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: Exchanging Thoughts With A Being From Another Planet, Part 1.

As some of the succeeding pestulates become quite involved, I decided to include this fanciful dialogue to help the reader gain a clearer understanding of Kurzberg’s views. In this dialogue, Kurzberg visualizes a being from another planet in which reason is the dominant force that motivates the being’s actions.

Tybol:  Let me say that it’s a pleasure to meet you, Mr. Kurzberg.  I’ve enjoyed wandering around the earth and studying its history.

Allan:  Please call me Allan.   Do you have names on your planet?  And, if so, what is your name?

Tybol:  Our sounds are not quite equivalent to yours, but if you call me Tybol, that will be a close approximation. Certain sounds predominate as they do in your languages.  However,  our language is quite precise, has one grammatical structure, and is devoid of the ambiguities and the figurative connotations that are part of your language system.

Allan:  Then, everyone on your planet speaks the same language?

Tybol:  Correct.  I know that on your planet you have quite a myriad of languages, so it is not surprising that communication is often difficult.  But, even if one selected one language, you would still have difficulty communicating, because of the imprecision of terms and the dependence of gesture.

Allan:  Then gesture is not a part of your language?

Tybol:  No.  The sounds that we make are understood by all without the need of gesture.

Allan:  With your emphasis on reason, do you consider yourself an advanced civilization?

Tybol:  We don’t use terms like advanced or backward, inferior or superior, for those are judgmental words that insult those whom we would designate as backward or inferior.  As you have written, judgments of comparison create an “Other” and an atmosphere of distance.  From such judgments anger and mistrust follow.  That situation is what our inner reason tells us to avoid.

Allan:  But isn’t it impossible to have a society that doesn’t use comparisons?

Tybol:  No.  Let me tell you something about life on our planet, Allan.  We view ourselves as a whole which every member of the planet is a piece of.   Each member has something wonderful to contribute to the life of our planet.  We use terms such as “discovery” and “exploration” in connection with our fellow beings.  We try to meet and learn from as many beings as we can, because this is what makes our lives so exciting and surprising.  We would never use terms that induce isolation or discontent, since we would be harming ourselves and depriving us of the joy of getting to know other beings.

Allan:  So you trust your fellow creatures?

Tybol:  Absolutely.  There is no reason not to.

Allan:  That type of thinking would be unthinkable on our planet.  As you probably know, our history is full of mayhem and destruction of our fellow humans.  Doesn’t anyone on your planet ever get the urge to harm or injure someone?

Tybol:  Why should we wish to harm or destroy that which we most admire and cherish?  It doesn’t make any sense.  Further, it would be a sheer act of masochism to do what you suggest, because we would be limiting our own growth.  I cannot understand why you allow such rampant destruction of human life on your planet, which might be depriving you of future medical researchers, astronomers, artists and individuals with great insight into the problems humanity faces.  And, it seems incredible to us that you would follow leaders who are clearly mentally unbalanced and carry out their nefarious orders.  Why do you do this?

Allan:  I really have no definite answer to your question, Tybol.  It is a puzzle to many of us as well.  That certain forms of mental illness are linked in many people’s minds to power and strength, cannot be denied.  Why there is such a strong attraction, yes, and fear to mentally unbalanced individuals, is something we don’t really understand.


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 Radical Philosophy Of Allan Kurzberg And His Fundamental Pestulates, Part 1.

I first became acquainted with Allan Kurzberg when I was a freshman at USC.  It was a time of immense turmoil and change, but also a time of great excitement and discovery.  Many college students were seeking alternative lifestyles other than those propounded by “The Establishment.”  The reason for this was simple:  the lifestyle emanating from “The Establishment” was producing a plethora of lies, bodies of prejudice, and the Vietnam War, resulting in countless injuries and deaths.  Many students thought of alternative lifestyles that encompassed communes, the Hippies of San Francisco, philosophies from the Far East, especially meditation as practiced by famous Maharishis.  Youths were also reading about the links between science and psychology, mathematics and computers.  To cope with the rigid mindset of “The Establishment”, young people smoked marijuana, took PCP and LSD to reach other mental states than were condoned by the AMA.  Families were not only torn by war, but by “the generation gap”, which led to a total breakdown in the family structure, the shock waves of which are still affecting the present.  It was a time when one person could change the world and each was encouraged to  “do your own thing”.  People used profanity as a rebellion against the norm and as a strike for human freedom.  Sex became far more casual and explicit.  The notion of premarital sex as a taboo was tossed out the window.  The Living Theater performed on streets and in parks.  And there were “sit-downs” and riots across college campuses as the emotions of anger engulfed the U.S.  The authoritarian approach that had for so long defined the hierarchy of professor and student broke apart, and closer, more meaningful relationships were developed and encouraged.  Especially, there was much talk of peace, while paradoxically, different factions were building.  It was during this epochal time that one of my girlfriends, Janet, suggested that I look at Allan Kurzberg’s essays on the Theory of Us.  I told her that I already had a full course load and had numerous books I wanted to read.  Why should I read Kurzberg?  She told me that in her opinion he was the only true radical, because he opposed both “The Establishment” and the youth.  She thus led me into a hitherto unknown world:  the mind of Allan Kurzberg.

His first essay was entitled:  The Fundamental Pestulates.  I started reading and found myself absorbed by a writer that was a curious mixture of strict reason and digressions.  “In this essay I have attempted to establish the cornerstone to the Theory of Us.  In my writing I use some principles of mathematical reasoning when applicable…  Different branches of psychology remind me of lonely subsets in search of a universal set.  Each is merely a limited, restricted set of elements…  I will use the term pestulate instead of postulate, because these fundamental principles by which humankind is denoted are rather like pests in the lives of humans…  The Main Pestulate, 1. reads:  There is no species on earth that lies, prevaricates or dissembles more than the human species.  Since to lie is to speak falsely and no other species can “speak” in the way that we define it;  as an assemblage of sounds so sequenced  and intoned as to give expression so broad it allows not only for denotation, but connotation as well, our case is proved.  We might also accept the pestulate, since no counter example can be proffered.  We do know that camouflage is widespread in the animal and insect kingdom, but this is only for survival.  Humans can use camouflage on, say, Halloween, and the object is sheer play, not survival…”

I must say that I found Kurzberg’s essays the most difficult of any essays I had ever read.  It was not on account of their intelligibility, but rather that I found myself being challenged, so I retaliated by writing NOs on many of the pages, and, in some cases, actually writing rebuttals.  Now, after almost fifty years have passed, I’m not sure I was altogether correct in my objections…

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.



Time In Thomas Mann’s The Magic Mountain, Part 2.

Time is unreal, because the whole of any picture cannot be perceived at once.  Although the hands of a watch tick away, they cannot be said to be measuring minutes.  No one can know what the hands in truth are measuring for they are unaware of the divisions they pass over.  Grass grows so unobservably that it seems not to be growing.  However, at some minute grass appears from the seed.  This reflects the Greek paradox of being coming from non-being and furthers the mysterious notion of time as motion.  According to Mann, time has moved to bring changes according to a “succession of dimensionless points”, but at any one point the momentary effect on the grass is imperceptible.  As Piet Hein said in one of his Grooks, “We glibly speak of nature’s laws, but do things have a natural cause?  Black earth turned into yellow crocus is undiluted hocus pocus.”  Furthermore, the unreality of time rests on the premise that the immediacy of now varies with each person, with each event and with each thing.  According to the way it is interpreted, the same interval can be exciting, monotonous, capable, wasteful or productive.  Time is there in the feeling of the beholder, which Marcel Proust explored in depth in A la recherche du temps perdu.  As an entity, as a circular function that cannot go anywhere, time is a “hastening while” that “streams silently and ceaselessly on.”  So Thomas Mann develops the magic quality of time as a background against which seven years in the life of Hans Castorp takes place.

Actual time in the novel is sometimes represented by the number seven.  The book has seven chapters and the plot interval covers the seven years from 1907-1914.  There are seven tables in the sanatorium dining room, each of which is occupied by Hans Castorp during his seven years there.  The room of Frau Chauchat, “the charmer” is numbered seven.  Of course, the number seven has symbolic Biblical meaning.  Mann may be implying that the “new” Hans Castorp emerges within the Biblically significant number of years.  The seven years do change Hans through his own efforts.  There is unmeasured time for self-education in depth.  Self-study in books opens widespread areas of learning in the structure and function of the body, in the structure of snowflakes, in the functions of government, in the beginnings of the world, in the preservation of food and in the efficiency of the x-ray.  Time permits Hans Castorp to acquire encyclopedic knowledge.  It is interesting to note that in our modern era, educator Howard Gardner has identified seven distinct intelligences and French topologist, Rene Thom writes about seven elementary geometric catastrophes, so central to his Catastrophe Theory.

Time also becomes a relative concept as Hans Castorp stays longer away from the flat-land, under the influence of the magic mountain.  And the magic of timelessness becomes operative.  The days go by and Hans Castorp’s stay is lengthened to a month, then to six months, then to a year, then to seven years.  Hans loses all sense of time and cannot remember how long he has been on the mountain.  He becomes so engrossed that he forgets the flat-land and becomes part of the timeless spirit of the mountain.

The circular quality of time affects Hans when he welcomes back his cousin, Joachim, from the army.  He completes the circle of his own arrival as he meets his cousin on the same train, at the same station, and at the same time of the year.  The plot development is also circular.  Hans Castorp finally returns to the very place from where he started;  the flat-land.  He disappears on the battlefield of World War I, having completed the circular journey up and down the mountain.  And the higher one goes on the mountain the more unreal a measured minute becomes.  In the snowy vastness there is only the magic of timelessness…