Math Drudge

Forget the ‘precautionary principle.’ The amount of risk to which the public should be exposed is greater than zero. –Michael Krauss, Financial Post, June 20, 2008.

Almost without exception the critical or contentious issues of our times involve numbers–even “intelligent design” advocates usually try to juggle inconvenient dates or data. Errors with numbers are ubiquitous. Sometimes these are amusing as with:

Ideal Toy Company stated on the package of the original Rubik cube that there were more than three billion possible states the cube could attain. It’s analogous to MacDonald’s proudly announcing that they’ve sold more than 120 hamburgers.

(Recorded by J. A. Paulos in Innumeracy.)

Sometimes they may damage the innocent error-maker:

Dear Sirs: We just bought six packages of your blueberries from our local Costco (in San Ramon, CA). On the label it says that the net weight is “2.75 pounds”, or “0.9 Kg”. Indeed, the 2.75 pounds figure appears to be correct — I weighed one of our packages and it weighs 2.93 pounds, including the plastic case, so 2.75 pounds net weight is entirely credible. But in that case, “0.9 Kg” is not correct. There are 2.2046 pounds in a kilogram, so 2.75 pounds converted to kilos is roughly 1.25 kg (actually 1.247 kg), not 0.9 kg. In other words, your packages have more kilograms of blueberries than your label says they have.

I thought you might appreciate the note. David H Bailey

and the response (which did not include a year’s supply of blueberries) that went:

Dear David,

Thank you for bringing this to our attention. We are making the change on our label now. Hope you are enjoying the blueberries!

Best Regards, [owner of the Hurst's blueberry farm]

Others numerical errors have resulted in the “pentium bug”, Mars missions crashing, Patriot missiles overshooting their targets, and much more.

That said, the lack of numerical sense is more pervasively damaging to all modern public policy debates. All such debates involve a need to grasp notions of relative risk, be they about: infrastructure renewal, health care costs, exposure to carcinogens, nuclear power and waste, regulating nutritonal supplements and organic products, climate remediation, or … mad cow disease. As described by Simon Jenkins in Boneless Wonders in the Times of London, Dec 6, 1997:

The giant finger whooshes out of the night sky and points at the dumbstruck face in the window. “It could be you,” says a voice. This week the Agriculture Minister Jack Cunningham impersonated the National Lottery advertiser. As the nation’s fork was poised with a T-bone steak on its way to the nation’s mouth, Dr Cunningham screamed: “Don’t touch it.” According to the great god science, new variant Creutzfeldt-Jakob disease (nvCJD) could be lurking in that mouthful. There is a small risk, and where there is risk, a government must ban.

Perhaps only mathematicians are aware of the enormity of what the Government did this week. It took a risk that is statistically negligible and exploited it as an act of insufferable nannying. Beef ribs, T-bones and oxtails present a public health risk publicised as “very small” and “a chance of one case per year” (though none of Britain’s 22 nvCJD cases has been positively linked to beef). Most newspapers cluelessly converted “a chance” into a certainty, and ridiculed the risk as a tiny one in 56 million. But that is not what the scientists said. They suggested the chance was “5 per cent”, so the risk is nearer to one in 1.1 billion, or one in 560 million among the half of the population that eats beef. There can have been no more tenuous basis for an infringement of personal liberty.

But given a populace without the tools to distinguish real solutions to real threats from flavours of the week (e.g., H1N1 most recently, anthrax, exploding shoes and plastic knives for plane meals since 9/11, and obesity) how much choice does a democratically elected government have? Especially given media either too willing to feed such confusion or too unschooled to dispel it.

It is possible with a fairly straight face to blame our genes. As Richard Dawkins and many others have observed, and as Kieran Egan writes in Getting it Wrong from the Beginning:

The bad news is that our evolution equipped us to live in small, stable, hunter-gatherer societies. We are Pleistocene people, but our language and brains have created massive, multicultural, technologically sophisticated and rapidly changing societies for us to live in.

He also notes that “The cement like learning of our early years can accommodate almost anything, then it fixes and becomes almost unmovable” but that “we can, as a result, change our earlier beliefs and commitments. We also know this is difficult for most people.”

In On Deep History and the Brain Daniel Lord Smail notes that “the large human brain evolved over the past 1.7 million years to allow individuals to negotiate the growing complexities posed by human social living.” In consequence we find various modes of argument more palatable than others, and are more prone to make certain kinds of errors than others. We are over impressed by coincidence, poor at dealing with very-large scale or small-scale events (spatial or temporal) , and entirely unprepared for Nassim Nicholas Taleb’s Black Swans or tail events, or lotteries.

In consequence, schools could simultaneously improve the general quality of both mathematics education and public policy debate by focusing less on abstract algebra and elementary calculus, and paying a great deal more attention to topics such as

• robust mental arithmetic: if one needs a calculator to compute 10% of 12 dollars, how meaningful is any discussion of subprime mortgages?
• orders of magnitude and scale conversions: as for Hurst’s Berry Farm above.
• approximate reasoning: Guesstimation or Fermi problem solving included.

Nonetheless, this will be to little avail unless the shared supply of common knowledge is also dramatically enhanced. A modern secular education should include a requirement that everyone know things like:

• the approximate population of Cairo and of Canberra.
• the distance to the moon and between Mumbai and Moscow (and where they are).
• the relative cost of a congressional junket to the annual federal US budget.
• how Google googles?
• what is a serotonin re-uptake inhibitor, cellulite, or restless-leg syndrome?
• whether having asbestos in your ceiling riskier than frequenting a tanning parlor?
• what is a DNA letter, gene, chromosome, telomere, stem cell, recombinant DNA, or for that matter un-recombinant DNA?
• what is a nanotube, terabyte, database, or multicore processor?

—rather than who starred in Marley and me, whether Michael Jackson fathered his own children, or how much the transfer of Renaldo from Manchester United cost Real Madrid?

Semiotic fiddling while a digital Rome burns

“So to summarise, according to the citation count, in order of descent, the authors are listening to themselves, dead philosophers, other specialists in semiotic work in mathematics education research, other mathematics education research researchers and then just occasionally to social scientists but almost never to other education researchers, including mathematics teacher education researchers, school teachers and teacher educators. The engagement with Peirce is being understood primarily through personal engagements with the original material rather than as a result of working through the filters of history, including those evidenced within mathematics education research reports in the immediate area. The reports, and the hierarchy of power relations implicit in them, marginalise links to education, policy implementation or the broader social sciences.” (Tony Brown)

From his article “Signifying “students”, “teachers” and “mathematics”: a reading of a special issue, published online: 28 May 2008, Springer Science + Business Media B.V. 2008. (Your access to this article will depend on what your instution—if such you have— has paid Springer. Springer is one of the biggest academic publishers but not one of the biggest electronic sinners.)

Contrast this devastating if accurate critique with the following:

Enter Don Tapscott, who is looking at the challenges the digital revolution poses to the fundamental aspects of the University.

“Universities are finally losing their monopoly on higher learning”, he writes. “There is fundamental challenge to the foundational modus operandi of the University, the model of pedagogy. Specifically, there is a widening gap between the model of learning offered by many big universities and the natural way that young people who have grown up digital best learn.”

The old-style lecture, with the professor standing at the podium in front of a large group of students, is still a fixture of university life on many campuses. It’s a model that is teacher-focused, one-way, one-size-fits-all and the student is isolated in the learning process. Yet the students, who have grown up in an interactive digital world, learn differently. Schooled on Google and Wikipedia, they want to inquire, not rely on the professor for a detailed roadmap. They want an animated conversation, not a lecture. They want an interactive education, not a broadcast one that might have been perfectly fine for the Industrial Age, or even for boomers. These students are making new demands of universities, and if the universities try to ignore them, they will do so at their peril.

Contrary to Nicholas Carr’s proposition that Google is making us stupid, Tapscott counters with the following:

My research suggests these critics are wrong. Growing up digital has changed the way their minds work in a manner that will help them handle the challenges of the digital age. They’re used to multi-tasking, and have learned to handle the information overload. They expect a two-way conversation. What’s more, growing up digital has encouraged this generation to be active and demanding enquirers. Rather than waiting for a trusted professor to tell them what’s going on, they find out on their own on everything from Google to Wikipedia.” (Don Tapscott)

This is excerpted from the Edge describing Tapscott’s article The impending demise of the university.

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.

Material and expressions of opinion on this site are provided for research interest only and do not necessarily reflect the views or policies of the authors’ respective institutions, funding agencies or any other organization. Please send any comments or questions for this site to Bailey or Borwein — see http://www.experimentalmath.info for email addresses.

Math Drudge is maintained by David Bailey and Jonathan Borwein and is intended to highlight thoughts on the nature of mathematics and of science more generally. Any similarity with the Drudge report is totally unintended.

Rather we are motivated by Samuel Johnson’s definition of a:

lexicographer, n., a writer of dictionaries; a harmless drudge, that busies himself in tracing the original, and detailing the signification of words.

We take as compass Norman Levitt’s observation that:

“The dictum that everything that people do is “cultural” … licenses the idea that every cultural critic can meaningfully analyze even the most intricate accomplishments of art and science. … It is distinctly weird to listen to pronouncements on the nature of mathematics from the lips of someone who cannot tell you what a complex number is!”

[In Norman Levitt, The Flight From Science and Reason, New York Academy of Science, quoted from Science, 11 Oct 1996, pg. 183.]

See also http://www.experimentalmath.info/quotations.html. Some of our posts will be entirely excerpts from others while some will delineate our own opinions. We welcome email contact.

Albert Einstein once wrote:

“On the other hand, I maintain that the cosmic religious feeling is the strongest and noblest motive for scientific research. Only those who realize the immense efforts and, above all, the devotion without which pioneer work in theoretical science cannot be achieved are able to grasp the strength of the emotion out of which alone such work, remote as it is from the immediate realities of life, can issue. What a deep conviction of the rationality of the universe and what a yearning to understand, were it but a feeble reflection of the mind revealed in this world, Kepler and Newton must have had to enable them to spend years of solitary labor in disentangling the principles of celestial mechanics! Those whose acquaintance with scientific research is derived chiefly from its practical results easily develop a completely false notion of the mentality of the men who, surrounded by a skeptical world, have shown the way to kindred spirits scattered wide through the world and through the centuries. Only one who has devoted his life to similar ends can have a vivid realization of what has inspired these men and given them the strength to remain true to their purpose in spite of countless failures. It is cosmic religious feeling that gives a man such strength. A contemporary has said, not unjustly, that in this materialistic age of ours the serious scientific workers are the only profoundly religious people.” [Albert Einstein, New York Times Magazine, 9 Nov 1930, pg 1-4, reprinted in Albert Einstein, Ideas and Opinions, Crown Publishers, Inc. 1954, pg. 36-40.]