The Liebers And The Anti-SAMites

The Liebers, Lillian Rosanoff(Rosenberg) Lieber and her husband, Hugh Gray Lieber, were pioneers in conceiving of mathematics in terms of human values.  They also sought to link the disciplines of science, mathematics and art through informal and often entertaining writing accompanied by creative drawings.  Lillian Lieber was the mathematician, and her husband Hugh was the artist.

Mathematician and Educator, Lillian R. Lieber, Courtesy of Robert Jantzen.

Lillian Lieber was born in Nikolaev, Russia, on July 26, 1886 and died less than a month from her 100th birthday on July 11, 1986.  She did not marry until 1926, quite unusual for a woman at that time.  Her husband, Hugh Gray Lieber, was about ten years younger and died in his mid-sixties in 1961.  Together, they wrote some innovative books mostly about mathematics with insightful social commentary.  Lillian often linked the development of modern mathematics with ethics, politics and humanity.  The Liebers encompassed non-Euclidean geometry, lattice theory, the theory of the infinite and Einstein’s theory of relativity.  They also wrote an entire volume on the nature of logic.  However, their most popular volume was The Education of T. C. MITS, (The Celebrated Man In The Street), which begins with problems intended to show that things are not always what they seem!  Lillian’s language ranges from the informal to the formal and then to a downright questioning manner intended for the reader:  “But what are “Truth”, “Justice”,”Freedom”, “Reason”?  Do these words really mean anything?  And how can we be loyal to them if their meaning is not clear?  Are they not just “fakes” invented so that some people can make slaves of others by fooling them with such meaningless abstractions?…”  To explore and investigate such terms is a major part of her and Hugh’s educational purpose.

The discovery of non-Euclidean geometry destroyed the notion that mathematical truths are eternal verities, for by changing one postulate(the parallel postulate), new geometries come into being such as a geometry based on a sphere, Riemannian geometry where the angles of a triangle are greater than 180 degrees and as much as 540 degrees!  But the Liebers stress that within the new freedom to create other geometries remains the recognition that such creations are systems with definite rules, which cannot allow for contradictions.  They then make a comparison between mathematical freedom and human freedom and warn that true human freedom does not imply unlimited license but careful responsibility.

The Anti-SAMite was a unique creation of the Liebers.  S/he was a person who opposed or was totally ignorant of the wonderful discoveries that had been made in science, art and mathematics.  They believed that these three subjects formed the building blocks of human culture and that all three were united through the passion of discovery, which encouraged further questioning and exploration.  As Joseph S. Alter states in his perceptive article on Lillian R. Lieber, “She called those intolerant of new ideas in these fields”anti-SAMites.”  Anti-SAMites were indifferent to “the good, the true, and the beautiful,” and there was a clear implication that anti-SAMites were responsible for prejudice and war.  To Lieber, war was the greatest danger facing humanity and SAM our greatest hope against its destructive forces.  Philosopher, Walter Kaufman, would have concurred. Allan Kurzberg, controversial thinker of the 1960s-1970s, would definitely not have.  In fact, he accused the Liebers of being just as intolerant as the anti-SAMites by using the latter as scapegoats.  Allan was not shy in including the Liebers as competent “Other” creators in his essay “Mathematics and World Peace.”  However, I will defer a more in-depth analysis of Kurzberg’s essay to another post.

On a personal note:  The Liebers influenced me greatly during my college days.  At that time I was reading Einstein’s theory of relativity, various studies in the philosophy of science and discussing all the above with Grandma Lillian.  It was an exciting time and people were considering all kinds of thought and alternate lifestyles.  I was caught in the brouhaha concerning the Vietnam War and voted for the Peace and Freedom party a few times.  The “establishment” and the “military industrial complex” were highly pejorative terms at that time.  Professors were open, and, with few exceptions, liked to be called by their first names.  I remember talking to my calculus professor, Charles Kalme, about the meaning of life and the importance of reason.  I remember him telling me with his Latvian accent:  “Who is to say that you’re born and you die, and what’s in-between doesn’t matter?”  Who indeed?  Compared to the dogmatic, but sometimes fun studies in high school, I felt an incredible freedom in college that I had never experienced before in an educational setting.  My freshman year was a blast and I enjoyed applying mathematics to linguistic structures and taking a course in semantics with an ex-judge at the Nuremberg trials, Wolf Helmut von-Rottkay. My comparative literature instructor, Al DiPippo gave stirring talks on Greek culture and Kierkegaard.  My young Russian professor, Edward Purcell, was one of the first to use computer exams.  Alas, the excitement of my freshman year would never be duplicated.

My last three posts bring a strong sense of deja vu.  Thomas Mann had a major impact on my concept of literature, especially though his knowledge and application of science, philosophy, music and time.  It was the art of literature that encompassed the whole human experience that engaged my curiosity.  Susanne K. Langer’s works on aesthetics and her pioneering study, Mind:  an Essay on Human Feeling in three volumes were close to my bed.  It is curious that in August Dover Publications has chosen to reissue Take a Number by the Liebers, a book written more than seventy years ago!  Also, they are reissuing The Development of Mathematics by E.T. Bell the same month.  This is an extensive volume, dealing with the history and evolution of mathematical thought.  The Liebers refer to Bell’s works on numerous occasions and Bell was effusive in his praise of the Liebers:  “I have been following the education adventures of T.C. Mits with absorbed interest, and in doing so have(I hope) acquired some education myself…”  For anyone interested in the growth of human though, I cannot recommend these two volumes too highly and I look forward to seeing them on my shelves.



Some Thoughts About Scrapbooks, The New Year And Writing

My baby scrapbook, published by Richard G. Krueger, Inc. and designed by Ditzy in 1946. It was a gift from my godparents Aunt Jackie and Uncle Ralph.

My baby scrapbook, published by Richard G. Krueger, Inc. and designed by Ditzy in 1946. It was a gift from my godparents Aunt Jackie and Uncle Ralph.  At that time my name was “Rodger” Weiss, but was soon changed to Robert Weiss.

“Life may be a stage, but I wish I didn’t have a reserved seat!”–Uncle John from Aunt Jane’s Nieces by L. Frank Baum

Usually in the month of January I peruse my many scrapbooks.  I begin by looking at my baby scrapbook with its satin sheen cover and remarks about me by my mother, Twyla.  It takes me back to my childhood days of the 1950s, when people left their doors open, kids had vacant lots and piles of sand to play in, and lemonade stands were plentiful with lemonade one cent a cup.

However, time goes on and memories begin to fade as new memories take their place.  The almost unbearable slowness of  early childhood is exchanged for the almost unbearable speed of late adulthood.  And New Year follows New Year.  I think of lines by Robert Clairmont from Forever X:

When wrinkles cut your brow

And love goes gaily by,

Sing:  Young, old, tiny, tall,

Whatever happens, happens to all

When we leave this Odd Old Ball.

Indeed, this earth truly is an “odd old ball”.  Events follow events, triggering other events.

Like any mathematical curve, life has points that mark a change of direction.  Some of these points are obvious:  marriage, the birth of children, the loss of a beloved family member.  However, other points are not so obvious and I must admit that I envy Truman Burbank for he is able to “rewind” his life from the time he escaped his set up world to his birth.  Thus, he can see how certain events changed his thinking and further actions.  I am not so fortunate.  And when I look through old scrapbooks only pieces of experiences remain, so I have to reflect and guess at events that might have caused my life to shift dramatically.  Such critical points mark the essence of theater, novels and other writings where an author can juggle them and insert them where s/he wills.  Perhaps, that sense of power and completeness is what attracts us to literature.  The writer plays God just as Christoff does with Truman.  However, the individual must depend on his/her own wavering memories to try to understand the meaning of his/her life.

How A Handful Of Pebbles Contained The Universe, Part 1.

For thousands and thousands of years, people had only two words for quantities:  “one” and “many”.  Even ancient languages reveal this fact.  When people saw an aggregate of stones, they referred to them as “many”.  And when they picked up a single stone they used the word “one”.  “I have one stone not many stones”, they would say.  It must have seemed obvious to them.  Little did they know that by using these terms they were blocking off the awesome universe which surrounded them.

At some mysterious point in time, someone must have picked up the stones individually and wondered how to designate them.  Perhaps, one person wanted to exchange his “many” for another person’s “many” in a business matter.  How could each determine that he wasn’t cheated except by counting.  Stones, then, were no longer simply a collection of “many”, but had special identities or designations.  Thus, from such primitive beginnings the world of number was born.  The simplest numbers uncovered were the “counting numbers” or “natural numbers”.  The numbers 2, 3, 4, 5, 6, 7, 8, 9 came into being, which followed 1.

The Babylonians and Egyptians needed numbers to measure fields and the shapes of the fields necessitated some rudimentary elements of geometry for buying and selling properties.  However, it was the ancient Greeks that studied number as something that had its own existence independent of human needs.  Indeed, to Pythagoras, all things were numbers.  Men and women were numbers and he assumed there were only ten heavenly bodies, since 10 had special significance.  In fact, Pythagoreans worshipped the “tetrakys”, a triangle composed of four rows of dots.  The first row had one dot, the second two dots, the third three dots and the fourth four dots.  When the four rows were added, the sacred 10 was the result.  Everything appeared to represent perfect balance and harmony until…  Yes, even in this well-constructed world, irrationality stuck out it’s ugly snout.  Someone constructed a square with sides one.  When a diagonal was added, the length of the diagonal had to be the square root of 2.  Pythagoras’s own theorem led to the unhappy result.  The irrational could be dealt with later.  But there was still one important number missing:  0.

The Greeks never could find 0, and this fact imposed strict limits on what they could do with numbers.  For zero, we have to go to another country, India.  The Indians had long had a concept of nullity.  It came from their philosophy.  It came from their religion.  When “0” joined the counting numbers, a major step was put in place for solving equations, the construction of the Cartestian plane, and, in today’s world, the binary system which is the basis of computer circuitry.

In the Middle Ages, the first algebraic equations were born, arising from Arabic countries.  The mysterious x and y of algebra represented an abstract way of thinking hitherto unknown in mathematics.  The Greeks may have been the philosophers of number, but the Arabs were not only philosophers, but active participants in extending the range of number to greater practical and theoretical heights.  However, algebra and geometry were still separated.  It required a major step to bring them together.

Some Thoughts About Mathematics And Life

The one thing that comes to mind when I think about mathematics and life is:  You can’t solve any mathematical problem with a confused or unfocused mind.  So, to do a math problem your mind needs to be clear and directed to the problem at hand.  The same could be said about any problem that arises in a life situation.  We are more likely to achieve a better solution if our mind is tranquil and rational.  In other words,  unsettling, spoiling emotions must be kept at bay.  For, a great disturbance in many life events is the spilling over of emotions that cause us to act in an irrational manner and to reach sometimes distorted and even absurd “solutions”

In the realm of life problem solving, mathematical problems form only a tiny subset of all the problems we must deal with.  Mathematicians have established clearly defined rules for solving mathematical problems.  In their special province they serve as architects, beginning with the simple counting numbers or natural numbers, and then including 0 and the rational numbers and stretching out to the irrational numbers to form the set of real numbers.  The real number line is created where all these numerical sets have their home.  And mathematicians begin with axioms and postulates(assumed truths) and from them derive theorems and corollaries to theorems.  Theorems and their corollaries must be subjected to the rigor of mathematical proof before they can be accepted as truths.  What can we use to prove a particular theorem?  Any definition(a definition is an agreement to use words, phrases or symbols as substitutes for other words, phrases, or symbols.), postulate or axiom, or previously proved theorem may be used in a proof.  The use of precedent is also essential to legal, medical and some forms of scientific problem solving.  And mathematics teaches us that to disprove a theorem it is sufficient to find only one example where the statement does not hold.  This latter statement applies to all life problems as well.  For, when we toss around generalizations, it is important to realize that it takes only one counterexample to destroy our generalization.

Mathematics also teaches us to think twice; to be careful before reaching a conclusion.  When graphing functions on the Cartesian plane, it’s not uncommon to have restricted domains, meaning the functions are defined on a certain interval.  And sometimes separate cases must be considered, for example, what does the graph look like when x is greater than zero and how does the graph change when x is less than zero.  Arguments in life may also have restricted domains and statements that may be true for an adult are utter nonsense when applied to a child.  So, we must be cognizant of our audience and know where to apply our argument.  Thus, the study of mathematics can and does help us to cope better and to grasp better the multitude of problems we encounter in life.

“Where’s The Moon? I Don’t See The Moon!” Or, Mathematics To The Rescue

I was dragging myself up the stairs of Founders Hall.  The cement steps and barren walls reflected the darkness of the time ahead.  For, my next class was Speech Communication with Professor B.  I was not doing well in the course.  As my current lady would say:  “You’re going down, down, down!”  And so I was.  But perhaps, I should tell you something about Miss B and how I got into trouble.

Miss B was a tall, wiry lady with sharp, unforgiving eyes and a total lack of manners.  We didn’t get along from the start.  I remember her saying with a sarcastic tone:  “Look at that!  A little boy wearing his tennies!”  She was frank, if nothing else.  And when I tried to act out a favorite childhood verse, she would yell out:  “Where’s the moon?  I don’t see the moon!”  At the time, that comment stunned and hurt me, because I was quite fond of the verse I was interpreting.  Later, Professor B told me that the only thing that could save me was the final, which was a monologue of at least ten minutes.  I thought and thought about possible selections.  I knew if I picked something well-known I could be compared with the greatest and I’d come up way short.  Fortunately, at that time, I was reading some wonderful mathematical stories from Clifton Fadiman’s Fantasia Mathematica.  Bruce Elliot’s story, “The Last Magician” really appealed to me.  The main character was an old man who was fond of a magician’s helper and commits murder because of the cruel way the magician treats her when a futurist society has condemned her to death for misceganation(She was Martian and became pregnant by the magician from Earth).  So, the story had intrigue, action build-up and the main character was an old man.  And, growing up next door to my Dad’s parents, I knew my Grandpa Johnny quite well, so I thought I could act out the part with some accuracy.  Also, the story dealt with the magician trying to escape from a supposedly real Klein bottle

Attempt to picture a Klein bottle, a three dimensional surface that has only one side, which is impossible.

An attempt at constructing a Klein bottle, a three dimensional surface that has only one side, which is impossible.

and was mathematical in nature, so probably few, if any, people had seen it performed.  When I thought about all the advantages, I thought it would be an excellent choice for a monologue.  I would need to trim some parts, though.

Finally, the long-awaited day arrived.  Everyone was busy rehearsing their lines and trying to get into character.  Wouldn’t you know it?  I was the first person Miss B called on.  I knew if I wanted to do well, I was going to have to become an old man in every way.  I tried hard to imagine my Grandpa Johnny and become him.  I tried to walk with difficulty, struggle to get some of my words out and look confused.  And as I reached the podium, the words did come out.  “The harder he worked the worse he treated Aydah…  It seemed as if every time I turned around I’d find her hiding in some corner, crying… I knew she would have to die.  That was why I had pressed the button that switched the bottles the first time, before she ever did…  I guess I must be getting old;  lately I’ve taken to wondering about King Solomon.  He knew so much, I wonder if he knew about Klein bottles…”  Then, a loud applause.

“Well, Bob just disappeared!  A feeble old man replaced him!”  Professor B’s eyes sparkled with admiration and respect.   Mathematics had come to the rescue.



Something To Think About: Two Mathematical Thought Problems From Russia

The Russians have a long tradition of mathematical thought problems which occupies a distinguished part of their elementary mathematics classes.  Here are two samples by J. I. Pearlman:

  1.  Who Counted More?  Two people counted the number of people that passed them on the sidewalk for a period of one hour.  One stood at the gates of a house, the other walked up and down the sidewalk.  Who counted more?
  2.   The Grandfather and his Grandson.  What I am going to tell you took place in 1932.  My age then was the same as the last two digits of the year I was born.  When I told my grandfather about this correlation, he surprised me by declaring that the same correlation was true for his age as well.  How old was each of us?

Something to Think About: Fifty Million People can be Wrong!

Hugh Lieber's drawing of Nazi Germany.

Hugh Lieber’s drawing of Nazi Germany from The Education of T.C.MITS by Lillian and Hugh Lieber.  More on the Liebers in a future post.