Abstract:

The mathematical research community is facing a great challenge to re-evaluate the role of proof in light of the growing power of current computer systems, of modern mathematical computing packages, and of the growing capacity to data-mine on the Internet. Add to that the enormous complexity of many modern capstone results such as the Poincaré conjecture, Fermat’s last theorem, and the Classification of finite simple groups. As the need and prospects for inductive mathematics blossom, the requirement to ensure the role of proof is properly founded remains undiminished. I shall look at the philosophical context with examples and then offer some of five bench-marking examples of the opportunities and challenges we face.

Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

• Unique focus on the functions themselves, rather than convex analysis
• Contains over 600 exercises showing theory and applications
• All material has been class-tested

Contents:

Preface; 1. Why convex?; 2. Convex functions on Euclidean spaces; 3. Finer structure of Euclidean spaces; 4. Convex functions on Banach spaces; 5. Duality between smoothness and strict convexity; 6. Further analytic topics; 7. Barriers and Legendre functions; 8. Convex functions and classifications of Banach spaces; 9. Monotone operators and the Fitzpatrick function; 10. Further remarks and notes; References; Index.

Abstract:

The latest state-of-the-art scientific computer systems have achieved over 1 “Pflop/s” (one million billion floating-point arithmetic operations per second). Scientists have capitalized on this computational power by developing a wide range of sophisticated programs that are becoming so effective that scientific computing is now widely regarded as the third mode of scientific discovery, after theory and experiment.

In other words, the computer has become a virtual laboratory, wherein “experiments” can be performed to explore phenomena that are too complicated, expensive or dangerous to explore by ordinary empirical experiment. Many examples of this methodology will be described, including studies in climate and environmental science, astrophysics, biology, engineering, and mathematics.

Among the examples of such computations, the author will describe some recent research wherein new formulas of mathematics have been discovered, using high-precision numerical computations on state-of-the-art computers. Perhaps the best-known such discovery is a new formula for the mathematical constant pi, which has the curious property that it permits one to calculate digits of pi beginning at an arbitrary starting position in the binary expansion.

• Elizabeth Landau, “On Pi Day, one number ‘reeks of mystery’,” CNN, 12 Mar 2010, available at Online article.
• Jacob Aron, “Pi day: Five tasty facts about the famous ratio,” New Scientist, 12 Mar 2010, available at Online article.

Also of interest for Pi Day are:

• Jonathan Borwein’s talk and accompanying article, “The Life of Pi”: Online presentation | Online article.
• The Pi Day site of the Mathematical Association of America: MAA site.

Similar claims have been made about modern evolutionary biology. The recent movie “Expelled” claimed that creationist and “intelligent design” writers have been systematically “shut out” due to a “conspiracy” among the scientific establishment. Now, the two movements have joined forces to some extent, as legislators in several U.S. states are introducing legislation to require that students be taught “all sides of evidence” on evolution and global warming [Kaufman].

We certainly do not intend, in this column, to launch into a full-scale analysis of the claims of either group. With regards to creationism and “intelligent design,” numerous scientists have amply demonstrated that their claims are either refuted by the findings of multiple peer-reviewed published studies, or, in any event, fail to rise to anywhere near the level needed to challenge existing theories of geology and biology [SMR-creationism; SMR-ID].

With regards to global warming, we see no reason not to accept the overall consensus of the climate science community, namely that the planet is warming, and that this warming is due, at least in part, to human causes and is likely to increase in the future. We note, for instance, the indisputable fact that CO2 is a greenhouse gas (this can be confirmed in simple laboratory experiments), that levels of CO2 have increased significantly in the past few decades as compared with previous centuries (this is well-established from analyses of ice core samples), and that the decade 2000-2009 had the highest overall average temperature of any decade since accurate measurements began (confirmed by recently-announced NASA satellite data) [RealClimate; Revkin; NASA]. Further, human effects on climate have been confirmed in peer-reviewed studies for [Marshall]:

• The rise in global surface air temperature;
• The rise in surface air temperature over every continent, including Antarctica;
• The rise in atmospheric humidity (caused by the higher air temperatures);
• The rise in precipitation (rain, snow, etc) around the world, as a result of the higher humidities;
• Shifts in precipitation: dry tropical regions are getting drier while wet regions closer to the poles are getting wetter;
• The huge losses of Arctic summer sea ice;
• The rise in surface ocean temperature;
• Increasing salinity in the Atlantic Ocean.

It is important to keep an open mind, since future findings might draw some of these conclusions into question, but we see no reason to doubt that the process of peer review in the climate science field is capable of resolving whatever issues arise.

We also note that leading figures in the global warming denial community have ties to conservative research institutes funded in large measure by large corporations and oil companies [Sachs]. Fred Singer, arguably the leading figure of the community, has only six peer-reviewed publications in the climate science field, and none since 1997 [Singer]. In a similar vein, several of the leading writers in the “intelligent design” community are funded by conservative research institutes closely connected to the evangelical world, drawing into question their claims that they are merely doing objective, secular science. Philip Johnson, a leading intelligent design writer, is a retired law professor with no scientific credentials or peer-reviewed scientific publications [SMR-ID].

In any event, it is clear to us that those who lead the currently trendy movement to deny global warming (or to deny human causation of global warming), like creationists, “intelligent design” writers and the many “mathematicians” who keep our mail boxes and Inboxes flooded with claims that pi is rational or other similar nonsense, are operating well outside the established boundaries of peer-reviewed science, and thus, from these reasons alone, are deserving of considerable skepticism.

To begin with, there is a proper forum for debating scientific controversies, one that has been established for centuries and is an essential part of what is properly known as modern science. This forum is most assuredly not lectures, blogs, twitters, do-it-yourself websites, newspaper columns, Fox TV News, or state, provincial or national legislative bodies. Instead the proper forum for scientific debate is the system of peer-reviewed scientific journals and conferences sponsored by major scientific societies. If you see these issues being “debated” in any other setting, you can be assured that the discussion is decidedly “bush league” and not worth taking seriously as scientific debate.

Think of it this way: the next time you see a global warming denier or a creationist or an “intelligent design” writer or even some self-styled “mathematician” making a convincing-sounding argument in a lecture, blog, website or news column, ask yourself the following question: “If he/she really had a solid argument, why isn’t he/she back in the office furiously writing up this material for submission to a leading journal, thereby assuring him/herself future worldwide fame and glory?”

After all, overturning some long-held paradigm is what science and mathematics is all about. Much of the day-to-day work of a real scientist or mathematician is, frankly, somewhat tedious. Every mathematician dreams of being the first to prove (or, perhaps better, disprove) some long-standing conjecture or result. Every physicist dreams of being the first to uncover evidence that counters some decades- or centuries-old theory, or to publish a new mathematical theory that will, like Einstein’s papers on relativity, open up new realms of understanding. Every biologist dreams of finding a completely new species or biochemical feature that refutes some long-held assumption about how the natural world operates.

In many cases, those who attempt to grasp the public attention through other means are themselves well aware that they are short-circuiting this process, and, if pressed, further recognize that they do not really have an argument that could withstand the withering scrutiny of scientific peer review. They often allude to conspiracy or malign forces, or they latch onto underwhelming scientific constructionist arguments [Brown] to impute the scientific enterprise in toto. Thus, when they press their views in public — to a populace that for the most part does not understand how the scientific enterprise operates — they are either being more than a little bit dishonest or else are hugely ignorant (and thus unqualified to be pressing their case).

As mentioned above, some have claimed that various sectors of the scientific community are engaged in a “conspiracy” to silence critics and to keep the “truth” from the public. To a real scientist or mathematician, such claims are most absurd nonsense. How, in a worldwide community of hundreds of thousands of competitive researchers, from every nation on earth and from countless different cultural backgrounds, could a secret “conspiracy” be maintained? As Ben Franklin wrote in his Poor Richard’s Almanac, “Three can keep a secret, provided two of them are dead.” Or as one of us quipped, tongue-in-cheek, in response to a state legislator who was skeptical of evolution, “You have no idea how humiliating this is to me — there is a secret conspiracy among leading scientists, but no one deemed me important enough to be included!”

Here is another way to think about such claims: There are tens of thousands of senior scientists in their late 50s or early 60s who have seen their retirement savings decimated by the recent stock market plunge, and who now wonder if the day will ever come when they are financially well off enough to do their research without the constant stress and distraction of applying for grants (the majority of which are never funded). All one of these scientists has to do to garner both worldwide fame and considerable fortune (e.g., book contracts) is to call a news conference and expose the “truth.” Why isn’t this happening?

The system of peer-reviewed journals and conferences sponsored by major professional societies is the only proper forum for the presentation and review of new ideas, in any field of science or mathematics. It has been stunningly successful: errors have been uncovered, fraud has been rooted out and bogus scientific claims (such as the 1903 N-ray claim, the 1989 cold fusion claim and the more recent assertion of an autism-vaccination link) have been rapidly debunked. This all occurs with a level of reliability and at a speed that is hard to imagine in other human endeavors. Those who attempt to short-circuit this system are doing potentially irreparable harm to the integrity of the system. They may enrich themselves or their friends, but they are doing grievous damage to society at large.

1. [Broder] John M. Broder, “Scientists Taking Steps to Defend Work on Climate,” New York Times, 2 Mar 2010, available at Online article.
2. [Brown] Richard C. Brown, Are Science And Mathematics Socially Constructed?: A Mathematician Encounters Postmodern Interpretations of Science, World Scientific Publications, New York, 2009.
3. [Kaufman] Leslie Kaufman, “Darwin Foes Add Warming to Targets,” New York Times, 3 Mar 2010, available at Online article.
4. [Marshall] Michael Marshall, “Which climate changes can be blamed on humans?” New Scientist, 5 Mar 2010, available at Online article.
5. [NASA] “2009: Second Warmest Year on Record; End of Warmest Decade,” 21 Jan 2010, available at Online article.
6. [RealClimate] “IPCC errors: facts and spin,” 14 Feb 2010, available at Online article.
7. [Revkin] Andrew C. Revkin, “Lacis at NASA on Role of CO2 in Warming,” New York Times, 17 Feb 2010, available at Online article.
8. [Sachs] Jeffrey Sachs, “Climate sceptics are recycled critics of controls on tobacco and acid rain,” Guardian, 19 Feb 2010, available at Online article.
9. [SMR-Creationism] “Have creationist writers uncovered significant technical issues that draw into question established theories of geology and evolution?” 9 Mar 2010, available at Online article.
10. [SMR-ID] “Have intelligent design writers uncovered significant technical issues that draw into question the established theories of geology and evolution?” 16 Jan 2010, available at Online article.
11. [Singer] “S. Fred Singer, Ph.D.: Recent Professional Activities,” Mar 1998, available at Online article.
12. [Vanderhooft] Christian Vanderhooft, “Bill to freeze greenhouse-gas controls clears another hurdle in Utah Legislature,” Salt Lake Tribune, 19 Feb 2010, available at Online article ]]> http://experimentalmath.info/blog/2010/03/creationism-global-warming-denial-and-scientific-integrity/feed/ 0 http://experimentalmath.info/blog/2010/02/the-greatest-mathematical-discovery/ http://experimentalmath.info/blog/2010/02/the-greatest-mathematical-discovery/#comments Sat, 06 Feb 2010 19:51:13 +0000 David Bailey http://experimentalmath.info/blog/?p=272 Introduction

    Question: What mathematical discovery more than 1500 years ago:

    • Is one of the greatest, if not the greatest, single discovery in the field of mathematics?
    • Involved three subtle ideas that evaded the greatest minds of antiquity, even including geniuses such as Archimedes?
    • Was fiercely resisted in Europe for hundreds of years after its discovery?
    • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source?

    Answer: Our modern system of positional decimal arithmetic with zero, which was discovered in India in the fourth or fifth century.

    Why?

    As recorded by George Dantzig’s father Tobias Dantzig, the 19th century mathematician Pierre-Simon Laplace explained:

    It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. [Dantzig2007, pg. 19]

    As Laplace noted, today we often take this scheme for granted as “trivial,” but it is anything but trivial (as any youngster in grade school will attest), since it eluded the best minds of the ancient world, even the genius Archimedes. Note that although Archimedes saw far beyond the mathematics of his time, even anticipating numerous key ideas of modern calculus and numerical analysis, and also even though he was very skilled in applying these mathematical principles to engineering and astronomy, nonetheless he used the traditional Greek-Roman numeral system for calculations [Netz2007,Marchant2008]. It is worth noting that Archimedes’ computation of pi was a tour de force of numerical interval analysis performed without either positional notation or trigonometry [Berggren2004,Netz2007].

    Perhaps one reason this discovery gets so little attention today is that it is very hard for us to appreciate the enormous difficulty of using Roman numerals, counting tables and abacuses. Michel de Montaigne, Mayor of Bordeaux and one of the most learned men of his day, confessed in 1588 (prior to the widespread adoption of decimal arithmetic in Europe) that in spite of his great education and erudition, “I cannot yet cast account either with penne or Counters.” That is, he could not do basic arithmetic [Ifrah2000, pg. 577]. In a similar vein, at about the same time a wealthy German merchant, consulting a scholar regarding which European university offered the best education for his son, was told the following:

    If you only want him to be able to cope with addition and subtraction, then any French or German university will do. But if you are intent on your son going on to multiplication and division—assuming that he has sufficient gifts—then you will have to send him to Italy. [Ifrah2000, pg. 577]

    We observe in passing that Claude Shannon (1916–2001) constructed a mechanical calculator wryly called
    Throback 1 at Bell Labs in 1953, which computed in Roman, so as to demonstrate that it was possible if very difficult to compute this way.

    To our knowledge, the best source currently available on the history of our modern number system is by French scholar Georges Ifrah [Ifrah2000]. He chronicles in encyclopedic detail the rise of modern numeration from its roots in primitive hand counting and tally schemes, to the Babylonian, Egyptian, Greek, Roman, Mayan, Indian and Chinese systems, and finally to the eventual discovery of full positional decimal arithmetic with zero in India, and its belated, kicking-and-screaming adoption in the West. Ifrah emphasizes more than once that the discovery of this system was by no means obvious or inevitable:

    The measure of the genius of Indian civilization, to which we owe our modern system, is all the greater in that it was the only one in all history to have achieved this triumph. … Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own. [Ifrah2000, pg. 346]

    Indeed, the development of this system hinged on three key abstract (and certainly non-intuitive) principles [Ifrah2000, pg. 346]: (a) the idea of attaching to each basic figure graphical signs which were removed from all intuitive associations, and did not visually evoke the units they represented; (b) the idea of adopting the principle according to which the basic figures have a value which depends on the position they occupy in the representation of a number; and (c) the idea of a fully operational zero, filling the empty spaces of missing units and at the same time having the meaning of a null number. Ifrah describes the significance of this discovery in these terms:

    This fundamental realization therefore profoundly changed human existence, by bringing a simple and perfectly coherent notation for all numbers and allowing anyone, even those most resistant to elementary arithmetic, the means to easily perform all sorts of calculations; also by henceforth making it possible to carry out operations which previously, since the dawn of time, had been inconceivable; and opening up thereby the path which led to the development of mathematics, science and technology. … Now that we can stand back from the story, the birth of our modern number-system seems a colossal event in the history of humanity, as momentous as the mastery of fire, the development of agriculture, or the invention of writing, of the wheel, or of the steam engine. [Ifrah2000, pg. 346-347]

    It is astonishing how many years passed before this system finally gained full acceptance in the rest of the world. There are indications that Indian numerals reached southern Europe perhaps as early as 500 CE, but with Europe mired in the Dark Ages, few paid any attention. Similarly, there is mention in Sui Dynasty (581-618 CE) records of Chinese translations of the “Brahman Arithmetical Classic,” although sadly none of these copies survived [Gupta1983].

    The Indian system (also known as the Indo-Arabic system) was introduced to Europeans by Gerbert of Aurillac in the tenth century. He traveled to Spain to learn about the system first-hand from Arab scholars, prior to being named Pope Sylvester II in 999 CE. However, his advocacy of the system encountered stiff resistance, in part from accountants who did not want their craft rendered obsolete, to clerics who were aghast to hear that he had traveled to Islamic lands to study the method. As a result, it was widely rumored that he was a sorcerer and that he must have sold his soul to Lucifer during his travels, an accusation that persisted until 1648, when papal authorities reopened his tomb to make sure that his body had not been infested by Satan! [Ifrah2000,pg. 579].

    The system was later reintroduced to Europe by Leonardo of Pisa, also known as Fibonacci, in his 1202 CE book Liber Abaci. However, usage of the system still remained limited for centuries, in part because the scheme continued to be considered “diabolical,” due to the mistaken impression that it originated in the Arab world (in spite of Fibonacci’s clear descriptions of the “nine Indian figures” plus zero) [Ifrah2000, pg. 361-362]. Indeed, our modern English word “cipher” (or “cypher”), which is derived from the Arabic zephirum for zero, and which alternately means “zero” or “secret code” in modern usage, is very likely a linguistic memory of the time when using decimal arithmetic was deemed evidence of dabbling in the occult, which was potentially punishable by death at the hands of the Inquisition [Ifrah2000, pg. 588-589]. Decimal arithmetic began to be widely used by scientists beginning in the 1500s, and was employed, for instance, by Copernicus, Galileo, Kepler and Newton, but it was not universally used in European commerce until after the French Revolution in 1793 [Ifrah2000, pg. 590]. In limited defense of the Roman system, it is harder to alter Roman entries in an account book or the sum payable in a cheque, but this does not excuse the continuing practice of performing arithmetic using Roman numerals and counting tables.

    The Arabic world, by comparison, was much more accepting of the Indian system — in fact, as mentioned briefly above, the West owes its knowledge of the scheme to Arab scholars. One of the first to popularize the method was al-Khuwarizmi, who in the ninth century wrote at length about the Indian place-value system and also described algebraic methods for the solution of quadratic equations. In 1424, Al-Kashi of Samarkand, “who could calculate as eagles can fly” computed 2 pi in sexagecimal (good to an equivalent of 16 decimal digits) using 3 x 2²⁸-gons and a base-60 variation of Indian positional arithmetic [Berggren2004, Appendix on Arab Mathematics]:

    2 pi ~ 6 + 16/60¹ + 59/60² + 28/60³ + 1/60⁴ + 34/60⁵ + 51/60⁶ + 46/60⁷ + 14/60⁸ + 50/60⁹

    This is a personal favourite of ours: re-entering it on a computer centuries later and getting the predicted answer still produces goose-bumps.

    So who exactly discovered the Indian system? Sadly, there is no record of the individual that first discovered the scheme, who, if known, would surely rank among the greatest mathematicians of all time. The very earliest document clearly describing positional decimal arithmetic with zero is an Indian astronomical work entitled Lokavibhaga (“Parts of the Universe”). Here, for example, the number 13,107,200,000 is written as

    panchabhyah khalu shunyebhyah param dve sapta chambaram ekam trini cha rupam cha

    (“five voids, then two and seven, the sky, one and three and the form”), i.e., 00000 2 7 0 1 3 1, which, when written in reverse order, is 13,107,200,000. One section of this same work gives detailed astronomical observations that confirm to modern scholars that this was written on the date it claimed to be written: 25 August 458 CE (Julian calendar). As Ifrah points out, this information not only allows us to date the document with precision, but it also proves its authenticity.

    Fifty-two years later, in 510 CE, Indian mathematician Aryabhata described schemes for various arithmetic operations, even including square roots and cube roots, which most likely were known even earlier than this date. Aryabhata’s actual scheme for computing square roots, as described in greater detail in a 628 CE manuscript by a faithful disciple named Bhaskara I, is presented in full in Ifrah’s book [Ifrah2000, pg. 361-362]. Aryabhata even gave a decimal value of pi = 3.1416 [Ifrah2000, pg. 361-362]. From these and other sources there can be no doubt that our modern system of arithmetic — differing only in the symbols used for the digits — originated in India at least by the year 458 CE and probably several decades earlier.

    Modern History

    It is both disappointing and perplexing that this seminal development in the history of mathematics is given such little attention in contemporary published histories. For example, in one popular work on the history of mathematics, although the author describes Arab and Chinese mathematics in significant detail, he mentions the discovery of decimal arithmetic in India only in one two-sentence passage [Burton2003, pg. 253]. Another popular history of mathematics mentions the discovery of the “Hindu-Arabic Numeral System,” but says only that

    Positional value and a zero must have been introduced in India sometime before A.D. 800, because the Persian mathematician al-Khowarizmi describes such a completed Hindu system in a book of A.D. 825. [Eves1990, pg. 23]

    A third historical work briefly mentions this discovery, but cites a 662 CE Indian manuscript as the earliest known source [Katz1998, pg. 221]. A fourth reference states that the combination of decimal and positional arithmetic “appears in China and then in India” [Struik1987, pg. 67]. None of these authors devotes more than a few sentences to the subject, and, more importantly, none suggests that this discovery is regarded as particularly significant.

    We entirely agree with Ifrah that this discovery is of the first magnitude. The mere fact that the system is now taught in grade schools worldwide, and is implemented (in binary) in every computer ever manufactured, should not detract from its historical significance — to the contrary, these same facts emphasize the enormous advance that this system represented over earlier systems, both in simplicity and efficiency, as well as the huge importance of this discovery in modern life.

    Perhaps some day we will finally learn the identity of this mysterious Indian mathematician. If we do, we surely must accord him or her the same honors that we have granted to Archimedes, Newton, Gauss and Ramanujan.

    References

    1. [Berggren2004] L. Berggren, J. M. Borwein and P. B. Borwein, Pi: a Source Book, Springer-Verlag, New York, third edition, 2004.
    2. [Burton2003] David M. Burton, The History of Mathematics: An Introduction, McGraw-Hill, New York, 2003.
    3. [Dantzig2007] Tobias Dantzig and Joseph Mazur, Number: The Language of Science, Plume, New York, 2007.
    4. [Eves1990} Howard Eves, An Introduction to the History of Mathematics, Holt, Rinehart and Winston, New York, 1990.
    5. [Gupta1983] R. C. Gupta, “Spread and triumph of Indian numerals,” Indian Journal of Historical Science, vol. 18 (1983), pg. 23-38, available at Online article.
    6. [Ifrah2000] Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French by David Vellos, E. F. Harding, Sophie Wood and Ian Monk, John Wiley and Sons, New York, 2000.
    7. [Katz1998] Victor J. Katz, A History of Mathematics: An Introduction, Addison Wesley, New York, 1998.
    8. [Netz2007] Reviel Netz and William Noel, The Archimedes Codex, Da Capo Press, 2007.
    9. [Marchant2008] Josephine Marchant, Decoding the Heavens: Solving the Mystery of the World’s First Computer, Arrow Books, New York, 2008.
    10. [Stillwell2002] John Stillwell, Mathematics and Its History, Springer, New York, 2002.
    11. [Struik1987] Dirk J. Struik, A Concise History of Mathematics, Dover, New York, 1987.
    ]]> http://experimentalmath.info/blog/2010/02/the-greatest-mathematical-discovery/feed/ 0 http://experimentalmath.info/blog/2010/01/the-confusing-morass-of-copyright-laws/ http://experimentalmath.info/blog/2010/01/the-confusing-morass-of-copyright-laws/#comments Thu, 28 Jan 2010 18:03:52 +0000 David Bailey http://experimentalmath.info/blog/?p=221 Copyright law has always been a confusing arena, but recent developments have grave future consequences.

    We begin by noting that most of the world lives under different copyright laws: European Union, Commonwealth, Japanese, and other dispensations differ widely. See the CEIC’s writings for a record of details relevant to mathematical publishing. For instance, under Canadian copyright law, known as “Cancopy,” library use often generates royalties which the government gave away to publishers without individual ability to demur. Margaret Atwood does see them; small fish do not. Originally copyright was the concern of printers eager to protect their investment. Copyright for authors for 14 years, once renewable, was granted by the UK’s 1709 “Act of Anne”. For most purposes we would be better off if this were reinstated.

    For example, as Robert Nagle observes in an online article, the international Berne Convention agreed that all images by artists who had died 70 years or more ago would be in the public domain. At the same time, as a result of the U.S. court decision Corel vs. Bridgeman, it is legal for people to copy any image of a public domain painting provided it is a “faithful” reproduction. Thus artists could lose control of their creations unless they are the first to publish a quality photograph of it. Also, if an artist died prior to 1937, then photographs or scans of this artist’s works are in the public domain no matter when the photograph was actually taken.

    However, in the U.S., a painting is automatically in the public domain if the artist died 70 years ago except in cases where the artwork was originally published between 1978 and 2003. In that case, the artwork will not belong to the public domain (at least in the U.S.) until the year 2047. So in other words, one must first verify that an artwork was photographed or reprinted prior to 1978. How can one possibly do that?

    Confusion over this issue has led to a serious dilemma for the Wikipedia project. Because Wikipedia’s servers are hosted in Florida, they must comply with the stricter regulations passed by the U.S. Congress. As a result, many images on Wikipedia pages (and Wiktionary, Wikibooks and other members of the Wiki family) are now unavailable in the USA, even though those images are freely available almost everywhere else in the world.

    Confusion over copyrights also plagues Google’s efforts to scan and index millions of books, as Lawrence Lessig observes in a recent New Republic article. Google originally planned that if the book was in the public domain, then Google would grant readers full access and even permit downloads, but if it was protected by copyright, then Google would only permit one to see a few lines near the search target. If the work was still in print, then Google would request from publishers instructions on how to handle the text. But the Authors Guild and the Association of American Publishers argued that when Google scans these books to build its index, it made a “copy” of them, requiring explicit permission from the copyright owner. They argued that it doesn’t matter that Google scans these works simply to index them, or that Google would never distribute copies — Google must have permission to scan, otherwise it is guilty of piracy.

    For the 16% of the 18,000,000 books Google wished to scan that are in the public domain, there is no issue, nor is there much of an issue (other than haggling about the price) for the 9% that are still in print. But for the 75% that are out of print but in many libraries, the rule claimed by these organizations is fatal to Google’s effort. Fortunately, Google was able to reach an out-of-court settlement — Google would pay for the right to make available the roughly 20% of books whose author could not be available; for others, the public would pay to access the full book, with funds granted to a new non-profit organization that would pay royalties to those authors who want them.

    But there are complications with this arrangement. Journal articles are handled differently than books, and the terms are very complex. Lessig relates that just after his wife had given birth to their third child, the child fell into a state of lethargy, with symptoms suggestive of jaundice. While waiting to see a doctor, he accessed through Google a journal article that discussed jaundice and complications. But in a critical part of the article, a table was missing; in its place was the notice “The rightsholder did not grant rights to reproduce this item in electronic media.” As Lessig asked, “Before we continue any further down this culturally asphyxiating road, can we think about it a little more? … Before we release a gaggle of lawyers to police every quotation appearing in any book, can we stop for a moment to consider whether this way of organizing access makes any sense?”

    It is clear that this morass of copyright rules must be greatly simplified. Lessig, for instance, recommends that we shift to copyright owners some of the burden of keeping the copyright up to date, by requiring them to register their work after an initial fixed period of time. Failure to explicitly register the work would permit it to pass to the public domain. Governments should not operate these registries, but instead simply establish protocols for services that compete to hold these registries (similar to the system of handling Internet domains). Lessig also recommends that for any compiled work more than 14 years old, the law should secure rights to preserve the work without burden to the owner. Other details of Lessig’s proposal are given in his
    New Republic article.

    In any event, the present bloggers agree that this issue must be addressed in the next few years, before an explosion of digital work greatly amplifies the current clumsy state of confusion. Authors and publishers deserve a clear-cut and transparent means of protecting legitimate work and value added (it is worth noting that for legal purposes you may not be the “author” of the article or book you slaved over). The public deserves an equally transparent means of accessing work at a reasonable cost. The longer we wait before acting, the nastier the mess will become.

    Even now, it is already far from simple. It took one of the present bloggers and his coauthors under a year to determine the content of Pi: a Sourcebook, but four additional years to trace copyrights, obtain permissions, haggle over fees, and so on. Luckily in most modern cases the copyright was held by journals, not individuals, so at least there was a clear place to look. In one case, the British museum had to be paid for an image of the Rhind papyrus, circa 1650 BCE. In two cases, copyright issues changed the content of the book, as the permission fee the copyright holder requested was too outrageous to pay. These disputes were exhausting and seriously detracted from the substance of the scholarship.

    Since we originally posted this on 28 Jan 2010, an interesting essay that deals with this topic (among several others) written by Charles Leadbetter appeared on the Edge.org website: Edge article. Here is a brief summary of his conclusions:

    If content in the cloud is entangled in copyright and other forms of intellectual property then it will become increasingly difficult to mingle, match and collaborate. The creative potential of the web, to create new mixes, will be vastly reduced. To promote more open cultural relations on the web we should focus on:

    1. Finding collaborative solutions to the problem of orphaned works, perhaps by allocating them to forms of collective ownership, which would make it far simpler for people seeking to enjoy or adapt the content to negotiate rights. The collective owners would own the rights and hold money for the original rights holders.
    2. Governments should resist attempts to extend copyright terms.
    3. The copyright regime should increasingly put the onus on rights holders to justify their need for copyright and to pay for extensions. Any work not re-copyrighted after the expiry of its original term would automatically fall into public ownership rather than being orphaned.
    4. The presumption should be that all cultural products are in the public domain after a basic period of copyright or intellectual protection has expired.
    5. New forms of creative licensing are required, modelled on open access and creative commons, which are designed to allow sharing but also to clearly apportion credit to original work and authors.
    6. Most media industries will need new business models, which are tailored to allow more interaction with content and more peer-to-peer distribution. Countries that experiment successfully with these models will lead the next wave of cultural and creative industries.
    7. Finding ways to create more Pro-Am cultural exchanges which bring together the best of professional and amateur content.

    Other references:

    1. John Ewing, “Copyright and Authors,” 22 Sep 2003, available at Online article.

    2. “The Statute of Anne,” 1710, available at Online article.

    ]]> http://experimentalmath.info/blog/2010/01/the-confusing-morass-of-copyright-laws/feed/ 0 http://experimentalmath.info/blog/2010/01/sad-state-of-math-and-science-education/ http://experimentalmath.info/blog/2010/01/sad-state-of-math-and-science-education/#comments Fri, 08 Jan 2010 01:49:27 +0000 David Bailey http://experimentalmath.info/blog/?p=200

    The latest results for math and science education in first-world nations such as the U.S., the major European nations, and Australia are not particularly encouraging. In the following table, the first two columns contain the latest results from the “Trends in International Mathematics and Science Study” (TIMSS) for Grade Four and Grade Eight, respectively [Institute2009], while the third column contains rankings of math performance among 15-year-olds in a separate study by the OECD [OECD2003]:

    
    

    Grade Four TIMSS Rankings Hong Kong (607) Singapore (599) Chinese Taipei (576) Japan (568) Kazakhstan (549) Russian Federation (544) England (541) Latvia (537) Netherlands (535) Lithuania (530) United States (529) Germany (525) Denmark (523) Australia (516) Hungary (510) Italy (507) Austria (505) Sweden (503) Slovenia (502) Armenia 9500) Slovak Republic (496) Scotland (494) New Zealand (492) Czech Republic (486) Norway (473) Ukraine (469) Georgia (438) Iran (402) Algeria (378) Colombia (355)

    Grade Eight TIMSS Rankings Chinese Taipei (598) Republic of Korea (597) Singapore (593) Hong Kong (572) Japan (570) Hungary (517) England (513) Russian Federation (512) United States (508) Lithuania (506) Czech Republic (504) Slovenia (501) Armenia (499) Australia (496) Sweden (491) Malta (488) Scotland (487) Serbia (486) Italy (480) Malaysia (474) Norway (469) Cyprus (465) Bulgaria (464) Israel (463) Ukraine (462) Romania (461) Bosnia/Herzegovina (456) Lebanon (449) Thailand (441) Turkey (432)

    15-Year-Olds OECD Math Rankings Hong Kong Finland South Korea Netherlands Liechtenstein Japan Canada Belgium Macao Switzerland Australia New Zealand Czech Republic Iceland Denmark France Sweden Austria Germany Ireland Slovak Republic Norway Luxembourg Poland Hungary Spain Latvia United States Russian Federation Portugal

    (Note: Not all nations participate in these studies, so, for instance, Canada ranks seventh in the third table, but did not participate in and thus does not appear in first two tables.)

    In comparing the above data, it is interesting to note that while first-world nations do fairly well among 15-year-old students, the Asian “tigers,” namely Hong Kong, Taiwan, Japan, Singapore and Korea, dominate the top positions for fourth and eighth graders. This seems to indicate that the Asian countries are on the rise, and that future rankings will show them superior even among 15-year-olds.

    U.S. performance is mediocre among fourth and eighth grade students, ranking below the Asian tigers, but is downright dismal among 15-year-olds. This by itself is not a reason to declare disaster. What is a disaster is that this disappointing performance is delivered by a nation whose economy, arguably more than another single nation, is dependent on a steady stream of top math- and science-educated workers. Perhaps even more worrisome for the U.S. is the fact that in spite of a greatly increased focus on education, especially K-12 education, for at least the past 10-15 years, improvement has only been modest. In fact, since 1995, U.S. grade four scores on the TIMSS study have increased only 11 points, as opposed to 57 points for England, 50 points for Hong Kong, and 40 points for Slovenia. Similarly, at grade eight, the U.S. improvement of 16 points ranks well behind Colombia (47 points) and Lithuania (34 points) [Institute2009]. Other U.S. studies have come to a similar conclusion. Average fourth-grade math scores on the National Assessment of Education Progress were flat between 2009 and 2008, and average eighth-grade math scores increased only two points. Another fact of major significance is that the scoring gaps between white students and Hispanic and African-American students have not changed much in recent years [Tomsho2009].

    The “threat” of the Asian tigers is real. China, for example, has made remarkable progress in scientific research. In 1998, China’s research output was 20,000 articles per year. In 2006, it reached 83,000, overtaking Japan, Germany and the U.K. Last year it reached 120,000 articles, second only to the U.S. at 350,000, and is on track to surpass the U.S. by 2020 [Adams2010]. Obviously, quantity is not the same as quality, and some have expressed concern that only a fraction of these papers truly contain top-tier results. But given that China is home to nearly 25% of the world’s population, it is only a question of when, rather than if, China will become the world’s most prolific producer of scientific knowledge [Adams2010]. In a similar vein, China graduates more engineers than the U.S., and thus is well poised to become the world’s high-tech manufacturing center for the 21st century [Friedman2007, pg. 257].

    India has made similar strides, although not quite as dramatic as China. In his book The World Is Flat, Thomas Friedman documents that in the same way that China is well-poised to be a leader of manufacturing, India is similarly well poised to become a dominant center for high-tech services. Already India has numerous centers for accounting (done on behalf of first-world clients) and even medical “tourism.” As Friedman notes, both of these two nations are not racing the first world to the bottom or the middle of the economic pyramid; instead they are racing to the top [Friedman2007, pg. 265].

    Also, while the many Asian scientists and engineers now laboring in first-world nations are a great blessing to the West, now that China and India, in particular, are making great strides forward economically, some of those same scientists and engineers are now being lured back to their home countries [LaFraniere2010]. Thus, first-world nations cannot rely exclusively on imported talent.

    So how serious are these problems? What should first-world nations do?

    This week U.S. President Barack Obama has announced a $250 million program to improve math and science education [Anderson2010]. Together with matching funding from various high-tech business such as Intel, and several universities and foundations, the program seeks to prepare more than 10,000 new well-qualified math and science school teachers over the next five years, and to upgrade the training for an additional 100,000. As Obama declared, “Passionate educators with deep content expertise can make all the difference, enabling hands-on learning that truly engages students — including girls and underrepresented minorities — and preparing them to tackle the ‘grand challenges’ of the 21st century such as increasing energy independence, improving people’s health, protecting the environment and strengthening national security.” [Anderson2010].

    In a related development, several U.S. Congressmen have introduced a bill that proposes making it easier for students who have received advanced degrees in science, technology, engineering, and mathematics from U.S. universities to obtain green card employment visas, rather than lingering in H-1B “visa limbo” [McGee2009]. Similar initiatives are advancing in some other first-world nations.

    Commendable as these developments are, we still believe that fundamental structural changes must be made not only in the U.S. but throughout the E.U. and the Commonwealth as well. To begin with, a good part of the reason that the Asian “tiger” countries do so well is that the students simply spend more time in school and more time doing homework. Euclid is said to have replied to King Ptolemy’s request for an easier way of learning mathematics that “there is no royal road to geometry.” The same could be said of numerous topics of modern mathematics and science. Harold Stevenson of the University of Michigan, after spending several years researching differences between U.S. schools and those in China, Japan and Taiwan, found that Asian pupils spend almost 50% more time per week in class, and their school year is about one-third longer (there is no such thing as a summer vacation). In addition, many Asian students enroll in additional private tutoring [Eskildson2010].

    Another significant difficulty is the persistence of a significant anti-science mentality, which is particularly stark in the U.S., but is growing in the U.K., Europe and the English-speaking commonwealth as well. In a 2004 poll, 45 percent of Americans agreed that “God created human beings pretty much in their present form at one time within the last 10,000 years or so.” [Gallup2004]. In a 2005 poll, 42 percent of Americans agreed that “humans and other living things have existed in their present form since the beginning of time.” [Pew2005]. In a similar 2006 poll in Great Britain, 22% selected “God created human kind pretty much in his/her present form at one time within the last 10,000 years” among four listed survey options [BBC2006]. These statistics are stupefying in an era where mutated diseases, such as the recent antibiotic-resistant Tuberculosis and drug-resistant HIV strains, are everyday news and threaten millions of people, and where petroleum engineers routinely drill through fossil layers laid down many millions of years ago. Such statistics indicate a fundamental hostility to the entire enterprise of scientific research, and are exhibited in reluctance to increase public funding for math and science education. Until this hostility is adequately dealt with, there is not much prospect for significantly improved educational performance in major first-world nations.

    References

    1. [Adams2010] Jonathan Adams, “Get Ready for China’s Domination of Science,” New Scientist, 06 Jan 2010, available at
       Online article.

    2. [Anderson2010] Nick Anderson, “White House Announces $250M Effort for Science and Math Teachers,” Washington Post, 6 Jan 2010, available at
       Online article.

    3. [BBC2006] BBC, “Britons Unconvinced on Evolution,” BBC World News, 26 Jan 2006, available at
       Online article.

    4. [Eskildson2010] Loyd Eskildson, “Asian Students Spend 50% More Time in Class, School Year 1/3 Longer,” 6 Jan 2010, available at
       Online article.

    5. [Friedman2007] Thomas L. Friedman, The World Is Flat: A Brief History of the Twenty-First Century, Picador, New York, 2007.
    6. [Gallup2004] Gallup Poll, 2004, available at
       Online article.

    7. [Institute2009] Institute for Education Sciences, “Trends in International Mathematics and Science Study”, U.S. Department of Education, 2007 study updated with a 2009 overview, available at
       Online article.

    8. [LaFraniere2010] Sharon LaFraniere, “Fighting Trend, China Is Luring Scientists Home,” New York Times, 6 Jan 2010, available at
       Online article.

    9. [McGee2009] Marianne Kolbasuk McGee, “Work Visas Back On Congressional Agenda,” Government Information Week, 16 Dec 2009, available at
       Online article.

    10. [OEDC2003] Organization for Economic Cooperation and Development (OECD), “International Comparison of Math, Reading and Science Skills Among 15-Year-Olds,” 2003, available at
        Online article.

    11. [Pew2005] Pew Forum survey, 2005, available at
        Online article.

    12. [Tomsho2009] Robert Tomsho, “U. S. Math Scores Hit a Wall: National Test Shows No Gains for Fourth-Graders, Slight Rise for Eighth-Graders,” Wall Street Journal, 15 Oct, 2009, pg. A3, available at:
        Online article.

    ]]> http://experimentalmath.info/blog/2010/01/sad-state-of-math-and-science-education/feed/ 0 http://experimentalmath.info/blog/2009/11/192/ http://experimentalmath.info/blog/2009/11/192/#comments Sun, 15 Nov 2009 22:07:00 +0000 David Bailey http://experimentalmath.info/blog/?p=192 [This is a condensed version of a paper written by one of the present bloggers (Borwein). For the full article, with references, see http://www.carma.newcastle.edu.au/~jb616/psychology.pdf.]

    Some years ago, my brother Peter surveyed other academic disciplines. He discovered that students who bitch mightily about calculus professors still prefer the relative certainty of how-and-what we teach-and-assess to the subjectivity of a creative writing course or the rigors of a physics or chemistry laboratory course. Similarly, while I have met my share of micro-managing Deans–who view mathematics with disdain when they look at the size of our research grants or the infrequency of our patents–I have encountered more obstacles to mathematical innovation within than without the discipline.

    Why do we produce so many unneeded results? In addition to the obvious pressure to publish and requirements to have something to present at the next conference, I suspect Irving Biederman’s observations below plays a significant role: “While you’re trying to understand a difficult theorem, it’s not fun,” said Biederman, professor of neuroscience in the USC College of Letters, Arts and Sciences, “But once you get it, you just feel fabulous.” The brain’s craving for a fix motivates humans to maximize the rate at which they absorb knowledge, he said. “I think we’re exquisitely tuned to this as if we’re junkies, second by second.”

    Sometimes we sit firmly and comfortably in the sciences, sometimes we are–as the Economist recently noted–the most inaccessible of the arts. Why should the non-mathematician care about things of this nature? The foremost reason is that mathematics is beautiful, even if it is, sadly, more inaccessible than other forms of art: “a supreme beauty–a beauty cold and austere” in Russell’s terms and sometimes we sit or feel we sit entirely alone. On the other hand, the human genome project, the burgeoning development of financial mathematics, finite element modeling, Google and much else has secured the role of mathematics within modern science and technology research and development as “the language of high technology.”

    One of the epochal events of my childhood as a faculty brat in St. Andrews, Scotland was when C. P. Snow (1905-1980) delivered an immediately influential 1959 Rede Lecture in Cambridge entitled “The Two Cultures.” Snow wrote:

    A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics, the law of entropy. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: `Have you read a work of Shakespeare’s?’

    I now believe that if I had asked an even simpler question – such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, ‘Can you read?’ – not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their Neolithic ancestors would have had.

    I doubt I have ever met a scientist who had not read (or at least watched on BBC) some Dickens, who never went to movies, art galleries or the theatre. It is, however, socially acceptable to be a scientific ignoramus or a mathematical dunce. It is largely allowed to boast “I was never any good at mathematics at school.” I was once told exactly that–in soto voce–by the then Canadian Governor General during a formal ceremony at his official residence in Ottawa.

    As Underwood Dudley has commented, no one apologizes for not being good at geology in school. Most folks understand that failing “Introduction to Rocks” in Grade Nine does not knock you off of a good career path. The outside world knows several truths: mathematics is important, it is hard, it is usually poorly taught in school, and the average middle-class parent is ill-prepared to redress the matter. I have become quite hard-line about this. When a traveling companion on a plane starts telling me that “Mathematics was my worst subject in school.” I will reply “And if you were illiterate would you tell me?” They usually take the riposte fairly gracefully.

    Consider two currently popular TV dramas, Numb3rs (mathematical) and House (medical). A few years ago a then colleague, a distinguished pediatrician, asked me whether I watched Num3rs. I replied “Do you watch House? Does it sometimes make you cringe?” He admitted that it did but he still watched it. I said the same was true for me with Numb3rs, that my wife loved it and that I liked lots about it. It made mathematics seem important and was rarely completely off base. The lead character, Charlie, was brilliant and good-looking with a cute smart girl friend. The resident space-cadet on the show was a physicist not a mathematician. What more could one ask for? Sadly for many of our colleagues the answer is “absolute fidelity to mathematical truth in every jot and title.” No wonder so many of us make a dog’s-breakfast of the opportunities given to publicize our work!

    I once wrote “It is certainly rarer to find a mathematician under thirty who is unfamiliar with at least one of Maple, Mathematica or Matlab, than it is to one over sixty five who is really fluent. As such fluency becomes ubiquitous, I expect a re-balancing of our community’s valuing of deductive proof over inductive knowledge.”

    At a more fundamental level, I see the discipline boundary as being best determined by answering the question as to whether the mathematics at issue is worth doing in its own right. If the answer is “yes” then it belongs in the discipline; if not then, however useful or important the outcome, it does not. The later would, for example, be the case of a lot of applied operations research, a good deal of numerical modeling and scientific computation, and most of statistics. All significant mathematics should be nourished within mathematics departments, but there are many important and useful applications that do not by that measure belong.

    We have a lot to catch up with. We have too few prizes. We are insufficiently adept at boosting our own cases for tenure, promotion or for prizes. We are frequently too honest in reference letters. We are often disgracefully terse–unaware of the need to make obvious what is for us obvious. I have seen a Field’s medalist recommend a talented colleague for promotion with the one line letter “Anne has done some quite interesting work.” Leaving aside the ambiguity of the use of the word “quite” when sent by a European currently based in the United States to a North American promotion committee, it is pretty lame when compared to a three page letter for an astrophysicist or chemist which almost always tells you the candidate is the top whatever-it-is in the field.

    I’m not encouraging dishonesty, but it is necessary to understand the ground rules of the enterprise and to make some attempt to adjust to them. When a good candidate for a Rhodes Scholarship turns up at ones office, it should be obvious that a pro forma “Johnny is smart and got an A+ in my advanced algebraic number theory class. You should give him a Rhodes scholarship.” is inadequate. Yet the only letters of that kind that I’ve seen in Rhodes scholarship dossiers have come from mathematicians.

    I became a mathematician largely because it satisfied three criteria. (i) I found it reasonably easy (ii) I liked understanding or working out how things function, but (iii) I was not much good with my hands and had limited physical intuition, I really disliked pipettes. That left mathematics. I have had several students whom I can not imagine following any other life path but I was not one of those. I would I imagine have been happily fulfilled in various careers of the mind; say as an historian or an academic lawyer. But I became a mathematician. It has been and continues to be a wonderful life.

    ]]> http://experimentalmath.info/blog/2009/11/192/feed/ 0 http://experimentalmath.info/blog/2009/10/john-d-barrows-new-theories-of-everything/ http://experimentalmath.info/blog/2009/10/john-d-barrows-new-theories-of-everything/#comments Mon, 12 Oct 2009 17:36:13 +0000 David Bailey http://experimentalmath.info/blog/?p=179 John D. Barrow, New Theories of Everything, Oxford University Press, 2007.

    Both of the present bloggers have enjoyed Barrow’s previous works. Bailey was so enthralled with Barrow and Tipler’s 1988 book The Anthropic Cosmological Principle that he read every word of its 736 pages multiple times. Borwein (with his brother Peter) wrote a favorable review of Barrow’s 1992 book Pi in the Sky for the publication Science.

    Barrow’s latest book, New Theories of Everything, does not disappoint. In this wide-ranging work, Barrow examines the notion of viewing science as the search for algorithmic compression of observed data. In other words, the best scientific theory is the one that explains the most data precisely in as crisp a manner as possible.

    Barrow examines this proposition from many different angles, including physics, cosmology, mathematics, mathematical logic, computer science, biology, history, philosophy and religion. This material is so well organized, and so lucidly written, that readers are bound to learn many things of interest, no matter what their backgrounds. Advanced knowledge of these fields is not required (although it will help).

    Here are a few excerpts from this fascinating work:

    [pg 11] On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting – the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic [pg 12] behind the Universe’s properties that can be written down in finite form by human beings.

    [pg 52] [quoting Freeman Dyson] Godel proved that the world of pure mathematics is inexhaustible; no finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. I hope that an analogous situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we got into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.

    [pg 113] [quoting Albert Einstein] however, we select from nature a complex [of phenomena] using the criterion of simplicity, in no case will its theoretical treatment turn out to be forever appropriate. … But I do not doubt that the day will come when that description [the general theory of relativity], too, will have to yield to another one, for reasons which at present we do not yet surmise. I believe that this process of deepening the theory has no limits.

    [pg 121] There is no reason why life has to evolve in the Universe. Such complex step-by-step processes are not predictable because of their very sensitive dependence upon the starting conditions and upon subtle interactions between the evolving state and the ambient environment. All we can assert with confidence is a negative: if the constants of Nature were not within one percent or so of their observed values, then the basic buildings blocks of life would not exist in sufficient profusion in the Universe. Moreover, changes like this would affect the very stability of the elements and prevent the existence of the required elements rather than merely suppress their abundance.

    [pg 138] Somehow the breathless world that we witness seems far removed from the timeless laws of Nature which govern the elementary particles and forces of Nature. The reason is clear. We do not observe the laws of Nature: we observe their outcomes. Since these laws find their most efficient representation as mathematical equations, we might say that we see only the solutions of those equations not the equations themselves. This is the secret which reconciles the complexity observed in Nature with the advertised simplicity of her laws.

    [pg 222] That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results.

    [pg 231] In practice, the intelligibility of the world amounts to the fact that we find it to be algorithmically compressible. We can replace sequences of facts and observational data by abbreviated statements which contain the same information content. These abbreviations we often call “laws of Nature.” If the world were not algorithmically compressible, then there would exist no simple laws of nature. Instead of using the law of gravitation to compute the orbits of the planets at whatever time in history we want to know them, we would have to keep precise records of the positions of the planets at all past times; yet this would still not help us one iota in predicting where they would be at any time in the future. This world is potentially and actually intelligible because at some level it is extensively algorithmically compressible. At root, this is why mathematics can work as a description of the physical world. It is the most [pg 232] expedient language that we have found in which to express those algorithmic compressions.

    ]]> http://experimentalmath.info/blog/2009/10/john-d-barrows-new-theories-of-everything/feed/ 0