—In the talk I’d had with Blacky his earnestness had sent a shiver thru my chest —it is ever so, and men are men —And is Blacky less a man because he never married and had no children and did not obey nature’s injunction to multiply corpses of himself?  With his brooding dark face and pout by the stove and lowered pious eyes, on some rainy night next winter, there will come diamond and lotus hands to ring a rose around his forehead (or I bust)  (to miss my guess) —

Desolation, Desolation,

wherefore have you

Earned your name?

–– Desolation Angels by Jack Karouac p 60


Even Hilbert, however, could not foresee how mathematics would grow in the 20th century.  There were more and more mathematicians, more and more journals, more and more professional societies.  The phenomenal growth that had begun in the 1800s [sic] continued, with mathematical knowledge doubling every twenty years or so.  More original mathematics has been produced after astronauts first walked on the moon than there had been in all previous history.  In fact, it is estimated that 95% of the mathematics known today has been produced since 1900.  Hundreds of periodicals published all over the world devote a major share of their space to mathematics.  Each year the abstracting journal Mathematical Reviews publishes many thousand synopses of recent articles containing new results.  The 20th century (and perhaps this new century, as well) is justifiably called the “golden age of mathematics.”

Quantity alone, however, is not the key to the unique position the current era occupies in mathematical history.  Beneath this astounding proliferation of knowledge, there is a fundamental trend toward unity.  The basis for this unity is abstraction.  This conceptual unity has led in two directions.  On the one hand, new, more abstract subfields of mathematics have emerged to become established ares of research in their own right.  On the other, researchers working on truly big classical problems, such as Fermat’s Last Theorem or Hilbert’s twenty-three problems, have become increasingly adept at using new techniques from one area of mathematics to answer old questions in another.  As a result, the 20th century has been a time when many old questions were finally answered and many new questions have come to the fore.

–– Math through the Ages (A Gentle History for Teachers and Others)” by William P Berlinghoff and Fernando Q Gouvêa pages 53-54


Self v Other re: Ideas

There will come many times as you progress through life where the other will present opposition to the self.  This is not surprising in the main, but it will potentially be in the specific.  It can come at any time and it can work to pull out that proverbial rug.  Proceed with caution.

I was taking guitar lessons from a guy I knew.  Not much of teacher, really, but he had his certain skills as a guitar player.  At any rate I informed him when we were first starting out about how I was developing my finger-picking style.  It was a style that involved all five fingers of the picking hand.  He dismissed this as not being finger-picking but perhaps merely some gimmick and proceeded to inform me of a two-to-three finger-picking style (such as he used).

Subsequently I read about other finger-picking styles, some of which included five-finger styles.  Perhaps he wasn’t wrong but he wasn’t all that right.

When I was a young driver I received a negligent driving ticket.  This is a criminal as opposed to a moving violation, and as such is subject to trial by (short=8 person) jury.  I told my mom I would be defending myself in court and she proceeded to inform me that I would be up against professional attorneys and would surely lose.

This is the only time I swore at my mother.  I scolded her for not believing in my abilities.  Subsequently I won in court.  The assistant prosecutor said to the prosecutor “his jury instructions are better than yours” when I arrived at court prepared for court the first day.  The prosecution motioned to dismiss.  No one objected.  The case was dismissed.  Perhaps she wasn’t wrong but she wasn’t all that right.

During my time studying architecture at the University of Washington I remarked to a professor that I had surmised architectural perspectival drawing was a type of geometry.  He assured me that was not the case at all.  I could not imagine a way to reconcile these disparate notions; surely it must be a kind of geometry.  We even built each line according to a set of rules!  Alas, he said, it was not.  Even Euclid, primarily famous for his Geometry, wrote one other book to secure fame and his place in history:  Optiks.  No, not geometry.

Subsequently, I read this passage in “Math through the Ages (A Gentle History for Teachers and Others)” by William P Berlinghoff and Fernando Q Gouvêa (page 36):

Somewhat related to all this was the discovery of perspective by Italian artists.  Figuring out how to draw a picture that gave the impression of three-dimensionality was quite difficult.  The rules for how to do it have real mathematical content.  Though the artists of the Renaissance did not subject these rules to a complete mathematical analysis, they understood that what they were doing was a form of geometry.  Some of them, such as Albrecht Dürer, were quite sophisticated in their understanding o the geometry involved.  In fact, Dürer wrote the first printed work dealing with higher plane curves, and his investigation of perspective and proportion is reflected both in his paintings and in the artistic work of his contemporaries.

As you can see, many did not know that what they were doing was geometry but yet some did.  Perhaps my professor wasn’t wrong but he wasn’t all that right.

Also at the UW, I had a philosophy professor who let me know that what he was doing was philosophy but he was unclear as to what I was doing.  I suggested my philosophical leanings were more narrative to his more analytical leanings.  He also insisted this was incomprehensible, assuring me that he was doing philosophy and having no idea what I was doing.  I could hear Nietzsche cringe, I’m sure of it.

It’s hard not to believe people, especially those whom you respect for any reason, when they tell you that your brilliant idea is shit.  It’s always a challenge to disagree with the other and all the more challenging in this kind of situation.  Often these folks are trying to help!  They are not nefarious.  They want you to succeed!

Of course, just because this pattern exists doesn’t guarantee the self is justified or correctly imaging the world at large.  But it may well be worth your time to scour the literature or other evidence to see if there isn’t a modicum of vindication available.  And, as always, proceed with caution.

As a peripheral tale, when I was in grade school and bored with the math homework we had been assigned (four-digit multiplication) I sought to make it more interesting by doing the assignment in Roman numerals.  I don’t know if you have ever tried doing any sort of math with Roman numerals but they are not well-suited to long multiplication, let me tell you!

Also from Math through the Ages (but page 70):

Of course, it’s not impossible to compute with Roman numerals.  It’s just complicated.  Logician Martin Davies tells that

“In 1953, I had a summer job at Bell Labs in New Jersey (now Lucent), and my supervisor was Claude Shannon [a computer pioneer and the creator of the mathematical theory of communication].  On his desk was a mechanical calculator that worked with Roman numerals.  Shannon had designed it and had it built in the little shop Bell Labs had put at his disposal.  On a name plate, one could read that the machine was to be called “Throback I.”

Though we still use Roman numerals for ornamental purposes, there is no chance we’ll ever abandon the compact, convenient, and useful Hindu-Arabic system.  The power of the Hindu-Arabic system stems from its efficient positional structure, which is based on powers of ten.  That’s why we call it a decimal place value system.

I managed to complete only one of the problems assigned.  The teacher made no mention of these efforts.  I was disappointed.  I thought “look what I did!” and no one cared.  Story of my life, I guess.


It is said to have been Cardinal Richelieu’s disgust with a frequent dinner guest’s habit of picking his teeth with the pointed end of his knife that drove the prelate to order all the points of his table knives ground down.  In 1669, as a measure to reduce violence, King Louis XIV made pointed knives illegal, whether at the table or on the street.  Such actions, coupled with the growing widespread use of forks, gave the table knife its now familiar blunt-tipped blade.  Toward the end of the seventeenth century, the blade curved into a scimitar shape, but this contour was to be modified over the next century to become less weaponlike.  The blunt end became more prominent, not merely to emphasize its bluntness but, since the paired fork was likely to be two-tined and so not an efficient scoop, to serve as a surface onto which food might be heaped for conveying to the mouth.  Peas and other small discrete foods, which had been eaten by being pierced one by one with a knife point or a fork tine, could now be eaten more efficiently by being piled on the knife blade, whose increasingly backward curve made it possible to insert the food-laden tip into the mouth with less contortion of the wrist.  During this time, the handles on some knife-and-fork sets became pistol-shaped, thus complementing the curve of the knife blade but making the fork look curiously asymmetrical.

––  The Evolution of Useful Things (How Everyday Artifacts–from Forks and Pins to Paper Clips and Zippers–Came to Be as They Are) by Henry Petroski  p 12