G. H. Hardy's "Apology"

I am certainly among the perhaps one million readers of this classic who could never master the mathematics that Hardy taught. But the continuous popularity of this book speaks to the fact that our common culture values pure mathematics. It must seem like hubris for me to say that Hardy was wrong about pure mathematics. I enjoyed the book nevertheless and took my time through it, and read parts of it again. I also skipped the introduction by C. P. Snow at first. I wanted to know Hardy, not Snow. But I went back purposefully and accepted Snow on his own terms. 

Hardy asks, "Is this important and am I the person to do it?" On the second point, Hardy is naturally demur.  Even though he makes the case for ego, he does not spend much time advancing his own to the reader. He does say that when he was very young, excelling at maths was a way to beat others, to best them at something difficult. ([29] p. 144) Only later did he discover a different pride, a different merit collaborating with Littlewood and Ramanujan.

A Mathematician’s Apology by G. H. Hardy
(with a Foreword by C. P. Snow).
Cambridge University Press,
1940, 1967; 23^rdPrinting 2016. 

“Good work is not done by ‘humble’ men.” – G. H. Hardy. (Math. Ap. [2] p. 66)

“I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well.—G. H. Hardy. (Math. Ap. [3] p. 67)

…perhaps five or even ten per cent of men can do something rather well.—G. H. Hardy. (Math. Ap. [3] p. 68; also, [5] p. 73)

… Poetry is more valuable than cricket, but Bradman would be a fool if he sacrificed his cricket in order to write second-rate minor poetry.”–G. H. Hardy. (Math. Ap. [3] p. 69)

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”–G. H. Hardy. (Math. Ap. [10] p. 84)

Hardy refers to cricket throughout the book. In the Foreword, C. P. Snow cites John Maynard Keynes who said that if Hardy had spent as much time with the stock market reports as he did with cricket scores, he would have retired a millionaire. The quote above was only one of very many allusions to the sport. It sent me (appropriately enough) googling. 

Sir Donald George Bradman, AC (27 August 1908 – 25 February 2001), often referred to as “The Don”, was an Australian international cricketer, widely acknowledged as the greatest batsman of all time. Bradman's career Test batting average of 99.94 has been cited as the greatest achievement by any sportsman in any major sport. -- Wikipedia.

“There are many highly respectable motives which may lead men to prosecute research, but three which are much more important than the rest. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one’s performance, and shame that overcomes any self-respecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings. It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it. If a mathematician, or a chemist, or even a physiologist were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him (nor should I think better of him if I did).” ([7] p. 79)

Godfrey Harold Hardy 1887-1947. 
https://www.maths.ox.ac.uk/node/13895

Hardy is eloquent, making a strong case for his field of theoretical mathematics as having greater value than applied mathematics. He begins by laying out four theorems. The first is Euclid’s proof that no end exists for prime numbers, that an infinity of prime numbers exists. The second is Pythagoras’s proof that the square root of two is irrational. Those he proves for the reader. 

Two others are left unfinished. The Fundamental Theorem of Arithmetic says that any integer can be resolved in only one way into a product of primes. Hardy shows some examples, but says that the proof might be tedious for the general reader. Similarly, he tells us of Fermat’s “Two Square” Theorem which says that all prime numbers can be divided by 4 with a remainder of either 1 or 3; and that all of the first class and none of the second can be written as the sum of two squares. Again, he lists some examples, but leaves the proof for the reader. 

From section [8] of [29] throughout, Hardy argues on behalf of the beauty of pure mathematics. He says that the practical work that benefits civil engineering and other advances in civilization is not beautiful and therefore not permanent. It does not matter that pure math is not practical. In fact, he says, it is better that it not be. I disagree.

I believe that all mathematics is potentially practical, only that we have not found a specific application. I do not know how to prove that. Hardy says (and I agree) that a proof by enumeration of cases is the least attractive. But I point to irrational, negative, and imaginary numbers, all of which were denied as real and all of which have practical applications today. 

PREVIOUSLY ON NECESSARY FACTS

Simply Dirac 

How the Martians Discovered Algebra

The Man Who Loved only Numbers

Elisha Loomis and the Pythagorean Theorem

Imaginary Numbers are Real; Pegasus is Not

Robert Leonard's "Curious Currency"

I was disappointed not to see an advance in the scholarship since the first edition of 2010. Make no mistake: this little book is very scholarly. An impressive 495 footnotes support the 126 octavo pages of text and about 250 illustrations (some are composites). But we all have our passions and prejudices. 

I am passionate about the research of Denise Schmandt-Besserat which tied the origins of writing and the invention of numbers greater than three to the creation of clay tokens in the Fertile Crescent of the Middle East circa 7500 BCE. Though not traded as money or gifts, the tokens served an economic purpose: they recorded debts. 

Curious Currency: The Story of Money
from the Stone Age to the Internet Age;
2^ndEdition by Robert D. Leonard, Jr.;
Whitman Publishing, 2019;
153+vi pages; $16.95.

That reinforced my prejudice for the research of David Graeber. Debt: The First 5,000 Years (Melville House, 2011) completely overturned our common imaginings about the origins of money. Those airy theories were shared by both Karl Marx and Ludwig von Mises. Marx at least relied on the best scholarship of his day. Mises just ignored the facts.

Trade did not originate with economic calculations of surplus. Money did not originate with trade for profit. Money did not evolve from barter. Coins did not evolve from money. 

Trade began as ritual gift exchange. Often it was the giving of a tangible to acknowledge an intangible based on social status. No example is known of a society that moved from barter to money, but many examples show that barter is what people resort to when money fails. Money as we understand it began with the payments of debts for torts. Coins began as honorary awards. Robert Leonard’s rich monograph supports those assertions. I am only sorry that he did not make them explicitly. 

I believe that Chapter 1, “What is Money?” is contradicted by the text. Leonard writes: “In simplest terms money is ‘anything used to make a payment that the recipient trusts can be reused to make another payment.’ This includes items used as money only for special purposes or situations, such as bride-price, funeral offerings, heiliges geld (offerings made to propitiate deities), trading with Westerners, or usage only by native chiefs. Among those bride-price is payment made to the bride’s parents as compensation for their loss of her valuable work services.” (Page 2) 

Obviously, of the items listed, none is an example of any expectation of further exchange. Bride-price is a case in point. The material offering only completed the social bonding of the families by the marriage. Calculating the labor of the bride eventually evolved thousands of generations later and in only in some places and times, not universally. Of course, the complement of bride-price is dowry. If bride-price is meant to be the result of an economic calculation that is carried out in money objects, what moneys are accepted as dowry; and if her labor is valuable, why is dowry being offered? Clearly, the complicated social context explains what appears to be a mere financial transaction. 

I also believe we all use the word “money” too readily to mean things that are not money. By analogy, I point to our confused speech about power, energy, and work, or velocity and speed. Generally, no harm is done, but physicists are not so casual. And as numismatists, we should communicate clearly about money, currency, and exchange.  Perhaps numismatists should convene an online standards committee to define our terms.

For a small book, Curious Currency delivers a lot to think about. We easily call it “coin collecting” even though numismatics is the art and science that studies all of the forms and uses of money. This dense little book is about the forms that “money” (exchange objects, ritual gifts) has taken over the thousands of years of human society. Of necessity, this is a broad topic, potentially encyclopedic in scope. Robert Leonard makes the information load manageable by wrapping the stories and narratives into convenient chapters based on those broad themes. 

After an introductory overview, the chapter titles are Raw Materials, Useful Articles, Ornaments, Customary Objects, and Money Substitutes. Coins fall under “raw materials” because they were valued as metal. But silver, gold, copper, bronze, and iron must take their place alongside obsidian and flint which also were money. Coins also appear under “customary objects” along with elephant tails, woodpecker scalps, and human skulls. 

Whiskey, tobacco, tea, cocaine, and postage stamps are considered “useful articles.” Beads of coral, jade, glass, clam shells, cowry shells, silver, and turquoise, arm rings, neck rings, anklets, and many kinds of necklaces are “ornaments” of course. 

That almost anything can be used as money underscores the broad extent of society and culture. Therefore, it may be perfectly fine that the book closes with examples of “nothing” as money. RFID transceiver chips that you wave at a gasoline pump, cellphones as proxies, and cybernetic cryptocurrencies bring the reader near to—but not at—the end of the story of money. 

Overall Curious Currencyis an excellent treatment of a complex and difficult subject. The book is easy to read and worth every minute. 

PREVIOUSLY ON NECESSARY FACTS

Acccounting for Civilization 

Debt: the Seed of Civilization 

Numismatics Informs Economics 

Numismatics: the Standard of Proof in Economics 

Supplies and Demands 

Two Hot Mamas Salsa

New Mexico is the home of chili peppers. In both Las Cruces and Alburqueque, at Christmas time, people hang red and green chilis from their doors, the way people up north put up holly wreaths.  When we were newlyweds in Las Cruces, I learned that it is too easy to make salsa too hot for anyone to eat.  Chili peppers have subtle flavors that burst and blend if you treat them right. Two Hot Mamas of Austin understands that.

Two Hot Mamas Contact 

Last Sunday, I met Johnna from Two Hot Mamas at the Wheatsville Co-op. I tried several and settled on the Mambo Combo Hot. It was perfect for my taste.

PREVIOUSLY ON NECESSARY FACTS

Coffee at the Co-op Tradition and Novelty
Shannon Beer of Keller, Texas
Sunday at the Co-op
Awesome Austin Foods at the Wheatsville Co-op