Dr. Steve J. Poulos, Professor of Soil Mechanics at Harvard University under Arthur Cassagrande, and the first person to formally define the steady state condition, graciously agreed to my interviewing him. I interviewed him on March 1, 2014 between 10:00 and 11:00 AM at his house in Lexington, MA, USA.
The sound recording of our interview is available by clicking on the picture below. (Note: it may take a few seconds to load into your browser.)
Note: After the interview I suddenly noticed that on the table next to Steve was James Gleick’s book Chaos. In the early 90’s this book had been on the New York Times best seller list in the non-fiction category, for several months. It was serendipity pure and simple because it was this book that had caused me to realize that soil deformation must be a dynamical system! I asked Steve if I could photograph him holding the book and he kindly agreed.
Steve has minored in mathematics and so I had naturally thought he was already aware of “dynamical systems” and that this was why he’d used the term “steady state” to describe the end condition of shear. Should you listen to the interview you will hear that I ask him if when he was at Harvard, he’d heard the term “steady-state” or knew of the field of “dynamical systems.” I have asked these questions of him three times at least, starting in 1995, and each time his answer has been consistently the same, and so I have to believe him. This is why I think that he is a true “shaman” which in my book, is the highest tribute I can give someone.
Why did I ask Steve this question in 1995? Because in 1995, I had read Al Gore’s classic book “Earth in the Balance.” (IMO, Al Gore ranks intellectually with the Founding Fathers of the US.) In Gore’s book for the first time, I saw the term “steady state” used in connection with something other than soil mechanics–Gore had used the term in connection with global climatic systems. I was amazed! I had thought that the “steady state” was a term that applied narrowly to soils, but here was Al Gore using it to refer to climate!
I looked at the back of the book and found that Gore had referenced James Gleick’s book “Chaos–Making a New Science.” I immediately got Gleick’s book from the library and to my astonishment learned from it that “steady states” were to be found everywhere in nature and that there was an entire sub-field of mathematics called dynamical systems theory that was devoted solely to studying systems that had steady states. I immediately realized that soils must behave as a dynamical system and sought to prove it. I tried for two years but had no success. Little did I know that I had started on a journey which would take about twenty years to complete. (Had I known it would take this long, I would probably have given up on the spot!) Yes, this online course represents a distillation of twenty years of deep introspection on soil behavior, much of which time was spent in fruitless dead ends.
I felt like a paleontologist who had stumbled on the fossilized toe of a creature (s)he knew would revolutionize her/his field, but who lacked the resources and knowledge to extract it from the ground. I started with a simple linear model (what you get when you take out the exponentials from the non-linear model). I gave it my best for two years, but after filling up three notebooks with mathematical explorations (basically junk) had got nowhere.
Note: One very important thing that I learned was that thermodynamically, soil deformation is a dissipative process and that there are proofs to show that all thermodynamic dissipative systems must have steady-states (see for example, Nicolis and Prigogine, 1977). Indeed, were soil deformation not to possess a steady-state, it would violate the fundamental laws of thermodynamics, falsifying them and forcing a reconsideration of the entire field of thermodynamics. This is very unlikely to be the case. This is why I am a bit fed up with naive understandings of the steady-state condition in soil, the most common one being that only soils with “needle or plate like” particles can “align” to a steady-state. These understandings are naive–the steady-state is a statistically constant structure, and even an assemblage of perfectly spherical ball-bearings made of steel and produced to extremely narrow tolerances, can reach steady-state conditions. So please (puh-leeze?!), before you email me that steady-states in soils are only to be found in clays with needle or plate-like structures, first introspect on what a “statistically constant structure” means; next attempt to visualize in your minds eye, a collection of steel ball-bearings, reaching a steady-state.
Back to my search though. It was Christmas of 1999 and I was visiting my sister in Texas. There sitting at her kitchen table, I opened my notebook and tried my quest again. My brother-in-law, a PhD in Electrical Engineering, and a few years older than me asked me what I was doing. I told him what I was trying to do, (trying not to be condescending!) and not expecting he would understand! It turns out that electrical engineers are very familiar with dynamical systems; I soon discovered that my brother-in-law had forgotten far more about dynamical systems than I knew at that time. In the span of just half an hour he was easily able to figure out what my goal was! Note: this is the power of a model that is rooted in basic science–it does not take much to educate someone in a different field, about it. I did not realize this, except in hindsight, thinking about this experience with my brother-in-law.
My brother-in-law told me that he knew a famous applied mathematician and that he would take me to meet him. About a year later, my brother-in-law introduced me to this famous applied mathematician who was actually a Professor of Electrical Engineering no less! He heard me out in great detail–again, the power of a theory based on fundamental science is that it can be easily communicated to those in other fields and so it wasn’t hard for me to describe my thinking (such as it was at that point) to him. After he heard me out, he asked several questions and then said that he thought I had a good case. He warned me though that if the system was non-linear, it would take me 10 years to figure out the model. By now, discouraged, I had stopped working on the linear model for a year, but meeting with the applied mathematician and hearing his encouraging answers, I came back to the problem and finally, I “bit the non-linear bullet.”
Note: the lesson I learned from this is that when you are stuck on a problem, then, try to get another person to help–the new information you get from them invariably takes you out of any rut you may have backed yourself into. Fifteen years later, when figuring out how to apply my model to finite element analysis (FEA), I got stuck again…but this time instead of waiting for years, after going nowhere for a few months, I asked my friends for input. They had none. So I widened my circle and asked the author of a well known (I consider it a classic) book on FEA. He didn’t quite understand what I was up to, but one of his remarks to me made me realize that I should be searching for a square stiffness matrix. With this jolt of information from the outside, I was able to climb out of the rut I had dug myself into, and within a few months was able to solve my problem. In hindsight, this may seem obvious. But then again, everything is obvious in hindsight.
Back to my search…sure enough it turned out that this mathematician was about right–in the end it took me about ten years as he had predicted–three years exploring the linear model (a dead end no doubt, but extremely educative in value nonetheless) and the next seven or eight years with the non-linear model. During these years I slowly accumulated the tools and knowledge I needed to “extract” my find. I felt as though I was in a pitch dark room filled with furniture, trying to find my way to the door at the opposite end from where I was, and stumbling and falling over and over again in the darkness, banging and hurting my shins and head against the sharp, hard edges and pointy corners of the furniture.
Then one day, January 6, 2006, while in the shower, I “found the light switch” in my dark room and turned it on, and lo, the way was clear! In short, suddenly that day, the thinking from the years of introspection suddenly came together and “clicked.” The result was the phenomenological dynamical systems model, described here in Chapter 2 1 .
Once you have a quest you are passionate about, no matter how dumb or ignorant you may be at the start, no matter that others tell you that you are a fool, remember what the poet William Blake said: “if a fool were to follow their folly, (s)he shall become wise!” Stumbling in that dark room for years, “following my folly,” slowly but surely made me wiser in soil mechanics. As Nietzsche famously said: “The doer alone learneth!” All told, it had taken me just over 10 years. And my own experience has confirmed all of what these wise people, through the centuries, have been telling us.
For example, because of this experience I fully agree with Einstein–far more important than knowledge or answers are imagination and questions. In school many of my classmates were people who knew their text books inside-out, but who could not, for the life of them, extend this information to anything outside the text book because they lacked that quality which I count as far more important than intelligence or even knowledge –of course I am talking about “imagination.” A person may have the biggest brain in the history of the human race, and may know all the textbooks cover to cover, but if (s)he lacks imagination then (s)he lacks the ability to ask new questions, will not be open to new possibilities, and the likelihood (s)he will make breakthrough discoveries is…yes, you guessed it…ZERO! Conversely, if one has imagination, is open to new possibilities, then regardless of “brain size,” with enough effort, breakthroughs are indeed possible.
In philosophy of science, there is a key finding by that tragic philosopher Edward Constant, known as “presumptive anomaly.” Briefly put, Constant suggested that new discoveries are made because the discover finds something interesting used in some field, that should work in his or her own field but finds that anomalously it is not used. Her/his presumption that it will work in her/his field is strong enough that (s)he then puts in the effort and makes it happen! Hence, Frank Whittle is thought to have got the idea of the airplane jet engine based on his learning about the steamship turbine of the 1900s. Likewise, supposedly, Frank Williams (yes, another Frank!) the famed F1 car designer, when boarding a plane suddenly realized that if he inverted the wing shape of a plane and used it on a car, he could get tremendous down-force. He did this, and his cars were unbeatable until such wing shapes were banned! Long story short–if you want to make breakthrough discoveries, look for “presumptive anomalies.”
So too for soils–it would have been anomalous if, possessing a steady-state, soil behavior was not a dynamical system. For me, the easiest way to find such anomalies is to read widely–no, not newspapers, magazines, text books, nor even journals (and no, sorry, I do not count watching documentaries as “reading”). I mean reading books, then too, non-fiction books, of all kinds (I also get the same benefit from listening to them on tape or CD or MP3 player when commuting ). Speaking for myself (you could well differ in what works for you, after all Williams found his anomaly when boarding a plane), I find the long and immersive experience of reading a non-fiction book as necessary to find lasting, fruitful presumptive anomalies. And even if a book leads to no presumptive anomaly, I find it gives often gives me new information, fertile ground for new associations and new ways of looking at old problems! I owe a huge debt to my mother and father, for encouraging in me a deep and abiding love for reading such books, and I read them continuously because I enjoy doing so. I try to average at least one non-fiction book a week and if I do not read at least 30 such books in a year I feel I am slacking.
In fact, I think that reading non-fiction books is a “secret” and powerful weapon, a secret that lies “hidden” in plain sight, available for all to use, but which in actuality, very few do. If I had one piece of advice it would be this: read books–ideally, non-fiction books outside of your field. Should you read only the journals or books belonging to your field then rest assured that you are highly likely to be one of T. S. Kuhn’s “foot-soldiers of science,” i.e., someone destined to only incrementally extend existing discoveries and not someone who will create new, deep understandings triggered by resolving presumptive anomalies discovered in the course of reading books outside of one’s field. By reading Gore’s book, I for one discovered one such presumptive anomaly, one that resulted in the creation of DSSM. This method of discovering and resolving presumptive anomalies in one’s field and thereby making deep discoveries is simple, but requires sincere effort.
Now say, that based on your extensive reading on other fields, you’ve identified a presumptive anomaly in your own field, worked on it for several years, and finally, after much effort, have successfully resolved it. You may then think, as I did, that you are done–that researchers in your field are going to welcome your findings and that all you need do is to write up your results and send it off for publication . Far from it! In fact, your work is just beginning, as I learned the hard way! Typically, resolving a “presumptive anomaly” results in significant disruption of existing practice. And the more disruptive your finding, the more it is likely to meet obstruction by the experts and other vested interests in the current paradigm. An essential (if difficult) book to read on this topic is T. S. Kuhn’s “The Structure of Scientific Revolutions.” Almost everyone who in the process of resolving a presumptive anomaly creates thereby a new paradigm runs immediately into a head on confrontation with the existing establishment in their field.
Edward Constant in his book The Origins of the Turbojet Revolution (a fascinating book, a must read!) reveals how, for example, Frank Whittle was asked to take his idea to leading aeronautical engine experts in the world, (experts in internal combustion engines, i.e., the then current paradigm), they scoffed at Frank’s idea of the turbojet, with one such pundit proclaiming dismissively to Frank that his engine would not have the power to “…pull the skin off a rice pudding.” The stresses resulting from these obstructionists led to Frank Whittle’s 1940 “nervous breakdown,” the first of three that he would have in the years to come! F1 racing car designer Frank Williams did not have to face this issue as he controlled his own company, had his own source of funds, and consequently could do as he pleased.
As Edward Constant points out in his book, when you have a NEW paradigm, the LAST person to take it to for validation is an expert in the CURRENT paradigm. The reason the is quite simple–practitioners seek to protect their investment (normally substantial) in the current paradigm and so they have a bias against new paradigms. Such a bias may be conscious or unconscious, and regardless of all the solemn talk about “the need for out-of-the-box thinking.”
I was lucky. In 2008 I was fortunate to present my first paper to Joe Labuz, then editor of ASCE’s Journal of Geotechnical and Geoenvironmental Engineering. Joe had the technical knowledge and self-confidence to override a negative recommendation from one of his reviewers, realizing that this reviewer knew nothing about dynamical systems theory. Joe also took me up on a suggestion I made to him to have the paper reviewed by a professor of applied mathematics. I made this suggestion because I knew that most soil mechanics people would know nothing about dynamical systems theory. The review by the mathematician that Joe found would turn out to be worth ten times its weight in gold.
This anonymous mathematician validated my model, but said that while two hundred years ago such a model might suffice, today the bar was higher–that what I had was really a phenomenological model (I didn’t understand what “phenomenological” meant when I first saw this term in the review), and that I really needed to find the physical basis of my phenomenological model. For this review, I owe Joe Labuz a big thank you because prior to Joe’s feedback I had been quite unaware of neither what a phenomenological model was nor the need to drive a phenomenological model down to its underlying physical basis, obvious though it now all seems in hindsight (again, hindsight makes all things seem “obvious”). Sure enough, even though initially I was convinced it would be impossible for me to find the physical basis of my model, two years later, because I had been primed for it by Joe’s feedback, the physical basis presented itself to me.
Like Joe Labuz, Peter Clayton at Geotechnique, Chandrakant Desai and Musharraf Zaman at the International Journal of Geomechanics were three other technically very strong, self-confident editors, able to override less than skillful assessments from one or two of their reviewers. Some of these reviewers, apart from an expected lack of knowledge of dynamical system theory occasionally betrayed what struck me as a surprising and completely unexpected lack of understanding of the basis of the scientific method (i.e., that in science, the only thing that counts is whether predictions from theory match experimental data and that one’s own subjective opinions, i.e., biases, play little to no role). A few reviewers also seemed surprisingly ignorant of basic calculus and probability theory (for example one reviewer did not even know the chain rule!).
Strangely, this seemed particularly true of some of the assessors at Geotechnique and the ASCE’s Journal of Geotechnical and Geoenvironmental Engineering. And unfortunately for me, due to a change, these journals were now piloted by editors (Alexander Puzrin and Patrick Fox respectively) who seem to lack the technical confidence needed to make their own independent decisions regardless of what their assessors said. Bureaucrat gate-keepers, both failed the acid test of a new idea.
In general, the reaction from the older academic community, with the exception of a few “stalwarts,” has been one of “silence.” The history of science shows that this reaction, unfortunately, is to be expected. As Max Planck said wryly: “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”
Sad but true! Nonetheless, I find Planck to be unduly pessimistic–he did not have the tools we have today to track changes in opinion. In my case, the usage logs for this site, and the email questions I receive reveal that the interest in DSSM is very strong, and as old Max correctly said, is primarily to be found in the next generation. For example, graduate students send me numerous emails, representative ones of which I have posted directly on this site, emails that show that they are keenly aware of what is going on! Kudos to them!
So though Max Planck was correct in that the next generation, having little vested in the existing/old paradigm is open to a new paradigm(in fact, they often feel sealed out of the existing paradigm and so actively seek out new ones), the process of change is continuous and does not wait for the older generation to completely disappear.
The web logs also indicate something unexpected–that yes, the Cold War still rages on, but silently, at the “grass-roots” level, and with a new wrinkle. Thus, on average, every month, after accounting for bots, spiders, and other non-human agents, the web logs for this site indicate that about 1,500 unique users visit the site. Between them, every month, they average a total of about 6,500 visits, in the course of which they download on average a total of 24,000 pages. More precisely, for all of 2014, a total of 18,352 visitors made 79,631 visits and downloaded 284,542 pages. Note: This indeed is the revolutionary power of the internet–that in one year, I, who have never once taught a single classroom based soil mechanics class (or any class for that matter), can suddenly reach out and teach far more students in one year than most people do in their lifetime in a classroom setting. Life is indeed strange.
But it is the distribution of visitors across countries startles me and leads me to the conclusion that a new Cold War, not that much different than the old one, is waging silently, at the grass-roots level, revealing its outlines in cyberspace, and defining what the 21st century is going to look like. Thus, the country with the most visitors and pages downloaded is the US–on average, the number of visitors from the US and the pages they download equals the sum of all the other visitors from the other countries and the pages that they download…but for one country–China! China is second in terms of the number of visitors and pages downloaded. In fact the enthusiasm from China is HUGE–one user even translated my table of contents to modern Chinese on his own and kindly sent it to me to use. And use it I did! Third in terms of usage are, cumulatively, the countries of the former Soviet Union–Ukraine first, followed by the Russian Federation.
Then come countries from the EU: Netherlands, Germany, France, and so on…strangely the UK hardly figures–I now think that sadly, the UK is no longer a “power house” in terms of new ideas in soil mechanics. Lost in dreams of the glories of yesteryear, it betrays no curiosity about new developments in other countries. Brazil and India also show very low usage. I understand why it is so for Brazil, after all, English is not their main language. However college education in India is exclusively in English and in modern India, English is often a first language (yes, first language) for many, myself included. I am beginning to think that India is perhaps at root, hostile to innovation and new ideas, a result maybe, of a residual caste system based mentality in which innovation is ever a threat.
So basically, the logs appear to tell me that the 21st century is going to be a struggle for dominance in the world of ideas between the US, China, and the countries of the former Soviet Union! Being brought up under old school rules of “cricket” all I have to say is: “may the best side win.” And hopefully, they will treat the rest of us with kindness.
Back to Steve though. It is a real tragedy he has not been recognized. But he is not alone –I believe Golamreza Mesri at the University of Illinois at Urbana-Champaign deserves formal recognition for his discovery that the ratio is nearly constant, and that the EOP curve under static loading is unique (see Chapter 7 of this course to see how both these conclusions can be reached from first principles using DSSM theory). Many people, long since forgotten, have been awarded Terzaghi and Rankine lectureships for reasons that, unlike the case of the Nobel prize, are murky and hard to figure out, captured by a generic “…contributions to the field.”
I was taught by no less than four Terzaghi lecturers, including one who was also a Rankine lecturer. These were intense semester-long interactions in small size classes in Graduate school, but today, almost 30 years later, despite being familiar with the important research in soil mechanics, I am yet to figure out what exactly is the specific innovation that each of these so called “Terzaghi lecturers” contributed. They were very charismatic people though with strong personalities and the ability to “meet and greet,” and with a huge network of devoted former students.
I recently had to interact with a Rankine lecturer, a professor at an esteemed college no less, who amazingly did not know that i) a key requirement for a theory to be scientific was for it to have a falsifiable hypothesis, ii) to teach a theory that was not scientific (lacks a falsifiable hypothesis) as though it were scientific, makes the teaching meet the definition of pseudoscience and c) a common meaning of the word “scholastic” is to “adhere to tradition and logic (Aristotelian) and to pay little heed to the empirical evidence.” Naively he seemed to think it was confined to mean “…someone with no practical work experience.”
This is why I now believe that the Terzaghi and Rankine lectureship awards are but the result of the smooth and seamless operation of incestuous “old boy/girl networks.” These networks are not “conspiracies,” but rather, a natural outgrowth of the system as it currently is. This is why I have now come to believe historians when they say, that we must wait till 50 years after the events have occurred and the dust has settled, before we can decide questions of true merit. Surely, then, Steve Poulos and Golamreza Mesri will be given the kudos they so richly deserve.
Because of this “old boy/girl network” based identification of awardees, the Terzaghi or Rankine lectureship is given for the reasons that are opaquely worded as “…contributions to the field”–the real contribution is unclear in most cases. It is precisely to guard against this kind of loss of credibility that the Nobel Prize specifically targets a specific achievement–for ex. the discovery of the structure of DNA–and consequently, the Nobel Committee can confidently award a 34 year old youngster the Nobel Prize for a discovery he made when he was 25. I am talking of course about young James Watson who at 25 made the key discovery that would win him the Nobel prize ten years later.
So if one were to take Golamreza Mesri for example, he would have earned his award about ten years after his discoveries in the mid-eighties. His award should have read…”for discovering that the ratio is approximately constant and that the EOP curve obtained from static, incremental, one-dimensional loading is unique.” Likewise Steve Poulos formally presented the steady state in courses at Harvard University starting in 1971 and ideally he should have got his award in the early 80’s. For Steve the reason for his award would read something like…”for formally describing the steady-state condition in soils and its impact on soil behavior.” Not in either case, the opaque “…for contributions to the field.” Even statements like “…for contributions to the development of the piezometer” (for example) do not quite cut it–exactly what contributions are we talking about?
Both Golamreza’s and Steve’s theories meet that acid test of a scientific theory–they are falsifiable hypotheses that yet remain unfalsified. As detailed in the Conclusion, with its many findings, DSSM is also strongly falsifiable. By contrast, metal theories of plasticity as applied to soils can be easily, routinely falsified–simply apply the model to any real soil with a variety of grain-sizes, and Ko consolidated. Yet proponents of metals based theories do not seek to falsify but rather to confirm their theory by modeling only “simple clays” with “simple” stress-strain curves. This kind of conformance testing is not the way to attempt to falsify a hypothesis! In short, metals theory of plasticity as applied to soils is a broken theory, readily falsified with stress-strain data from real soils. Yet, no one seems to call to account those that push these theories!
Experience has taught me that the best way to judge someone’s geotechnical merit is to ignore the number of publications they have, ignore their degrees, ignore their titles, their prizes…ignore everything about them, including how they smell, look, feel, talk, or (horror of horrors) whether they kick their dog when alone at home. They may even look green, drive what looks like a flying saucer, and appear to be from Mars. Ignore all this! Instead, simply ask them to tell you in one sentence restricted to twenty-five words or less (ideally ten words or less), what their most important contribution to soil mechanics has been. Judge their soil mechanics merit accordingly. You will find that many of these so called Terzaghi and Rankine lecturers will simply disappear, while people like Mesri and Poulos will come to the fore. Note: The fewer the words, the more powerful the finding–Watson’s discovery (“discovered the structure of DNA”) takes just five words; Poulos’s (“formalized the steady-state condition”) takes just four!
NOTE: I am happy to report that six months after I posted this, in September 2014, the ASCE suddenly presented Mesri with his long overdue Terzaghi Award. Nonetheless, note how the citation remains as always, opaque and does not mention his key contribution, one that will last for the forseeable future–finding that the ratio is approximately constant and that the EOP curve obtained from static, incremental, one-dimensional loading is unique. Why might this be the case? Because to specify his key contribution would be to highlight the fact that almost every other Terazaghi Award winner, has made NO such fundamental contribution, the four such awardees who taught me for example. Anyway, perhaps the less said the better. Now, its Steve Poulos’s turn.
Looking back on my journey in soil mechanics, it seems nothing but the same old story–The Three Metamorphoses of the Spirit–that Nietzsche powerfully described in his classic Thus Spake Zarathustra. Thus at Purdue and MIT, I was a camel, loading myself with knowledge. After MIT I wandered in the desert until at GEI, thanks to the nurturing kindness of Gonzalo Castro and Steve Poulos, I turned into a lion. For the next ten years I fought with the great dragon: “‘Thou-shalt,’ is the great dragon called. But the spirit of the lion saith, ‘I will’.” And then, in the course of this struggle, I became like a child in soil mechanics: “Innocence is the child, and forgetfulness, a new beginning, a game, a self-rolling wheel, a first movement, a holy Yea.” I wonder how many of you will understand how exactly Nietzsche described this, my long (three decades) journey in soil mechanics–I myself realized it only after my journey was long done.
Anyay, that’s it folks! As always, I love hearing from readers, so do email me (email@example.com) your thoughts/opinions; rest assured I shall reply so long as I am physically able to.
- For those with some background in neuroscience, it may be of interest to know that when I formulated the equations for the phenomenological model, my thinking till then had been predominantly in terms of visual images–grains tumbling and rearranging under the action of small force vectors that I pictured in my mind. In late 2006, I sent the equations for the phenomenological model to my undergraduate professor, Prof. S. V. Ramaswamy, who had introduced me to soils in 1980. He then asked me what the equations signified. It was only when he asked that I “translated” the equations into words for the first time, even to myself (“the rate of change of p’ and q is proportional….etc.”). In other words, the verbal description in ordinary English statements and words came only about six months after I had first described the mechanism mathematically. Now both mathematics and the higher stages of visual imagery involve the PFC and my solving the problem without using language, must imply it was largely a right brain hemisphere and PFC activity. I hypothesize now that the left hemisphere, with significant parts of it used up by the Wernicke and Broca areas that deal with language, may actually be handicapped in terms of introspective problem solving, and that contrary to common thinking, the right hemisphere may be actually more powerful in solving deep problems using imagery together with the mathematical functionality of the PFC. In short, the basis of so called “creativity” or “intuition” (commonly associated with the right hemisphere) may but be but genuine, logical understandings that are only not expressible in conventional language. Looking back, the three hardest problems that I have solved in my life have all been solved outside of language. Perhaps in this case the three notebooks I filled up with “junk mathematical explorations” were not a waste at all but rather drove the mathematics past the intermediary barrier of language, right down to the actual mechanisms being modeled? Perhaps this deep immersion even though in what turned out to be dead ends made my brain more powerful in its approach to the basic problem as now language became “superficial” allowing my brain to discard “mere words” and to tackle the problem directly? ↩