Sunday, May 12, 2019

Regimental Public Affairs Officer

On May 4 and-5 May 2019, I attended training for public affairs officers in the Texas Military Department. The training was presented by public affairs professionals from the news media and from the military. Of necessity, some presenters wore both hats. Of course, the panel discussions and lectures were focused on news reporting for military public affairs. In addition, I picked up good pointers about writing for any market or audience.

The Introduction by LTC David Spanton set the context. “We have three goals,” he said. 
1.    Meet your peers. Make friends. 
2.   Recognition of our work.
3.   Professional development.
The morning sessions were on Crisis Management, Media Relations, and Social Media. In the afternoon, we covered Photo & Video, and Cracking the Awards. On the second day, we were reminded of the fundamental truths in career development. The seminar ended with an awards ceremony to recognize the writers, photographers, and videographers in the Texas Army National Guard, the Texas Air National Guard, and the Texas State Guard.
 
Texas State Guard Public Affairs
Maritime (left) and Army (right)
The first panel discussion on the first day was about Crisis Communications. The presenters were Bryce Dunbee (Austin Bergstrom International Airport), Natalie Bidnick Andreas (public relations consultant),  Caitlin Rourk (crisis management consultant).

Bryce recommended having a tagline that closes every statement: “As always, safety is our number one priority.”  Natalie said that crisis communication begins with an apology. She also emphasized the need for clear social media policies that apply to everyone from the CEO to interns. Caitlin encouraged us to find case studies of actual events. “Pick a crisis and then wordsmith a statement for it during your monthly drill as an exercise.” Look for bookmarkable case studies in crisis management.

They all insisted that public affairs officers need to reach out early and often to news media.  “Get your buy-in before the crisis…. Make your contacts in times of peace… Become part of the command staff equation. … Know your contacts in your legal affairs department.” And remember that today everyone is on a national stage.

Panel Sessions were held in 
the Audie Murphy Building
Natalie warned us to watch out for scope creep. You will have a client or stakeholder, a commander of some kind, who wants more services, whether that means more of your time or creating a new and better spin to the story. She recommended holding your ground. “You tell them: ‘I am duty bound and this is what I can do.’”  So, how do you push back to a client or a commander? You outline for them the reputational risk and give them a threat analysis. 

Crisis preparation is defining the “so-what” in a story. The numbers do not lie. Metrics speak. Tell the media not only why your organization is out there doing these things, but why your communications directors or spokespersons are saying these things. “We are continuing to investigate.” Tell them when you will have the information they seek.  

Have your speakers, your commanders, be prepared for the next question.  Reporters know that you are prepped for one or two responses. So, they keep asking more questions until they get to the one you are not prepared for.

At the end of the first day, the sixth panel was titled “Cracking the Awards” (SSG Will Reinier, Jordyn Fetter, and MSG Thomas Wheeler presenting). They did not say this, but it is a known cliché (perhaps from Napoleon) that a soldier will do anything for a piece of colored silk. Awards matter for the same reasons that publicity itself matters: recognition is important. So, how do you get recognized? 

Stick to the criteria. Follow the instructions. Look for categories with fewer entrants. It is pretty easy. 
 
Class Photo: Texas Military Department
Public Affairs Officers Seminar
I found value in their outline for any story. We are writing non-fiction. These are reports to the public. But it is best told as a story. 
Five elements of any story.
1.    Set the scene. Take a wide shot.
2.   Introduce the character. Root for or against.
3.   Find the conflict. Find the challenge.
4.   Conflict changes the character.
5.   Resolution.

Similarly, the first panel of the second day on Career Management (LTC David Spanton, MAJ Jose Perez) underscored truths that apply to all professions. Look for opportunities in education, training, promotions, and assignment. Follow up on continuing education whether that is the completion of certifications or accreditations or maintaining your society memberships.  

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Sunday, May 5, 2019

Asteroid Hunters by Carrie Nugent

This is an inspired little book with a lot of problems. I pulled it from the stacks at my city library neighborhood branch because I wanted something casual to read and because I wanted to find a book to review for my local astronomy club newsletter. Asteroid hunting is one of the activities of amateur astronomers that intersects the work of professionals. Comet hunting is another. I was pleasantly surprised by Dr. Nugent’s easy explanation of why, contrary to our common assumptions, asteroids and comets are often the same. That was one of several interesting facts in a flawed presentation. 

With a lot of hope, I expected to read about amateur and professional astronomers, including photographs of the instruments used by my unpaid colleagues who work for the love of it. Instead, I soon met the first of over 20 errors of fact. It started with the vernacular American style which includes jarring grammatical lapses and sentence fragments.
  • “When I tell people I’m a space scientist studying asteroids, they sometimes assume I’m a super-smart math whiz. The kind of person who skilled a bunch of grades and went to college when they were sixteen.” (page 2)
  • “There wasn’t going to be any surprises.” (page 15)
  • “Soy latte, check e-mail, wait for the caffeine to kick in.” (page 29)
  • “… there’s a few…” (page 73)

Nugent writes, “… sometimes artifacts of the telescope can masquerade as an asteroid moving across the sky. These artifacts can be a series of cosmic rays, or the edge of a flare from a bright star.” (page 29).That is not what an “artifact” is in a viewing instrument. When you see your eyelashes in a microscope because of internal reflections in the lens system, that is an artifact. 
Asteroid Hunters by Carrie Nugent, 
TED Books Simon & Schuster, 2017, 
108 pages, $16.99.
https://www.crnugent.com
Nugent wrongly claims that astronomers in 1800 thought that the planets orbit in circles. “Existing methods of the time [1800-1802] used the assumption that the planets traveled around the Sun in circular paths, when in reality they traveled on a specific geometric path called an ellipse.” (page 43) But it was Kepler who first fit the orbits of the planets into elliptical paths about 1605.

At that point, Nugent already referenced Kepler, though she never mentions Newton. Kepler showed that the planets travel in ellipses and then that was proved as mathematically necessary by Newton. Newton’s calculus demonstrated that objects moving under the influence of a central force do so in paths that are conic sections. It is called a “necessary fact” something that is both required by logic and observed in experience. The shape of the orbit (line, hyperbola, parabola, ellipse, circle) depends on the velocity of the object. 

An underlying theme of this book is that getting money to search for near-Earth asteroids has been difficult because the political agencies that fund such research consider the possibility of catastrophe to be remote. The kind of asteroid that could end life as we know it comes only every 65 million years. However, two asteroids dramatically became meteorites in recent times: the Sudan 2008 TC3; and Siberia 2012DA14 (Chelyabinsk). The Sudan fall was predicted a few hours ahead of impact. Siberia was a complete surprise.

Nugent explains the difficulties in spotting asteroids. For one thing, the Sun blocks our view. But she also explains the work-arounds of observation and orbit plotting. At least, she says that such tools exist.  In fact, Nugent presents orbit plotting as extremely complex, difficult mathematics that only a genius could master. If not for Carl Gauss, we would be working in the dark, so to speak. But it only takes three points, three observations, to define a conic section. 

That being so, what makes orbit plotting and asteroid hunting a challenge is the many perturbations that change the neat conic sections into wobbly, wonky drunk walks. Nonetheless, celestial bodies travel in very predictable, mathematically definable paths. Even the perturbations are knowable and predictable. That being so, it remains that even now, 300 years after Newton’s Principia, for the most exacting predictions, we depend on tables of previous observations, rather than applying undergraduate calculus to three observations.

Nugent that “Space is cold.” (page 64 para 6), It is not cold if you are in the sun. Then, it is hot. Whether space is hot or cold depends on the definitions. Statistical mechanics defines temperature by the number of particles with a significant velocity. If you have one or two traveling at the speed of light,  you have “cold.” If you have thousands going 10 mph, it is hot: just rub your hands together as fast as you can. (Don’t burn yourself.) 

Nugent discusses the important statistical method of her team without naming it. Writing about p-values, (page 76) she calls it “debiasing” (pgs-75-77). “They also knew the time and location in the sky of every image NEOWISE had ever taken. The computer simulation exactly modeled how NEOWISE observed the sky and what it would be able to see. Then, they simulated hundreds of thousands of ‘synthetic’ asteroids and ran them through the simulation to see how many asteroids NEOWISE would have seen. The result was compared to what NEOWISE actually saw.” Nugent then gives an example using 10, 12, and 18 samples.  “Of course, the actual implementation is more complicated than that, and many more asteroids are simulated so that the results have statistical significance. But you get the idea. With this method, we know what we don’t know.” (page 77).  

Dr. Nugent delivered her TED Talk in February 2016. This book came out in 2017. But p-values and “statistical significance” were being questioned for the very misuse and abuse of statistical methods by scientists such as Dr. Nugent. 
  • “Why Most Published Research Findings are False,” John P. A. Ioannidis, PLoS Medicine, August 2005. https://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124
  • “How to Use p-Values Correctly,” Kerry Grens, March 9, 2016,The Scientist. https://www.the-scientist.com/the-nutshell/how-to-use-p-values-correctly-33910
  • “The ASA’s statement on p-values: context, process, and purpose,” by Ronald L. Wasserman, and Nicole A. Lazar, The American Statistician, March 9, 2016. https://www.tandfonline.com/doi/full/10.1080/00031305.2016.1154108

It is a serious fact that keeping up with all of the sciences not related to your own is a challenge. However, this controversy was first opened for discussion in 2005 and just as Dr. Nugent was approaching TED Talks, it burst out. The ASA Statement can be found on many websites for university undergraduate classes in statistics. She should have known. They all should have, rather than surging forward with their millions of computer-simulated asteroids.

And yet, there is much here, even though the details may not motivate anyone else. I always accepted the easy statement that a thousand Earths could fit inside Jupiter. The giant planet’s diameter is about ten times our own’s. Ten cubed is a thousand. (In fact, it works out to Jupiter’s volume being about 1381 times that of Earth.) In a footnote, Dr. Nugent explains that packing spheres leaves space between them. Only about 800 Terras could be fit into Jove. I learned something that I should have figured out on my own. So, I am appreciative.

Jupiter is discussed in the context of asteroid Shoemaker-Levy 9 striking Jupiter. “Unfortunately, the impact was going to hit the side of Jupiter that was facing away from Earth, so astronomers with telescopes wouldn’t have a direct view. … A fleet of spacecraft was trained on Jupiter, including the Hubble Space Telescope, the ROSAT X-ray satellite, and …” (page 81) Neither of those was in any position to see the far side of Jupiter. Both Hubble and ROSAT orbited Earth. In fact, Hubble did send images from after the impact. 
This NASA Hubble Space Telescope image of Jupiter's cloudtops was taken at 5:32 EDT on July 16, 1994, shortly after the impact of the first fragment (A) of comet Shoemaker-Levy 9. A violet (410 nanometer) filter of the Wide Field Planetary Camera 2 was used to make the image 1.5 hours after the impact.  http://hubblesite.org/image/170
And it was not a single event but a series of impacts. The comet had broken up two years earlier in a previous pass-by in July 1992. The fragment stream impacted the planet over six days, July 16-22, 1994. As Jupiter rotates on its axis with a period of about 10 hours (9 hr 55 min 30 sec), the effects of the fall could still be detected. In fact, “ripples” on Jupiter out to its thin ring were recorded as late as 2002.  (See https://en.wikipedia.org/wiki/Comet_Shoemaker–Levy_9) and see https://www.skyandtelescope.com/astronomy-news/comet-shoemaker-levy-9-20-years-later-07162014/). So, the actual moments of these impacts may have been missed by all, but being in Earth orbit would not have made any difference.

When you watch Prof. Carrie Nugent’s TED Talk here, it is obvious that she is excited about her work and the opportunity to share it with an audience that will care. Those six minutes are fine as far as they go. The difference is that print captures everything except (perhaps) exuberance. 

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Thursday, April 25, 2019

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; 23rdPrinting 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. 

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Saturday, April 20, 2019

Gratsi Old Country Wine

New to Austin, Gratsi Old Country Wine was at the Wheatsville Co-op Friday night. I do not know how Italian they really are, but I am pretty sure that on Good Friday, most Catholics do their wine drinking at communion. But that's not me; so I tried it. And it was pretty good. Their sales representative, Amber, said that it is a blend of Cabernet Sauvignon, Petite Sirah, and Zinfandel. I liked the tart and fruity confluences. 


https://gratsi.com
(Still not much content on the website.)
















Cabernet Sauvignon is itself a cross between a red grape and a white grape. Wine connoisseurs give merlot several mid-range adjectives. So, it is easy to understand the success of this blend.

Quickly popular at Wheatsville Co-op


















Gratsi Old Country wine is available in about 30 retail stores around Austin. At our store demo, they happened to be given space in front of an end cap for chocolate bars with a sample tray.

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Awesome Austin Foods 
Awesomes Austin Foods at the Wheatsville Co-Op 
Sunday at the Co-op
Jerry Emanuelson's Algebraic Proof of Ricardo's Law of Association 

Friday, April 19, 2019

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
;
2ndEdition 
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. 

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Tuesday, April 2, 2019

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.

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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

Saturday, March 30, 2019

Simply Dirac by Helge Kragh

Paul Adrien Maurice Dirac predicted the existence of the positron and the magnetic monopole. He received a Nobel Prize in 1933. Born in 1902, the consensus is that Dirac did his best work between 1927 and 1937, which is typical for physics and mathematics. He devoted his very long later years to chasing philosophical questions, though, like other physicists, he denied the value of philosophy.
Simply Dirac by Helge Kragh
New York: Simply Charly (2016)

I read elsewhere that each equation costs 100,000 readers. So, popular books about science tend not to have them. This monograph has several, but none of them is the eponymous Dirac Equation. The author, Helge Kragh, was a professor at the Niels Bohr Institute. He wrote Dirac: A Scientific Biography as well as textbooks on physics. On that basis, Kragh brings Dirac to life for the general reader. To achieve that, Kragh also explains how quantum mechanics developed in the early 20thcentury as the accepted  theory of atomic physics. 

As a general reader, I believe that no “grand unified theory” ties QM to relativity. However, according to Kragh, among Dirac’s accomplishments was just that, at least for the electron. 

“Having no faith in the Klein-Gordon equation, at the end of 1927 Dirac decided to find a better solution for a wave equation in accordance with the theory of relativity. Based on the relativity requirements and the general requirements of quantum mechanics, he was convinced that the equation he looked for must be linear, not only in the general Schrödinger equation, but also in momentum.” (page 23) 

Dirac’s struggle with metaphysics was inherent in his work. How he viewed those problems is not clear from this biography. Dirac kept to himself. But this thin book quietly opens a very large and cumbersome door. Dirac was attracted to challenges from pure mathematics. If an equation is simple and beautiful, then it must have some physical reality, be of some empirical value, as both description and prediction. 
Dirac Equation From the BBC
The author apparently sides with the positivists who insist that the perception of truth begins with tangible experience. Moreover, strict positivists segregate truths: what is true in chemistry may not be true in zoology or aesthetics. That being as it may, Kragh admits: “Nonetheless, in the summer of 1926 Schrödinger published the relativistic version of his wave equation and so did half a dozen other physicists. Pauli called it “the equation with many fathers.”  Because two of the fathers were Oskar Klein and Walter Gordon a Swedish and German physicist respectively, it became known as the Klein-Gordon equation. It was a nice but apparently useless formula. The problem with relativity was not exclusive to wave mechanics for it also appeared in matrix mechanics and Dirac’s q-number algebra. After all, the three formulations were just different versions of the same theory, quantum mechanics.” (page 22)

Necessary facts are undeniable truths, not isolated chocolate-versus-vanilla preferences. Therefore, it is not surprising, but expected that many derivations exist for QM. Over 300 proofs are known for the Pythagorean Theorem.

University of California at San Diego
Dirac’s work explained electron spin. He began by not assuming it, working without the required spin in a simplified mathematical model. But as Dirac developed the equations, spin appeared as needed. Moreover, Dirac’s theory explained the fine structure of the hydrogen spectrum, an important empirical validation. Firmly established in sensible reality, Dirac’s 4x4 matrix coefficients also admitted of seemingly impossible alternatives. Some particles exist beyond the speed of light and the more energy they obtain, the slower they move. George Gamow called them “donkey electrons.” (The more you beat it, the less it moves.) Kragh makes no mention of tachyons which (it seems) come into our awareness only as they decelerate below the speed of light.  

Through his mature years, and to the end of his life, Dirac rejected the renormalized quantum electrodynamics advanced by Richard Feynman (among others). To Dirac QED was inelegant. Kragh defends Dirac by citing Einstein. Question: “What would you say if the experiment did not prove your theory?” Answer: “I would tell them to run the experiment again.” 

Wikipedia
The false divide between reason and reality has been the grave error of modern philosophy. The British empiricists and the continental rationalists argued chickens and eggs. And it continues today. Electricity carried messages and powered railroad locomotives before nuclear power heated water to light up our cities. James Watt and Mathew Boulton literally made money by making money at the Soho Mint a lifetime before statistical mechanics explained thermodynamics. Those theoretical explanations must be internally consistent because contradictions cannot exist. 

Paul Adrien Maurice Dirac discovered and pursued necessary factual truths. Experimental physics and applied engineering are the tools that extend our senses to perceive to the beauty in truth.

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