Tony’s Take November 2022

This month’s topics:

Math education in The New Yorker.

The New Yorker staff writer Jay Caspian Kang has two recent pieces on the magazine’s website about mathematics education in the United States. How Math Became an Object of the Culture Wars (Nov. 15, 2022) and What Do We Really Know about Teaching Kids Math? (Nov. 18).

In the first installment, Kang starts in 1915 and follows the ebb and flow of various progressive math education movements. Initially it seemed that Euclid would be jettisoned along with Caesar and Cicero as abstract material once deemed healthy for developing critical thought, but irrelevant to modern life. Then, during and after World War II, technology (computers, for example) showed mathematical skills to be important after all; the problem was how to implant them in the population. In response, the “new math” movement, started in the early 1950’s (Robert Hayden’s A history of the “new math” movement in the United States covers its genesis in detail) and greatly accelerated by the intellectual panic following the Sputnik launch, emerged as an attempted solution. As Kang observes, “The same fight has repeated itself on several occasions since then.”

In the current iteration, the discussion has expanded from pedagogy to include equity, because of the realization that the American student body is not homogeneous. Changes in curriculum and delivery will impact students from different segments of society in different, and sometimes inadvertently harmful, ways. For example, the draft plans for California’s elementary and secondary math education guidelines were “criticized by the usual suspects … but also many equity-focussed educators who worry that the program may be seen as a slackening of expectations for minority and low-income students.”

The impetus for Kang’s second piece was the announcement, dated October 18 by the Gates Foundation of an ambitious program to improve K-12 mathematics education, starting, according to Education Week with a $1.1 billion 4-year investment. (The program has an explicit equity component.)

The Gates Foundation plans to use technology and experimentation to identify education techniques that work. But there’s a more fundamental core to the problem of improving U. S. mathematics education. Kang evokes a collision between two uncontroversial facts: “The first is that a society has a duty to educate all of its citizens, regardless of race, socioeconomic background, or whatever else. The second is that parents will almost always do what they think is best for their children.”

It is disappointing that in this discussion the word “math” is a completely opaque token: nothing is said at all about how mathematics is different from other subjects, or why so many find it hard to learn.

The geometry of logical arguments.

The website DailyNous (“News for & about the Philosophy profession”) posted “The Artful Geometry of Logic” by Justin Weinberg (South Carolina), on November 11, 2022. Weinberg is bringing to our attention the online availability of The Leuven Ontology for Aristotelian Diagrams Database, a searchable collection of some 3200 logical diagrams. He shows some examples, including the Square of Opposition as rendered in a manuscript copy of Peri Hermeneia (“On interpretation” — the title is Greek but the text is Latin) by Apuleius of Madaura (born c. 124 CE). The copy dates from about 1000 CE.

A rectangular diagram with text written in Greek.
Apuleius’s Square of Opposition from a 10th-century manuscript. Image source, Bibliothèque nationale de France.

The concepts in this diagram go back to Aristotle (384-382 BCE), but as far as we know Apuleius was the first to present them in a 2-dimensional arrangement. (He did not leave us an actual picture, but instructions: the square form, what should be on the top line, what should be on the bottom line, and so forth).

The Square of Opposition is part of the Aristotelian study of syllogisms. These are three-part arguments like the sequence of propositions “Socrates is human; all humans are mortal; therefore Socrates is mortal” (the traditional example). The propositions occurring in a syllogism can be in one of four logical forms:

  • Universal Affirmation: “all $A$ are $B$”
  • Universal Negation: “all $A$ are not $B$”
  • Particular Affirmation: “some $A$ are $B$”
  • Particular Negation: “some $A$ are not $B$”

The Square of Opposition shows the logical relations between the four forms. In this translation, Apuleius’s example for each form is shown in italics and the names of the relations between the forms appear in blue.

A rectangle. In the upper left corner is written "Universal Affirmation, every pleasure is good." In the upper right corner, "Universal Negation, no pleasure is good." In the bottom left corner, "Some pleasures are good, particular affirmation." In the bottom right, "Some pleasures are not good, particular negation." Blue text reading "contradictory" stretches between opposing corners (from Universal Affirmation to Particular Negation, and from Universal Negation to Particular Affirmation). Along the top is written "Contrary or Incompatible", along the bottom "subcontrary: subequivalent or subneutral" and along the sides "subaltern".
Apuleius’s Square of Opposition diagram with text in approximate English. Thanks to my colleague Rosabel Ansari for advice on the translations.

The names of the relations can be deciphered as follows, following the Stanford Encyclopedia of Philosophy. Propositions are contrary if they cannot both be true but can both be false; subcontrary if they can both be true but cannot both be false; contradictory if they cannot both be true and cannot both be false. Finally, proposition $Q$ is a subaltern of proposition $P$ if $Q$ must be true if $P$ is true, and $P$ must be false if $Q$ is false.

Number theory breakthrough, in Nature.

A news item in the November 24, 2022 issue has Davide Castelvecchi commenting on a recent ArXiV posting by Yitang Zhang (University of California, Santa Barbara), which would prove a weakened version of the Landau-Siegel conjecture. Terence Tao has observed, as quoted in John Baez’s Twitter feed, that the as yet unrefereed (111-page) article has typographical errors and missing pieces which make it impossible to evaluate; Tao recommends patience.

Meanwhile, what is this conjecture and why does it matter? The Laudau-Siegel conjecture derives its importance in part from its proximity to the Riemann hypothesis. That hypothesis concerns the (Riemann) zeta-function, defined for real $s>1$ by
$$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}$$
and extended to the complex plane by analytic continuation. It can be shown (here, for example) that $\zeta(-2), \zeta(-4)$, etc. are all equal to zero. These are the “trivial zeroes” of the zeta-function. It is also known that all the other zeroes must lie in the critical strip of complex numbers with real part strictly between 0 and 1. The Riemann hypothesis states that these other zeroes must in fact have real part exactly $\frac{1}{2}$ (the dashed line in this diagram).

Plot in the xy-plane. Along the x-axis, numbers –4 and –2 are marked as "trivial zeroes" of the Riemann zeta function. The strip between x = 0 and x = 1 is shaded in and labeled "Critical strip: all non-trivial zeroes must be in here." A dotted vertical line at x = 1/2 is labeled "Riemann hypothesis: all non-trivial zeroes lie on this line." Near x = 1 is marked "Landau-Siegel zeroes would be here."

The Generalized Riemann hypothesis is about a larger class of zeta-type-functions $L(s, \chi)$ (which includes $\zeta(s)$), and asserts that all their non-trivial zeroes (defined as before) lie on the dashed line where the real part of $s$ is $\frac{1}{2}$. This class includes a family of generalized zeta-functions, $L(s,\chi_D)$, that depend on an integer $D$. A Siegel zero would be a zero for one of these functions $L(s,\chi_D)$ located on the real line, strictly between $\frac{1}{2}$ and 1, so it would be a counterexample to the generalized Riemann hypothesis.

The Landau-Siegel conjecture, in one formulation from Wikipedia, states that there exists a constant $\delta$ such that no zero of $L(s, \chi_D)$ can occur in the interval $(1-\frac{\delta}{\log D}, 1)$. Zhang’s Theorem 2 states that there exists a constant $\delta$ such that no zero of $L(s, \chi_D)$ can occur in the interval $(1-\frac{\delta}{(\log D)^{2024}}, 1)$. This is a much smaller interval! Presumably that large exponent can be whittled away, as happened with an earlier estimate of his.