Quote of the Week...from René Descartes

I only speak one language really, mathematics, but I dabble in others like English just enough to answer questions and buy groceries.        ---René Descartes

René Descartes (1596-1650) was a famous French philosopher and mathematician...we have him to thank for the Cartesian coordinate system named after him.  I encountered his above quote through a cryptogram I solved the other day, from a mixed puzzle magazine I bought at Publix while, incidentally, buying groceries. Being an amateur, dabbling student in foreign languages, I was intrigued by his likening of mathematics as a language in its own right. Also, I just finished reading Foo Fighters creator and front man Dave Grohl's autobiography The Storyteller and derived from his experiences with music that it, too, is a sort of language.  So, is learning the languages of mathematics and music essentially different in nature from learning Chinese, Russian, German, English...or any other tongue?  I wonder...maybe Steve Kaufmann's advocacy of attaining passive comprehension in them works similarly to what he promotes with foreign language acquisition.  After all, he says that the human brain is naturally wired to pick up languages and that we should focus ourselves on reading and listening on a grand scale, discerning the linguistic patterns as we go along on a deliberate journey of learning that necessarily takes patience with the results we first achieve.  I want to become "proficient" in both math and music, although with the former I scored a perfect 800 on my SAT and with the latter I have a mental library of thousands of songs and pieces I've heard over the years...albeit without the detailed analytical music theory behind them or musical skill with instruments or voice.  I'd written before on this blog that I'd like to be able to play a guitar or keyboard, if for no other reason other than to be able to share some of that music I already mentally possess with others.  Yes, I wonder whether the linguistic theory of "comprehensible input" can apply to those two areas as well...I'll bet Descartes knew the answer...

Let's Talk about Math

 Let's talk about mathematics...with apologies to Salt-N-Pepa...

---What do you think about mathematics?  

---Do you still do any math, and if so, what?  

---Does your job/career involve doing math?  

---When you went to school, what mathematics courses did you take?  

---What do you think about how they were taught?  

---How easy or hard were they?  

---How hard did you work in them?  

---Was/is mathematics interesting to you? 

---What is your favorite area in mathematics? What is your least favorite?

---What was the most "advanced" course of math that you took?

---How much of the mathematics you learned in school do you still remember?

---How much of it have you used since leaving school?

---Do you think that mathematical proficiency puffs up some people's egos while difficulty in the field deflates others?  

---Do you believe that there is absolutely no ego within mathematics itself?  

---Do you see people who bring up mathematical ideas as trying to impress others?

---Can you "speak" math, that is express concepts, equations and diagrams verbally with a fair degree of approximation?  

---If you spotted somebody out in a coffee shop studying math and the two of you struck up a conversation, would you automatically ask them what it was for?  

---Do you think people have to be studying "for" something when they study something like mathematics?  

---If people respect the etiquette of not interrupting someone when they are watching a two-hour movie, should they also respect the wish of that same person to be left alone for a half hour studying mathematics? 

---If you randomly brought up mathematics in conversation with others, would you feel awkward?  

---Do you feel awkward reading this article about mathematics? 

---Do you think this is too provocative a topic for Salt-N-Pepa to sing about?

Of course, I have opinions about all these questions and more and plan to give my own answers sometime down the line...

The Awkwardness of Talking About Mathematics

Many decades ago, back in late 1973, I was a teenager working at a Miami Shores firm as a records clerk.  Hank, one of the auditors there, one day randomly asked me if I knew about trigonometry and could explain to him the meaning of sine and cosine.  Having studied the basics of it all a year earlier, I patiently presented the concepts as I had learned them: starting with a circle of radius 1 and center (0,0) and then explaining cosines and sines as the x and y points on that circle where the line intersecting said circle goes through the origin and creates an angle with the x-axis (of which cosines and sines are functions).  But I didn't speak any of this by itself...I got pen and paper and illustrated it.  Ol' Hank then got very angry and said he only wanted to know about solving triangles using trigonometry (which I was on the way to covering) ...and I walked away, not in the least bit willing to accommodate that anger: screw him.  Hank was a contentious little so-and-so who would take history classes at the local community college just to argue with the instructor...a real ditto-head long before Rush Limbaugh would coin it and turn it into a dubious badge of honor.  I mention all this not because I carry any grudge or offense against this forgettable character, but because it happens to be the last time anyone brought up anything about mathematics to me away from school that went beyond simple arithmetical problems.  And since I did not become a mathematics teacher, engineer or scientist, I have never professionally used anything I have ever learned in this field beyond early elementary school, although in college I studied calculus, differential equations, probability and statistics, and linear algebra. Yet within the academic corridors mathematics reigns supreme, although it's not exactly a conversational icebreaker.  In fact, I find bringing up mathematics in conversation with anyone extremely awkward and random...after all, mathematics is the antithesis of ego: it exists on its own terms and cares not whether folks like it, understand it, or even find it interesting.  But I think I'm going to experiment around a bit and see whether or not I can get people to open up about how they feel about mathematics, and even delve lightly into its contents if that's possible.  My initial results seem to indicate that people are a lot more interested in talking about how math made them feel and affected their lives than discussing it on its own terms...

Just Finished Reading A Beautiful Mind by Sylvia Nasar

A Beautiful Mind, published in 1998, is Sylvia Nasar's biography of American mathematician John Forbes Nash, Jr. (1928-2015).  It is better known for its 2002 Ron Howard-directed movie adaptation with the same title, starring Russell Crowe as Nash and which won the Academy Award for best picture that year.  Yet the movie diverges in major ways from the book that took me aback.  In both versions we are presented with the life story of a brilliant, accomplished academic whose young adult life was turn upside-down...as well as that of his loved ones...when he became afflicted with paranoid schizophrenia.  The film version, however, takes great liberties with Nasar's account, weaving its own imaginary tale about John Nash suffering from visual hallucinations (which he didn't) and imaginary friends and foes (again, he didn't).  According to the book, in the late 1950s Nash began to have grandiose delusions of his role in the world with conspiracy theories abounding...in today's dumbass world that would have got him elected president but back then it just got him committed...and released...and committed...and released.  Ron Howard's movie makes the story of Nash's life very compact and simple, omitting the Noble Prize-winning mathematician's earlier affair and love child, for which he denied financial support.  It also omitted the fact that even following his mental breakdown he continued to produce mathematical works, many of which Nasar bravely attempted to describe for us nonmathematical readers: in the film you'd think that all he did of significance was that prize-winning paper on economic equilibrium.  The book and the film each sought to sell a narrative about the life of John Forbes Nash, Jr., but even though the print version was obviously closer to the truth about him, by its very structure and nature it was still far from what really happened.  A life is lived one minute, one hour...one day at a time and attempts to make a narrative tale out of it that conforms to traditional storytelling patterns is bound to distort it.  Although I found the movie version to be much more suspenseful, endearing and entertaining (I even bought the DVD), it's a bit unsettling to see how far it diverged from the true life of this extraordinary and flawed human being after reading Sylvia Nasar's excellent account.  If you saw the film...or didn't, for that matter...A Beautiful Mind will be a great ride through history...not only of one special individual's life, but also of the social and historical context in which he lived.  Sadly, in 2015 both John Nash and his wife Alicia perished in an auto crash in Newark after returning from Norway, where he had been honored with an academic prize...he was 86... 

I got two significant take-aways from the book and film.  One is that it's not prudent to automatically accept the truth and accuracy of a work's narrative and characterizations, just because the author or producer claims it to be nonfiction or based on real events: the very nature of the story structure with the presentation of a moral theme and the aim to be interesting is bound to twist it.  Two is that I think almost all of us, like Nash, believe in nonsensical, irrational...and sometimes farfetched ideas but that for "healthy" people they understand the extreme nature of those beliefs and thereby tend to avoid communicating them to others...unlike Nash, whose narcissistic sense of self-importance precluded this.  But if within society certain wacky ideas seem to become more mainstream then an explosion can ensue with masses of the population expressing adherence to them...do you know what I'm talking about?

Just Finished Reading Calculating the Cosmos by Ian Stewart

Ian Stewart is a prolific popular science writer...he's a British mathematician and has also dabbled in science fiction writing.  I just finished reading his 2016 book Calculating the Cosmos, in which he goes through selected interesting topics concerning our solar system, galaxy, the universe...and their respective origins, laying out the different historical viewpoints as well as the current scientific perspective, including ongoing disputes.  Like the other popular science book I just read, David Sumpter's The Ten Equations That Rule the World, Calculating the Cosmos...while generally easy to follow...does dip into some rather technical jargon that loses me for a bit until I can regain my footing.  More than about math, though, this book is an excellent introduction to the fields of astronomy, astrophysics and cosmology, discussing interesting (to me) topics such as how Earth got its Moon, the various planets and their satellites, the Kuiper Belt, Oort Cloud, the nature of black holes, the big bang theory, the cosmic radiation background, dark matter and dark energy.  Along the way Stewart surprised me by claiming that some of these concepts aren't necessarily established truths and that his colleagues are coming up with alternative theories to explain observations without involving them, especially dark matter and dark energy...I found that pretty provocative.  And, as was the case with Sumpter's book, I intend to reread Calculating the Cosmos...nonfiction that has a lot of technical content has to be approached differently than with regular fiction.  I'm also interested in checking out some of Stewart's other works...

Just Finished Reading The Ten Equations That Rule the World by David Sumpter

David Sumpter is a professor of mathematics at a Sweden's Uppsala University, born and raised in Great Britain. His The Ten Equations That Rule the World, from 2020, is about the secret society of TEN, a loose group over the course of modern history comprised of mathematicians and those who have capitalized on ten fundamental formulas that describe aspects of our world and social organization. Sumpter goes chapter by chapter in laying out each equation, its history, and applications while inevitably slipping into rather technical mathematics. Still, he always returns to straightforward nontechnical prose illustrating principles that even those not inclined toward mathematics can understand and use to analyze the world around themselves and draw some benefit. If you're not well-versed in mathematics...and I certainly fit the bill here...you may feel a little lost in some parts of this book. But I'm confident that you'll come away from the reading experience with a greater respect for how things work and how to assess one's situation in diverse settings more reasonably and accurately. I wrote out the ten equations in the above picture...below are their corresponding names... 

1 JUDGMENT EQUATION (Baye's Rule)

2 BETTING EQUATION (Logistic Regression)

3 CONFIDENCE EQUATION (Confidence Intervals)

4 SKILL EQUATION (Markov Assumption)

5 INFLUENCER EQUATION (Stationary Distribution)

6 MARKET EQUATION (Stochastic Differential Equations)

7 ADVERTISING EQUATION (Correlations)

8 REWARD EQUATION

9 LEARNING EQUATION

10 UNIVERSAL EQUATION

Starbucks, Running and Math

Today promises to be pretty dreary weatherwise here in Gainesville...I'm sitting here inside the Starbucks in northern Gainesville on 43rd Street for the first time since early in the year.  Some Starbucks have a more intimate, welcoming quality to them, but this one is more like a mall food court with noises amplified by the acoustics of the spacious room...that's all right, the other Starbucks I usually frequent was off-line for my order and I've had it with standing in line anymore.  Once again I'm sitting along one side facing the electric substation just south of Talbot Elementary School...connected by a walking path I would traverse during my marathon training phase back in 2010-11 while on one of my many long runs back then: ah, those were the days!  I'd like to return to that wonderful era, and yesterday's measured run around my house totaling 14.4 miles was a good step in that direction.  There is a specific type of lifestyle that I am seeking which mixes family, a small number of friends, learning, occasional travel, and running/walking into a pleasant-yet-challenging routine combining interaction, growth, and new memories.  Part of it I am already living...much more is yet to come.  I have been stepping out tentatively into the vast, wild ocean of mathematics by watching special programming and reading books on the subject...my goal is to be in deep waters by this spring and then see where it all takes me.  The book I'm currently reading, The Ten Equations That Rule the World, by mathematician David Sumpter, illustrates the conundrum I find myself in whenever such an expert in the field introduces various concepts.  It starts out quite clearly as he lays out the basics, but then suddenly it's as if he has taken off with his discussion and left me lost, standing on land while he is frolicking way out there, far offshore.  This also happens with the Great Courses DVD set The Joy of Mathematics as Arthur Benjamin takes on the role there as teacher.  I don't blame either Sumpter or Benjamin for me falling behind their respective narratives, but rather see this as a challenge that is similar to that which getting back to running imposed itself on me in the weeks following my July surgery: at first progress is slow and I can clearly see my own inadequacy in the area and where I want to go for improvement, anticipating a breakthrough down the line if only I just stick with my efforts.  For me, whether it's running or mathematics or any other area, this is fun stuff to pursue...

An Upcoming Personal Project for 2022

I've been watching the Beatles' Get Back movie on Disney+, which depicts its members going through a series of grueling all-day sessions of song creation, rehearsing and jamming. My initial reaction was to question why anyone would subject themselves to this kind of regimen. Then it occurred to me that for Paul, John, George and Ringo this was their great love, where their skills and interests coincided and they were the most accomplished and recognized: that's why it worked for them (although they still quarreled).  The question I ask is for myself is what in my life is comparable to that which the Beatles experienced back in January, 1969 when they embarked on that project?  And I think I got the answer...

Engaging in cognitive reverie...which is what Russell Crowe's character, mathematician John Nash, often does in the movie A Beautiful Mind...is something I often find myself doing as well.  Sometimes it takes the form of regretful, retroactive daydreaming when I think of times in my distant past and wish that I had adopted different priorities and habits.  One of these areas involves my studying in high school...my attention to homework assignments was atrocious, especially with regard to mathematics.  This is lamentable since every standardized test I ever took showed an exceptional ability in the area...I even scored a perfect 800 in it on my SAT.  Well, they say folks shouldn't bury themselves in their pasts, and I agree...but with an important caveat.  If what I'm regretting can neither be resurrected or made relevant to the present, then I should let it go.  But if there is a way in which I can learn from my past experiences and apply those lessons to the current setting, then this kind of cognitive reverie can be useful and a positive force in my life, here and now.  To this extent I've decided to make the upcoming year 2022 as one themed with intense study of mathematics...I can definitely think of an application or two for the years to come. Much will be simple review and much will be a reexamination in greater depth of what I have earlier covered, but only shallowly.  I have to credit a birthday present I received a few weeks ago: The Great Courses DVD pack of The Joy of Mathematics, authored and presented by Arthur Benjamin...wish I'd had him as a teacher!  Of course, I'll keep all ya'll updated on my progress with my upcoming math adventures...to that end I think it's time to load up my computer with some basic mathematics software...

Quote of the Week...from Carl Friedrich Gauss

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.                                          ---Carl Friedrich Gauss

Carl Friedrich Gauss (1777-1855), from Germany, was the greatest mathematician of his time.  I picked his above quote because it illustrates the duality...and apparent contradiction...involved in areas like goal-planning and self-motivation.  You want to learn or accomplish something that currently is beyond your present means, be it not enough skill, connections, money...or whatever.  Fulfilling that goal requires a focused and consistent investment of your time and energy...with your future achievement always being forefront in your mind.  Yet when it has been attained, most likely you will look back and see that the journey to the goal's end was the meaningful aspect of it all...yet without the goal you would not have set out on that journey.  This also dovetails nicely with the message from Paolo Coelho's novel The Alchemist, which I read recently.  Gauss is referring to learning and indicates that while it's important to amass knowledge and enlightenment, the road to them is the true jewel of the process.  I get what he's saying: I've always liked to study stuff...yet my problem has always been a lack of focused, clear end goals.  Regarding mathematics, the true greats in that field...such as the late John Nash...realized that behind all of its complexity, mathematical reasoning can be broken down into a relativity simple...but sometimes elusive...pattern of principles known as "governing dynamics".  People like Gauss and Nash always seemed to have clear goals and projects before them, but it was getting down within those dynamics and learning that gave them their true joy.  So set clear goals, work to accomplish them...and revel in that process of work...

Quote of the Week...from E.T. Bell

 'Obvious' is the most dangerous word in mathematics.                E.T. Bell

I'm not sure what mathematician E.T. Bell exactly meant here by the word "obvious", but when I read this quote earlier this week on the BrainyQuote website it brought to my mind a pet peeve regarding how this field is often taught in schools.  It's more annoying than dangerous when a teacher is rambling through an obtuse presentation at the front of a class and decides to pepper his or her lesson with "of course this" and "obviously that"...if it were all so obvious, we poor students wouldn't need you up front to explain it to us!  Here's another "mathematics" quote I read from the same site, by the late economist Paul Samuelson: "My belief is that nothing can be expressed by mathematics that cannot be expressed by careful use of literary terms".  This quote seems to fly in the face of how we're accustomed to perceive mathematical concepts, in a more two-or-higher-dimensional presentation instead of with ordinary language.  Have you ever heard or read a purely verbal, literary exposition of a mathematical course or even lesson?  I haven't, but I wonder whether the capacity to accurately verbalize mathematical concepts shouldn't be a valuable, sought-after skill for mathematicians as well as mathematics teachers.  Wouldn't it be great, instead of tuning in on the radio to political hacks like Rush Limbaugh, Michael Savage or Mark Levin, you had a station with a talk show whose host spoke Math...has it even ever been done before?  What if society educated people to be able to speak out detailed principles in mathematics and technology in ordinary conversational language with one another...I might find the listening experience akin to discerning meaning from a foreign language I only have rudimentary knowledge of, but I also know that many of us process our thoughts out verbally and such an added component to our communications would doubtless raise the educational and reasoning levels of our general population, knowing such abilities existed and were attainable.  Maybe if some of us achieved a verbal proficiency in math we also could cockily ramble on like everything in it was "obvious", although the opposite effect may instead be the result.  I think Bell may have meant his quote to stress that every step in a mathematical progression of reasoning has to be rigorously attended to and that carelessly glossing over a point as being "obvious" could well unravel the entire solution or theory being examined.  So the upshot of all this is that when anyone is engaged in the sometimes torturous endeavor of mathematical discussion it would be better to just preemptively omit that pesky word from their vocabulary...

Quote of the Week...from Maya Lin

Math...it's a puzzle to me.  I love figuring out puzzles.                 ---Maya Lin

Maya Lin is the architect and artist responsible for the design of the Vietnam Memorial wall in Washington, D.C., which was initially considered controversial but is now widely esteemed as one of the greatest memorials anywhere...Melissa and I were very impressed when we visited it last October. Lin's above quote resonates with me as I have always thought that mathematics was taught in schools incorrectly.  If the puzzle-like nature of this field were emphasized in school instead of making it all a solemn judgment of the poor student's abilities and work ethic, than I believe we'd see a lot fewer people...including those with a high intellect...suffering mental blocks over the subject.  As for me, whenever I do pick up one of my several Shaum's Outline Series books on mathematics...and this series covers just about every course imaginable...I regard their "problems" as fun puzzles to work out, and the answers are provided along with explanations as to how to solve them.  And since I'm not going to school, there's no verdict coming down on me being a dumbass because I got a wrong answer.  So many people nowadays like to do puzzles such as Jumble, Sudoku, Kakuro (my favorite), Hidato, Crosswords, and others... I'm also a fan of jigsaw puzzles.  But believe it or not, picking up a Schaum's book about a particular field in math and going through it can be its own kind of fun puzzle experience...and who knows, you may discover that you're not as mathematically-challenged as you thought.  But keep it secret if you do this, otherwise folks will be asking you what you're going to do with your new mathematical learning while if you stick with the more "standard" kinds of puzzles they won't bug you over it...

Quote of the Week...from Albert Einstein

The difference between stupidity and genius is that genius has its limits.            ---Albert Einstein.

Can we all agree that Albert Einstein, the author of the incredibly complex and paradigm-shifting General Theory of Relativity, was a genius?  And that a lot of the folks we see around us (not our own selves, naturally) seem pretty stupid? Well, that usage of "genius" and "stupidity" isn't what the groundbreaking physicist had in mind, in my opinion.  Rather I think he was making a profound point: each of us has our "genius" moments and our "stupid" stretches, whether we're deemed to be smart or not.  That's certainly true for me as I look back on my life.  My "Eureka!" experiences pale in significance and scale compared to all of the dunderhead blunders I've committed...yet there's something almost divine or supernatural during those rare instances when everything seems to come together into focus and a new, foundational idea or principle occurs to me.  I've noticed this probably the most in the past during my usually frustrating experiences in school, especially during different math classes in which the teacher would stand at the front of the students presenting the material and suddenly, out of the blue, I would "get it".  Like Einstein, I was generally a mediocre-to-average math student (although I consistently scored high in it on the various standardized tests we students were subjected to) ...and I'm not too happy with the quality level of teaching I was dealt in school regarding this field.  But I digress...

Sometimes a genius moment doesn't involve understanding or creating a complex idea...it can be the opposite when I figure out how to simplify things to a level that makes my life more manageable.  Being a person of routine, I am looking for ways to integrate the various things I want to accomplish within the framework of a day or week...not abstract physics, but still something that involves the management of chaos, to a degree.  And that's where I think most of the genius of humanity is expressed: not in great works of art, music, literature, engineering, or science, but rather in the multitude of adjustments, discoveries, and innovations people make in their daily lives, ingenious little things that generally go unnoticed by the rest of the world. Yet I sadly acknowledge that, in the midst of all this good stuff, the stupidity I see in that world seems limitless...   

Weekly Short Story: A Problem of Numbers by Isaac Asimov

If Dean Karnazes can make the claim as the most famous ultramarathon runner around (I haven't heard him do this), then the late Isaac Asimov could have also boasted of ultramarathon accomplishments within his field...he authored or contributed to more than 500 books in his lifetime and wrote countless short stories.  His fiction usually has a strong intellectual bent to it, but the characters can often seem a little shallow and undeveloped.  However, although Asimov isn't my absolute favorite author, his stories are always interesting and leave me with something to think about when I've finished them.  A case in point: his short story A Problem of Numbers, published in 1970 and appearing in the collection The Best Mysteries of Isaac Asimov (Ballantine Books, 1986)...

A chemistry graduate student approaches his accomplished professor and asks for his approval to marry his daughter.  The professor, knowing that the two have been romantically involved, replies that his blessing isn't necessary but that he will give it if the young man can decipher a numerical code he writes down...and then explain how he reached his solution.  I'm not exactly spoiling the story's plot by noting that the student does solve the problem and gives a reasonable explanation as well...it's what he says at the very end that is the "punch line" for this intriguing story, revealing a lot about how the great scientists and mathematicians (or writers, composers, artists, etc.) think, commonly-held public impressions to the contrary...

Intuition, or as some might called it, "gut feeling", accounts for a great deal of thinking in areas that you might believe are completely bound to precise, provable reason. Starting at the beginning and going from line to line is the standard way that logical arguments, mathematical proofs...and literature and music...are usually presented.  But when creating those arguments, proofs, and stories, the thinking is nothing but linear.  There seems to be a kind of well that great thinkers have learned to draw from, a source of enlightenment so to speak, that enables them to see around problems and detect the patterns that point toward their solution...this certainly was the case in A Problem of Numbers.  There are few things in this world as sweet as the "eureka" moment when everything comes together and becomes clear...or when a very simple solution to a seemingly intractable problem suddenly makes itself known.  I live for these moments...and recommend Asimov's clever short story...

Weekly Short Story: The Feeling of Power by Isaac Asimov

In many science fiction stories from the mid-twentieth century, interstellar travel is presented as a  certainty, but computers are also pictured as filling up entire rooms, completely failing to anticipate microtechnology and how much it would pervade nearly every aspect of our lives.  A notable exception to this misjudgment of the future is Isaac Asimov's 1958 short story The Feeling of Power, appearing in the anthology Isaac Asimov Presents The Great SF Stories 20 (1958) (DAW Books). 

The Feeling of Power is only eleven pages long but it packs a big punch, with great application to what's going on today.  It is in the future and space travel has humanity going to the stars, settling there, trading...and, sadly, going to war among themselves.  The computer technology has developed to the point where scientists and engineers just push buttons...the machines themselves come up with new innovations.  A lowly clerk has discovered how to mentally multiply numbers, which he demonstrates to the skeptical top brass by quickly answering 9 x 7 = 63.  That gets their attention...but then he amazingly demonstrates that he can multiply any two numbers ON PAPER with a system of his...of course, this is stuff we (well, at least my generation) learned in early elementary school.  The generals and officials are flabbergasted...eventually one of them realizes that this could be the breakthrough to free themselves from the total dependence on computers that have placed both sides in an interstellar war into a stalemate.  The story's ending, which you can read for yourself, is not a happy one...

When I read about the clerk's arithmetical demonstration, I had to chuckle...but there's some profound truth here.  There's no doubt that having sophisticated computers and software has lightened the mental burden of computing that heretofore was tedious but necessary, either mentally, on paper, or by slide rule.  But now we have incredibly powerful machines that do the tough work for us, be it arithmetic, correcting our shoddy grammar and spelling, instantly translating other languages to our own (no need to learn a new one, right?), telling us where we are by GPS without us ever having to orient ourselves through maps, and just about any other area where people went to school to learn certain skills.  With this super-technology we may be falling into a trap of collectively losing old skills after they have been deemed "obsolete"...there was even a push in schools to eliminate the teaching of writing in cursive!  As our digital gadgets make us appear to be smarter, without them to supply all the right answers we're becoming a society of dunces.  Maybe we should take a moment to reassess where we're going with all this...

Tuesday's List: Fundamental Units of Physics and Derived Formulas

A few decades ago I was studying basic college physics when I tired of having to remember all of the formulas in mechanics and electricity...and decided to develop numerical strings for each type of quantity that, for me at least, were easier to retain.  Now in physics there are seven fundamental units, of which everything else can be derived (back then I only used four):

1 mass.......................kilogram (kg)  
2 length.....................meter (m)
3 electric current........ampere (A)
4 time........................second (s)    
5 thermodynamic temperature...degree Kelvin scale, or kelvin (K)
6 amount of substance...mole (mol)
7 luminous intensity.....candela (cd)

For the purposes of this article, I'll leave off the last three units, although you might find it interesting that thermodynamic temperature does not figure into the formula for energy...heat energy or otherwise.  What I did was to write out four-number strings, each number representing the exponential powers for the units or combinations thereof.  So...

mass:          1  0  0  0  kg
length:        0  1  0  0  m
current:      0  0  1  0  A
time:           0  0  0  1  s

From this we can derive more strings:

area             0  2  0  0   length-squared
volume        0  3  0  0   length-cubed
velocity       0  1  0 -1   length per time
acceleration 0  1  0 -2  length per time-squared
density         1 -3 0   0  mass per length-cubed
force            1  1  0 -2   mass multiplied by length per time-squared
energy         1  2  0 -2   mass mult. by length-squared per time-squared
power          1  2  0 -3   mass mult. by length-squared per time-cubed
pressure       1 -1 0 -2   mass per length per time-squared
charge          0  0  1  1  current mult. by time
voltage         1  2 -1 -3  mass mult. by length-squared per current per time-cubed

Each number of the string represents the exponential power of that fundamental unit within the formula...and you multiply across the string.  A unit represented by "0" simply isn't present within that formula.  I found this numerical mnemonic system, once learned, to be easier to remember...

And it's also easier to see relationships between the different formulas, for example:

We're taught that force=mass x acceleration...or:

      1   0   0   0    mass
      0   1   0  -2    acceleration
      1   1   0  -2   force     (note that when multiplying exponents you add the powers)

If you're interested, you can go through the whole physics textbook like this and the formula derivations can get to be almost a matter of second nature after a while...

Two Presidents: Dr. Teleprompter and Mr. Spontaneous

I remember the Oscar-winning 2001 movie A Beautiful Mind in which mathematics genius John Nash hits upon a life-changing revelation: in economic and conflict scenarios the best result does not come about with each party working solely in their own best interests, but rather with each party working both in their own best interests AND in those of the group as a whole.  This principle of duality, which Dr. Nash worked out with mathematical precision, eventually won him a Nobel Prize.  Yet how many of us practice it in our lives?  But today I'm looking up at the leadership of my country, and wonder how one elected by a particular constituency can remain loyal to them and their interests while at the same time administering to the interests of the country as a whole: in other words, duality practiced on a national level.  To this end I am examining the presidency of Donald Trump, now a little more than seven months old...

On some level I think Mr. Trump...as well as his predecessors...understands the principle of duality described above.  But instead of integrating the general national interests with those of his supportive voting base, he has apparently undergone a bizarre schism...much like Robert Louis Stevenson's Dr. Jekyll and Mr. Hyde.  On one hand, "Dr. Teleprompter", with his prepared texts, delivers relatively reasonable and rational messages that tend to persuade and unify the entire country behind his policies.  But then he'll slip into a different personality: "Mr. Spontaneous", expressing himself through Twitter, rallies, and interviews, emotionally panders to "his" people and rips into anyone or any institution that he even remotely suspects opposes him...in other words, a thoroughly divisive persona...and in that mode he often contradicts his "Dr. Teleprompter" speeches, in spirit if not also in essence.  So I have to ask myself: Who's the President?

That's the question an old friend asked me when I pulled up at his Checkers drive-through window one evening way back in late November of 2000.  Although his intention was to elicit whether I thought Bush or Gore had won the recent controversial and then still-disputed election, I had a quick, smart-ass reply: "Why, Bill Clinton of course!"  Now I'm asking the same question to myself and, looking at the answer, I see nothing at all funny...

Weekly Short Story Review: A Subway Named Mobius by A.J. Deutsch

A few years ago I was a very enthusiastic reader of classic science fiction short stories, amassing many collections of "year's best" books.  In one such book, Isaac Asimov Presents The Great SF Stories 12 (1950) (DAW Books, 1984), there appeared A Subway Named Mobius by A.J. Deutsch, an astronomer not known for his writing...at least in the realm of fiction.  Deutsch, who lived from 1918 to 1969, would later be honored for his astronomical work with a moon crater named after him.  But you can all put your telescopes back into storage...the Deutsch crater is on the far side of the moon...

In the mathematical field of topology, the Möbius strip, represented by a strip of paper twisted back on itself and fastened to form a continuous loop, is often given as a seemingly impossible example of an object with only one surface...continue going along the paper and you will eventually cover both "sides".  With A Subway Named Mobius, Deutsch postulated that a sufficiently complicated system, such as the subway in a large city (in his story it's Boston), can also create an apparent physical paradox with the addition of an element that changes its mathematical structure.  And this indeed happens after a new shuttle run is added to the system and one of the trains is discovered to be missing in transit.  Where did it go, how are its passengers, and what will happen to it? The investigators had better find out quickly, for time is running out...

This story is not your typical science fiction tale, for we're not talking about future dystopian societies, alien encounters, or space exporation here: it is this immediacy, this sense that something so bizarre could occur in our own everyday, humdrum lives that gave it such an appeal to me.  And the ending was nothing short of brilliant, making me wonder why Deutsch didn't feel motivated to write more fiction...who knows, maybe they would have honored him with a crater on "our" side!  I noticed that this story is on the Internet on a couple of sites in PDF format.  I don't know whether his copyright expired, but I have a feeling that A Subway Named Mobius has slipped into public domain...so why not Google it and read it for yourself?

Quote of the Week...from James R. Newman

The most painful thing about mathematics is how far away you are from being able to use it after you have learned it.                                              James R. Newman

I found the above quote and decided to use it, although I hadn't heard of James R. Newman before.  He was a mathematician as well as a popularizer of the field, writing a number of books.  What attracted it to me was how strongly I have agreed with its message over the course of my lifetime...well, at least that lifetime after my mathematics education in school passed from arithmetic to geometry, algebra, theorems, and proofs.  Seeing how my life veered in a direction from a career that directly used mathematics...careers like engineering, physics...and of course, teaching mathematics...I've found no occasion whatsoever to employ anything I've learned beyond simple arithmetic...not even basic algebra...and at college I've taken courses in calculus, differential equations, linear algebra, and mathematical statistics.  This has to be the one subject in school for which the grand majority of graduates will find little or no application in life...beyond that basic grammar school arithmetic competency.  Yet I find the subject intriguing and from time to time delve into it...

In a way, mathematics is an area that anyone can do...even one untrained in it.  If you're not trying to get something extraneous from it, such as a passing grade in school or math-related job, then you don't have to worry about impressing others or being as fast or accurate as them.  Ultimately, removing the competition and time pressure angles from studying mathematics can make this field more attractive for life-long learning.  Of course, it also helps to have a good amount of free time at your disposal!  Still, YouTube is loaded with free math courses on just about any level, and the Internet provides many resources that in former eras would have been more difficult and expensive to amass...

I've decided to go "back in time" and approach mathematics as something to embrace instead of fearing...in the process hoping that I will better grasp some of the principles contained within it than when I was "under the gun" in school...

Just Finished Reading Letters to a Young Mathematician by Ian Stewart

Ian Stewart is an English Professor of Mathematics at the University of Warwick. In the last several years, he has written a number of books popularizing the study of mathematics...a subject I fear needs a heavy dose of popularizing.  The book of his that I just read, Letters to a Young Mathematician, gradually walks a hypothetical aspiring mathematician all the way from high school through college, graduate school, and then through the stages of her ensuing professional career in the field in the form of letters he imagines he would have written her. Along the way, he shows the importance of mathematics in our lives, how mathematicians think, how to learn mathematics, how proofs...the bedrock of mathematics...are very similar to stories in their structure, and some important career tips.  The book is short, only 203 pages, and I read it very quickly.  I left it feeling encouraged about mathematics, although I personally am not at all in the same position that the imaginary "Meg" or Ian Stewart are: their paths through this field were rather conventional, spanning the greater parts of their lives and going through the standard educational and career stages at certain chronological ages, whereas I am sitting here at age 58 looking at it from a slightly different perspective.  Still, I don't think it would hurt me to delve back into this field a little...I've just begun studying linear algebra and am practicing matrix operations.  They're actually quite fun, very much like Sudoku or Kakuro...only here I actually do need the "answer key" to tell if I got the right answer.  From time to time I'll report on the progress I've been making with my mathematical adventures...

Letters to a Young Mathematician is just one of a number of mentoring books out there which perform the twofold purpose of popularizing a particular field of endeavor for everyone to better understand while at the same time encouraging interested people among them to enter it...even if it is at a more modest level than that attained by the books' writers.  I can think of a couple more books like this: Ultramarathon Man by Dean Karanazes and On Writing by Stephen King.  As with mathematics, there are many people who would never take up running or writing because they fear they would look stupid to others...but with anything, you're going to stumble a bit in the beginning, so a little bit of humility is a needed ingredient in any such undertaking.  I happen to do all three to various degrees of success: running, writing, and once again after many years, mathematics.  I'll always find people around me better in them, but that's not the point: pursuing these fields is fulfilling in itself...

Getting "Good" at Math vs. Kindling a Passion For It

I was just listening on my car radio to a PBS show featuring an interview with a young man who, as the result of a street mugging that left him with a concussion, suddenly began to discover that he was inclined to view his reality in mathematical terms, focused on geometry.  I don't know how his story ended, because at the time a squall line was about to strike, with me racing into my Magnolia Parke Starbucks parking lot to get inside before all hell broke loose...so I left the program unfinished and rushed indoors to escape the deluge.  Now that I'm inside having taken the last available table...with that hell now breaking loose outside...I'm reflecting a bit on what it takes to be "good" in mathematics...

I'm not being arrogant or prideful when I state that I have always registered extremely high in mathematical ability on aptitude tests (e.g. my SAT math score was the maximum 800).  However, that aptitude never translated well to the classroom, where I consistently underachieved.  Why?  I have always felt turned off by the combined hyper-competitive and moralistic tone of most math classes.  By this I mean that the focus was rarely on the fascinating field of mathematics itself, but rather on the performance of the math students.  We were supposedly in some kind of de facto competition with each other to see who was "best", and failure to answer questions correctly always seemed to have a moral sense of condemnation.  The irony is that whenever I undertake the study of something for its own sake, free of competitive pressures, then I naturally tend to float to a higher level than others.  The young  mathematics scholar on the radio program had independently developed his own high level of competence in the field, relatively free of the aforementioned distractions and obstacles imposed in formal math instruction.  He envisioned mathematics ultimately in its geometric form, according to how his brain's functioning had changed following the assault...

I don't believe I need a bop on my head in order to get good at mathematics, but then again, that young man wasn't necessarily trying to "get good" at it, either.  No, he was able to kindle a passion for mathematics based on the mental equipment he was furnished according to his circumstances.  I see no reason why I couldn't follow the same principle, albeit with the particular gifts toward the field that I have been provided...