Thinking about fundamental questions: the madman theory of education

Someone once introduced me to the madman theory of education. It says it is good for a university to have a faculty member who is mad because opposition to his crazy opinions stimulates students into thinking seriously about fundamental questions. Adler was the University of Chicago's madman.

Martin Gardner, before 2013.

Who is the madman at your university? Adler, by the way, is Mortimer Adler, who initiated the Great Books movement at the University of Chicago in Illinois, along with Robert Hutchins and Richard Peter McKeon (see Understanding something about science). Mathematics and science writer Martin Gardner (1914-2010)—in his posthumously published autobiography—has a lot to say about those three controversial and interesting characters.

Reference
Martin Gardner with Persi Diaconis and James Rand: Undiluted Hocus-Pocus. Princeton University Press, Princeton, New Jersey, 2013; page(s) 70±.

Understanding something about science: how to approach a domain of interest

To understand something about science the best plan is to read a short history of science, and popular works on relativity and quantum mechanics.

Martin Gardner, before 2013.

Mathematics and science writer Martin Gardner (1914-2010) suggested this “strategy” in his autobiography, introduced with a foreword by statistician Persi Diaconis (see Diaconis' blurb). Gardner wrote this while discussing the Great Books movement initiated by Robert Hutchins, Mortimer Adler and Richard Peter McKeon at the University of Chicago in Illinois. The idea of the Great Books scheme was that a list of preselected, original books—classics of Western culture and breakthrough literature—would be the best educational approach in advancing an academic career. Certainly, any such undertaking is culturally biased. In a world of fast, dynamically changing priorities and progress, learning goals are best achieved within (inter)disciplinary and community context: as outlined in the quotation, a targeted short history, scholarly overview or review will be most inspiring and introductory before diving deeper into the domain of interest. The detour through precursors and classics may then be taken at a later stage, when time frames allow in-depth studies and divergent curiosity.

Keywords: didactics, education, history.

Reference
Martin Gardner with Persi Diaconis and James Rand: Undiluted Hocus-Pocus. Princeton University Press, Princeton, New Jersey, 2013; page 50.

Disliking history classes, yet liking history

Like many of us, the popular mathematics and science writer Martin Gardner (1914-2010) disliked history the way the subject was (and often still is) taught in classes. But there are so many interesting facets of history. In his postumously published autobiography, Gardner points to the history of science and technology: 

The really important history, it seemed to me, was the history of science. Of all the vast changes in human life, most are the result of the steady progress of science and technology.

Martin Gardner, before 2013.

Let's add natural history (strongly overlapping with science), the history of languages and terminology, and the history of music, arts and crafts to further suggest that diving into history can be enlightening and personally empowering.

Reference
Martin Gardner with Persi Diaconis and James Rand: Undiluted Hocus-Pocus. Princeton University Press, Princeton, New Jersey, 2013; page 21.

Being an innocent youngster and a math professor at the same time

Martin Gardner's autobiography has posthumously been published [1,2]. Known for his Scientific American math column, Martin Gardner was fascinated by recreational mathematics, magic tricks and scientific research and he knew how to fascinate others. Persi Diaconis tells us that he wrote the following blurb for one of Gardner's books [2]:  

Warning: Martin Gardner has turned dozens of innocent youngsters into math professors and thousands of math professors into innocent youngsters.

Persi Diaconis, 2013.

Diaconis admits that he was one of those youngsters. Who else? I assume, the number of Gardner enthusiasts is growing exponentially.

Keywords: inspiration, recreation, magic, puzzle solving, mathematics.

References and more to explore
[1] David Singmaster: Master puzzler. Nature, September 19, 2013, 501 (7467), pp. 314-315. doi: 10.1038/501314a.
[2] Martin Gardner with Persi Diaconis and James Rand: Undiluted Hocus-Pocus. Princeton University Press, Princeton, New Jersey, 2013; page xvii.

Molecular chatterboxes: genes talking to genes

When genes became damaged or mutated, their hosting cells may turn malignant. Such cancerous cells can grow into tumors over time. Understanding of cancer, a group of diseases medically known as malignant neoplasm, relies on  the basic concept of uncontrolled, misregulated growth of cells into nearby parts of a patient's body. In addition to genetic changes, many other factors contribute to the biochemistry of malignancy, including body-occupying bacteria and the complex biomolecular interactions switching certain genes on and off. George Johnson summarize the new insight into the physics and informatics of cancer as follows: 

In the end, all biology comes down to genes talking to genes—within the cell or from cell to cell—in a constant molecular chatter. I had not considered, however, that the genes in human tissues can also exchange information with the genes residing in the microbes that occupy our bodies. Cancer is a disease of information, of mixed-up cellular signaling. Now there is another realm to explore.

George Johnson, 2013.

The new realm goes beyond the cell-centric mechanism of repeated mutation acquirement stimulating abnormal growth. A new paradigm that hopefully provides the needed insight to come forward with new treatments and advances in curing cancer.  

Keywords: cell biology, oncology, medicine, epigenetics, cancer treatment.

Reference
George Johnson: The long trail of cancer's. Scientific American, November 2013, 309 (5), 2012; pp. 60-63 [www.scientificamerican.com/article.cfm?id=book-excerpt-george-johnson-explores-the-latest-discoveries-about-cancer].

As well as we like: accurate(ly) versus efficient(ly)

Doing something well can mean to do it accurately or efficiently, or both. Julian Havil is asking the question how well an irrational number can be approximated by a rational number; and—within this context—he discusses the distinction between the adjectives accurate and efficient or, to be more accurate, the adverbs accurately and efficiently:

If by well we mean accurately, then the answer is as well as we like. It is intuitively clear that the accuracy of rational approximation can, in theory, be chosen to be what we will: there are plenty of rationals and as many as we could desire as close as we desire to our chosen number; consider the decimal expansion of the irrational number, truncated as we please. Yet, there is a hidden cost, as we shall see. Alternatively, if by well we mean efficiently the story is more complex since some numbers are more amenable to rational approximation than others - and from this relative compliance we can draw important distinctions [...]

Julian Havil, 2012.

Well done! 

Keywords: semantics, adjectives, adverbs, word disambiguation, approximation.

Reference
Julian Havil: The Irrationals. Princeton University Press, Princeton and Oxford, California, 2012; page154.

Elementary versus simple

Mathematics makes a nice distinction between the usually synonymous terms elementary and simple, with elementary taken to mean that not much mathematical knowledge is needed to read the material and simple to mean that not much mathematical ability is needed to understand it.

Julian Havil, 2012.

When the mathematically different meanings of these two adjectives collapse into one, the adjective complex makes for a suitable antonym. Julian Havil discusses the distinct meanings of the adjectives elementary and simple within mathematical context. He also provides an example in which their distinct meanings become difficult to grasp or where both words even regain identical meaning: the proof that ζ(3) is an irrational number, which has been achieved in quite different ways by (1) Roger Apéry, (2) Frits Beukers, (3) Wadim Zudilin, and others. An interesting, yet complex topic—even when illustrated in simple terms. 

Keywords: semantics, adjectives, antonym, word disambiguation.

Reference
Julian Havil: The Irrationals. Princeton University Press, Princeton and Oxford, California, 2012; pp. 152-153.

Named and unnamed mathematical constants: from anonymous to famous

Mathematical constants are either anonymous or famous, with fame a reflection of the constant's importance.

Julian Havil, 2012.

Along his compellingly illustrated path through the history of irrational numbers—delivering insights for mathematicians and non-mathematicians—Julian Havil introduces readers to interesting constants beyond the “famous constants” π and e [1]: the Conway Constant (also written Conway's Constant [2]), for instance, which isn't exactly famous. Neither is it anonymous, as it is named after the English mathematician John Horton Conway, who introduced and analyzed the look-and-say sequence leading to the discovery of the Conway Constant [3].

Anonymous constants may become famous. The Conway Constant and its look-and-say sequence should be of interest in the study of self-descriptive processes such as molecular self-replication; and, thus, will contribute to biomolecular modeling and advances in macromolecular chemistry and biochemistry [4].

Keywords: mathematics, special numbers, rationals, irrationals, transcendentals.

References and more to explore
[1] Julian Havil: The Irrationals. Princeton University Press, Princeton and Oxford, California, 2012; pages 136 and 137.
[2] Wolfram MathWorld: Conway's Constant [mathworld.wolfram.com/ConwaysConstant.html].
[3] John H. Conway: The Weird and Wonderful Chemistry of Audioactive Decay. Eureka 1986, 46, pp. 5-18 (see TOC on www.archim.org.uk/archives/eureka/#46).

by Óscar Martín

[4] Óscar Martín: Look-and-say biochemistry: Exponential RNA and Multistranded DNA. American Mathematical Monthly 2006, 113(4), pp. 289-307 [www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-april-2006].

Óscar Martín

by Ó Óscar Martín