Math Drudge

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.

Review/synopsis of Unscientific America: How Scientific Illiteracy Threatens Our Future, by Chris Mooney and Sheril Kirschenbaum, Basic Books, NY, 2009:

Carl Sagan, in his 1995 book The Demon-Haunted World issued this sober warning:

We’ve arranged a global civilization in which most crucial elements profoundly depend on science and technology. We have also arranged things so that almost no one understands science and technology. This is a prescription for disaster. We might get away with it for a while, but sooner or later this combustible mixture of ignorance and power is going to blow up in our faces.

As Mooney and Kirschenbaum observe, “At present we’re marching steadily toward that outcome.”

The authors note that on the plus side, Americans are surrounded by technology and see first-hand the fruits of scientific research. The Internet and other information technologies have made it much easier to disseminate accurate scientific information. But this same technology also makes it possible to disseminate all sorts of nonsense, and to amplify the fallacy-ridden writings of the creation science and global warming denial communities.

In other arenas, the authors lament that only minor progress has been made. Long gone are the glory days of Sagan’s “Cosmos” series on PBS. Nowadays Hollywood and network TV mostly avoid scientists and scientific themes, and when they do deal with a scientific theme or scientific undercurrent, they prefer paranormal topics (UFOs, time travel, etc) and frequently depict scientists as villains, geeks, and jerks. There are a few bright spots — Numbers, a few of the CSI epsiodes and others — but by and large the Hollywood/TV picture is pretty discouraging.

Mooney and Kirschenbaum criticize writers such as Sam Harris, Christopher Hitchens, Richard Dawkins and Daniel Dennett for going out of their way to attack religious belief and believers. In the authors’ opinion,

If the goal is to create an America more friendly toward science and reason, the combativeness of the New Atheists is strongly counterproductive. If anything, they work in ironic combination with their dire enemies, the anti-science conservative Christians who populate the creation science and intelligent design movements, to ensure we’ll continue to be polarized over subjects like the teaching of evolution when we don’t have to be.

One of the more interesting parts of this book is the authors’ account of how Carl Sagan, who arguably was the most successful communicator of science in history, was himself snubbed by the scientific establishment for his efforts. His 1992 nomination to the National Academy of Sciences was rejected, mainly on grounds of his public endeavors. The message was clear to any current or aspiring scientist: engage the public at the peril of your career.

To address these problems, Mooney and Kirschenbaum call for nothing less than a fundamental restructuring of the scientific establishment. First of all, scientists themselves must squarely face the abysmal job they have done in communicating their research to the public. They cannot rely on advanced technology such as Internet blogs here, since these same technologies are just as effective in spreading nonsense. Instead, training in and encouragement for the dissemination of results to the public must be incorporated into the curriculum of every scientific field.

Mooney and Kirschenbaum further argue that society must rethink the financial reward system for those who pursue scientific careers. As a recent blog entry cited in the book laments, “Had I to do it over again, I would not choose a PhD, at least not a general science degree. I would have gone to medical or law school, or perhaps a PhD in public health (a very rapidly growing field). At least after training in these programs your skill set is clearly defined, and you can be confident that you will have a job post-graduation”.

In their conclusion, the authors quote C.P. Snow, “We require a common culture in which science is an essential component. Otherwise we shall never see the possibilities, either for evil or good.”

See also Chris Mooney’s recent essay at HuffingtonPost.com.

Richard C. Brown, Are Science and Mathematics Socially Constructed?: A Mathematician Encounters Postmodern Interpretations of Science, World Scientific, 2009.

In this book, Brown recounts the rise of what is now known as the “postmodern interpretations of science” (PIS) or “sociology of scientific knowledge” (SSK) movement. In addition to pioneers Karl Popper and Thomas Kuhn (the latter of whom Brown personally knew), the author describes the contributions of Berkeley philosopher Paul Feyerabend; Harry Collins and Trevor Pinch at Bath University; Steve Woolgar at Brunel; Michel Callon and Bruno Latour in Paris; a group of scholars at the University of Edinburgh; and numerous others.

Brown emphasizes that many of these writers start with a premise that is basically sound (and with which the present reviewer fully agrees): both science in general and mathematics in particular are unavoidably human enterprises, and are subject to all the varieties of human weakness. There are numerous instances of major errors in mathematical proofs (for example, in the original proof of Fermat’s Last Theorem by Andrew Wiles). Some scientific “discoveries”, such as “N”-rays and “cold fusion”, proved short-lived and vacuous (although some still hold hope that some form of cold fusion will prove real). Other scientific discoveries were founded or bolstered by experimental evidence that was later found to be flawed (such as the original observations of the bending of light around the sun, in tests of general relativity). In some instances, it appears that new theories initially prevailed as much because of the persuasiveness and personality of a leading researcher as the scientific merits of the claims. In this sense, mathematics and science are indeed “socially constructed”.

Some SSK writers, such as Karl Popper (who emphasized the value of highly falsifiable theories) and Thomas Kuhn (who analyzed the phenomenon of paradigm shift) have been quite influential and accepted, and their teachings (with some notable exceptions) have become part of the scientific enterprise. But many of the more recent SSK writers go further. They argue that claims of mathematics and science are strongly determined by the ideology and economic class of the actors, and in many, if not most, cases not because of compelling logic or experimental evidence. Thus nature cannot be interpreted independently of the theories and conceptual paradigms that scientists bring to their tasks. As a consequence, the vaunted objectivity of science, and the claimed progress of science through the years, are illusions, and science is merely an ideology on a par with various religions, the myths of native cultures, astrology, and even the “creation science” and “intelligent design” theories advanced by modern-day evangelicals.

Many of these writers emphasize the downside of technology and argue that mathematics and science are tools of western corporate capitalism and have been employed in the repression of women and minorities. SSK scholar Sandra Harding once described Newton’s Principia as a “rape manual”. Many have a rather low opinion of the scientific enterprise. SSK scholar Andrew Ross contemptuously dedicated one of his books to “all the science teachers I did not have; it could only have been written without them”.

Latour and Woolgar argue that laboratory procedures, experimental results and scientific theories should be regarded as rituals, no different from the rituals of some tribal shaman; they may be reported and analyzed, but certainly should not be believed by the “anthropologist”. They also argue, for instance, that the Thyrotropin Releasing Factor (TRF) hormone was “constructed” by Guillemin’s laboratory in 1977; that Robert Koch “constructed” the tuberculosis bacillus in 1882 — it had no real existence prior to that date; and, similarly, Pasteur’s microorganisms did not exist in a strict sense before Pasteur “discovered” them.

Until the 1990s, few mathematicians or scientists were even aware of the SSK movement, even though articles of the SSK genre were widely published in some leading humanities and social studies journals. During that decade, some scientists (Gross, Levitt and others) launched a counter-offensive. They highlighted errors of scientific fact and common sense, noted passages of meaningless technical jargon, and accused the SSK community of greatly exaggerating various scientific controversies. The resulting “science wars” came to a head in 1996 when NYU physicist Alan Sokal wrote the parody “Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity”. Sokal’s article included approving quotes from the writings of SSK scholars, profuse usage of erudite scientific jargon and flawed references to scientific theories, all wrapped in leftist political rhetoric. One sample paragraph is the following:

“In this way the infinite-dimensional invariance group erodes the distinction between the observer and observed; the pi of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity; and the putative observer becomes fatally de-centered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone.”

Note, in addition to the gratuitous technical jargon, Sokal’s assertion that pi and G are not constants! In spite of these flaws (deliberately inserted by Sokal so that any knowledgeable scientist could spot them), the article was accepted and published in Social Text, a leading postmodern science studies journal (in fact in a special issue devoted to the “science wars”). When shortly afterward he exposed the hoax, the episode drew worldwide attention, including front-page coverage in the New York Times. Sokal emphasized that he composed the hoax out of a sincere attempt to steer the SSK community, many of whose ideals he shares, away from nonsense and irrelevance: “Theorizing about the social construction of reality‚ won’t help us find an effective treatment for AIDS or devise strategies for preventing global warming. Nor can we combat false ideas in history, sociology, economics, and politics if we reject the notions of truth and falsity.”

In the present book under review, mathematician Richard C. Brown briefly summarizes the history of the “science wars”, although he does not attempt to rehash these debates, referring the reader to books by Gross and Levitt, Sokal and Bricmont, and others. Instead, Brown’s objective, as stated in the Preface, is to examine in some detail the philosophical and political genesis of the SSK movement, and then discuss how these debates relate to the field of research mathematics. His account is often deeply personal, for example when he gives a first-hand account of the political clashes (which resulted in at least one death) at the Mathematical Research Center at the University of Wisconsin-Madison during the early 1970s.

Brown’s most interesting and most useful material is his discussion of how these issues relate to mathematics. This material begins in Chapter 10, which he appropriately titles, “The Deconstruction of Mathematics”, followed by chapters on “Epistemic Issues” and “The Fallibility of Conventionalism and Fallibilism”. Here he points out some significant errors of reasoning in some of the SSK literature, such as when philosopher Paul Ernest denies that mathematical proof “has the absolute and extra-human basis of certainty presumed by absolutism”. Brown points out, for instance, that Ernest evidently does not clearly distinguish between mathematics as a formal game based on axioms and mathematics as a tool that permits approximate applications in the real world. In any event, Brown points out that few, if any, professional mathematicians truly hold an “absolutist” view in Ernest’s sense — the results of mathematics are not necessarily tied to the real world, and the mathematical literature has numerous mistakes (Brown acknowledges that two of his own papers were later found to have significant errors). Brown further points out that the proof of a mathematical proposition from certain axioms can be “absolutely” valid, whether or not the axioms are consistent. In summary, he observes: “The fearmongering, therefore, concerning mathematics of supporters of SSK seems an exaggeration, and considering the real contradictions within their system, calling mathematics ‘fallible’ on the basis of Godel’s theorems is especially ill-conceived”. [pg. 234]. Brown concludes,

“Whatever the philosophical problems about the status of mathematical ‘truth’, settled areas of mathematics — the pureed kind found in undergraduate or graduate level textbooks as distinct from the frontier — appear more certain than the claims of almost any other human discipline, including the hard sciences such as physics or chemistry.” [pg. 235]….

“Like Ol’ Man River, mathematics just keeps rolling along and produces at an accelerating rate ‘200,000 mathematical theorems of the traditional handcrafted variety … annually’. [quoting Davis and Hersh's book The Mathematical Experience, pg. 24]. Although sometimes proofs can be mistaken — sometimes spectacularly — and it is a matter of contention as to what exactly a ‘proof’ is — there is absolutely no doubt that the bulk of this output is correct (though probably uninteresting) mathematics.” [pg. 239].

In this reviewer’s view, Brown has made a valuable contribution to the philosophy of mathematics with this book. Even readers quite familiar with the “science wars” will find many new insights here into the history of these movements and their potential to further influence the scientific enterprise. The book is decidedly even-handed, offering as close to an objective view of both sides as one could hope for in a single book, even though the author, towards the end, clearly points out some of the weaknesses in the SSK reasoning. And Brown’s analysis of the interplay of these issues with mathematical research is very well done, and commands careful analysis. All of this is crafted very skillfully. It is clear that Brown has a real talent for writing to a general audience, which is unfortunately rare among practicing research mathematicians. The book is well worth the purchase price.