions of ancestral relics in our DNA — all undeniably attest to our common evolutionary origin with the rest of life. If someone can believe that all living organisms share the same creator, why not consider that all living organisms share a common genetic heritage?

[This was previously posted at SMR blog.]

Image from the Turing archive

Turing's cryptographic 'bombe' as rebuilt at Bletchley Park

 
Turing (see http://www.turing.org.uk) was many other things: a world class marathon runner, a troubled homosexual, and an atheist who famously said “The universe is a differential equation. Religion is an initial condition.”  Those who knew him savour Feynman-like anecdotes. His achievements are perhaps most succinctly summarized by Harvard scholar Steven Pinker, who declared:

It would be an exaggeration to say that the British mathematician Alan Turing explained the nature of logical and mathematical reasoning, invented the digital computer, solved the mind-body problem, and saved Western civilization. But it would not be much of an exaggeration.

One of Turing’s many signal contributions was a 1950 article that defined what is now known as the “Turning test.” In this article, he proposed a test in which a human “converses” with two entities — one human and one computer program — over a text-only channel (i.e., a computer keyboard/screen), and then attempts to determine which is the human and which is the computer. If, after say five minutes of testing, the majority of human interrogators are unable to determine which is which, Turing said that we could claim that the computer system has achieved a certain level of intelligence.

Turing’s article even anticipated several possible objections to his test, including mathematical and philosophical objections, which continue to be debated to the present day. For example, some potential questions might not be “fair” to a computer. And we all have human acquaintances who might be judged “computer” in such a test.

In the decades since 1950, when Turing proposed the test, it has been widely influential in directing progress in the computing field in general and in artificial intelligence in particular. Some early attempts at Turing test programs pointed out both the promise and the perils of this enterprise. In 1966, Joseph Weizenbaum created a program, known as ELIZA, which identified keywords in text typed by a human, and then responded with some sort of clever but inquiring response, in the style of a psychologist interviewing a patient. Although some subjects were genuinely surprised to learn that the “psychologist” was a computer, to more skeptical testers its weaknesses quickly became evident. The present authors do remember enjoying playing with it when personal computers first allowed for relaxed therapy sessions.

Progress in the field languished somewhat during the 1970s and 1980s, but since 1991 there has been an annual Loebner Prize in artificial intelligence in which ELIZA’s children — now called “chat-bots” — compete to pass the Turing test. Two recent advances have dramatically enhanced interest: (a) the ready availability of many terabytes of data, from technical documents on every conceivable topic to the growing personal databases of “lifeloggers”; and (b) sophisticated statistical techniques for organizing and classifying this data.

This technology was perhaps most publicly brought to the public eye with the recent defeat of two champion human contestants on the American quiz show “Jeopardy!” by an IBM-developed computer system known as “Watson.” Additional details on Watson (both its Jeopardy! achievement and future commercialization plans) are available from the IBM Watson website and also in a New York Times article. See also our previous blogs on Watson: Math Drudge #1 and Math Drudge #2 and HuffPost article on Games computers play with us.

Watson is now rapidly moving into specializations for medicine and voice recognition among other things. IBM clearly views Apple Computers’ Siri assistant, available with the iPhone 4S, as a main target of competition. Meanwhile Google and AT&T are working on similar systems, according to a recent UK report. Among other things, Watson-type technology offers amazing opportunities as an intelligent assistant for mathematical research.

So far no computer system has passed the Turing test, according to the strict rules of the Loebner prize competition, but they are getting close. The 2010 and 2011 competitions have been won by a chat-bot computer system known as “CHAT-L,” programmed by Bruce Wilcox. In 2010 this program actually fooled one of the four human judges into thinking it was human.

All this raises the question of whether a computer system that finally passes the Turing test is really “conscious” or “human” in any sense. These issues were summarized by Robert M. French in a recent Science article:

All of this brings us squarely back to the question first posed by Turing at the dawn of the computer age, one that has generated a flood of philosophical and scientific commentary ever since. No one would argue that computer-simulated chess playing, regardless of how it is achieved, is not chess playing. Is there something fundamentally different about computer-simulated intelligence?

French is among the more pessimistic observers. Others, such as futurist Ray Kurzweil are much more expansive. He predicts that in roughly the year 2045, machine intelligence will match then transcend human intelligence, resulting in a dizzying advance of technology that we can only dimly foresee at the present time. Kurzweil outlines this vision in his recent book The Singularity Is Near.

Only time will tell when Turing’s vision will be achieved. But civilization will never be the same once it is.

Our first posting comes from a very interesting set of charts just released by the U.S. Treasury entitled “The Financial Crisis Response in Charts”. This is chock-full of intriguing data on the recent financial crash of 2007-2009, as well as programs (such as the Troubled Asset Recovery Program (TARP)) instituted under the Bush administration and in the first few months of the Obama administration to ameliorate its potentially devastating impact. Together these charts and data destroy numerous myths and misconceptions that have been circulated during the past two or three years, both by “liberal” and “conservative” political commentators.

Here are a few highlights:

1. The financial crisis of 2007-2009 was a shock larger than that which lead to the Great Depression. The resulting fall in U.S. GDP was roughly double that of any recession since 1974.
2. The recession can hardly be blamed on the current administration, because the most of the decline took place before Jan 2009, and further most of the countermeasures were enacted just prior to the start of the present administration.
3. The passage of the TARP program and other financial programs coincides almost exactly with the start of the recovery. Largely as a result of these programs, job losses during the current recession were only about 1/3 the job losses of the Great Depression.
4. In spite of early estimates of the cost of TARP and other financial stability programs (ranging from 341 billion to several trillion U.S. dollars), at the present time the total net cost is about $60 billion, and it is expected that over the next few years even that remaining deficit will be fully repaid, resulting in a NET PROFIT to the U.S. taxpayer.
5. Nonetheless, the U.S. debt has ballooned in the past decade (mostly due to the cost of the two Middle Eastern wars and the 2001 tax cuts), and the nation must get serious about reducing its deficits (and eventually even paying down part of its debt), as painful as that might be.

Some of the changes are obvious, some are less so.

Computer games humans play

It is both useful and sobering to consider the enormous progress that has been made in computer technology over the past 50 years. Back in 1965 Intel co-founder Gordon Moore observed in a little-noticed article that the complexity of integrated circuits had increased at a rate of roughly a factor of two per year for several years, and “there is no reason to believe it will not remain nearly constant for at least 10 years.”

That was arguably the world’s greatest understatement: the trend of computer devices roughly doubling in complexity every 18 months or so (not quite every 12 months as originally predicted) has continued unabated for nearly 50 years, resulting in devices, such as the latest microprocessor and memory chips, that incorporate billions of components and are billions of times more capable than the primitive items that once were considered the pinnacle of modern technology.  This doubling is now called Moore’s law, and when it comes to games, “the Moore the merrier”.

What are we doing with these devices? Among other things, many millions of persons worldwide hold in their hands a smartphone with features, such as voice recognition and precise GPS positioning, not available on any device at any price just a few years ago.

And, as anyone with a `teenager’ in the house will attest, computers are indeed playing games with us. The latest systems from Sony, Nintendo and others have more computational horsepower (over one trillion arithmetic operations per second) than the world’s fastest supercomputers just 15 years or so ago, and the real-time graphics they generate are truly remarkable. One consequence is the emergence of multi-player real-time immersive games.

Human games computers play

But computers are also playing games that require something else — significant  intelligence, at least of the artificial variety.  Let us look at various famous human games. State-of-the-art computer programs never lose when playing Checkers and Backgammon. They can play a very good hand of Bridge; indeed, human players often practice against Bridge-playing programs, as they do with chess. Likewise brute force Scrabble will impassively demolish most human payers.

Computer programs for Poker have significantly improved in the past few years (bluffing is less of a big-deal than it might appear), and one program recently defeated some professional players in a tournament. Computer programs for Go have also become much stronger in recent years, although there is still a significant gap between the best computer programs and strong human players. (Go has many many more positions than chess, and so raw power is less effective.)

Many are familiar with the 1997 defeat of Garry Kasparov, the world’s reigning chess champion, by IBM’s “Deep Blue” computer. Commenting on his experience, Kasparov later reflected, “It was my luck (perhaps my bad luck) to be the world chess champion during the critical years in which computers challenged, then surpassed, human chess players.”

Deep Blue employed some new techniques, but for the most it simply applied enormous computing power, so that it could store many openings, look ahead many moves, and — except when the programmers erred — never make mistakes. Cheap laptop programs now beat all human comers. So we have `solved chess’. But as John McCarthy wrote,

In 1965 the Russian mathematician Alexander Konrod said “Chess is the Drosophila of artificial intelligence.” However, computer chess has developed as genetics might have if the geneticists had concentrated their efforts starting in 1910 on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies.

Deep Fritz, a humble work station, beat world champion Vladimir Kramnik in 2006.   Marty Newborn, one of the match organizers, said, ”If you are interested in programming computers so that they compete in games, the two interesting ones are poker and go. That is where the action is.”

In a fundamental way, human games that computers can always win at are no longer human games. It is hard to see checkers, bridge or chess making a serious resurgence in this century.