During the century between about 600 BCE to 500 BCE, the first school of Greek philosophy flourished in Ionia. This, arguably, is the first historical record of philosophy as a reasoned attempt to explain things without recourse to the gods or out-and-out magic. But where on earth was Ionia? Wherever it was it’s now long gone. Wikipedia, of course, supplies an answer. If one sails east from the body of Greece for around 150 miles, passing many islands in the Aegean Sea, one reaches the mainland of what is now Turkey. Along this coast at about the same latitude as the north coast of the Peloponnesus (37.7 degrees N) one finds the island of Samos, a mile or so from the mainland; and just to the north is a long peninsula poking west which in ancient times held the city-state of Ionia. Wikipedia tells us that this city-state, along with many others along the coast nearby formed the Ionian League, which in those days, was an influential part of ancient Greece, allying with Athens and contributing heavily, later on, to the defeat of the Persians when they tried to conquer Greece. One can look at Google Earth and zoom in on these islands and in particular on Samos, seeing what is now likely a tourist destination with beaches and an interesting, rocky, green interior. On the coast to the east and somewhat south of Samos was the large city of Miletus, home to Thales, Anaximander, Heraclitus and the rest of the Ionian philosophers. At around 570 BCE on the Island of Samos Pythagoras was born. Nothing Pythagoras possibly might have written has survived, but his life and influence became the stuff of conflicting myths interspersed with more plausible history. His father was supposedly a merchant and sailed around the Mediterranean. Legend has it that Pythagoras traveled to Egypt, was captured in a war with Babylonia and while imprisoned there picked up much of the mathematical lore of Babylon, especially in its more mystical aspects. Later freed, he came home to Samos, but after a few years had some kind of falling out with its rulers and left, sailing past Greece to Croton on the foot of Italy which in those days was part of a greater Greek hegemony. There he founded a cult whose secret mystic knowledge included some genuine mathematics such as how musical harmony depended on the length of a plucked string and the proof of the Pythagorean theorem, a result apparently known to the Babylonians for a thousand years previously, but possibly never before proved. Pythagoras was said to have magic powers, could be at two places simultaneously, and had a thigh of pure gold. This latter “fact” is mentioned in passing by Aristotle who lived 150 years later and is celebrated in lines from the Yeats poem, Among School Children:

…

Plato thought nature but a spume that plays

Upon a ghostly paradigm of things;

Solider Aristotle played the taws

Upon the bottom of a king of kings;

World-famous golden-thighed Pythagoras

Fingered upon a fiddle-stick or strings

What a star sang and careless Muses heard:

Yeats finishes the stanza with one more line summing up the significance of these great thinkers: “Old clothes upon old sticks to scare a bird.” Although one may doubt the golden thigh, quite possibly Pythagoras did have a birthmark on his leg.

I became interested in Ionia and then curious about its history and significance because I recently wondered what kind of notation the Greeks had for numbers. Was their notation like Roman numerals or something else? I found an internet link, http://www.math.tamu.edu/~dallen/history/gr_count/gr_count.html which explained that the “Ionian” system displaced an earlier “Attic” notation throughout Greece, and then went on to explain the Ionian system. In the old days when a classic education was part of every educated person’s knowledge, this would be completely clear as an explanation. Although I am old enough to have had inflicted upon me three years of Latin in high school, since then I had been exposed to no systematic knowledge of the classical world so was entirely ignorant of Ionia, or at least of its location. I had heard of the Ionian philosophers and had dismissed their philosophy as being of no importance as indeed is the case, EXCEPT for their invention of the whole idea of philosophy itself. And, of course, without the rationalism of philosophy, it is indeed arguable that there would never have been the scientific revolution of the seventeenth century in the West. (Perhaps that revolution was premature without similar advances in human governance and will yet lead to disaster beyond imagining in our remaining lifetimes. Yet we are now stuck with it and might as well celebrate.)

The Ionian numbering system uses Greek letters for numerals from 1 to 9, then uses further letters for 10, 20, 30 through 90, and more letters yet for 100, 200, 300, etc. The total number of symbols is 27, quite a brain full. The important point about this notation along with that of the Egyptian, Attic, Roman and other ancient Western systems is that position within a string of numerals has no significance except for that of relative position with Roman numerals. This relative positioning helps by reducing the number of symbols needed in a numeric notation, but is a dead end compared to an absolute meaning for position which we will go into below. The lack of meaning for position in a string of digits is similar to written words where the pattern of letters within a word has significance but not the place of a letter within the word, except for things like capitalizing the first letter or putting a punctuation mark after the last. As an example of the Ionian system, consider the number 304 which would be τδ, τ being the symbol for 300 and δ being 4. There is no need for zero, and, in fact, these could be written in reverse order δτ and carry the same meaning. In thinking about this fact and the significance of rational numbers in the Greek system I came to understand some of the long history with the sparks of genius that led in India to OUR numbers. In comparison with the old systems ours is incredibly powerful but with some complexity to it. I can see how with unenlightened methods of teaching, trying to learn it by rote can lead to early math revulsion and anxiety rather than to an appreciation of its remarkable beauty, economy and power.

In the ancient Western systems there is no decimal point and nothing corresponding to the way we write decimal fractions to the right of the decimal point. What we call rational numbers (fractions) were to Pythagoras and the Greeks all there was. They were “numbers”, period, and “obviously” any quantity whatever could be expressed using them. Pythagoras died around 495 BCE, but his cult lived on. Sometime during the next hundred years, one of his followers disproved the “obvious”, showing that no “number” could express the square root of 2. This quantity, √2, by the Pythagorean theorem, is the hypotenuse of a right triangle whose legs are of length 1, so it certainly has a definite length, and is thus a quantity but to the Greeks was not a “number”. Apparently, this shocking fact about root 2 was kept secret by the Pythagoreans, but was supposedly betrayed by Hippasus, one of them. Or perhaps it was Hippasus who discovered the irrationality. Myth has it that he was drowned (either by accident or deliberately) for his impiety towards the gods. The proof of the irrationality of root 2 is quite simple, nowadays, using easy algebra and Aristotelian logic. If a and b are integers, assume a/b = √2. We may further assume that a and b have no common factor, because we may remove them all, if any. Squaring and rearranging, we get a²/2 = b². Since b is an integer, a²/2 must also be an integer, and thus “a” itself is divisible by 2. Substituting 2c for a in the last equation and then rearranging, we find that b is also divisible by 2. This contradicts our assumption that a and b shared no common factor. Now we apply Aristotelian logic, whose key property is the “law of the excluded middle”: if a proposition is false, its contrary is necessarily true, there is no “weaseling” out. In this case where √2 is either a fraction or isn’t, Aristotelian logic applies, which proves that a/b can’t be √2. The kind of proof we have used here is called “proof by contradiction”. Assume something and prove it false. Then by the law of the excluded middle, the contrary of what we assumed must be true. In the early twentieth century, a small coterie of mathematicians, called “intuitionists”, arose who distrusted proof by contradiction. Mathematics had become so complex during the nineteenth century that these folks suspected that there might, after all, be a way of “weaseling” out of the excluded middle. In that case only direct proofs could be trusted. The intuitionist idea did not sit well with most mathematicians who were quite happy with one of their favorite weapons.

Getting back to the Greeks and the fifth century BCE one realizes that after discovering the puzzling character of √2, the Pythagoreans were relatively helpless, in part because of inadequacies in their number notation. I haven’t tried to research when and how progress was made in resolving their conundrum during the 25 centuries since Hippasus lived and died, but WE are not helpless and with the help of our marvelous number system and a spreadsheet such as Excel, we can show how the Greeks could have possibly found some relief from their dilemma. The answer comes by way of what are called Pythagorean Triplets, three integers like 3,4,5 which satisfy the Pythagorean Law. With 3,4,5 one has 3² + 4² = 5². Other triplets are 8,15,17 and 5,12,13. There is a simple way of finding these triplets. Consider two integers p and q where q is larger than p, where if p is even, q is odd (or vice-versa) and where p and q have no common factor. Then let f = q² + p², d = q² – p², and e = 2pq. One finds that d² + e² = f². Some examples: p = 1, q = 2 leads to 3,4,5; p = 2, q = 3 leads to 5,12,13. These triplets have a geometrical meaning in that there exist right triangles who sides have lengths whose ratios are Pythagorean triplets. Now consider p = 2, q = 5 which leads to the triplet 20,21,29. If we consider a right triangle with these lengths, we notice that the sides 20 and 21 are pretty close to each other in length, so that the shape of the triangle is almost the same as one with sides 1,1 and hypotenuse √2. We can infer that 29/21 should be less than √2 and 29/20 should be greater than √2. Furthermore, if we double the triangle to 40,42,58, and note that 41 lies halfway between 42 and 40, the ratio 58/41 should be pretty darn close to √2. We can check our suspicion about 58/41 by using a spreadsheet and find that the 58/41 is 1.41463 to 5 places, while √2 to 5 places is 1.41421. The difference is 0.00042. The approximation 58/41 is off by 42 parts in 100,000 or 0.042%. The ancient Greeks had no way of doing what we have just done; but they could have squared 58 and 41 to see if the square of 58 was about twice the square of 41. What they would have found is that 58² is 3364 while 2 X 41² is 3362, so the fraction 58/41 is indeed a darn good approximation. Would the Greeks have been satisfied? Almost certainly not. In those days Idealism reigned, as it still does in modern mathematics. What is demanded is an exact answer, not an approximation.

While there is no exact fraction equal to √2, we can find fractions that get closer, closer and forever closer. Start by noticing that a 3,4,5 triangle has legs 3,4 which though not as close in length as 20, 21, are only 1 apart. Double the 3,4,5 triangle to 6,8,10 and consider an “average” leg of 7 relative to the hypotenuse of 10. The fraction 10/7 = 1.428 to 3 places while √2 = 1.414. So, 10/7 is off by only 1.4%, remarkably close. Furthermore, squaring 10 and 7, one obtains 100, 49 while 2 = 100/50. The Pythagoreans could easily have found this approximation and might have been impressed though certainly not satisfied.

I discovered these results about a month or so ago when I began to play with an Excel spread sheet. Playing with numbers for me is relaxing and fun; and is a pure game whether or not I find anything of interest. I suspect that this kind of “playing” is how “real” mathematicians do find genuinely interesting results, and if lucky, may come up with something worthy of a Fields prize, equivalent in mathematics to a Nobel prize in other fields. While my playing is pretty much innocent of any significance, it is still fun, throws some light on the ancient Greek dilemma, and for those of you still reading, shows how a sophisticated idea from modern mathematics is simple enough to be easily understood.

With spreadsheet in hand what I wondered was this: p,q = 1,2 and p,q = 2,5 lead to approximations of √2 via Pythagorean triplets. Are there other p,q’s that lead to even better approximations? To find such I adopted the most powerful method in all of mathematics: trial and error. With a spreadsheet it is easy to try many p,q’s and I found that p = 5, q = 12 led to another, even better, approximation, off by 1 part in 100,000. With 3 p,q’s in hand I could refine my guesswork and soon came up with p = 12, q = 29. I noticed that in the sequence 1,2,5,12,29,… successive pairs gave increasingly better p,q’s. This was an “aha” moment and led to a question. Could I find a rule and extend this sequence indefinitely?

In my life there is a long history of trying to find a rule for sequences of numbers. In elementary school at Hanahauoli, a private school in the Makiki area of Honolulu, I learned elementary arithmetic fairly easily, but found it profoundly uninteresting if not quite boring. Seventh grade at Punahou was not much better, but was interrupted part way through the year by the Pearl Harbor attack of December 7, 1941. The Punahou campus was taken over by the Army Corps of Engineers and our class relocated to an open pavilion on the University of Hawaii campus in lower Manoa Valley. I mostly remember enjoying games of everyone trying to tackle whoever could grab and run with a football even if I was one of the smaller children in the class. Desks were brought in and we had classes in groups while the rain poured down outside the pavilion. Probably, it was during this year that we began to learn how fractions could be expressed as decimals. In the eighth grade we moved into an actual building on the main part of the University campus and had Miss Hall as our math teacher. The math was still pretty boring, but Miss Hall was an inspiring teacher, one of those legendary types with a fierce aspect, but a heart of gold. We learned how to extract square roots, a process I could actually enjoy, and Miss Hall told us about the fascinating things we would learn as we progressed in math. There would be two years of algebra, geometry, trigonometry and if we progressed through all of these, the magic of “calculus”. It was the first time I had heard the word and, of course, I had no idea of what it might be about, but I began to find math interesting. In the ninth grade we moved back to the Punahou campus and our algebra teacher was Mr. Slade, the school principal, who had decided to get back to teaching for a year. At first, we were all put off a bit by having the fearsome principal as a teacher, but we all learned quickly that Mr. Slade was actually a gentle person and a gifted teacher. As we learned the manipulations of algebra and how to solve “word problems”, Mr. Slade would, fairly often, write a list of numbers on the board and ask us to find a formula for the sequence. I thoroughly enjoyed this exercise and learned to take differences or even second differences of pairs in a sequence. If the second differences were all the same, the expression would be a quadratic and could easily be found by trial and error. Mr. Slade also tried to make us appreciate the power of algebra by explaining what was meant by the word “abstraction”. I recall that I didn’t have the slightest understanding of what he was driving at, but my intuition could easily deal with an actual abstraction without understanding the general idea: that in place of concrete numbers we were using symbols which could stand for any number. Later when I did move on to calculus which involves another step up in abstraction, I at first had difficulty in the notation f(x), called a “function” of x, an abstract notation for any formula; or indeed a representation of a mapping that could occur without a formula. I soon got this idea straight and had little trouble later with a next step of abstraction to the idea used in quantum mechanics of an abstract “operator” that changes one function into another.

Getting back to the sequence 1,2,5,12,29,… I quickly found that taking differences didn’t work; the differences never seemed to get much smaller because the sequence turns out to have an exponential character. I soon discovered, however, using the spreadsheet that quotients worked: take 2/1, 5/2, 12/5, 29/12, all of which become more and more similar. Then multiplying 29 by the last quotient, I got 70.08. Since 29 was odd, I needed an even number for the next q so 70 looked good and indeed I confirmed that the triplet resulting from 29, 70 was 4059, 4060, 5741 with an estimate for √2 that was off by only 1 part in a 100 million. After 70 I found the next few members of the sequence, 169, 408, 985. The multiplier to try for the next member seemed to be closing in on 2.4142 or 1 + √2. At this point I stopped short of trying for a proof of that possibility, both because I am lazy and because the possible result seemed uninteresting. What is interesting is that the sequence of p,q’s goes on forever and that approximations for √2 by using the resulting triplets will converge on √2 as a limit. The ideas of a sequence converging to a limit was only rigorously defined in the 19th century. Possibly it might have provided satisfaction to the ancient Greeks. Instead, the idea of irrational numbers that were beyond fractions became clear only with the invention by the Hindu’s in India of our place based numerical notation and the number 0.

Place based number notation was developed separately in several places, in ancient Babylon, in the Maya civilization of Central America, in China and in India. A place based system with a base of 10 is the one we now use. Somewhere in one’s education one has learned about the 1’s column just to the left of a decimal point, then the 10’s column, the 100’s column and so forth. When the ancient Hindu’s and the other civilizations began to develop the idea of a place based system, there was no concept of zero. Presumably the thought was the idea that symbols should stand for something. Why would one possibly need a symbol that stood for nothing? So, one would begin with symbols 1 through 9 and designate 10 by ”1·”. The dot “·” is called a “place holder”. It has no meaning as a numeral, serving instead as a kind of punctuation mark which shows that one has “10”, not 1. Using the place holder in the example above of Ionian numbers, the τδ would be 3·4, the dot holding the 10’s place open. The story with “place holders” is that the Babylonians and Mayans never went beyond, but the Hindu’s gradually realized the dot could have a numerical meaning within its own right and “0” was discovered (invented?). Recently on September 13 or 14th, 2017, there was a flurry of reports that carbon dating of an ancient Indian document, the Bakhshali manuscript revealed that some of its birch bark pages were 500 years older than previously estimated, dating to a time between 224 – 383 AD. The place holder symbol occurring ubiquitously in the manuscript was called shunya-bindu in the ancient Sanskrit, translated in the Wikipedia article about the manuscript as “the dot of the empty place”. (Note that in Buddhism shunyata refers to the “great emptiness” a mystic concept which we might take as the profound absence of being logically prior to the “big bang”) A readable reference to the recent discovery is https://www.smithsonianmag.com/smart-news/dating-ancient-indian-text-gives-new-timeline-history-zero-180964896/. According to the Wikipedia article the Bakhshali manuscript is full of mathematics including algebraic equations and negative numbers in the form of debts. As a habitual skeptic I wondered when I first heard about the new dating whether Indian mathematicians with their brilliant intuition hadn’t immediately realized the numerical meaning of their place holder. Probably they did not. An easy way to see the necessity of zero as a number is to consider negative numbers as they join to the positives. In thinking and teaching about math I believe that using concrete examples is the best road leading to an abstract understanding. The example of debts is a compelling example of this. At first one might consider one’s debts as a list of positive numbers, amounts owed. One would also have another list of positive numbers, one’s assets, amounts owned. The idea might then occur of putting the two lists together, using “-“ signs in front of the debts. As income comes in one’s worth goes, for example, -3, then -2, -1. Then what? Before going positive, there is a time when one owes nothing and has nothing. The number 0 signifies this time before the next increment of income sends one’s worth to 1. The combined list would then be …, -3, -2, -1, 0, 1, 2, 3, … . Doing arithmetic, using properly extended arithmetic rules, when one wants to combine various sources of debt and income becomes completely consistent, but only because 0 was used.

If the above seems as if I’m belaboring the obvious, let me then ask you why when considering dates, the next year after 1 BCE is not 0, but 1 AD? Our dating system was made up during an early time before we had adopted “0” in the West. Historians have to subtract 1 when calculating intervals in years between BCE and AD and centuries end in hundreds, not 99’s. This example is a good one for showing that if one gets locked in to a convention, it becomes difficult if not impossible to change. I was quietly amused at the outcry as Y2K, the year 2000 came along with many insistent voices pointing out the ignorance of we who considered the 21^st century to have begun. The idea of zero is not obvious and I hope I’ve shown in considering the Pythagorean’s and their dilemma with square roots, just how crippled one is trying to get along without it. Back to Top

My last post was on 8/11/17 shortly before we needed to prepare for a big road trip from Bend, Oregon to the Maritime Provinces of Canada, followed by visits to Sue’s family in Lake George, New York and my daughter’s family in Annapolis, Maryland. Preparations for the trip had to be made early because just before the trip there was the total solar eclipse of 2017 on Monday, the 21st, the shadow passing 25 or so miles north of us. In the days before the eclipse our house filled with family. We had made viewing plans and they worked out well. On Monday before dawn we drove to an open field Northwest of Prineville, saw the sky darken, leaf shadows sharpen, and felt the temperature fall by 12 degrees or so. We then watched as a black shadow fell on Gray’s Butte 10 miles to the West and rushed towards us at 1700 MPH. The last bright spark on the sun’s rim flickered out; and there was the corona and Bailey’s shining diamonds along the rim of the shadowed sun. The entire experience was as stunning as advertised and brought home to us the reality of cosmic events. There really is a moon out there, a sun and an entire cosmos whose very existence is an impenetrable mystery that we can experience during our brief stay in conscious awareness.

After the eclipse we waited a day for the traffic to clear, took my computer to the shop, finding out the mother board was dead, then headed out across the continent after taking Sue’s sister Nancy to the Portland airport. The trip was long and accomplished what travel should. We saw new country and discovered that some Canadians were more concerned with the possible shortcomings of their prime minister than with those of Trump. As a child I’d read about the tides of the Bay of Fundy but had no idea even where it was. Now we saw the 45-foot tide come in (record some 50 odd feet), finally got a good look at a tidal bore and added three provinces to our list. (We’ve traveled in all 50 US states so are now adding Canadian provinces and territories to our travel deeds.) We had been somewhat leisurely going East to the Maritimes. But then, after our family visits, drove across the US in 6 days, seeing some new territory on the way and being moved by a visit to the California Trail Interpretive Center on I80 in Nevada. One reads about the hardships and heartbreaks of the Westward migration and understands intellectually, but seeing the exhibits and dioramas makes for a much deeper emotional understanding. Arriving home on September 30th, we settled in for a week or two before going to the Stanford Alpine Club reunion. Now we’re really back home with a new computer fired up and it’s time to write.

In previous posts I’ve expressed the theme that Western thought would be more satisfying if informed by the spirituality of the East, especially Zen Buddhism. Now I want to turn a somewhat skeptical eye on the foundations of that idea but later move away from the skepticism to try find a clearer and deeper exposition. I begin by considering what seems to be an unbridgeable gap between the Western idea, that in philosophy, science and humanism meaning can only be apprehended in words; and the Eastern idea, in Zen, that the deepest meaning is totally beyond direct expression in language.

Let me first be skeptical about extreme claims for language. I’ve already talked about Plato and Wittgenstein with their thoughts on the limits of what can be said. Some humanists not only ignore possible limits to what language can express, but claim that only with language can there even be thinking. That idea seemed absurd to me the first time I heard it and has so seemed ever since. Perhaps it makes sense if one replaces “thinking” by “intellectualizing”. To me “thinking” is simply conscious mental processing and, at least, for me can occur in an entirely wordless manner. For example, when out hiking one often comes to a stream without a bridge but with rocks that will provide stepping stones if one doesn’t slip and take a fall into the water. When I arrive at such a place, I take in the scene, sketching out possible paths and making a wordless judgement about the slipperiness and stability of the rocks along each possible route. If one path seems feasible and best, I concentrate, get balanced and begin to hop. There has clearly been “thought” here, but none of it has been put into words. Of course, it could have been, and on some occasions, the hiking party might well discuss the matter, analyzing verbally the various possibilities before making a decision about the crossing. Another example, concerns a bear in Yosemite Valley who presumably lacked language, but through experience and awareness learned about canned goods. In one instance, during the night at Camp 4, a less experienced member of our group had left a rucksack full of canned food, out in the open. The next morning, we found the rucksack torn apart and cans scattered about. Some of the cans had been ripped open and the contents eaten. Others were untouched except for a single tooth hole in one end. The bear knew that some cans might have less desirable contents and saved energy by a “puncture and sniff” methodology whose existence to me implied “thought”.

While thought clearly can be nonverbal, it seems to me that Zen seemingly goes further. Let me postulate that for Zen the deepest awareness about life and the emotional reconciliation with our non-existence and loss of awareness in death, is not only wordless, but, unlike the experience of stream crossing, is necessarily completely nonverbal. Further, that attempting to understand this experience through language is not only a distraction, but is counterproductive, a false path, that hinders rather than helps.

Having not had the ultimate Zen experience I am in an excellent position to be skeptical about this postulate. This skepticism can operate on several fronts.

First, though I’m unwilling to doubt the authenticity of the ultimate enlightenment for people who have claimed to have had this experience, I can doubt that it will ever happen to me. The fact of the matter is that other people having the experience is irrelevant to my spiritual understanding. Furthermore, if in the future I claim to have finally achieved satori, that should be irrelevant to you who read this blog.

Second, I do think the Soto Zen insight is true and relevant. One can gradually gain deeper understanding of life and the world. One asks, “What is the alternative?” Just give up? Abandon the struggle to understand? Gradualism has its attractions in that there is at least the experience of “being in the zone” not only athletically, but philosophically and artistically. I definitely HAVE experienced being in the zone so know that it can contribute to almost any life activity. It may not be satori, but may well be a way station on the path and, in any case is well worth experiencing.

Third, if the ultimate experience is totally unreachable through language, why write about it at all? There are countless books about Zen. The standard conclusion is that one must join an Ashram of some sort and devote one’s entire life to practices that will possibly bring about enlightenment. From the beginning I have been skeptical about joining a spiritual community. There are too many frauds about and even sincere gurus have no magic touch for bringing about the desired result. As I’ve said earlier in this blog concerning spiritual matters, “The buck stops here” with you and me. Spiritual support can possibly be of help but quite possibly also contribute to self-delusion.

So why do I write this blog? Simply because I have an irresistible urge to try “get things straight”, to understand as much as possible about everything, to share my ideas, and to become a skillful enough writer to be worth reading. Concerning Zen, I feel that there is a paradox involved. Being as skeptical as possible advances Zen. Smash it. Stomp it. Deny it sincerely.

Such a denial of the basic postulate could be considered a Western approach to Zen. A fundamental trait of Western culture is the idea of “speaking out”, of not holding back. Accompanying this is a certain lack of respect for authority. The Eastern tendency, on the other hand, is to remain quiet and humble in the face of what likely cannot be said or understood. Besides a deep respect for authority, there is the idea that being forward is being egotistical by being “showy” to no end but self-aggrandizement. A Western approach to Zen would be a tradition-denying attempt to actually spell out what “cannot be said”, weaving a magic potion in words. A potion that not only makes perfectly clear but also carries to its reader an emotional acceptance of why one should be content and happy in the thought that the uniqueness that each of us possesses vanishes with our death forever into the emptiness of non-being. To attempt this kind of verbal depth and clarity is not only very Western, but paradoxically very Zen. “Let’s not grasp at the idea that nothing can be said.” At root Zen is neither Eastern nor Western. It is about such a complete letting go, that one mustn’t get hung up even on the idea of letting go.

As I continue in a possibly too-outspoken Western manner, consider that in what I’ve said above is an explicit acceptance of the idea that our awareness does indeed vanish with our death. There is no consciousness after death. Perhaps the mind functions briefly after the heart is stilled, but such functioning is brief and comes to an end. In rejecting the idea of “eternal life” I’m applying the spiritual postulate that there be no acceptance of belief simply because it seems comforting. It certainly would be extremely meaningful and exciting to be reconciled with all one’s family and friends who have passed away. Whether one could be happily conscious for an eternity is another question, but still it seems that any awareness might well be better than none. As a friend of my wife said talking about accidents and sickness, painful medical treatment, and long boring recoveries while incapacitated, “Any kind of living you can live with; it’s the dying you can’t stand.” And I think that is the way most of us instinctively feel. Certainly, although there is no certainty about what happens after death, the weight of the evidence, seems to me, to favor oblivion. Whether or not that is the case, if oblivion is what we really fear, that fear is what we need to grapple with spiritually in order to find understanding and peace.

When I use the word “spiritually”, it brings to mind traditional Western religion; in particular Christianity and the belief in God. What are my thoughts on this matter? Here I’ll deal with them briefly. It seems that there may well be the possibility of a deeper consideration in future posts. So… Am I an atheist? Well, no. Do I believe in God? Well, no. Am I an agnostic? Well, no. Surely either one believes in God or is an Atheist. Well, no. The problem as I’ve said before is Aristotelian logic, the curse of Western Philosophy, and, I might add, Western thought in general. When formalized, logic is tremendously useful in mathematics, theoretical physics, generally in science and in many areas of life. When applied elsewhere, its denial of any possibility beyond true and false, black and white, is untrue to reality. In most areas of life there are “shades of grey” which Aristotelian logic simply can’t deal with. In the distinction between atheism and belief, there is, as well, another problem. The entire distinction, seems to me to be stuck in spiritual shallows. Getting lost in controversy about a dichotomy which may well be meaningless instead of attempting to dive more deeply into spiritual awareness seems to me a waste of time and life. Let us consider belief in “God”. When one uses a word to characterize the deepest experience of spirituality, one inevitably comes to think of God as Something, in particular Something apart from the remainder of existence, having all sorts of contradictory properties. He (certainly not “She” or “It”) is all powerful and all controlling, but tolerates “evil” and the “devil” as a necessary part of existence. And I have mentioned only one muddle. The problem lies in Naming an ultimate which is beyond what we can possibly know. In Judaism and the Old Testament of Christianity, there is a tradition of revulsion in making images of gods or of even speaking God’s name except once a year. The sin involved is called idolatry, a belittling of the ultimate mystery, belief in a false image of God. It seems clear to me, however, that simply in treating the ultimate as a concept and calling it God, one is close to committing idolatry. Whether idolatry is the deep sin claimed by the Old Testament is possibly questionable, but one can well imagine that the ancients had a sound and provocative insight. The idolatry of Naming the ultimate is likely the root cause of religious conflict.

One begins with the Name. From the Name comes the tenets. From the tenets Belief. From Belief comes fanaticism and we all know where fanaticism leads. Of course, this sequence is by no means logically necessary and most thoughtful believers realize that “God” is simply a convenient word for what they apprehend in their deepest religious experience. A word that simply spells out an ultimate mystery whose properties are beyond our understanding. For example, the theologian Paul Tillich is very aware of assigning false attributes to the deity and uses the phase “the ground of being” instead of “God”. Nevertheless, there have been many “believers”, past and present, who HAVE followed the sequence from the concept of God to tenets to a tight grip of belief that can only be labeled as fanaticism. Fanaticism demands the death of all apostates and war against other religions or even other branches of one’s own religion. Every thoughtful person should know about the “Thirty Years War, 1618-1648” to say nothing of the horrors occurring in the name of Christianity before that period and understand the potential for fanaticism which lurks in “belief”.

So where does this leave us? It seems to me that modern, mainstream Western thought, especially in the sciences, but also in philosophy and the humanities, in realizing the trap of belief, has accepted the unspoken idea that any spirituality involves false beliefs about the deity and a lack of critical thinking which leads to an acceptance of SUPERSTITIONS from astrology to witchcraft to evolution by intelligent design; leads in fact to a rejection of the fundamental skepticism which drives science and, above all, to a total abandonment of reason. Any acceptance of spirituality threatens a new dark age.

What I’m pointing out in this blog is not only that there is no necessary link between spirituality and mindless superstition, but that the extreme skepticism of the spirituality which I’m advocating is completely in line with that informing science and modern thinking in general. For lack of a better name and to emphasize its doctrine that ungrasping from all belief leads to depths of meaning and understanding, free from all superstition, I have called it Zen. This label emphasizes and pays respect to the long historical development in the East of the realization that belief is unnecessary for spiritual well-being. Unfortunately, Zen carries the connotation of Eastern thought, of the quietism mentioned earlier in this post. A form of what I’ve called Western Zen would comfortably fit with our Western science, philosophy and humanism. Based on “radical ungrasping” it would take up the idea that our spiritual ignorance can drive a quest for spiritual knowledge and answers, growing out of the deep mysteries that have arisen from our secular science and knowledge. Although we have made remarkable progress in science and in other fields in the past several centuries, our remaining ignorance is not only infinite in extent but concerns the questions most significant to our spiritual well-being.

For the deep questions are not going away. What is the meaning of your life or my life? What is the meaning, if any, of our deaths? What is this universe all about anyway? Can one live in a spiritual vacuum? Is one to suppress the urgency of these questions and lose oneself in the anodynes of work, pleasure, sex, sports and consumerism, resisting of course, the threat of addiction to these as well as to less heathy activities such as drinking, drugs and gambling? Or is one to seek answers in the superstitions mentioned above or in shallow forms of Fundamentalism, stilling any doubts by an ever tighter grasping at unreasonable beliefs? It seems to me that Western thought in ignoring its spiritual vacuum is helping to bring about the very evils it fears.

A final word. What I’m proposing falls short in that it lacks specificity. That fact must be accepted in all humility. Nevertheless, I do think that I’ve made a showing that there is a path towards a Western spirituality which does not violate the integrity of our thought and that such a path is would fill an important gap. Back to Top

In this post I want to take a path that starts with some thoughts about classical Buddhism. These thoughts are far from being based on extensive knowledge or scholarship, but this very lack enables, I hope, a freedom to break free from tradition, and seek a meaningful relevance for our times. Consider the following:

He whose desires have been throttled,
who is independent of root,
whose pasture is emptiness—
signless and free—
his path is as unknowable
as that of birds across the heavens.

I came across this verse in the heading of the first chapter in a sci-fi fantasy book, Lord of Light by Roger Zelasny. The book, incidentally, won both the Hugo and Nebula science fiction awards the year it came out. The hero, a reincarnation of the Buddha, who goes by the name of Sam, is “great souled”, but is also a crafty, scheming fighter for a cause of freedom, which involves defying the Lords of Creation. If there were such a concept as “heresy” in Buddhism, he would perhaps be eligible. But that is somewhat beside the point. I’m concerned here with the verse itself, not the book; though the book is one I love and read repeatedly.

In the book the verse is credited as Dhammapada (93). Consulting Wikipedia one finds that the Dhammapada is part of the Pali Canon, the extensive writings from the early days of Buddhism, forming a tradition called Theravada Buddhism, the Buddhism of South East Asia. A second great Buddhist tradition which developed later is that of Mahayana Buddhism, the Buddhism of Northern India, Tibet, China, Korea and Japan. Zen Buddhism is part of this latter tradition.

It seems to me that the Dhammapada verse summarizes many of the great themes of Buddhism. The first line is echoed in a great poem of Yeats, Sailing to Byzantium whose

3rd stanza reads,

O sages standing in God’s holy fire
As in the gold mosaic of a wall,
Come from the holy fire, perne in a gyre,
And be the singing-masters of my soul.
Consume my heart away; sick with desire
And fastened to a dying animal
It knows not what it is; and gather me
Into the artifice of eternity.

In both of these selections “desire” is a word for all the negative emotions that beset us as human beings: longings, fear, rage, depression, selfishness, egotism; and perhaps too, emotions considered as positive: joy, happiness, complacency, satisfaction. The question arises: Does Buddhist discipline involve trying to “throttle” all of these emotions head on, by leading a disciplined, saintly acetic life, devoid of pleasures? My answer is no, and I have the feeling that an affirmative answer to this question involves a misunderstanding, a putting the cart before the horse so to speak. Certainly discipline is required in following any spiritual path, but discipline, if misdirected is futile and ultimately frustrating. My view is that effective Buddhist discipline lies in an indirect approach to dealing with “desire”; a direction of becoming aware of one’s beliefs and addictions, and of trying to relax ones grasp on them. Ideally one would have no beliefs whatever, and would be totally free in the universe. That is, however, for most of us a distant goal. As unenlightened humans we can’t help having beliefs and addictions. What we can do is to try become aware of them, to relax our grip on our beliefs, holding them lightly, recognize our addictions, and work on bringing them under control.

In our culture a very common attitude is to tighten our grip when one of our beliefs is challenged, to never admit a mistake, and to “double down” if our judgement has proven faulty. If we eye such behavior dispassionately, we see that it is egotistical and basically immoral, a rejection of “truth”; nevertheless, owning up to fault can be very discomforting. If one has ever been a professor, lecturing to a class, one has inevitably been in a position of having a sharp student who is closely following, raise his or her hand and point out a mistake in one’s reasoning. I remember several occasions when this situation happened in a math class taught at Stanford by Professor George Pólya, a distinguished mathematician of the early 20th century (see Wikipedia). Pólya’s lectures were a model of clarity and he always payed close attention to how his students were reacting. When a blunder was pointed out, he would exclaim in his Hungarian accent, “Oh! How stupid of me!”, and then correct his error. I wondered at times if he deliberately made mistakes to keep his students alert, but think it more likely that in concentrating on clarity, he sometimes lost track of a logical connection. Later, when I was a professor, often while teaching elementary physics to future engineers, Pólya’s example stood me in good stead. I would admit to screwing up, congratulate the student who pointed out my blunder, go back over what I had done and correct the error. I did experience some intellectual discomfort in doing this and I’ve noticed that many professors are simply unable to admit their mistakes and try to weasel out of them.

In trying to guard our beliefs when they are challenged, we are obviously hoping to protect our egos and sense of self-worth. However, I think there is more going on than simply ego protection. Our beliefs, especially those which are only partially conscious and which we take for granted, form a foundation for our life, a comfort zone, a cozy nest into which we can relax, the very basis of our being. When these beliefs are questioned, the underlying floor of our security is threatened with break-up. Such beliefs are the psychic equivalent of the safety net which protects a trapeze artist or tight rope walker. Letting go of such beliefs or even relaxing one’s grip on them, is similar to a performer abandoning his or her safety net and moving to the next level where a fall would likely be fatal. The big difference, of course, is that letting go of one’s beliefs is not fatal, but can actually give one a sense of freedom. Such freedom is not a license to act without restraint, but is, instead an openness to see and reason clearly and act with creativity. One does become aware of the difference between social conventions and a deeper fundamental reality. This doesn’t mean that one necessarily defies convention, but simply that one understands that conventions are constructs of the society one lives in, not absolute moral dictates. One role of meditation, besides its calming effects, is to help become aware of our unconscious beliefs and loosen our grip on them. One guide to meditation that I’ve long ago lost track of mentions that as thoughts begin to fade away, a chasm looms in front of us, what Oliver Sacks in Musicophilia calls “an abyss of non-being”. The guide recommends that one mentally hop over it and continue to meditate. Later one hopes to float in this abyss while meditating and lose one’s fear of it in ordinary life by so loosening the grip on one’s beliefs that they no longer act as a support of one’s being above a meaningless void.

If one attains such a free and easy state in one’s life, does that imply a cessation of desire? Or should it? I think that the key to understanding such a question involves the concept of addiction and the relationship between addiction and a tight grip. Surely Buddhism doesn’t forbid joy in life or reveling in pleasure. The problem arises when a pleasure becomes addictive. In dealing with such matters I think that present day psychotherapy has as much to offer as ancient Buddhist ideas, though relaxing one’s grip would seem helpful when undergoing psychotherapy for both the patient and the therapist. See the interesting book Tales from the Couch by Bob Wendorf, a clinical psychologist with 36 years of experience. Trained in behavior modification therapy, Dr. Wendorf discovered that to be successful in practice he needed to relate directly to his patients using whatever theoretical psychological basis seemed appropriate for a particular patient. In other words he was able to see that humans are more complex than any current psychological theory, relaxing his grip on beliefs formed during his training. I’ll have more to say on this subject in a later post. For the moment the point to be made is that while Buddhist ideas may be helpful in this and other areas, they need not, indeed must not, necessarily replace Western ideas. Instead one hopes for a creative amalgam of West and East.

Does a meditative practice help one to loosen one’s grip? Perhaps. What loosening one’s grip really means is allowing a questioning of beliefs that one thinks are correct, a step up from admitting one’s blunders. Such a questioning is the ideal of scientific thought, but seldom actually practiced by scientists when their own beliefs are concerned. Fortunately, in science, one’s colleagues are ready and eager to fill in with doubts and skepticism about one’s latest pet theory. One can claim that this is why science works, to the extent that it actually does. Outside of science, however, we are left to our own awareness and resources.

So where does this leave us? I use the word “us” advisedly, for I assume that you, the reader, have been following along with your own understanding and questions.

Or to put the question another way: What parts of Buddhism in all of its different manifestations can be taken over into our Western culture, helping us to think clearly and ultimately to give us a deep religious understanding which is harmonious with the path our culture has been taking?

Or still another contrary way: Shouldn’t Eastern religion in general and Buddhism in particular be totally rejected as being incompatible with the direction our Western culture should be going?

In considering these questions, in what I’ve written so far in this piece and in the entire blog, it is clear that there is considerable complexity and a danger of getting bogged down in details. There are many directions I could take and enough material for considerable writing.

For now I’ll consider just one area that is amendable to a relaxing of one’s grip. This is the matter of the so-called “two cultures”, the culture of science and of the humanities. There has been a conflict and so much writing over such a long time about these “two cultures” that one would think that there is not much left to say on either side. I became aware that there was supposedly a conflict between humanities and science back around 1960 when I was a graduate student in physics at the University of Virginia while my wife, Barbara, was studying for a degree in English literature. At about the same time the British scientist and novelist C.P. Snow had given his influential 1959 Rede Lecture, “The Two Cultures”, which pointed out this gulf in our scholastic culture. Snow came down hard on the “side” of science. I quote C.P. Snow from the Wikipedia Article, “The Two Cultures.”

“A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is the scientific equivalent of: Have you read a work of Shakespeare’s?

“I now believe that if I had asked an even simpler question — such as, What do you  mean by mass, or acceleration, which is the scientific equivalent of saying, Can you read? — not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their neolithic ancestors would have had.”

On the other hand many of my lit major friends at the University of Virginia referred me to writers in the humanities who pointed out that as far as they were concerned, beginning with the 17th century scientific revolution, much of the rich meaning of our culture, constructs such as “the great chain of being”, had been destroyed by science and a richness had been reduced to joyless gray empty facts without meaning.
I was shown John Donne’s famous lines from An Anatomy of the World (1611):
……..

And new philosophy calls all in doubt,
The element of fire is quite put out,
The sun is lost, and th’earth, and no man’s wit
Can well direct him where to look for it.

………

John Donne lived from 1573 to 1631. He was a contemporary of Shakespeare (1564-1616) and, more to the point, with Kepler (1571 -1630) and Galileo (1564 – 1642). Galileo was born two months before Shakespeare and died the year that Isaac Newton was born. During his life ideas of the new natural philosophy, later called “the scientific revolution” spread throughout Europe.

At the time rather than getting very upset about any of this, I felt that I understood both cultures and luxuriated in being a “bridge” between them without feeling any compulsion to take sides. Surely, this conflict was simply the aptly called “tempest in a teapot” and would go away with time.

In many ways the conflict has subsided. Awareness of quantum mechanics, though not always very well understood, has spread to humanists who have taken it as deep and fascinating. Many scientists are well read and take great joy in the poetic, in the arts and in music. Nevertheless, in some respects the conflict is worse than ever. From the science side, it is no longer simply annoyance with the ignorance of the educated, an ignorance much reduced, but the growing contempt for science and the emergence in magical thinking among people at large. For the key point of the scientific revolution was the rejection of magic as an explanation of what went on in the physical world. Now people seem to be totally ignorant of the facts which have led to their cell phones, tablets and TV, and take superstitions such as astrology as having serious meaning. Are we at the point of descending into a new dark age?

From the humanities side the concern is “scientism”, the belief that valid knowledge and meaning comes about only through the application of scientific methodology. Ray Monk, Wittgenstein’s biographer, writes

“Scientism takes many forms. In the humanities, it takes the form of pretending that philosophy, literature, history, music and art can be studied as if they were sciences, with “researchers” compelled to spell out their “methodologies”—a pretense which has led to huge quantities of bad academic writing, characterized by bogus theorizing, spurious specialization and the development of pseudo-technical vocabularies. Wittgenstein would have looked upon these developments and wept.” https://www.prospectmagazine.co.uk/magazine/ray-monk-wittgenstein

In my view these quotes, on both “sides”, reveal grasping motivated by two things: fear and laziness. The fear is that our treasured world view is under attack. Not only is defense needed, but also an attack on the “other side”. The laziness comes about because it would take work to relax our grip and grapple with the task of feeling and understanding the meanings of a wider, multi-cultural world including all branches of science, the humanities and arts, history, economics and popular culture. It is easier to relax and allow one’s grip to stroke one’s ego.

The rewards of a more relaxed, aware view would include a flourishing of creativity, a combination of ideas that seem antithetical. Consider the thought of humanizing science, mythologizing about its meaning and mystery. (In future posts I will try to understand the proper place of myth in our culture.) The best science writers are already close to this mythologizing. Does such constitute an attack, a belittling, or even a refutation of the “scientific method”? Certainly not. Actually, “scientific method” itself is far from being a set of cut-and-dried formulaic rules that can be applied blindly in any situation. A beautifully clear exposition of this fact is Richard Feynman’s essay “Cargo Cult Science” in Surely You’re Joking, Mr. Feynman! “Cargo Cults” arose among the natives in certain South Pacific islands as WWII drew to a close. These people had seen giant airplanes land on the newly made runways and disgorge an incredible array of “cargo”: weapons, living quarters, food, bulldozers, and other amazing materials. Then, suddenly, it all stopped; the people left, the islands were deserted and the runways disintegrated. The people wondered, “How could the largess be restored?” And cargo cults resulted. Quoting Feynman:

“So they’ve arranged to make things like runways, to put fires along the sides of the runways, to make a wooden hut for a man to sit in, with two wooden pieces on his head like headphones and bars of bamboo sticking out like antennas – he’s the controller – and they wait for the airplanes to land.”

They’ve recreated the form, but with a naïve theory about the relation of form to “reality”. Lest one feel a smug superiority about these natives, I should point out that this sort of mistaken understanding occurs all the time in science; not only in the pseudo-science noted by Ray Monk, nor in the soft sciences, but in the so-called hard sciences as well. Much of the time one’s ideas are just not right. Even when they are right and experimentally verified, “truth” is not established for all time, but only provisionally and subject to a future scientific revolution. This is not to denigrate science, but rather the opposite. Science is truly difficult and theories usually not obvious when first conceived. When scientific theories do finally become well-established, they really work, and they do bring the miracles and the nightmares of “progress”. Clearly, “scientism” is nonsense, but so is a lack of awareness on humanist’s part for the deep humanistic meanings which may arise from science.

The Buddhist idea that is useful here is that of relaxing one’s grip on all kinds of belief, not simply credulous belief, but skeptical belief as well. One takes in the whole panoply, miracle and craziness of modern life with clear-headedness, joy and awareness, reveling in its diversity, taking action against what seems like mistaken ideas, but without “attachment” to any of it. If one cares to go further on a Buddhist path seeking the “great peace” in the “emptiness” beyond words and beyond the panoply, one should do so in a relaxed manner, with a loosened grip and without attachment or expectations. Back to Top

My last post, talking about Wittgenstein, (January 11^th) was followed by a wonderful trip to the “big island” of Hawaii, enjoying the tropical climate with high temperatures above 20 ͦ F, large breaking waves, a mongoose with teeth bared, lava flowing into the sea, and thoughts of the Polynesians journeying across the trackless ocean. Then after a short flight to Oahu our group experienced the touristy ambience of Waikiki, so different from what it was when I was a child.

After returning I have been slow to settle in for one reason or another: visitors here in Bend, excellent cross country skiing, and most recently a trip to Seattle to stay with my son and his wife in Bellevue and attend an American Alpine Club homage to an old friend, Nick Clinch, who recently passed away. With all of these doings there has been little time for idle thought. Perhaps, too, “bone laziness” has been a factor. Not bone laziness as sloth, but a principled bone laziness conducive to emptiness of thought and experience. However, now as some time has opened up, thoughts have occurred that I should at least mention before getting back to philosophy, physics, and the mysteries of existence.

A concern is that readers might think that I am proselytizing; that in this blog, I am pushing the idea that one should become some kind of Zen Buddhist, whatever that is. In fact, a primary principle of the blog is that I am pushing nothing whatever. However, I am putting forward ideas which seem out of the mainstream of our times, but which seem cogent to me.

I am suggesting that, as a personal matter, if one has our present age view of rationality, and can’t pursue a traditional religion based on belief in a mystic vision which one doesn’t have, then one might consider a religious view which is totally opposed to that of the traditional great religions, holding no “beliefs” of any kind, but which can be experienced and understood outside of language and which becomes an underlying foundation for one’s life journey whatever that journey may entail in a concrete way. I am not suggesting that one necessarily take up the exercises and practices of such a religion, but am simply putting out there that such a possibility exists. One can delve into the deepest understandings without the least compromise of one’s integrity in pursuing “truth”.

I am furthermore suggesting, not simply as a personal matter, but as a matter for society as a whole, that there can be an underlying substrate that encompasses and deepens all of our pursuits, in every science, in every art, in every humanistic discipline, in every traditional religion, in politics or the law, in medicine, in every athletic endeavor, in every fight against ignorance or evil, in the making of any object, in building roads, in welding, in every climb of a mountain or wandering in a wilderness, a substrate that subsumes all antagonisms between these pursuits and that leads to a respect for everyone in society leading a sincere life of integrity.

This is the vision I have, the vision that underlies everything posted here.

Does this vision ignore the darker side of history and human nature? I don’t think so. It is a vision of awareness. Awareness of all the possible positive human pursuits with appreciation and understanding rather than hostility; but also a full awareness of the “fury and the mire of human veins” as Yeats puts it. An awareness of how an entire society can go off the tracks, with demagogues arousing insecurities that lead to hatred, to simplistic, evil policies and ultimately terrible wreckage. This awareness can help us avoid our own hatreds and our own counterproductive blind opposition as we intelligently become active, opposing where it will do some good and seeking areas of leverage where we can actually have a positive effect and perhaps help avoid the worst of the wreckage.

So much for this interlude. I feel that I’ve made somewhat more explicit what this blog is always about even when it seems to wander far afield. Back to Top

Interlude I

This will be a short post because I do want something to happen on the blog while I work on a more difficult, longer post. Sometimes working on the next post gets interrupted by doings such as skiing around here with the first snow. Then there are difficulties with making the writing clear, then guests, Thanksgiving away and so forth. Since I’m not a professional writer and have no obligation to write, I sometimes simply put pen to paper or pixels to screen as Susan says and save the results for later thought or for the bit bucket.

Let’s start with a comment on my last post “Into the Morass, Part I” about different kinds of epiphanies. This is from a friend, MER, who follows this blog.

“I have been thinking a lot about what you wrote, mainly, ”epiphany”. I’ve been trying to figure out, have I ever had an epiphany? I wasn’t sure if I had….but then when you wrote about music, I remembered. I had gone to a John Hyatt concert in Rutland, Vt. And he had two guitarists, a base, and a drummer with him. One of the guitarists, he did a solo. It was so good, it was if the music you could see floating out from his guitar and into the air. I was stunned. Never before had I actually seen the music.

Then thinking of art, I went to the Boston Museum of Art to see the exhibit of Monet. It was the first time for me to go to an actual museum of art and I was in my early 40’s. There I stood, looking at the different paintings of Haystacks. It took my breath away. Same haystacks, but different time of day, different season, every one, everything different. So much color, so much paint, layer upon layer, turning each into works of pure enjoyment.

Poetry…..Bill writes poetry….and his words are written down on anything he can find to write on. Cardboard to yellow legal pad. I guess I have an epiphany when I read his poems too. They express so much of his soul, again, his heartfelt words take my breath away and i am in awe. Those so far as I can think through, are my epiphanies.

I am so grateful that, little by little, I can have these experiences. I don’t look for them, I have no bucket lists, (don’t care to think of lists to do before I die) I live for the moment and when something wonderful happens….an epiphany! Awesome!!!”

In talking about epiphanies there are many directions one may go. I’m struck by MER’s comment that she wasn’t sure that she’d ever had an epiphany until she began to think about it. Then she realized that indeed she had had that kind of experience which is not simply enjoyment, but a feeling about being “more alive” and that life has meaning. Although one can’t program epiphanies, just becoming aware that one has had them and can have more of them in the future is, I think, the beginning of a light-hearted but serious religious practice. One looks for joy and meaning and finds oneself becoming aware of the depths of existence and the sacredness of everything, particularly one’s fellow beings. And one becomes grateful.

This feeling of gratitude is a key point. I remember a trip to the mountains, which at one point ended up with two of us climbing up Middle Sister from a camp at Upper Chambers Lake lying to the south of the peak. It was a day of blowing clouds with glimpses of sky and on the summit one could see down through deep rifts in the clouds to depths below and across to South Sister. This was some time ago before the summit register on top of Middle Sister was removed. I started idly reading through some of the entries, many of which were simply on scraps of paper loose in the aluminum register. Then I came across one that affected me deeply. It was written by a women who had climbed Middle Sister for the first time and expressed how grateful she was to have been given the experience of being on the summit. I immediately felt the same way; that I had been given an incredible gift, the kind of gift that makes life deeply meaningful. This time it was MY eyes that filled with tears.

On the way down the two of us glissaded using our ice axes. I worked my way more and more to the east where the slope steepened towards 45 degrees (which seems when looking down almost vertical). I was in some kind of easy relaxed zone. The glissade was a joy, but I warned my less experienced comrade to stay on the less steep ridge which was safer for him.  Being in the “zone” is another kind of epiphany which I will go into later, but for now I want to consider how epiphany or, indeed, “mystical experience” relates to philosophy and language.

Ah, philosophy! The morass deepens and I’m floundering. But I will thrash around and hope to extricate myself and hope to have more to say besides incomprehensible garbage. Back to Top

I am very unhappy about the last post: “A Zen Practice.” I feel that it over-promises and is dishonest. It being in a WordPress blog I could delete it, but I will not do that. Let it be a warning about how easy it is to be deluded when on a spiritual journey. Such a journey is difficult, must be pursued with concentration and with absolute honesty. I am nowhere near where I need to be, to give advice to anyone about their own journey. If you become a reader of this blog, then I need your insights at least as much as you might need mine. Please comment when you think I’ve overdone it or if you have an insight to share.

All this being said I still think that there needs to be a Western path towards a deep understanding. Calling it Zen is probably over-promising. I will certainly continue these posts because I do have some things to say and it has been helpful to me to try write with intensity and honesty. If you become a reader, I hope that you gain some insights and perhaps commit to a lifetime journey of your own.

After rereading this on Sept 21, 2016, I decided that it wasn’t all that bad except for my feeling of revulsion at the statement that “I actually feel an obligation that I must share some insights”. That is a lie, lifted directly from Brad Warner’s book. There is no obligation, just a compulsion to write as clearly and engagingly as I can hoping that some insights do emerge now and then. Back to Top

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. Back to Top