Inspired by Prof. Petr Beckmann’s classic 1970 book “*A History of Pi*”, Eli Maor in 1994 wrote “*e: The Story of a Number*”. ISBN 0-691-03390-0. Maor’s view is that the teaching of mathematics suffers from a lack of context. Including the historical background and the characters whose struggles led to advances would, he thinks, make mathematics more interesting and accessible to students and the general population. He certainly makes that case in this book which spans the ages from Babylonians in 1700 BC to modern times, digging into the mysterious number 2.7182818284… which keeps on cropping up in everything from interest rates to nuclear explosions. Math will always be math, causing fear & loathing in the kind of non-numerate person who ends up devising national budgets in Western democracies, but putting a human face on it may make it more intriguing.

One of the many fascinating elements in this book is the story from the 1600s of Sir Isaac Newton (upon whose head the apple fell) and Baron Gottfried Wilhelm von Leibniz. Back in high school math class, most of us were told that these two gentlemen independently invented calculus; there was some debate at the time about who should get the credit, but nowadays no-one really cares. Then it was on to memorizing formulae.

As Maor tells the story, it is much more interesting. Newton came from an ordinary rural English family which had been crossed by misfortune. He was an archetypal nerd student at Cambridge U. in England when, in 1665, Europe suffered a genuine plague – truly deadly, unlike today’s CovidScam. The young Newton withdrew to his family’s rural home and, working in isolation over a period of 2 years, developed the central ideas of calculus – the mathematics of change. Newton seems to have been a rather reclusive untrusting type, reluctant to publish his work. When not breaking new ground in mathematics or physics, Newton spent a lot of energy on astrology.

In contrast, Baron von Liebniz was the gifted son of a German philosophy professor. He had wide interests in philosophy, law, languages, and literature as well as mathematics. A highly sociable man, Liebniz spent much of his life as a diplomat. In the 1670s, he fully developed both differential and integral calculus – including the notation and rules which are still used today. (Newton’s more awkward notation is now of historical interest only). Liebniz openly shared his work with Newton, but Newton refused to reciprocate.

Liebniz published his development of calculus in 1684, sharing it with the world. Newton did not publish his own work until 1704, but did so with an implication that Liebniz had seen this almost 30 years previously before publishing his 1684 book. The resulting controversy over who invented calculus dragged on for decades, outliving both men.

Curiously, the touchy & unsociable Newton became a national hero in England. When he died in 1727, he was given a State funeral and buried in Westminster Abbey. In contrast, by the time Liebniz, the man of the world who gave us the form of calculus we still use today, died about 10 years earlier, he had almost been forgotten in his native land; only his secretary attended his funeral. Life is unfair!

OK, knowing all that does not make mathematics any easier – but it may tweak the occasional person’s interest in the topic.

It’s difficult to peer into Newton’s famously opaque mind, but I’ve suspected that one of the reasons he kept calculus private for so long was his desire to use it as a secret weapon to solve problems others could not. In the

Principia, there are a number of places in which he works out problems, such as the motions of the planets, which he clearly solved using calculus (fluxions) but expressed in the book in the much more arcane and cumbersome, but widely-known technique of conic sections.That he had a secret weapon was apparent to some. When he “anonymously” solved Bernoulli’s challenge of the brachistochrone, Bernoulli remarked, “tanquam ex ungue leonem” (we recognise the lion by his claw mark), identifying Newton.

Newton may have been one of the first scientists in the modern mould, but was also one of the last sorcerers. He valued the secrecy of his techniques.

I recall a book that was helpful to me when I was in an advanced algebra class in high school. It was

Realm of Numbersby Isaac Asimov. He wrote a couple of other books that were great companions to students in the sciences.This also puts me in mind of books we used while we were homeschooling. There is a series of textbooks by Jay Wile that are intended for homeschoolers, but I think they would be great supplemental study books for any high school student. I especially have recommended his physics book, even to university students, for clear, concise writing and a lack of errors.

These books generally feature a page about a scientist or mathematician in each chapter. These give a short thumbnail bio and then a paragraph that relates the person to the contents of the chapter. That helps the books become more accessible for struggling students.

https://www.home-school.com/eblasts/Berean/20200728BereanBldrs.php

The book that persuaded me, in junior high, that there was a world of wonder beyond the tedium of the weapons of math instruction targeted upon us was George Gamow’s

One Two Three … Infinity. Seventy-three years after its publication it remains in print, and there is a Kindle edition. Because mathematics is eternal, little in the book has become dated, although some of the problems cited as unsolved have been solved over the years.It is a great gift to a youth who is bored with and about to give up on math.

Another book I encountered many years later but also highly recommend is Rudy Rucker’s

Mind Tools. It presents mathematics as ways of thinking about things in new ways, not about getting the answer out of tedious problems. Years later, I was to collaborate with Rudy on several projects, includingCellular Automata Laboratory(which is how I earned my Erdős number).Gamow has been a hero of mine ever since I learned that he added Hans Bethe to the names of the authors of a paper on the Big Bang, so that it would be by Alpher, Bethe, Gamow.

And then, when the theory fell out of favour, Hans Bethe said he was thinking of changing his name to “Zacharias”.

I remember that.

Thanks.

Do you have a review of

Mind Tools? It looks like something that would make a good Christmas gift for a nephew.No. It’s a book I read (in the late 1980s) before I started reviewing all the books I read. It is aimed at a more mature audience than

One Two Three … Infinity, but is entirely accessible to a bright middle schooler.My moment arrived when a substitute teacher filled in for ill Mr. Chazen in geometry. First, we proved a triangle’s three internal angles summed to exactly 180°, SOP. Then the substitute pulled out a globe of the Earth and showed that was never true.

There was hope.

Newton boldly stated the Equivalence Principle in preamble, and then boldly set (knowingly or otherwise) four constants from which to derive all of physics. Give the man credit for getting big G right, though not valued. Put your money on the U/Wash quadrupole torsion pendulum, for valued big G is still up in the air.

https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201900013

Weak postulates, weak conclusions. Physics should be more humble (bad for funding) or better at what it cannot do.

An Introduction to Mathematics by Alfred North Whitehead is worth a look

From Uncle Al’s link:

“

Progress in Precise Measurements of the Gravitational ConstantJunfei Wu, Qing Li, Jianping Liu, Chao Xue, Shanqing Yang, Chenggang Shao, Liangcheng Tu, Zhongkun Hu, Jun Luo”I do not worry about the long-term future of the human race, but I am deeply concerned about the near-term future of a decadent West which is ignoring the technological foundation on which it built its former glory.

It seems that the best advice for smart young people in the West today would be — after you have learned enough math to be competitive, buckle down and start learning Chinese.

https://www.researchgate.net/publication/329930975_Test_of_the_Equivalence_Principle_with_Chiral_Masses_Using_a_Rotating_Torsion_Pendulum

PHYSICAL REVIEW LETTERS 121, 261101 (2018)

DOI:10.1103/PhysRevLett.121.261101

… Jun Luo is a cool dude! His name at the top, mine at the bottom.

Alas, the single crystal quartz test masses were not up to structural purity spec: Water molecules occluded along the crystallographic c-axis, aluminate plus counterion contamination, some 100 dislocations/cm² (surface etch pits). For all that, the guy has big brass clangers. We could have gotten another decimal place or two.