Monthly Archives: March 2009

A Mathematician’s Mathematicians: Professor Herb Keller, and his Notation for QED

I decided to change my major from Physics to Mathematics at the end of my junior year at Caltech. When I returned to the campus in September, 1965, I was told that my advisor would be Prof. Richard diPrima. I learned as I started to write this post that there is an award named in his honor, The SIAM Richard C. DiPrima Prize.

Prof. diPrima was the first person I met who was also a member of NYU’s Courant Institute of Mathematical Sciences (CIMS), then and now the world’s foremost school of applied mathematics.

He was a wonderful man. I recall vividly our first conversation, in which I said I had decided to change my major since I had learned how to program, and found that, especially when done in applied mathematics, programming was much more fun than physics.

He said that was very important, for if you found something that you enjoyed doing, then all the rest would take care of itself.

We didn’t do much mathematics together. He suggested I do an independent study project with a former colleague of his from CIMS, Prof. Herbert “Herb” Keller, who was spending the year visiting Caltech.

Herb was a wonderful teacher and mentor. He suggested as a project that I wrote programs related to the Kruskal Solitrons, and I had a lot of fun working with him.

Fun. It is ALL ABOUT FUN. Never forget that, and the details will take care of themselves.

Herb returned to CIMS in September, 1966, the same month I began my graduate studies there.

I took his course on Ordinary Differential Equations in the first, or perhaps second, year.

His classes were a great joy. He would start with a substantial theorem or topic, and then spend the rest of a two-hour class proving the theorem, or digging into the details. His passion for mathematics was evident in every sentence he spoke, or every equation he wrote on the board.

Whenever Herb finished a proof, he would not write QED but a symbol of his own devising.

QED stands for “Quod Erat Demonstrandum” and is the standard way of marking the end of a proof.

Herb instead wrote one of the following symbols:

– o –
o – o

I of course copied this into my notes, and continued to do so for all the other proofs I went through during my remaining years as a student of mathematics

I have continued to use this notation to this day, to mark the end of an important paragraph, or the end of a first draft, and so forth

Of course, each and every time I do I think of him, and what a fine man he was.

Herb left CIMS to become a professor at Caltech within a year or so. I read recently that he had died at the age of 83, and that he had been an avid bicyclist in his later years.

I also had the great good fortune to have Eugene Isaacson as my professor for at least one, and I think more, courses.

The good fortune is doubled in that I count among my many great professors both authors of a classic work on numerical analysis, Analysis of Numerical Methods, known to mathematicians as “Isaacson and Keller.”

Prof. Isaacon was also a wonderful teacher. He also had a striking resemblance to my favorite author, the man after whom this blog is named, A. J. Liebling. I often think of the other when I think of one of them.

Professor Herbert “Herb” Keller – May His Memory Be a Blessing.

– o – o – o – o

The World is Flat and Small: Bruce Robinson

This past Thursday I attended an event for Caltech alumni in New York City. This was only the third or so time I had done so. The most recent was a few years back when the alumni attended as a group a performance of a play about R. P. Feynman in which Alan Alda played the title role.

I was sipping a glass of wine when I asked the man standing to me if he had gone to Caltech. He said that he had not, but had attended a choral school near Princeton, New Jersey.

When I asked him what he did, he said he was an agent for playwrights.

I said that one my relatives was the playwright Bruce Robinson.

He said he was Bruce’s agent and reached over to shake my hand.


– o –

The World is Flat and Small

Several recent incidents have convinced me that only do we live in a Flat World, we also live in a small world.

The world is both Flat and Small, as I shall try to demonstrate in a series of posts on this topic.

I will begin with two incidents. Both happened within the last few days. They led to my claim that the world, viewed as a social network, is much smaller than most people think.

Dr. Barbara Cooper: Work to Keep Anxiety At Bay

One of my wife’s oldest friends is Dr. Barbara Cooper. My wife recently told me that Barbara had suggested it was important to work, because working would help keep anxiety at bay, especially in these troubled times when so many of us, as am I, are unemployed and seeking employment.

This is a profound observation, and so I am posting it here to share the thought with others.

Barbara’a web site is College Counseling Info.

Barbara has asked my help in updating her web page, and I have already created a Ning site to help her do so. I’ll let you know when the new version is available.

Thanks Barbara!

On Technology: Charles Babbage and the Invention of the Cowcatcher

During my years as a graduate student at NYU’s Courant Institute of Mathematical Sciences (CIMS), I spent many a day in the CIMS library on the twelfth floor. From time to time I would take a break by looking for a book that was about mathematics, not an exposition of it.

Among my favorites was the work of Augustus De Morgan. De Morgan was a famed mathematician. He was also a wonderful writer.

He spent some of his spare moments debunking circle-squares and pi-finders, as shown by the following quote from the Wikipedia entry about him:

How can the sound paradoxer be distinguished from the false paradoxer? De Morgan supplies the following test:
“The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself… New knowledge, when to any purpose, must come by contemplation of old knowledge, in every matter which concerns thought; mechanical contrivance sometimes, not very often, escapes this rule. All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors and learned in what had been before them. There is not one exception.”
“I remember that just before the American Association met at Indianapolis in 1890, the local newspapers heralded a great discovery which was to be laid before the assembled savants — a young man living somewhere in the country had squared the circle. While the meeting was in progress I observed a young man going about with a roll of paper in his hand. He spoke to me and complained that the paper containing his discovery had not been received. I asked him whether his object in presenting the paper was not to get it read, printed and published so that everyone might inform himself of the result; to all of which he assented readily. But, said I, many men have worked at this question, and their results have been tested fully, and they are printed for the benefit of anyone who can read; have you informed yourself of their results? To this there was no assent, but the sickly smile of the false paradoxer”

He also wrote many anecdotes about mathematicians and scientists, Among my favorite is the story of how Charles Babbage, best known as the creator of the Analytical Engine, came to invent what is now known as the Cowcatcher:

Charles Babbage and his accomplice, Lady Lovelace,
came very close to inventing the computer more than a century before American engineers produced ENIAC. The story of the “Analytical Engine” is

a tale of two extraordinarily gifted and ill-fated British eccentrics whose
biographies might have been fabrications of Babbage’s friend Charles Dickens, if Dickens had been a
science-fiction writer.

Like many contemporary software characters, these computer pioneers of the Victorian age attracted as
much attention with their unorthodox personal lives as they did with their inventions.

One of Babbage’s biographies is entitled Irascible Genius.. He was indeed a genius, to judge by what
he planned to achieve as well as what he did achieve. His irascibility was notorious. Babbage was
thoroughly British, stubbornly eccentric, tenaciously visionary, sometimes scatterbrained, and quite
wealthy until he sank his fortune into his dream of building a calculating engine.

Babbage invented the cowcatcher–that metal device on the front of steam locomotives that sweeps errant
cattle out of the way. He also devised a means of analyzing entire industries, a method for studying
complex systems that became the foundation of the field of operational research a hundred years
later. When he applied his new method of analysis to a study of the printing trade, his publishers were so
offended that they refused to accept any more of his books.

Undaunted, he applied his new method to the analysis of the postal system of his day, and proved that the
cost of accepting and assigning a value to every piece of mail according to the distance it had to travel was
far more expensive than the cost of transporting it. The British Post Office boosted its capabilities
instantly and economically by charging a flat rate, independent of the distance each piece had to travel–the
“penny post” that persists around the world to this day.

Babbage devised the first speedometer for railroads, and he published the first comprehensive treatise on
actuarial theory (thus helping to create the insurance industry). He invented and solved ciphers and made
skeleton keys for “unpickable locks”–an interest in cryptanalysis that he shared with later computer
builders. He was the first to propose that the weather of past years could be discovered by observing
cycles of tree rings. And he was passionate about more than a few crackpot ideas that history has since
proved to be nothing more than crackpot ideas.

His human relationships were as erratic as his intellectual adventures, to judge from the number of lifelong
public feuds Babbage was known to have engaged in. Along with his running battles with the Royal
Societies, Babbage carried on a long polemic against organ-grinders and street musicians. Babbage would
write letters to editors about street noise, and half the organ-grinders in London took to serenading under
Babbage’s window when they were in their cups. One biographer, B. V. Bowden, noted that “It was the
tragedy of the man that, although his imagination and vision were unbounded, his judgment by no means
matched them, and his impatience made him intolerant of those who failed to sympathize with his projects.”

Babbage dabbled in half a dozen sciences and traveled with a portable

He was also a supreme nit-picker, sharp-eyed and cranky, known to write outraged letters to publishers of
mathematical tables, upbraiding them for obscure inaccuracies he had uncovered in the pursuit of his own
calculations. A mistake in navigational table, after all, was a matter of life and death for a seafarer. And a
mistake in a table of logarithms could seriously impede the work of a great mind such as his own.

His nit-picking indirectly led Babbage to invent the ancestor of today’s computers. As a mathematician and
astronomer of no small repute, he resented the time he had to spend poring over logarithm tables, culling all
the errors he knew were being perpetuated upon him by “elderly Cornish Clergymen, who lived on seven
figure logarithms, did all their work by hand, and were only too apt to make mistakes.”

Babbage left a cranky memoir entitled Passages from the Life of a Philosopher–a work
described by computer pioneer Herman Goldstine as “a set of papers ranging from the sublime to the
ridiculous, from profundities to nonsense in plain bad taste. Indeed much of Babbage’s career is of this sort.
It is a wonder that he had as many good and loyal friends when his behavior was so peculiar.”

In Passages, Babbage noted this about the original inspiration for his computing machines:

The earliest idea that I can trace in my own mind of calculating arithmetical tables by machinery rose in this

One evening I was sitting in the rooms of the Analytical society at Cambridge, my head leaning forward on
the table in a kind of dreamy mood, with a Table of logarithms lying open before me. Another member,
coming into the room, and seeing me half asleep, called out, “Well, Babbage, what are you dreaming about?”
To which I replied, “I am thinking that all these Tables (pointing to the logarithms) might be calculated by

In 1822, Babbage triumphantly demonstrated at the Royal Astronomical Society a small working model of a
machine, consisting of cogs and wheels and shafts. The device was capable of performing polynomial
equations by calculating successive differences between sets of numbers. He was awarded the society’s
first gold medal for the paper that accompanied the presentation.

In that paper, Babbage described his plans for a much more ambitious “Difference Engine.” In 1823, the
British government awarded him the first of many grants that were to continue sporadically and
controversially for years to come. Babbage hired a master machinist, set up shop on his estate, and began
to learn at first hand how far ahead of his epoch’s technological capabilities his dreams were running.

The Difference Engine commissioned by the British government was quite a bit larger and more complex
than the model demonstrated before the Royal Astronomical Society. But the toolmaking art of the time
was not yet up to the level of precision demanded by Babbage’s design. Work continued for years,
unsuccessfully. The triumphal demonstration at the beginning of his enterprise looked as if it had been the
high point of Babbage’s career, followed by stubborn and prolonged decline. The British government finally
gave up financing the scheme.

Babbage, never one to shy away from conflict with unbelievers over one of his cherished ideas, feuded over
the Difference Engine with the government and with his contemporaries, many of whom began to make sport
of mad old Charley Babbage. While he was struggling to prove them all wrong, he conceived an idea for an
even more ambitious invention. Babbage, already ridiculously deep in one visionary development project,
began to dream up another one. In 1833 he came up with something far more complex than the device he
had failed to build in years of expensive effort.

If one could construct a machine for performing one kind of calculation, Babbage reasoned, would it be
possible to construct a machine capable of performing any kind of calculation? Instead of building
many small machines to perform different kinds of calculation, would it be possible to make the parts of
one large machine perform different tasks at different times, by changing the order in which the
parts interact?

Babbage had stumbled upon the idea of a universal calculating machine,

an idea that was to have momentous consequences when Alan Turing–another brilliant, eccentric British
mathematician who was tragically ahead of his time–considered it again in the 1930s. Babbage called his
hypothetical master calculator the “Analytical Engine.” The same internal parts were to be made to
perform different calculations, through the use of different “patterns of action” to reconfigure the order in
which the parts were to move for each calculation. A detailed plan was made, and redrawn, and redrawn
once again.

The central unit was the “mill,” a calculating engine capable of adding numbers to an accuracy of 50 decimal
places, with speed and reliability guaranteed to lay the Cornish clergymen calculators to rest. Up to one
thousand different 50-digit numbers could be stored for later reference in the memory unit Babbage called
the “store.” To display the result, Babbage designed the first automated typesetter.

Numbers could be put into the store from the mill or from the punched-card input system Babbage adapted
from French weaving machines. In addition, cards could be used to enter numbers into the mill and specify
the calculations to be performed on the numbers as well. By using the cards properly, the mill could be
instructed to temporarily place the results in the store, then return the stored numbers to the mill for later
procedures. The final component of the Analytical Engine was a card-reading device that was, in effect, a
control and decision-making unit.

A working model was eventually built by Babbage’s son. Babbage himself never lived to see the Analytical
Engine. Toward the end of his life, a visitor found that Babbage had filled nearly all the rooms of his large
house with abandoned models of his engine. As soon as it looked as if one means of constructing his device
might actually work–Babbage thought of a new and better way of doing it.

The four subassemblies of the Analytical Engine functioned very much like analogous units in modern
computing machinery. The mill was the analog of the central processing unit of a digital computer and the
store was the memory device. Twentieth-century programmers would recognize the printer as a standard
output device. It was the input device and the control unit, however, that made it possible to move beyond
calculation toward true computation.

This invention shows that Babbage, like my late friend, colleague and thesis advisor, Jacob T “Jack” Schwartz, had a very wide-ranging mind. He did not limit his skills, or his vision, to a single field, but worked in many fields.

This is also a reminder that it pays to attempt to expand one’s own reach. You should not be afraid to try new things, and to take risks in doing so, for that is the best way to grow, both professionally and personally.

Marketing Yourself on the Web: Give the Gift of a Name

One of the nicest presents you can give two members of your network is to give each The Gift of a Name.

With one email you can turn two relationhips into four:

  • Yours with one of the two members;
  • Yours with the other member ;
  • The new relationship of the two members;
  • Your relationship with yourself, for taking the time to introduce them.

Not bad? Two relationships turned into four, something especially appealing to me as a progorammer who thus thinks in powers of two.

The time to write the initial email is small, but to be most effective you need to write many more emails, as follows:

Each and every time you accept someone into your network, or they accept you into theirs, you need to review all the membership, looking for two people known to you who would benefit from the give of each other’s name.

Thus the cost of adding a new member grows exponentially as the size of your network grows,

This is perhaps the strongest form, in terms of the value added by each new member, of what is known as the Network Effect.

Even though the cost is high, the benefits of creating new relationships will more than pay for itself, as I have written in my post, On Volunteerism: Cast thy bread upon the waters for thou shalt find it after many days.

The Gift of a Name

A few days back I realized that two women in my network should get to know each other, so I sent the folowing email:

To: Fran Allen, Susanna Kass
From: Dave Shields

You two should get to know each other. The nicest present I can give each of you is the name of the other. Enjoy!


Fran is a legendary figure in programming and compiler optimization. She is the “Allen” in “Cocke/Allen” She was IBM’s first woman Fellow (IBM’s most senior technical postion), and has worked for *decades* to encourage women to make careers in science and technology. She is also one of those rare people, like yourself, about whom it is absolutely impossible to say anything bad.


I met Susanna when she was at Palamida. The OSSC did an evaluation of Palamida back in 2006. I spent a week in Tucson working with Storage folks on the evaluation. On her own initiative, Susanna flew from San Fran to Tuscon and brought along the CTO of Palamida to work with our engineers. I have since met with her several times when she wanted to find a way to help folks engage with IBM, notably Scott Collison, CEO of

Scott Tolliver, Palamida’s CEO, said Susanna was in the top 1% of the people he had worked with, and I agree.

I consider you both to be among my mentors, and am very happy that I can say that.


On the Meeting of the Business School Deans in a Closed Room

I have heard a rumor, though I’m confident it is a fact, that someone recently invited the Deans and senior professors of the world’s major business schools to a one-day closed meeting in a nearby state.

Here is what I was told, presented in the format of a Q and A:

Q1: Why did they come?

A1: They know that the ONLY thing heading south faster than the Dow is the value of MBA degrees from their institutions.

Q2: What public action did they feel obligated to do? Hint: None of them have.

A2: Apologize for training the generation of graduates of their schools who were so effective at using the education they received that they almost destroyed the world economy.

Q3: What is their biggest problem going forward?

A3: They know they will have to rebuild their curriculum, and their faculty, from the ground up, and that will take at least a generation.

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