Comfortable Belief

I almost made this first post about wine, partly to emphasize that this blog is not confined entirely to spiritual matters. Of course, there might be some peripheral relation between wine and spirituality, but mostly I think wine is simply a joy of life and good company. It happens that it can be indulged in not only by drinking but by travel to great wine growing areas near where I live. I have, however, resisted wine because this is the very first post to the blog. Wine will be saved until later. Since, in fact, the idea of a spiritual journey is the main theme here, I should start with that.

So… Let’s begin with some fundamentals. Here is a paragraph I wrote Jan. 8, 2002.
1. About the time I got out of college I came up with some ground rules for a spiritual-religious quest. A basic idea is that one should try as hard as possible to be free of superstition in the spiritual quest. This is much more important for religion than for science simply because science has a built in mechanism for eliminating error that works pretty well most of the time. (The basis for this idea of a superstition free spirituality is the ancient idea that one should not worship false gods. Idolatry is perhaps the most serious religious sin. We are seeking the truth at any cost.) A rule that follows from this idea is that we should never try to believe something simply because it offers comfort.

This theme of rejecting beliefs of comfort began for me back in college days during a ride from the Bay Area to Yosemite to go climbing. It was dark and we were riding along in the Sierra foothills on the way to Camp 4 when somehow the talk turned to religious beliefs. One young woman said that she didn’t have any idea whether one survived after death. However, she found that idea comforting so that is what she believed. I was totally amazed and appalled by her remark. I don’t remember saying anything, being too polite in those days, nor do I recall that there was much if any discussion on the part of anyone else. This incident made a big impression on me and led to the idea that pursuing the “truth” at all costs as a basic spiritual postulate. All of this occurred back in the early 1950’s as I was just beginning to realize the importance of spiritual answers in my life. Back now to Jan 8, 2002.

2. This morning I had the thought that one should expand the rule. Rather than using it merely to reject ideas that have no basis but their “comfort”, one can use it to choose between competing possibilities about spiritual matters. One should base ones religious ideas on the seemingly “bleakest” possibility.

At first blush this idea seems somewhat perverse. Why go out of the way to consider what seems worst to one? I suppose the idea is to guard against the seduction of attractive ideas. We humans have a great capacity for rationalizing emotionally attractive ideas into the “truth”. If one cuts through what one thinks is the worst possibility and finds comfort anyway, the loss of the worst possibility isn’t likely to be too devastating. I note as a sideline that the comfort is the theme here, not the ideas themselves. I could imagine, for example, that immortality might get to be monumentally boring after a few thousand years, especially if one knew there was no escape. The saying “after the first death there is no other” might carry an appalling meaning in these circumstances. At the risk of being repetitive I’ll close this post from a note from Feb. 19, 2003.

7. One might ask, “Isn’t the whole point of religion to offer spiritual comfort?” “If so, choosing bleakness seems perverse.” My answer to this idea is first, I am unable to find any comfort in an idea that seems false on its face. Or to put it bluntly: Truth is more important than comfort. In seeking truth we hope that comfort might come as a by-product, but finding spiritual meaning is a more important goal.

For my Future Wife

Our relationship will never go to that completion I desire unless you are
as crazy as I am. Our human condition is to be trapped in an animal body
aspiring for the stars. Our consciousness non-existant for an eternity of
the past and to be non-existant for an eternity to come. Meanwhile, for a
brief instant we are here. If one truly realizes our condition one must
be crazy, at least by conventional standards. This is not a sick,
destructive craziness, but a creative, tragic, open and aware craziness.
This kind of craziness I’m talking about is the only sanity. Are you that
crazy? A humorous saying is, “Never sleep with anyone crazier than
yourself.” I would turn that around and say, “Never sleep with anyone who
isn’t as crazy as you are.” If one is in touch with another at that level,
nothing else matters; not differences of age, of personality, of
temperament, of wealth, of fame, or of position in society. If one
doesn’t have a deep bond at that level, the relationship may be nice but
will never be complete. And can never bear the name of true love.

Spiritual Quest, 1953-1954

What compels one to sail on a spiritual journey? Fundamentally there is no real answer. It is a mystery similar to that arising when considering the question of why some people climb mountains. If someone who has no interest in mountains asks why I climb, I’m really at a loss to explain. It is obvious to me that often I feel more alive and full of joy on a mountain, seem to feel a sacredness in great mountains, seem to “see” further, and feel a sense of spiritual well-being and insight, indefinable in concrete terms. If a person is immune to such feelings, there is really nothing to be said that would give her or him a real, personal understanding of why one would feel compelled to climb a mountain.

As a child I was fortunate in having parents who were not conventionally religious. Neither ever went to church. There was no attempted brain washing. I was curious about religion and on one or two occasions actually went to Sunday school at Central Union Church in Honolulu. Sunday school there was obviously simply a place to harbor children while their parents were in church. Theology was conspicuously absent. No help there. Later, in high school, we had a Christian non-denominational chapel which I found exquisitely boring. I was already an unthinking atheist. So, later in life, when I felt the need for an understanding that was deeper than that provided by a scientific and humanistic education, I could approach the search, more or less unbiased by childhood experiences. I was open to any spirituality based on beliefs that seemed reasonable to a person who had unconsciously committed to a scientific world view.

In an earlier post I’ve already set out one criterion for my journey: it must be superstition-free and beliefs should not be based on their comfort level. Now, in the years after college as I worked as a mathematician in Pasadena for a branch of the Naval Ordinance Test Station, climbed and skied in the Sierra, and met the friend working on his Ph.D. in physics at Cal Tech, I gradually formed another criterion: Any spiritual outlook should be totally comfortable and compatible with a scientific world view. This is not to say that I thought a scientific world view the be all and end all of knowledge and life. I had heard about logical positivism in college and had taken it to claim (probably mistakenly) that the world of experience outside of science lacked meaning or validity. I felt that this view was ridiculous though I didn’t know enough about positivism to argue against it. I knew that I loved poetry, good poetry and kitchy bad poetry, was carried away by classical music, and felt impelled to try understand all of life, scientific and otherwise. Somewhere along the line I had run into the world of myth in Joseph Campbell’s books and found exciting his ideas about how myth gives meaning to life.


Although I was captivated by the “worlds” outside of science, I decided that any religious claims that attempted to contradict the stories of the scientific world were unworthy of consideration as serious truth, spiritual or otherwise. At the time this left a big question in my mind. Was there already existing somewhere in all of mainstream world religion or in any existing sect, a spiritual path that satisfied the criteria I had developed?

Dream and Myth

Sometime ago I had an unusually vivid dream. I was swimming underwater, perhaps ten or fifteen feet below the surface. I was swimming along with little effort, not holding my breath. Looking to one side I saw that I was swimming alongside a huge shark. I had a moment of panic being so close to such a fearsome beast. The shark’s side was 5 feet from top to bottom, it was about 30 feet long, but it wasn’t a harmless whale shark. Then I looked at the shark’s large eye which was staring at me and noticed that the shark’s gaze was intelligent and somehow caring. Immediately, in the dream I felt not only comfortable but deeply peaceful. When I woke up, I remembered the dream clearly.

Among Hawaiians there is a tradition of having a personal totem animal called one’s ʻaumākua.

The wehewehe dictionary’s definition:
nvt. Family or personal gods, deified ancestors who might assume the shape of sharks (all islands except Kauaʻi), owls (as at Mānoa, O’ahu and Kaʻū and Puna, Hawaiʻi), hawks (Hawaiʻi), ʻelepaio, ʻiwi, mudhens, octopuses, eels, mice, rats, dogs, caterpillars, rocks, cowries, clouds, or plants. A symbiotic relationship existed; mortals did not harm or eat ʻaumākua (they fed sharks), and ʻaumākua warned and reprimanded mortals in dreams, visions, and calls. (Beckwith, 1970, pp. 124–43, 559; Nānā 38.) Fig.., a trustworthy person. (Probably lit.., ʻau 4, group, + makua, parent.) See pulapula 2. hō.ʻau.makua To acquire or contact ʻaumākua.

Growing up in Honolulu one heard much of Hawaiian traditions. At my elementary school we heard Hawaiian stories of how Maui raised the islands out of the sea and, on occasion, learned a Hawaiian chant celebrating the god Lono. So it was clear to me that I had a giant shark god as my `aumakua. This was a source of satisfaction; however, if you ask how seriously I took this as protection if I swam in shark infested waters, the answer is “not very.” And, in fact, there is no way I would voluntarily swim in such waters. I knew of and felt the power of some Hawaiian myths, but remained a geeky haole (white person), quite afraid of sharks. I remember body surfing at a great place near Portlock place at Koko head with a crowd of others. As we waited for waves a fishing boat came by and warned us that they had just seen a shark. Everyone moved in closer to the break line, but after missing a wave or two moved back to where we could catch the waves. I was definitely fearful and on guard. True Hawaiians, on the other hand, were not all that afraid of sharks.

My uncle who had come to Hawaii a little prior to 1920 told a story he had heard about the time when dynamite had first come to the Islands. The Hawaiians would go to the outer reef in their canoes, set off a stick of dynamite under water, leap into the water and throw the stunned fish into their canoes. Sharks would show up and go into a feeding frenzy. The Hawaiians would pretty much ignore them except for an occasional push or kick and keep throwing fish into their canoes. I’m not sure how much of this I believe, but certainly the story has it that Hawaiians were very respectful, but not afraid of sharks. One must of course distinguish between reef sharks and the pelagic Tiger shark. Spear fishermen know that they should keep their speared fish at the end of a long line rather than tied to their waist because reef sharks would go after the speared fish. Actual shark attacks, on the other hand, seem to come from Tiger sharks who mostly can reach shore only where the reef is missing or not very wide.

My `aumakua experience of Hawaiian culture was a dream. Others have had vivid, waking experiences. On the Big Island of Hawaii where there are active volcanoes, there is a tradition of Madame Pele, the goddess who controls volcanoes. Many people native or otherwise believe that Madame Pele actually exists. She supposedly is fond of gin so people living near the volcano leave offerings of gin near their houses. It seems to be a good idea to be respectful and try to be in her good graces whether or not she really exists. My mother told me, probably in the 1970’s, of an older couple (haoles), friends of her and my dad, who happened to be in the parking lot of the Crater rim road at the Kilauea Iki view point during the great eruption of 1959. It was at night, but hardly dark, as a fountain of lava about a mile down the crater was illuminating the landscape. A mile sounds like a considerable distance. However, the lava fountain, on one occasion reached almost 1900 feet high (see Wikipedia). The Empire State building in Manhattan is 1454 feet high at its very tip top, while the new World Trade Center’s spire tops out at 1776 feet. So while my folk’s friends stood near the edge of the crater one can imagine a garish, flickering light, intense radiating heat, dark shadows and a loud roar as molten gobs of lava fell into the lava lake at the fountain’s edge. The couple took in the awesome scene for a while, then noticed a Hawaiian women nearby. She had reddish hair and wore a red holuku (Hawaiian mother Hubbard dress). They exchanged a few remarks with her about the incredible scene they were witnessing, then turned back to look at the fountain. Finally, satisfied, they turned to go back to their car and noticed the Hawaiian women rise a couple feet off the ground, drift away to the edge of the parking lot, and vanish.

When I heard this story, I felt a tensing of muscles in my back and my hair felt like it was standing on end. Could I have had this same experience? My emotions said so, but my reason said unlikely. The couple had an hallucination. But just what is the status of hallucinations? One notes that our entire experience of the world is entirely inside our heads, with considerable mental processing of our raw sense impressions. Could not the processing of neuronal pathways, influenced by one’s cultural background, and an incredible, dreamlike scene, construct an image that appears in consciousness as real? It would seem so. Hearing this story made me think that stories, for example, of the Greek gods were likely based on genuine hallucinations rather than simply literary imagination. Probably literary imagination was involved in the elaboration of such hallucinations into an entire cosmology, but behind the stories were experiences of the gods and goddesses that passed for real.

For the Hawaiians living before Captain Cook’s discovery, the myths were integrated into their natural surroundings and constituted an entire world view. Besides being the background to hard won practical knowledge and lore, the myths told of the place of humans in the universe. Their meaning carried religious significance, practical significance and social significance. In the modern world, West and East, reality has been split into “areas” and thereby degraded.

This post has gone on long enough and I will doubtless get later into an inquiry about the concept of “reality”. For now let me just say that if someone explains to you what “virtual reality” is all about, the proper reply is “I didn’t know there was any other kind.”

Transition to Spiritual Quest, 1958-1959

I’m back home in Oregon again after almost 2 months on the road. My wife, Susan, and I drove our small (16 foot) camper van to Bellingham, Washington, took the ferry to Haines, Alaska, drove many of the roads in Alaska, to the coast, to Anchorage, to Denali, to Fairbanks, then drove the Alaska highway down through the Yukon, British Columbia, the Rockies, Montana and back to Oregon. It was my first time in Alaska which became state # 50. Now back home it is time to write and post a blog entry.

Let me continue by talking about spiritual quest, next stage, 1958-1959. This is going to involve talking about Zen Buddhism so let me say a few words on that subject.

First I should address my concern that to many the word Zen rightly conjures up some kind of new-age, hippy, Eastern garbage. Indeed, often when one runs into people who talk or write about Zen one hears what I would characterize as “pop Zen”, supposedly stunningly wise statements that if understood will turn ones life into enlightened bliss. Such pop Zen seems “self-indulgent [and] egoistic” to quote Owen Flanagan, professor at Duke University, talking about “New Age style religions” in general. See (Link valid as of 8/1/16.) I hope here to strenuously avoid pop Zen.

Another concern is that Zen arose out of Buddhism, an Asian religion, and seems irredeemably associated with Eastern cultures, especially Japanese culture. It is clear to me that Zen, if indeed there is such a thing, is independent of culture. So, although I grew up in Hawaii and am by no means hostile to Chinese or Japanese culture, I will try to emphasize Western ideas in this blog, at least at first. It seems to me that Western analytic philosophy of the 20th century associated with the analysis of language is relevant, as well as 20th and 21st century physics and cosmology; and that the kind of spiritual quest I want to talk about is completely rational.

Still another concern is that nothing can actually be said directly about Zen and that in fact, strictly speaking, there is no such thing as Zen. This fact smacks of deliberate obfuscation so anyone who is going to say anything about the subject has a lot of explaining on their hands and will necessarily be, in an ultimate sense, talking nonsense. The hope is that the nonsense will be interesting or even a fun kind of nonsense so that it will be enjoyable reading and may perhaps lead to some kind of worthwhile understanding.

Note to self: Avoid “strict speaking”, lighten up, avoid the word “Zen” as much as possible.

So, though the “Z” word will, I hope, appear infrequently, if at all, when I talk about various subjects, it will lurk like some kind of insubstantial ghost in the background.

All of this being said, let me now go back in history to 1958, skipping lightly for the moment, over the years 1954-1957 when I was in the army and then in Innsbruck, Austria, for a year on the GI bill, where I met and married my first wife, Barbara.

Spiritual Quest, 1958-1959

It is fall, 1957. I am newly married to Barbara, my first wife. We are in Auburn, Alabama where I am a Temporary Instructor in the Physics Department at the Alabama Polytechnic Institute, later to become Auburn University. We have ended up in Auburn because Barbara was acquainted with Dr. Howard Carr, head of the Physics Department and knew that the department badly needed people to teach elementary physics to engineering students and liberal arts majors. Letters written from Innsbruck plus a positive reference from my graduate adviser at Stanford (Georg Polya, a well-known mathematician) sealed the deal. Of course, I had pretty much forgotten the little physics I had ever known so I would need to learn the subject from the text I was teaching and try to stay a week or so ahead of my students. Since I was planning to become a physicist anyway, this was a fun challenge and I didn’t do too badly in meeting it. Certainly, I could appreciate and relate to the difficulties my students were having with the subject. Meanwhile, Barbara had decided to switch her major from mathematics to English literature so was taking graduate courses in the English Department.

That fall we were totally absorbed in life. I was passionately in love with Barbara and working hard on learning elementary physics and doing well with my teaching. As a faculty member I had easy access to football tickets and enjoyed going to games. The Auburn team that year was winning all their games, a new experience for me after watching games in high school and at Stanford. In high school I watched Punahou lose 64 – 0 to Kamehameha in their first game and lose every subsequent game thereafter. Stanford had a similarly bad season my freshman year. In retrospect I think that the Auburn team was the best college team I’ve ever seen. They had an overwhelming defense often holding opponents to negative yardage on the ground. Their games were not exciting because they did not seem to be very fired up. They would get a lead of a few points, shut down their opponents, and play out the rest of the game in a boring manner. There was only the suspense of wondering if the opposition would score on a fluke play. When it came time to play the last game of the season against arch-rival Alabama, the press was wondering if there would be an upset because Auburn’s wins had been less than dramatic while Alabama hadn’t done all that badly. The game started in a usual manner. Auburn won the toss and, as they always did in such circumstances, elected to kick. As the kickoff sailed down the field I suddenly realized I was looking at a different, fired up, team. The Alabama receiver took the ball in the end zone and started up the field, making little progress as flying tackles narrowly missed their target. The runner was shortly overwhelmed at about the 15 yard line. In the next few plays Alabama lost yardage and finally fumbled after a hard hit in their end zone. Auburn 7, Alabama 0. Subsequently Auburn finally displayed their offense. They did have an all-American end, Jimmy Phillips, who played sensationally and their ground game became effective. Final score 40 – 0. What impressed me about that Auburn team was the philosophy of doing the minimum necessary to win, in a relaxed manner, never playing to potential unless necessary or in a game with Alabama. This attitude, with its suggestion of power held in reserve, smacked of the Zen I would later encounter.

Also in that Fall Quarter I was becoming acquainted with Barbara’s family and numerous relatives, taking in the friendly Southern atmosphere, which overlay a terrible racism, seldom explicitly on display to me. However, I knew it was there. The first morning in Auburn I was awake at dawn, still not adjusted to the time change, so got up in the early light and headed to town up the main street. A black man came down the side walk in front of me, began to hesitate when about thirty feet away, then stepped off the sidewalk three or four feet into the street and cowered, half turned away from me with head bowed, as I walked by. I was totally appalled, having grown up in Hawaii where there are too many races and racial mixtures for serious prejudice though people other than haoles (whites) had been quite subjugated in the days before I grew up. By the time I was in high school, however, one could be taunted for being a haole and perhaps beaten up, so what prejudice there was operated in all directions. In Alabama, because I am a realist and definitely a coward as well, I never openly challenged the mores of that time, but tried to treat black people with respect.

A late poem of Wallace Stevens has the title “A Child Asleep in its Own Life”, suggesting the image of a child totally engaged and walking around with concentrated purpose, unaware of what was going on in its surroundings. It seems to me that I and probably Barbara were in a similar state of being that fall. We were fully engaged with our lives and with each other but I had certainly dropped all thoughts of a spiritual journey, of religion, or of anything else beyond our immediate circumstances. Winter Quarter came and things began to change. That quarter I was able to sit in on a class in quantum mechanics taught by Ernest Ikenberry. Professor Ikenberry was in the process of writing a text, loosely based on the quantum mechanics text by David Bohm. At the start of each class he would hand out three or four pages of mimeographed notes and then lecture, explicating the notes. Quantum mechanics for me was a revelation. I loved the mathematics and slowly came to realize that quantum physics totally demolished the deterministic world view of classical physics, a view that I had heartily disliked. I was delighted that physics had opened up in a way that could sit easily beside all the other worlds of human experience. Physics was going to be very exciting.

Another happening that made a huge impression on me was my first encounter with Zen. I can’t remember exactly when this was. Probably it occurred sometime during the Winter Quarter in 1958, or perhaps as late as early Spring Quarter the same year. In any case it came about because that year we were subscribing to Harper’s Magazine. My favorite writer appearing in the magazine at the time was Gilbert Highet whose essays were always interesting and beautifully written. So, it was with interest that I read Highet’s review of Zen in the Art of Archery by Eugen Herrigal, which had been translated from German to English a few years before. Besides talking about the book Highet also had some additional thoughts about Zen Buddhism, which, at the time, was practically unknown in the US and completely unknown to me. Intrigued by Highet’s piece I went to the Auburn library and found not only Herrigal’s book but one or two others by D.T. Suzuki who in the late 1920’s had thought to introduce Zen to the West. Suzuki’s Essays in Zen Buddhism particularly impressed me. Here was a religion with no doctrines or beliefs. One was to discover one’s own insights through meditative practice, aided perhaps, by concentrating on crazy riddles, called Koans or Mondos.

Three things about Zen were apparent to me. First, it was a serious religion. By that I mean a religion that becomes the center of one’s life. Second, because Zen makes no concrete specific claims about the world, it could be the deep, wordless philosophical center encompassing all other more concrete subjects without conflicting with any of them. Finally, if pursued successfully, it had the potential to bring spiritual peace to one’s life in the face of death.

At the time I was by no means converted to the religion. Life was far too busy and I simply didn’t know enough about it. I loved the riddle characteristic of the Koans. They seemed like the problems in physics that I enjoyed struggling with. However, a Koan did not have a rational, mathematical answer as in physics, but a deep non-verbal answer enveloped with understanding – rather like a joke whose point one gets instantly or not at all. (Of course I “got” none of the Koans at the time.) It was clear to me, however, that I would have an enjoyable time ahead learning more about Zen and about the Buddhism it grew out of before I would understand enough for a possible commitment.

As the academic year at Auburn drew to an end there was much on my mind. I had been accepted at the University of Virginia as a graduate student in physics and I was off for a summer trip to the Tetons with Barbara and her young sister Mary, driving an old Ford and camping along the way.

Fall 1958 found us living in a dank basement apartment in downtown Charlottesville, parking the old Ford on hilly streets for more assured starting in the morning, and attending our classes at the University. My most exciting class in classical electricity and magnetism was taught by John Plaskett, English and only a few years older than myself. Starting from Maxwell’s equations he would develop the story of how they came to be and their consequences in a lucid manner. In the afternoon I would spend three to four hours going over my notes, being sure that I understood every nuance of the derivations. In class I always sat in the front row because Plaskett had the charming habit when coming to a key conclusion at the end of a long, mathematical argument of announcing the dénouement in a voice that sank to barely above a whisper. How embarrassingly immodest it would be to be blatant about a beautiful result. I imagined Plaskett as a Zen master who was bringing enlightenment through electromagnetism. If the mastery of archery could bring Herrigal to the brink of transcendent understanding, why wouldn’t theoretical physics serve just as well? Of course I suspect that Plaskett hadn’t the slightest idea he was practicing a Zen art, but I, in my naivety was stumbling upon the idea that depths can lurk anywhere, even in the beauties of a science devoted to understanding the most mundane materialistic reality.

Of course, in my spare time I checked out the University of Virginia library to see what it contained in books about Zen. There was more than at the Auburn library but not so much that I couldn’t read it all during the year. Besides studying physics and reading about Zen I was exposed vicariously to English studies. My fellow physics students remained acquaintances whereas Barbara’s fellow English students became our friends. People studying literature tend to be more articulate and socially interesting than scientists and engineers so we ended up running with the English crowd. Barbara’s most exciting studies were in classes taught by Fredson Bowers, an authority on the textual criticism of Shakespeare. She also had a class in which every poem written by Yeats was read and discussed. In earlier years I had discovered how moving poetry could be so I thoroughly enjoyed the discussions among our friends though my contributions were minimal.

Classes continued during the winter and through the spring and came to an end in June at the beginning of a miserably hot humid summer. That summer we would spend in Charlottesville where I had a job studying and hopefully contributing to centrifuge science. Luckily there was swimming pool, surrounded by a shady lawn where we could hang out and be comfortable in the hot weather. And one afternoon while sitting in the shade with Barbara, cooling off after a swim, I became conscious that I was committing to Zen. Such a decision is not made consciously or rationally. I was very aware that I had little or no understanding and that the whole thing was a big mystery. However, it felt good that I intended to spend the rest of my life attempting to understand what it was all about.

Whimsical Math

These last few days I started to write some more about Zen, but the ideas were ill-conceived and turned to mush. Maybe someday they can be resurrected though I doubt it. To clear my brain I started to write something just for fun. Perhaps it is worth posting.

Whimsical Math

In the series 1,2,3,… what is the ultimate number? Not the largest number, there being no such thing, but the ultimate number. Answer: The last yianh. As 10 or 11 year olds, my brother and I, playing in our upstairs bedroom, speculated about numbers that were so large that it would be pointless to consider anything larger among the infinity of further numbers. There was an entire series of these, starting with the first yianh, a number so large that a googolplex would fade into an insignificant blob near zero. Of course we had never heard of a googol or a googolplex nor did we know anything about exponential notation, but such specifics are pointless when imagining REALLY large numbers. After a huge series of further numbers one would come to the second yianh. And so on. Finally, after many, many more yianh‘s we would reach the last yianh, a super fabulous number that ended our imaginings. I think now that we had some inchoate idea that the entire tail of the infinite integers would somehow be condensed into this ultimate number. Perhaps this is how mathematics gets developed. One plays around and then some idea like making an infinite series finite by coalescing numbers comes up. Then one sees this as a problem. Can the idea be made logical and coherent? If one succeeds, one has created some new math. If the effort fails, maybe the idea can be deployed somewhere else.

In the nineteenth century there lived a very great mathematician named Bernhard Riemann. Actually there were many very great mathematicians in the nineteenth century, but Riemann was one of the immortals, like Beethoven in music or Shakespeare in drama. (Well maybe not quite as uniquely great as Shakespeare.) Anyway Riemann studied a function, now called the Riemann Zeta function. A function is like a meat grinder. With a meat grinder, one feeds in meat and hamburger comes out. With a function one feeds a number in and another number comes out. The number coming out depends on the one fed in so that one always gets the same second number from the same first. Of course, when talking about the Zeta function there is a little complication that is likely to scare you away if you are a math phobic reader. But be reassured. I will explain this complication with the utmost lucidity and make it completely clear or at least translucent by telling a story about numbers.

Since Greek times there have been many developments about what numbers are. Numbers started as the 1,2,3’s. Then someone discovered a big shortcut, called multiplication, when the same number was added up many times. Immediately, problems arose where one needed to go backwards, so division and with it fractions came into being. The integers and fractions together were called rationals. But horror upon horror, the Greeks found that there were crazy numbers, like the square root of 2, that couldn’t be expressed as fractions. These were called irrationals, disturbing because the Greeks were wedded to being rational. Then when considering such things as debts, people realized that negative numbers were useful, and, more important, could be introduced in such a way that no logical contradictions arose. New math had been created. Later the idea arose that the gaps that still existed among the rational and irrational numbers could be filled to make a continuous stream of numbers with again no contradictions. But here trouble arose. One theme of this blog will be the difficulties we get into by misunderstanding language. The new numbers were called “real” numbers although they were the product of human imagination and could easily have been called “numbers of the imagination” or imaginary numbers for short. Of course, when the numbers we now call imaginary were introduced, this introduction caused all sorts of trouble, not only at the time, but subsequently to generations of math students. I certainly was very dubious about imaginary numbers. If they weren’t real, how could they even exist? Well how can any number exist? It exists because it can be used to calculate things without contradictions arising. So, it turns out we can have what are called complex numbers, consisting of a pair of real numbers, though the second of the pair is called imaginary. Consider imaginary as simply a label used to designate the second of the pair. (It is true, however, that squaring this second number results in a negative “real” number, but that’s really no big deal.) These complex numbers can be used without logical contradictions; but raise the question: What about a triple of numbers? Can a triple be made to act like other numbers? The answer is no. The number pairs are as far as numbers can go. (There are more complicated sets of numbers called vectors, but they work differently from numbers.)

Of course, the reason I’ve gone through this song and dance about numbers is that the numbers fed into the Zeta function are complex numbers, as are the numbers coming out. The Zeta function is called a function of a complex variable and such are studied in graduate level courses in math. (When I took the course at Stanford, I flunked, but later picked up some of the subject on my own.) Since complex numbers are pairs of numbers they can be plotted on a 2 dimensional sheet. We plot the first number on a horizontal x axis and the second, so-called imaginary, number on the vertical y axis. An important situation arises when we feed the first number into the Riemann meat grinder and get 0 out of the function. Such numbers are called “zeros of the Zeta function” although they themselves are not in fact zero, but produce zero when fed into the function. They should have been called zero producers, but that is too long-winded for mathematicians. Meat goes in the grinder, but nothing comes out. So call the meat a zero. Anyway, the Zeta function has many zeros some of whose location turns out to be connected to the distribution of prime numbers. Mathematicians call the others “trivial zeros” and study the ones that matter. Riemann calculated a few of the non-trivial ones (turning the crank of the Zeta function is not easy) and found that they lay on a line with the real part = ½ and the imaginary part on a vertical line rising up from ½ on the horizontal axis. Riemann speculated that all of the important zeros would lie on that vertical line with real part ½. He couldn’t prove it. Nor has anyone proved it in the 156 years or so since, though not for want of trying. (Whether true or false the hypothesis has Yuuge consequences.) Zeros in the billions have been shown to lie on that imaginary line, but billions aren’t equal infinity and a proof would show that all of the infinite number of zeros lay there.

I have a whimsical notion that if Riemann’s hypothesis is incorrect, somewhere up in the far reaches of the imaginary line there is a zero whose real part is not ½. I’ll call this zero the first yianh; other violators the second, third, etc. yianh. One wonders if there is a last yianh or does the sequence of violations never end? Note: I strongly suspect that Riemann’s hypothesis is true, in which case my last remark is even more whimsical.

Prime Obsession: Bernard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire is a fascinating book. A history of the times and a biography of Riemann alternate with chapters that go into the math seldom going beyond high school level. Or go to Wikipedia.

A final note: There are hints that the distribution of the primes for large numbers (primes up there with the yianh’s) has a connection with certain physical properties of the universe. It is difficult to keep “pure” mathematics pure.