My Dad, Atrophy And Mathematics

To bring back and extend Dad’s concept of numbers, my sister, Nancy, and I have been giving him simple math problems.  My sister is giving him addition problems that require carrying. I’ve been trying to get him to relearn the multiplication table.  Both my sister and I realize that part of his brain has atrophied.  But we believe that a better knowledge of numbers will not only help him solve basic mathematics problems, but also improve his ability to reason and strengthen his confidence.   I was astounded yesterday morning to see my Dad at 95 solve all my problems in ten seconds!  He was definitely proud of his accomplishment and so was I!

My Dad at work.


The Radical Philosophy Of Allan Kurzberg And His Fundamental Pestulates, Part 3.

In this post, P2 is discussed along with its consequences:

By means of the Corollary of Human Existence, which follows from P2, Kurzberg proposes another theory of evolution:  “…  I call my 2nd Pestulate:  Reason developed late in human existence.  Thus, humans were affected by strong emotions and irrational tendencies long before reason appeared.  The corollary from this pestulate is, I believe, a most important corollary, because through it we can gain a true understanding of humanity.  All human events be they historical, personal, or otherwise should be revealed through the corollary.  I term this corollary:  Corollary of Human Existence.   What it means is that those forces that shaped humans before reason arose are like large emotional magnets that pull us in different directions.  I call these emotional magnets OE- and OE+.  They stand for overwhelming negative emotion and overwhelming positive emotion.  By overwhelming, I mean that they are strong enough to overcome our sense of reason.  Of course, we possess E- and E+, negative and positive emotions, respectively, but these are not strong enough to overcome reason and do not cause major problems.  We enjoy them as simply negative or positive sensations.  Hence, I will concentrate on OE- and OE+, for they are the central forces that govern human behavior.  The Corollary of Human Existence:  Human behavior is fundamentally irrational and is governed by OE- and OE+.  Thus, when we read that man is a rational being, we are forced to admit the falseness of such a statement.  The statement should read that man is an irrational being that is capable of rational thought.  This raises interesting questions about evolution and humanity’s true place in the universe.  For, when we conceive of the countless planetary bodies that are scattered throughout the universe and apply the principle of probability, which works so well in quantum mechanics, we are compelled to concede that there may be beings in which reason developed earlier than us.  If so, then reason would become the powerful magnet that keeps OE- and OE+ in check, or keeps E+ and E- from becoming OE+ and OE-.  If such a civilization exists, how would it differ from our own?  Could we learn valuable information from such a civilization and prevent annihilating our species through reckless, irrational behavior patterns?  These questions continued to occupy my thinking, so I composed an interview between myself and another being, “Exchanging Thoughts with a Being from Another Planet.”  I also realized that there might be civilizations in which reason came into being at a later stage than ours.  In this case, OE- and OE+ would have even more more power over them than they do over us.  If we let small IR denote a completely irrational civilization, then we are somewhere between it and a completely rational civilization.  By rational, however, I do not mean devoid of emotion.  I do mean that such a civilization would be spared many of the problems we face due to a lack of reason.

If we are to survive, we must undergo some evolution away from menacing destructive behavior towards more rational behavior.  It seems we are just beginning to “know ourselves” and that must be our great adventure.  A catalog of parts of OE- seems overwhelming, but there is one aspect of OE- that dwarfs all others and that will be the subject of my 3rd Pestulate. 

A barred spiral galaxy that contains who knows how many stars with planetary bodies circling them. “How instructive is a star.  It can tell us from afar just how small each other are.”–Piet Hein from Grooks

The Radical Philosophy Of Allan Kurzberg And His Fundamental Pestulates, Part 2.

What follows are Allan’s  thoughts on the implications of the First Pestulate:  “…  Since mathematical reasoning is the highest form of reasoning that we humans have developed, and since, according to P1, we distort the truth more than any other species, we have the main reason for a universal study of mathematics:  to undo false reasoning through careful mathematical reasoning.  Indeed, I would go so far as to say that the more mathematical reasoning is applied to every facet of our lives, especially to our personal, the less contradictions will occur in our lives.  The reader might wonder why.  The answer lies in the kind of language that mathematics represents:  It is an objective language that seeks to prove statements through a series of conditional statements using precise definitions or previously proved theorems.  Mathematics does have synonyms and does use symbols that have different contextual meanings, but never foregoes consistency and brevity whenever possible.  In addition, mathematics involves a kind of generalizing that often leads to universals.  Most importantly, no mathematical system allows for contradiction, which is definitely not the case with other human-contrived systems such as political or social…  In my Theory of Us, I try to locate universals which will subsume all possible human interactive behaviors…  Thus, I feel that the primary reason for studying mathematics is the universal need to apply mathematical reasoning to disprove false statements, whatever field they arise from.  Such a universal need should be the first thing listed in any preface about mathematics.  Perhaps, some of the resistance many feel and fear about mathematics is due to the intrinsic awareness that mathematics is hostile and unmerciful towards human falsehoods and negative states of mind that so often engulf us.  I will call such overwhelming negative states OE-(negative overwhelming energy), which I will expand on later.  One thing I will say is that to attain world peace we must learn to detect, define and minimize OE-.  Our very survival may depend on our ability to do so…  People might say that not every one is capable of mathematics and on a certain level this is true.  Humans may vary enormously in their capacity for abstract reasoning and not everyone can prove limit theorems so essential for understanding calculus.  However, if we state simply that calculus enables us to delve into the infinite, helping us to study instantaneous motion, quantum mechanics, and the theory of relativity, the reader would at least gain some understanding of the enormous scope mathematics has.  I would add the above facts to our mathematical preface in a purely descriptive way so that many would understand the implication of mathematical reasoning.  I would also include some of the magic of the Cartesian graph, which enables us to view the behavior of simple and complex equations, in our preface.  However, no such preface has ever been written…  Mathematics is a series of carefully defined and proven steps that lead to further growth in its carefully built structure.    Postulates and theorems have led to many branches of mathematics, which it would be ludicrous to ignore in any preface that purports to describe the purpose of mathematical thought.  But it is just as ludicrous not to describe mathematics as being reason’s most essential tool for dislodging falsehood, deception and misrepresentation…”

In the next post, Allan Kurzberg reveals his 2nd Pestulate and what he calls the Corollary of Human Existence.

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.