A Guide to Previous Posts: January 2020

Here is a guide to posts so far: I am writing this for myself and anyone else that does not like the WordPress navigation system for older posts. This makes is easier to quickly browse through past articles. The first is the most recent …

  1. Yunfeng Hu in Seattle My former student is now in Seattle and I wrote a short note about it.
  2. A Sort of Synesthesia Thoughts on the difficulty in writing about deeper things.
  3. Silence and Beauty Inspired by Makoto Fujimura’s paintings seen at the Jundt Museum (at Gonzaga University) and videos the experience led me to.

  4. Ode to My Father Poetry.
  5. Soul Vibrations Poetry.
  6. Goodbye Twitter Thoughts about deleting my twitter account.
  7. Finding Depth, Seeing Clearly A meditation on Judgment an nuance in the age or Trump, with a thread connecting to Shoshana Zuboff’s new book, The age of Surveillance Capitalism.

  8. Finding and Following Your Own Path The title is pretty descriptive — what I wrote is pretty autobiographical, inspired by my finding my own path and the power of words.
  9. Fun with simple analysis problems I: the rest of the story A continuation of the earlier post with the same title, containing an exploration of an elementary problem in analysis.
  10. The Colors of Memory and Wisdom Reflection on Zeyn Joukhadar’s novel, The Map of Salt and Stars.
  11. Cultures of Creativity and Innovation Thoughts on Daniel Coyle’s book, The Culture Code.
  12. Letting Go Poetry.
  13. Median Shapes A short invitation to explore the paper, Median Shapes, that I wrote with collaborators.
  14. A Silence, Rich with Inspiration  Something inspired by Glynne Robinson Betts’ 1981 Writers in Residence
  15. Everything is Illuminated Poetry.
  16. Dual Tyrannies of Data and Democracy (and what to do about it) Being data driven is almost always assumed to be equivalent to correct or right. But the assumptions or axioms that one must have in place to use data are very often unexamined, without nuance, shallow or in some other way deficient. And since when did the majority have an inside track on the truth?
  17. Other Planets Poetry
  18. Animals and Empathy My reflections on why I do not support experimentation on animals, focusing on what happens to us when we allow ourselves to participate in these acts of cruelty.
  19. Freedom and Writing terse notes on writing
  20. The Space Between Poetry
  21. Faith Is Connection Reflections on the nature of faith.
  22. Obi Requiem for our late dog Obi
  23. Disrupting Digital Delusions Reflections on David Sax’s book, Revenge of Analog.
  24. Metrics and Inequality Thoughts on how we mislead ourselves in the practice of being obsessed with metrics and idea that this makes us fair.
  25. Fun with simple analysis problems I An exploration of where a simple analysis problem can take you if you sit with it and listen to it speak to you.
  26. Finding Quietness Also inspired by Glynne Robinson Betts’ 1981 Writers in Residence.
  27. Connection vs Attention A meditation on quietness and inspiration that is unleashed by attention.
  28. Doing Mathematics What is means to do mathematics, why I do mathematics, how to do this in a way generous to others.
  29. Learning to think and to act Thoughts inspired by William Deresiewicz’ book  Excellent Sheep: The Miseducation of the American Elite and the Way to a Meaningful Life.
  30. Heresy and Freedom Thoughts inspired by the Epilogue of Albert Schweitzer’s autobiography, Out of  My Life and Thoughts, Schweitzer’s autobiography and by Roger Williams’ life.
  31. Using Photography On my beginning to take photogtaphs and my perspective about photography
  32. Beginning Again Poetry
  33. An Invitation to Geometric Measure Theory: Part 1 The beginning of a book on geometric analysis — this piece is about differentiation.
  34. Thoughts on receiving a negative review Inspired by a rude review of a paper.
  35. Geometric Measure Theory by the Book A review of 9 books on geometric measure theory, an area I work in.
  36. Higher Education: the real problem is not the cost Short thoughts on four mistaken assumptions about higher education.
  37. Connection Poetry.
  38. The Power of solitude … and Social Connection Meditation on the power of a life tha combines time to think and deep connection with others.
  39. Rage Poetry.
  40. Stillness Thoughts on stillness and the power of walkabouts
  41. Anarchy as Optimal Versatility In this perhaps too provocatively titled article, I talk about the advantages of not tying yourself to an authoritarian system and what the import of the phrase Ye are the salt of the earth means to me.
  42. By the Light of the Moon in Broad Daylight Review of the movie Moonrise Kingdom.
  43. Cultures of Disrespect Reaction to commonly used phrases in mathematics that are not helpful.
  44. Scream Poetry.
  45. Brilliance and Renaissance My reactions to the the documentary The Philosopher Kings.
  46. First Post self-explanatory …

 

Yunfeng Hu in Seattle

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View From Yunfeng’s New Office in Seattle

Yunfeng Hu was recently moved from EMSI in Moscow, Idaho to a new research team at Amazon in Seattle. He is understandably very excited. Not only is the pay good, but the work is challenging and interesting and he has an excellent, inspiring team leader, Dennis Craig.

As readers of this blog know, Yunfeng was a PhD Student of mine that graduated in the spring of 2018. His work with myself and Bala (and a couple of other collaborators) is talked about in this post.

Yunfeng deserves this — he worked hard as an undergraduate, becoming an expert problem solver and as a graduate student, becoming an accomplished analyst and programmer as well. His internship and following year and a half at EMSI has prepared him very well for this new challenge.

Congratulations Yunfeng — you deserve this!

 

A Sort of Synesthesia

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Experiences of seeing, hearing, feeling, and knowing, often compel me to try to communicate those experiences in some way.  A sort of obsession with the beauty, with the experience of being in the moment, undulating, flowing, singing, vibarting, simultaneously opens all the senses and quiets the mind in a way that, at least for me, makes translation into words extremely hard. Sometimes I find the words to transmit something of value, but very often I find what I have written is unconvincing or even completely mute.

I need a sort of synesthesia, making translations from what I sense to words more natural or perhaps even involuntary.

Talking or writing about the experience directly, as though it were a story or a play is something I cannot do.  The full experience is so rich, so overflowing, so infinite in possibilities that direct representation is clearly unattainable. But, like visual subtleties  easier to see with your peripheral vision, something of the experience can be captured indirectly, by way of analogies, of shadows and impressionistic portraits, in reflection, after the experience.

This is why some of the most effective, powerful art is abstract or impressionistic. To transmit infinity, direct representational art, creating an expectation of finiteness, must be abandoned. Minimalism in music, moving us into rhythms and flows that slowly shift us to different states, is again, a sort of indirect encompassing, carrying us somewhere, but not directly. Experiences in nature align with this method of illumination, gently soaking in, moving us, so that we gradually become aware of the fact that we have been transformed, our attention has been shifted, profoundly altering what we see and hear and know.

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Quietness — rich, vibrating, living, infinite — finds its way from our experiences to the experience of others as they immerse themselves in our art.

We have, together, attained a sort synesthesia.

 

 

 

 

Silence and Beauty

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Silence and Beauty – Makoto Fujimura (Jundt Museum)

Immersing myself in the light and color and feeling of Mako Fujimura’s paintings, I listened over and over to Bach’s “Erbarme dich, mein Gott”, as though somehow this experience could open my eyes to the words communicating what I was feeling and seeing.

 

immersing, drawn into deep stillness, the quietness sings.
time stops to listen, to know color and feeling

light shines through the brokenness

 

Words feel clumsy, infinitely poor in comparison to the visual experience. But words can tell my own story of brokenness opening me to light, to color and feeling, to quietness that sings.

It becomes clear. The deep drive to express, to illuminate the experience, can only be satisfied by taking others by the hand and leading them to their own experience of listening, of seeing, of feeling. I can invite others to “come and see”, to know why their brokenness is the beginning and not the end.

For there was One broken for them and that One is ready to shine His light through their brokenness, to pour Himself into their darkness and trauma, to heal them with his Quietness and Beauty.

 

come and see 
   the quietness and beauty

in brokenness

 

 

 

 

Goodbye Twitter

I really did not have a lot going on in my twitter account. With 27 tweets over a few months, 20 followers and a collection 87 I followed, I was certainly not making any waves. But I spent a fair bit of time collecting those 87 threads to follow and found immersion in twitter threads to be oppressive and distracting, though this sense was more of an aftertaste than an in-the-moment realization. I also found that Twitter did not encourage habits of thought, attention and focus.

I had read Hamlet’s Blackberry by William Powers and Deep Work by Cal Newport, and had rifled through things written by Jaron Lanier (Jaron and I were both hanging around NMSU at about the same time back in the 1970’s, he in computer science, I in the music world, though I do not remember meeting him, if I ever did). Even before this, I had read The Shallows by Nicholas Carr and listened to (and had many students listen to) the Google Tech Talk, No Time To Think, given by David Levy at Google in 2008.

These books and Levy’s talk had in fact inspired habits of taking breaks from the internet and email, something that came naturally for me because I grew up keeping a pretty strict Sabbath one day a week.

Given the experiences with Twitter and the fact that I was now reading Shoshana Zuboff’s book, The Age of Surveillance Capitalism, I began having an internal debate as to whether or not I should just get off Twitter. What argued for getting off twitter was the state of mind I seemed to always edge towards (or even run towards) when focusing on twitter — a restless, distracted state that was very far from quiet. What argued against exiting Twitter was the fact that it seemed that every once in awhile, interesting people  would announce something using Twitter. You could discover cool things by browsing Twitter.

Action came as a result of a combination of the internal debate, slowly moving to a Quit Twitter stance, and the part of Shoshana’s book about Pentland’s Lab at MIT. (I was acquainted with Pentland and his lab — in fact, some of my early scientific work was connected to his, in the ares of face recognition.) The description of the Lab and his funders and his position in the minds of many that access his expertise, somehow, pushed me across the decision boundary.

And so, a few days ago, I deactivated my Twitter account. I believe it will be deleted in 30 days.


 

I would like to have something like Twitter, only slower, deeper, and much easier to tune or customize. But it also seems to me that if I succeeded in getting what I wanted from Twitter, I would be operating in an asymmetric fashion, one that expected others to behave in a way that I would not agree to act.

For the time being I have decided to focus on internet enabled tools and activities naturally co-existing with quietness, with taking time to think, with slowness-of-response enabling time to think. And of course whatever gets my attention and repeated use must be surveillance-capitalism free.

For now, this set of places and activities will be this blog, my arts blog (http://viksekrarts.com), my website (http://geometricanalysis.org), email (several accounts) and things like github and Google Scholar and LinkedIn (for contacts — I never read the LinkedIn posts).


 

A requiem for quietness can be seen and heard and felt beneath the noise of the mobile device generation. Yet, like the Requiem of requiems, it is also a door to renewal.

Slowing down to be embraced by that requiem, our own responding stillness opens new paths to explore. Dwelling there, listening to the music of quietness and stillness, alternatives to a slide into a shallow, subhuman future emerge.

 

 

 

Finding Depth, Seeing Clearly

The frequent presence of a serious thread of self-righteousness in the opinions and speeches and exhortations to save the earth, to stop hating, to become accepting, to love everyone, to be inclusive, to stop being a racist, to take responsibility, to work hard — you know, to be righteous — corrupts the conversations we need to have, blinding us to where we actually are and what we need to do.

Little that I hear that strikes me as coming from a simple, humble spirit of deeply honest, good will. Perhaps this is because most who have that authentic goodness — that simple approach to making the world a better place — do little talking and no preaching.

Instead, in the opinions and speeches and exhortations (and tweets and posts) there is often a clear dose of hypocrisy or arrogance or self-righteousness (or all three). Sometimes they are muted. Sometimes it is difficult to see anything but these three. The intensity with which this is impressed on me can be overwhelming, perhaps because I am myself particularly vulnerable to the temptation to self-righteousness.

Take, for example, the ongoing circus in Washington DC.

 

I am deeply opposed to almost all that Trump is doing. From a distance he seems to be a deeply narcissistic bully, to be into precisely one thing: himself. Yet the rhetoric of Trump’s many enemies is almost always very distasteful because of the self-righteousness, the hypocrisy, and sometimes even hatefulness.

This phenomena is not new with Trump – politics has always had these elements.

But in the era of Trump, it is out of control. While Trump has descended to a level that was unimaginable by most before he became president, his opponents have unwittingly become the flip side of the same debased coin they so despise, though this can only be seen in the nuance and the subtle details of the fantasy that is unfolding in Washington DC. This ugly show began as soon as Trump became a threat to the elites who were used to running things. Starting with the mistakes they made in thinking it was impossible for him to win, they quickly settled into the role of full out attack, with little attention to depth and nuance and detail and fairness. (It is not without merit to wonder what would have been if Trump’s enemies had not opposed him with such contempt and hatred, for does this not feed his worst instincts?)

This dearth of nuance, patience and depth is pervasive, extending far outside of the DC fantasy. The shallowness of TED talks, thought leaders and talking-head experts lulling elites into a self-satisfied state of cozy superiority is rarely interrupted by wisdom, deep thought or fearless attention to detail.

Stories in the news, podcast/televised interviews and broadcast discussions all suffer, sometimes to a great degree, from a lack of details and nuance that would put events in a very different light. This is intentional at some level — if there were a real will to produce deep reporting, the money aimed at truly thorough reporting would not be so microscopic. In the battle between thick data and big data, between clever TED talks and deep wisdom, between “faster, better, cheaper” and and taking the time to think, it seems that the big data, the TED talks, and the “faster, better, cheaper” is winning.

One hot topic suffering deeply from the lack of nuanced, detailed, deep examinations is the nature and uses of big data, machine learning and AI. I know that some would claim that O’Neil’s  Weapons of Math Destruction, or Broussard’s Artificial Unintelligence: How Computers Misunderstand the World, or Lanier’s Ten Arguments for Deleting Your Social Media Accounts Right Now, and others like them fill that gap. While these admittedly are a small start, these books neither plumb the depths nor exhaust the subject. In fact, they are rather unsatisfying as an answer to the threat that these hitherto undisciplined forces represent.

More generally, non-fiction literature is now dominated by book-length expansions of TED talk or Atlantic articles. There is an idea or two, inflated into a book, often with haste, with little attention to nuance. The time to think and the discipline of patient observation leading to a mastery of the art of waiting, to finding a “feeling for the organism”, is not often evident.

Every once in awhile though, you run across someone who has taken the time to really think, and feel, and see, and then write from a place of true depth and intense focus. While I have not finished the book, I am convinced the new book by Shoshana Zuboff, The Age of Surveillance Capitalism: the fight for a human future at the new frontier of power, is one such book. There is a depth and a passion flowing from the immersive focus that created the book. This intensity in combination with the focus on the threat to the very soul of civilization posed by surveillance capitalism is unique.

 

I first heard of this book when I listened to Shoshana being interviewed on the NPR show, On Point. I was very impressed. But a much deeper impression was made when I started reading the book.

I quickly came to the realization that I was dealing with the results of deep, focused thoughts I was accustomed to seeing in mathematics (for example, Federer’s Geometric Measure Theory) or in philosophy (for example, Josef Pieper’s Leisure – the basis of culture).   An example from Shoshana’s book: though I had thought about the parallels between the ruin brought on the natural world by industrialization and the ruin the information age promises to bring on the human mind and spirit, Shoshana’s book is the first time I have seen someone else put these thoughts on paper. And it was not only the fact that these observations were there, it was the way in which they appeared and the care with which she examined and stated things. Yet is is not a dead, scholastic work, though it is very deeply researched.

Somehow, it is also alive.

While I have not yet finished the book, the parts I have read so far only confirm the initial impressions of depth and thoroughness and even wisdom.While I am sure that there will be things I quibble with, I am certain they will not be because she is being hasty, careless or thoughtless in some way. I am convinced enough of its value to have chosen this to be the next book my own graduate students have to read, the next book for the book discussion group I lead, and the next book that I buy multiple copies of to give away.

 

In order for us to do something, we have to see things correctly and deeply — that is the thesis behind Shoshana’s book. If we are to do this in the political arena, we must first recognize that things were extremely corrupt, even completely bankrupt at the deeper levels long before Trump came along.

While Obama put a good face on things, he was not getting in the way of the massive transfer of wealth from the poor to the super-rich, nor was he into the truth, if that truth cost him power. Witness how he dealt with Ed Snowden.  (Those that think the Democrats are somehow more righteous than the Republicans are dangerously deluded. It is not an accident that “House of Cards” has the psychopathic president in the democratic party.) Listening, for example, to Jame Risen (at http://theintercept.com), you begin to understand that both Bush and Obama were doing deeply disturbing things, that they were precursors to Trump in very important, often ignored ways.

What allows almost all of the elite-bubble inhabitants to miss these facts is the haste with which they seek to know and the elitism that blinds them.  Their devotion to the big, fast, crowd and the unwillingness to spend the time to think and see and feel,  to wait patiently for wisdom, doom them to a darkness they think is light.

The humility that comes from realizing how all of us are actually susceptible to these errors opens us to the pursuit of depth, to investment in thick data, and, above all else, to a fundamental reorientation towards wisdom. When we prioritize the time to think and attention to the patient search, we begin to understand the intense power of quietness. Slowing down,  our eyes are opened to a rich, living path in and through everything. We begin to avoid labels. We abandon the naming so detrimental to our ability to see and to the desire to search more deeply.  Where we are and what we see becomes rich with information and nuance.

In this place of deeper awareness, armed with clearer pictures of the complexities and grounded truth, we begin to have a chance of making progress on real problems.


 

Returning to the work of Shoshana Zuboff, it is this universal lesson that comes as a corollary to a careful reading of her work. Though it seems obvious — that for real solutions, time to see, to feel, to think, to know, must be invested, the fact that books like Shoshana’s are not the norm tells us that the lesson needs much more emphasis.

In fact, I believe it is this deeper universal message that can have the biggest impact on thoughtful readers of the book. And this, in turn, compels me to do my part to make the number of such readers larger.

The stakes could not be higher.

 

Finding and Following Your Own Path

When my brother succeeded in persuading me to join an Alanon group in 1995 or 1996 I had little understanding of the healing for mind and spirit I would find among those gentle, powerful souls. They opened the door to healing simply by listening to me and speaking of their own paths in a way that made it clear I could take what I found healing and leave the rest.

I did know I was deeply afraid of others trying to tell me how to think, how to live, or even who to be. But I did not know the boundary violations I had experienced when I was young had created some very large traumas that were only increased by living through the slow deaths of both of our parents when my brother and I were teenagers.

Healing and a deep inspiration flowed from a combination of those illuminated spaces for listening and the walkabouts in the forests and mountains, first in Oregon, and then in New Mexico. In those experiences I found an understanding that no one had the answer for me, no one had the right to tell me what I should do, that only by that personal walk with God in those places of quietness and stillness could I hear and see and feel and find my own muse, my own path, my own unique way of creating and connecting.

I began to experience the power of the right words, at the right time.

 

I am still learning to understand the enormous power of words and the extent to which the misuse of words has created large swaths of humanity and society with seriously reduced capacity for sensing reality, for understanding the negative power of  words and images thrown around carelessly or even maliciously.

So much of what we say to each other is either powerless, without inspiration or filled with power to damage and limit those who accept the words. This comes either as a result of ignorance (the most common case) or intentional malevolence. I have been guilty of using words in ways that were not respectful of the need for others to find their own way. Phrases like “you should do …” are rarely helpful or useful and are often damaging. I would now argue that they never belong in print because, when they are appropriate, it is always very situation dependent. When they do appear with well intentioned people, I believe it is most often due to enthusiasm for discovered insights that have worked well for them.

We discover something that works for us and we immediately evangelize others, certain we know the way, that we have the answer for them as well. This is most pronounced in those that have an undeveloped gift for teaching, but it seems to effect everyone who has made discoveries they think others might need. So often we speak these words and add force of our own, lest those listening (or who are forced to listen) not get the importance of what we are saying.

Yet this betrays a misunderstanding of the power of truth and inspiration. It shows we are not sufficiently aware of how others find their muse, their path of creativity and connection.

It is arrogance, blind as it always is, that leads us to think we can find the path for others. Sometimes that arrogance is a subtle, cultural type of arrogance. Other times it is overt and obnoxious.

When we begin to see clearly and deeply, the humility that must accompany this leads us to get out of the way of others in their quest to discover who they are, where they can go and what they are privileged to create. We discover that the path of the creative teacher and mentor, collaborating alongside those involved in the joy of finding and following their own paths, is an experience full of living energy and fresh discovery.

 

Balanced, wholistic truth contains in itself all the power needed to take root and grow. In growing, it adapts to the soil it finds, encouraging the uniqueness it finds, illuminating the creativity that results. The creativity that results in turn illuminates new and original facets of the truth.

A narrative containing a truth, told simply, without force, transmits the truth in a way most likely to be accepted, though sometimes lying, like seeds in the ground,  awaiting just the right conditions to take hold and grow.

A telling of parts of our stories, encouraging listeners to find, explore, hear the stillness themselves, to become adventurers and participants in that deepest conversation with their Teacher — this becomes the most powerful thing we can do for others.

In their seeing what we have learned, what we have found, what is part of our story, they are inspired to begin their own journey, to find their own muse.

 

Fun with simple analysis problems I: the rest of the story

In an earlier post with the same title (and without the subtitle) I introduced some thoughts that were triggered by this simple problem:


Suppose that

     |\frac{df}{dx}(x)| \leq \lambda |f(x)|                    (1)

for all x, that f is continuous and differentiable, and that f(0) = 0.

Prove that f(x) = 0 everywhere.


In that post (which you can find here  Fun with simple analysis problems I ),  I started by presenting three solutions and then generalized and explored further.

What I did not reveal in that post, was that writing it, gave me an idea for a more advanced problem. Not too long afterwards, Laramie Paxton joined my group and I gave him this problem to work on for his dissertation. We collaborated in solving the problem, since that is how I mentor all my students — their dissertations are collaborations with me. This resulted in a paper we wrote together: A Singular Integral Measure for C^{1,1} and C^1 Boundaries that can be found here.


Laramie Paxton arrived at WSU quite naive with respect to analysis, having completed an online masters in mathematics that did not give him a good foundation in analysis. But he very quickly he adopted habits that led to rapid progress. He started by studying intensely the summer before arriving and passing the qualifying exam on his first try.  Then he took my challenging undergraduate analysis course (I used Fleming’s Functions of Several Variables), pushed through courses in advanced analysis, and geometric measure theory, and worked on applications in image analysis (generating papers he actually led) and finished his dissertation, all in the space of two years. After a year of postdoc, he landed the job he is about to start, at Marian University in Wisconsin. I believe that both the University and Laramie are lucky to have each other.

In general, I believe that small universities are good places to be nowadays, but from everything I hear, this place is better than good — it is perfect for Laramie’s talents and skills. (In addition to his impressively growing mathematical skills, he was already phenomenally skilled in logistics and organization which can be seen in his highly effective help in making the events listed here, from April 2017 to July 2018, a reality.)


A major point of both the original post on the problem and this present post, is that the paper with Laramie, as well as the results in the first post, flowed from taking time to think about a simple analysis problem that would usually be viewed as a not-too-hard exercise, not worthy of more thought than it takes to find one solution.

While I am sure that there are other undiscovered aspects of the problem that launched these two posts and Laramie’s dissertation problem, I believe that what has been explored illustrates why it makes sense to treat simple problems as invitations to playful exploration and creativity.

 

The Colors of Memory and Wisdom

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Reading Zeyn Joukhadar’s novel, The Map of Salt and Stars, has taught me once again that fiction can be more truthful than non-fiction. And even though the vast majority of fiction is to reading what junk food is to eating, there are novels that inspire even the pickiest of readers, with the highest (or most peculiar) standards for what is inspiring or illuminating.

What we know is a such a minuscule particle in a vast infinite universe of what could be known, that the skeptical inquirer is doomed to a rather poorly illuminated reflection of tiny bits of what is known. But skepticism is not the only option. Those willing to use all the tools at the disposal of an aware, enlightened human being, can embark on a voyage filled with light and a rich, ever-unfolding life.

In the living experience and fable woven together in Zeyn’s novel, the human spirit and the Infinite meet in an explosion of life and color and light and dark, moving us to a place where we can see and feel far beyond the narrow confines of overly rigorous, reductionistic thinking and experience. The deeper truths in the stories, sometimes stated very plainly, other times only seen in the wholistic experience of the story, are profound, demanding a stillness and quietness before they open to our view.

The overwhelming energy moving through the story, illuminating my response, was one of light and color and memory and feeling, reinforced by the synesthesia of Nour, the little girl through which we see the story. While a few might consider Nour’s synesthesia to be an unnecessary device, I found it completely natural, even essential. For me it was a door anyone can enter if they will but take the time to listen to the music and feel the color in the stillness and quietness, to see the light shining through the broken places, to experience the infinity between the ticks and tocs of a clock.

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When I taught 7th and 8th grade science during graduate school, I used to take my students out into nature with notebooks in hand and ask them to see and feel and hear, and then write. Most had a very difficult time finding the stillness necessary to do this. I know they had a hard time connecting with my descriptions of what happened on my walkabouts, when I moved into that living path mode of seeing and hearing. It was also my first time trying to describe this mode and inspire others to try it for themselves. After those experiences I often simply shared the insights I found in that state, realizing that it is a very hard thing to actually move somebody into that mode of being.

Nevertheless, over the years, I remained as hopeful as I was when I tried guiding the students, that this mode of seeing and hearing and feeling is open to anyone willing to listen to stillness.

Lately, though, I had started losing hope in the power of words to actually enlighten or inspire or even prompt others to begin a journey. I could find lots of examples that supported my growing doubt. But when I finished this book I was struck by the strong sense that I was wrong. Some written words are still very powerful, inspiring and healing, opening readers to that infinity I began to experience so many years ago in my walkabouts in the desert and later in the forests. Immersing myself in this story, I find again, in yet another form, that stillness containing infinity.

I was also reminded that when you have passed through extreme crisis, you learn what is important and what is not, you learn to choose the simple life and connections with those that love you and those that can benefit from your simple help. You remember that so many things in our surroundings, considered so important, cannot compare with the song of an insect, or connection with a friend, or peace of encompassing sunshine. You realize there is nothing to prove, that simple things contain everything you need because they are doors to infinity. You see helping those who struggle, easing the path of those who have very little and seek simply to live in peace, is an integral part of finding and sharing the depth and beauty we are wired to seek, to explore. One cannot truly have depth and beauty without the healing and compassion.

What remains for me, as I write these words in the afterglow of the story, is a sense of living stillness and remembering and color, and the deep peace when we remember the intense richness of knowing what is important.

 

 

 

Cultures of Creativity and Innovation

Books reliably inspiring enthusiastic conversations are books worthy of close attention. When Beata read and recommended Daniel Coyle’s book, “The Culture Code”, encouraging me by reading bits and pieces of it to me, it was not long before I knew that I had been introduced to just such a book. Soon I was buying copies and giving them away. Over the course of 2-3 months I gave away a bunch of copies and organized an evening in the top floor of the Monarch Motel in Moscow, Idaho devoted to discussion of the book.

The present article is part of my evolving reaction to the stories and theories in Coyle’s book, prompted by the reading-inspired, barehanded combat with those ideas.


The stories of remarkable environments for creativity and productivity, as well as the stories of studies and research aimed at understanding cultures of creativity and productivity, are brilliantly chosen. For this reason alone, I can, and do, recommend the book to everyone.

If those stories are listened to, and felt and thought about, and experimented with, the effect on the reader is large.

When I get a (non-mathematical) book and read it carefully, it means I have chosen to engage rather deeply. Usually I write in the margins, in a sort of hand to hand combat with the details and nuances.

There is a fair bit now written in the margins of this book.

While I sometimes have issues with the theories used to explain things – mostly nitpicky like the fact that nonlinearilty does not equal non-logical, see the story of the Allen Curve – the quality of the inspiration affected by the book completely outweighs any concern about the book’s shortcomings.


The core of this book is the threefold cord of (1) safety, (2) vulnerability and (3) purpose which, expanded a bit, becomes:

  1. Safety and belonging – taxing existential questions are never the lot of individuals in highly creative, productive environments. The growing scarcity of safety and belonging in many workplaces should be a source of deep concern. The gig economy is an indication that we are eating our seed corn and have ceased to pay even lip service to wisdom and a sustainable future.
  2. An empathetically evolved environment enabled by vulnerability powered connection. A status flat environment in which creative energy flows easily is an environment in which truth and kindness (together!) are common, even foundational. Empathy, in its nuanced and expanded incarnations, is at the root of all highly effective, sustainable environments.
  3. Purpose and vision – a bold, omnipresent clarity on the deeper foundational laws of being as well as the aims, the goals and the lofty visions that drive everything. An environment filled with signals keeping these principles and visions in constant view, is an environment whose vision is sustainable. Opposing the natural trend towards higher organizational entropy, these signals are an energy that enables the culture to remain inspired and organized for innovation and collaboration.

These three threads are the pillars of environments that have no trouble retaining those entering their influence. We visit and never want to leave – quite literally. In fact, Daniel himself admitted that when he was doing the research for the book, he found himself making excuses why he needed to stay in the environments he was investigating, even after he had the information he needed for his book.

Of course, some of the research was historical, visible only through the stories of those who were lucky enough to be part of those past places. Take for instance, Bell Labs in its heyday and Harry Nyquist.

In trying to understand the smaller group of super-innovators at Bell Labs, every possible factor was eliminated until it was discovered that all of these super-innovators ate lunch with Harry. He would draw out and listen to his lunch-mates with interest and curiosity, quietly giving them inspiring ideas and questions to go away and think about. Though Harry was also well known and influential because of his own research and innovation, neither this fact, nor his ability to spark innovation in others, seemed to effect his gentle, fatherly demeanor or tranquil reliability. In fact, these characteristics seemed to be significant part of the reason for his power. Disarmed by his demeanor, they opened up to his relentless curiosity.

At IDEO, the design company responsible for a large number of design innovations,  Roshi Givechi plays a similar role, roaming from one design group to another, helping them to overcome obstacles and find new creative grooves through a powerful ability to listen and ask questions. In fact, when Daniel Coyle told her the title of the book he was doing the research for, it was not long before he had a new subtitle after she asked a question about his choice of subtitle.


The other stories and anecdotes are very well selected and wide ranging. Some illustrate principles of collaboration. The Allen curve, showing that effectiveness of collaboration is inversely proportional to the distance between desks of those collaborating, is another striking story of discovery that is both surprising when you hear it for the first time and sensible, even intuitively reasonable, when you take it in and think about it for awhile. While it is not an illogical relationship, as Coyle asserts, it is a non-linear one that will nonetheless make sense to anyone whose intuitions include some instincts for physics and chemistry and interactions and reactions.

Other stories are rich with insight, a sort of living book waiting to be read more and more deeply. Coyle starts his book with such a story, of kindergartners outdoing, by a factor of two, groups of business students and professionals in a challenge to build the highest tower with a piece of tape, a string, a few dried spaghetti  and a single marshmallow. And for me at least, this set the tone of the book.


As noted above, I ended up with a book full of marginal notes (in pencil!) and a lot of thoughts that were discussed with others. If I had to select a phrase that captured the influence of the book on me I think it would be:

… brilliantly selected stories and simple principles that were even more compelling because they were validated by my own experiences in trying to build highly effective teams of innovators …


And the effect of the book does not end with the sharing of the book and discussions.

The histories of places like Bell Labs and Xerox PARC and Los Alamos and the Rad Labs in Boston were already part of my own context, either through direct experience or through careful histories that I had read and internalized, but something about the combination of this book and my own struggles with getting groups together that were sometimes partly or mostly successful and other times were pretty clear failures, created in me a deeper openness and readiness to put these principles into action.

While the experiment that is now underway is a topic for another article, I can say that the timing for the discovery that Beata made and passed on to me was remarkable.

I give the book my highest recommendation.

Median Shapes

When I wrote the paper with Simon Morgan pointing out the L^1\text{TV} functional was actually computing the flat norm for boundaries, we suggested this gave us a computational route to statistics in spaces of shapes. While earlier work certainly touched on this idea of using the flat norm for inference in shape spaces — see this paper on shape recognition, it was not until my student Yunfeng Hu collaborated with myself and Bala Krishnamoorthy (my collaborator, also a co-mentor of Yunfeng’s), that we started addressing the idea of statistics in shape spaces in the original paper with Simon Morgan.

The results can be found here: https://arxiv.org/abs/1802.04968 , in a paper with the title Median Shapes, with authors Yunfeng Hu, Matthew Hudelson, Bala Krishnamoorthy, Altansuren Tumurbaatar, Kevin R. Vixie. Tumurbaatar wrote the first complete version of the code used, and Matthew Hudelson contributed a pivotal new result on graphs inspired by a problem in the paper, while Bala led the computational end of things and I led, in collaboration with Yunfeng Hu (and Bala keeping us honest!), the theoretical parts of the paper. It was a fair bit of work.

We went over the more difficult results a few times, finding improvements and corrections. Of course, there may be a few things here and there to improve, but for now, it is done.

Yunfeng probably spent the most time writing up the piece proving that near regular points on the median, the collection of minimal surfaces meeting the median have a tangent structure we describe as a book.  While this is clear to experienced geometric analysts,  there are lots of little details and we wanted most of the paper to be more accessible to a wider audience. There are lots of other pieces here and there that took time to think about and write up (and rewrite). For example, when showing the set of medians need not contain any regular members,  the part where we show that we need only consider graphs when searching for a minimizer was not easy. And of course, as in most all of geometric analysis, there are problems you solve without too much effort at a high level, but find that writing down is tedious, though at times enlightening due to the fact that those little details turn out to be hard and illuminating.

Because the problem of computing the median reduces to a linear program, while the mean reduces to a quadratic program, we focused on the median problem. Some parts of the paper are a bit long winded, for the reason that we wanted it to have more details that would usually be in a paper communicating to others that understand geometric analysis.

Anyway, have a look. If you find yourself interested, there is already code you can use to compute medians, though we hope eventually to have faster code.

 

A Silence, Rich with Inspiration

Glynne Robinson Betts’ 1981 Writers in Residence is for me a lyrical invitation to quietness. The photo illuminated essays on places, in time and space, where writers lived and wrote,  invoke a strong sense of life lived with wide spaces for thought and creativity. The sections on Carl Sandburg, Anne Dillard, Robinson Jeffers and several others, recall daily rhythms friendly to depth.

These passages somehow bring back my life in the late 60’s and 70’s, when computers were rare and time to think was not hard to find. In college in the early 80’s, there was still time to think, time to read through a book on a weekend or master complex ideas in quietness, with a devotion to deep comprehension. I wonder, how many now feel that call to stillness, to a silence that is rich in inspiration?

Where do we find the wide spaces today? And who lives there? Those that tasted the thrill of illumination through immersion in quietness remember those spaces, but what of those addicted to their mobile devices, what of those who believe social media connects, wikipedia illuminates and TED talks are the pinnacle of inspiration? While wikipedia is useful, TED talks are sometimes narrowly inspiring, the intense illumination of bare-handed, personal discovery leaves you changed, forever.

Seeking the simplicity of those wide spaces, listening till we hear the quietness sing, we find the same places the writers found, the same illumination that is never forgotten … we walk through an open door to the infinity that lives between the ticks of the clock, between the words on a page, between the breaths we breath.

 

 

 

Dual Tyrannies of Data and Democracy (and what to do about it)

In this new age of extremes, celebrity and elitism without bounds, those that pride themselves on their enlightenment often make a big deal about being democratic in their ambitions and data driven in their thinking and reasoning.

This is also a new age of openness and deception. The increase in both is of course coupled. As openness is supported or exhibited, some of what is exposed resists and retaliates with deeper forms of deception.

And in ths new age, the old illusions also persist — like the illusions of rationality or unbiased examination or study without preconceived ideas — illusions that have a great impact on inference and our abilities to draw conclusions from observations. Democracy enters when we attempt to create a cooperative or civil society based in some semblance of truth or grasp of reality.

The problems I am focused on in this perhaps too provocatively titled article, are those caused by the use of data and democracy as tools of forceful persuasion or even hammers of coercion. While the idea that democracy is a system predisposed to tyranny is far from a new idea, the dangers in the new bandwagon of data-driven thinking seem to be less well known or thought about (even though Cathy O’Neil’s, Weapons of Math Destruction is a good start). So we will begin there.


It might be seem strange for someone who is a mathematician, with a great deal of experience in data science, who even founded the Data Driven Modeling and Analysis team at Los Alamos National Laboratory, to be concerned with or ambivalent about data driven anything.

Yet I am.

In fact I am very concerned. And the source of the concern is the inescapable fact that every inference, every conclusion and policy that is derived from data, is extracted from the data through the use of prior assumptions, many of which are unacknowledged or even very difficult to see. We can begin with the fact that we believe that rationality is the way in which truth is determined. But this is just not the case. Everything we do is framed in the deeper emotional/spiritual context in which “we live and move and have our being”.

As a result, even the decision of what data to collect is determined by our prior assumptions and preconceptions, and as a result, we can, often unconsciously, predetermine our conclusions before we even begin looking at the data.

The other problem I have with data driven scholarship is that, in its current forms, it only tells about what is, about the systems that have gained ascendancy and majority control of whatever it is that we are studying. It can say very little about what is possible. As a result, I believe that the industry of data driven scholarship and decision making will tend to reinforce what is, and limit diversity and real progress and innovation. (And by innovation, I am not talking technological innovation, but something much deeper and far reaching.)

This type of data science determines truth by, in essence, going with the majority vote in which the data doing the voting, has not only been selected by unseen and unacknowledged assumptions and biases, but is also, by its very nature, without an imagination.

What kind of data and data driven inference do I believe in? To begin with, I should say that I am very much for careful observation of the natural world, of human activity and behavior, and of the larger “inner” spiritual world on which everything is based. I think that the art of observation is a deeply neglected art, the rewards of which are little known and sorely needed. The problem lies in the fact that observation — data collection — is too often colored by stiff systems of preconception and unseen prior models of reality, influencing both choice of what to look at and what to do with the observations that are made.

Quietness and stillness as disciplines are not cultivated as ways to begin to really see beyond our current positions and perspectives. The fundamentally spiritual decision to let go and open to stillness is blocked by a complex of fear and fear inspired prejudice which are in turn based on previous experience with violation and force. Those experiences causing so many to relinquish child-like openness to reality, block them from full entry into the “kingdom of heaven”. As a result, those former children grow into adults that create systems that prevent them from entry into that illuminated kingdom and which they then use to block others from entering.

There are of course many flashes of insight that make it through the web of self-defense based preconceptions. But far too many of these quickly become part of that system that then blocks other illumination, blocking even the ability to understand the original insights correctly. The illumination falls prey to the temptations of greed for impact or fame or prestige or even simple fear, and the vision that could have been, fades.


There is of course the question of what to do about preconceptions and biases that are often predetermining results, especially in the case of data driven analyses. I believe the answer begins with opening your eyes, with taking the time to think:

  1. Take the time to think. The drive for bigger, better, faster has moved many people to abandon the discipline of taking time to think, to see, to feel. As a result, this basic first step to moving beyond our operating assumptions to something bigger and richer, to inspiration and growth, is severely limited.
  2. Time to think allows us to cultivate quietness and stillness as ways to let go and hear and see.
  3. We are only willing to see and hear and feel  what that quietness and stillness tells us in a state of emotional safety. This implies emotional wholeness underlies this whole project. Anyone who tells you differently is misguided at best. (This place of emotional safety is not external — this is a thing of the heart, not of “safe places” or elimination of harsh environments. The world is crazy and unsafe in which the emotionally whole still find a way to thrive, without expecting the world to be kept at bay.)
  4. Emotional wholeness requires cultivation of connection with others opened by the understanding that differences, instead of threatening us, enrich us.
  5. Connecting with others, observing their bits and pieces of illumination in a state of quietness, we are enabled to take the useful bits and let the rest go. (Because of emotional wholeness, we filter and thrive. No need for trigger warnings or cocoons that wall us away from reality.)
  6. Data is always filtered — quietness allows you to be aware of exactly what those filters are and to replace those filters if you find better ones.

While this approach to data (observation) and the inferences from that data is not new — it has always been the path of those taking the time to think and seek illuminated inferences, this path is becoming rarer. The noisy, overconfident bubble of thought leaders, influencers and celebrities are drowning out the careful thinkers and doers.


Democracy, even in its most beneficial forms, is only as good as the data driving it. Because of the difficulty in determining what is biased and what is not, the safest route is always to promote maximal freedom, opting for mild regulation only in the cases in which to not do so would harm the principles on which the democracy is founded. When freedom and compassion and safety and a healthy economic/social ecosystem are the principles, then this job is far from easy. But as soon as the regulation is influenced by entities that do not share those values, the whole enterprise is in peril. And if, in addition to this, the data that is used is twisted by the values that do not align with the goals above, it becomes hard to see what is and is not happening.

Of course, there are macro-measurements that reveal problems. When the divide between the rich and poor threatens to engulf us, we know something is wrong. When prisons are overflowing, when the rich are rarely held accountable, while the poor have difficulty for even small offenses or even simply because they are poor or nonwhite or both — when those things become impossible to ignore, we know the system is deeply broken.  And when the data is screaming and subtlety and nuance is no longer needed, when the data overwhelms preconception and prior assumptions, we know we are near a precipice.

But all is not lost.

As we learn quietness and openness to change, the data we gather and use will inform and illuminate, and the collective projects we embark on will reflect a synergy between freedom and cooperation. Time to think, quietness,  and the observations made in that frame of mind will supply the light and progress that keeps the biggest collective project — the democracy we live in — alive and headed in the direction of sustainable progress.

Other Planets

when
you set
my 
heart 
free

---

free from this 
and 
anti-this

escaping 
binary
compressions
and
bondage 
to narrow orbits

choosing 
the third
way 
that opens 
infinity

the living 
path
inviting 
me to flights
without limit

---

life

rich
illuminated
overflowing
unbounded

bright with 
the light
of 
countless
suns

constrained
by love
yet
unlimited

infinite
freedom
yet
running
in 
the
paths
of
Your
command

---

other planets
on which 
we 
were 
meant 
to live
to create
to unfold

no longer
traded
for a few shekels
for mere crusts of bread

for enough
is a feast
that 
opens infinity

in the light 
of other 
suns

on other planets 
without
number

 

Animals and Empathy

Many years ago, I ran a lab that used hamsters to study micro-circulation.  It is something that I eventually could no longer let myself do.

In recent years, I have come to be completely against the use of animals in experiments of any kind. I believe the cruelty that humans inflict on animals reduces, even removes their own humanity.

The lack of empathy shown in experimenting on animals should be frightening to us.

I propose that if we were fully conscious, fully mindful, that lack of empathy would horrify us. We would see the direct link between cruelty to animals and cruelty to humans. We would understand the lack of humanity, lack of conscience, lack of connection that enables us to kill other humans.

If we were to let ourselves feel with the whole heart, to understand that “the end justifies the means” is the most evil principle of all, to understand that mans inhumanity to animal and man alike is his greatest sin,  we would seek to heal instead of wound, to love instead of fear, to create instead of destroy.

Beneath all this cruelty is fear that drives us to deny our humanity, deny our empathy for other living things, deny other living things their right to life and peace. Believing that we must preserve our lives at the cost of anybody and anything that gets in the way, we attempt to save ourselves and in so doing, we condemn humanity to endless war and suffering. Only when we accept that “whosoever shall seek to save his life shall lose it; and whosoever shall lose his life shall preserve it” will we ever bring the cruelty to an end.

Empathy will guide us through darkness to light, if only we open our hearts to it. And when their suffering ends, so will ours.

Freedom and Writing

  1. Communicate in person often, one on one, one on few, spontaneously (even if you have prepared for hours).
  2. Write papers and books that you are inspired to write … if you stick to true inspiration, there might not be that many papers and books, but what there is will be very good.
  3. Write ideas compulsively, write, write, write … but do not feel the necessity of publishing most of it, even if you make most or all of it available as notes, online.
  4. See publication, or even posting online, less as carving something into stone and more as an invitation for others to join in your explorations.
  5. Never let the writing replace thinking — take the time to think (and plenty of it).
  6.  Don’t be afraid of mistakes and don’t edit as you write — edit by iteration, by revision.
  7. Polish your writing guided by one simple rule — to make your thoughts clearly visible to your readers. Forget all the other rules for proper writing, for how to write.
  8. Creativity is heightened by simple playfulness. Cultivate it.
  9. Share generously. Spread your inspiration and passion around. Make your environment rich!
  10. Sustain your inspiration by a commitment to freedom (with kindness) + connection (with generosity).

Faith Is Connection

Some use faith to connect with finite, comfortable places, perhaps in response to fears of one kind or another. Little boldness is required. Others courageously connect to the ragged edge, but without the calm quietness that warns of errors, omissions or even danger. Yet others refuse to acknowledge the fundamental, critical role of faith, making the mistake of identifying this primitive human faculty with something that implies a belief in God.

Faith is fundamentally neutral, having nothing to do with belief in dogma or God. “True faith” (or “false faith”) is therefore nonsensical: the words cannot sensibly be used together. Faith is simply the human faculty to connect and it is by that connection that we know, we experience, we see beyond. The great other, the great wholeness that dwarfs the tiny sliver of the universe that we believe we understand, can be opened by that connection we call faith.


In their abdication to a narrow vision of science and philosophy, many have surrendered an openness to paradox and a truly rich spiritual-mental-physical universe. Respectability tempts and seduces, but enslavement follows.

There is a radical faith not shying away from paradox, connecting us with a rich, bold and yet brilliantly simple vision. This faith draws us into a vibrant, living environment, for it is an explicit connection with Infinite Life.

In the patient, illuminated stillness flowing from connection, mystery and paradox, rather than being indicators of fuzzy or wishful thinking, become marks of clear vision. Quietly insisting on both horns of a dilemma, we are driven deeper until we find the transforming resolution opening us to new wavelengths of light, new worlds of thought and action.


Moving on from second hand knowledge of God to the vision through the torn veil, we find the connection that heals and illuminates.

Obi

Dogs are simple angels, disguised by their dogness, but ministering spirits all the same.

Filled with unconditional love and devotion, they are also sprinkled with flaws so as not to give away the fact they are working, under-cover, single-mindedly, to save as many humans as they can.

Their living fills us with joy, if we will but open to their joy. Their dying gives us such intense grief, it saves us from arrogance, if we permit ourselves to feel honestly their passing. And their simple, complete devotion and love while they are living disarms us so reliably, that we almost always permit ourselves to feel honestly, their passing.

Remembering them, we finally understand their mission in our lives.

We understand that love and humility are the channels through which all life and healing flow. We know that these simple, faithful friends never wavered, never faltered. We see the power of a pure heart.

Dogs are simple angels, disguised by their dogness, but ministering spirits all the same.

 

 

Disrupting Digital Delusions

A great deal is made now of inventions and ideas that will disrupt the usual way of doing things. “Thought Leaders”, eager technologists and the newly rich digital class are alive with a buzz that leaps from idea to idea and innovation to innovation, with scarcely a moment left for reflection or contemplation. Whatever is not the way it was yesterday, technologically speaking, is seen as the key to a glowing future.

Of course, there are naysayers, those that warn of the dangers and point out the signs that not everything is rosy, but humanity keeps accelerating, pressed on by their ubiquitous mobile devices. Email, Twitter, Facebook, Slack, Google, and thousands of other digital tools compete to capture and command our attention. Music in the form of mp3 files, books in the form of eBooks, anything (and everything) from Amazon, and a host of other digital replacements for what we used to see and feel and experience in “real life”, seduce us into thinking that  non-digital things — things that we can see and touch and hold and smell and own with no ambiguity — that those analog things are passe. Thus Vinyl Records, real books, real stores, and real jobs where things are made with our own hands are considered relics from a bygone era.

But anyone that slows down enough to think and to feel begins to question this idolization of technology and speed. Looking for books to read along these lines, they might find The Shallows: What the internet is doing to our brains by Carr, or You are not a Gadget: A manifesto by Lanier or Hamlet’s Blackberry by Powers, all of which are good. In Revenge of Analog: Real Things and Why They Matter, David Sax takes another approach, focusing on an argument for things thought obsolete, making the argument, often eloquently, that those analog things are not at all obsolete. He asks us to consider the possibility that analog things are far from dead, that in fact they might rescue us from the dangerous cliff that everything digital has lured us towards. I think he is on to something.

In the book, we find vinyl record companies like United Recording Pressing and Third Man Records, real film photography companies like FILM Ferrania, Lomography and The Impossible Project, real books made of paper and bookstores made of brick that thrive because they know their books and their customers, magazines like Stack — a meta magazine that send out a different, new independent magazine every month, and Delayed Gratification — a slow news magazine that you cannot read electronically. In a chapter in work, we discover Shinola, the luxury watch company in Detroit the is employing hundreds of formerly unemployed workers that construct watches and other distinctive products in an environment committed to making things in the USA. Another story I was intrigued and inspired by, was the story of the Newspaper Club that enables anyone with an idea to generate a small scale paper or magazine.

Even though, as Sax points out, real things are often still the way to make money, this does not explain why customers should prefer analog to digital, as is becoming clear is the case. Serendipity of finding a book you were not looking for when you go to a real bookstore stocked by real and knowledgeable staff or of meeting people in real places like cafes and brick and mortar stores, that we would not meet online, the importance of putting pen to paper for the purposes of remembering and recruiting the entire mind in the creative process, the nuances and range of response that real film offers that digital cannot match — these are a few of the reasons that digital cannot replace analog. In particular,  it seems that face to face connection, free of digital mediation is incredibly important for sustaining a network of real human connection, so important for mental and emotional health.


I finished the book before we traveled to Chicago on the Amtrak Empire Builder, in a sleeper compartment. The train seemed a fitting response to my decision to disconnect and slow down. The slow pace, the shared meals with other travelers we had never met,  conspired to engage Beata and I in conversations with several fascinating couples and individuals, some of whom may in fact become long term friends. The slow pace also, somehow, prepared me for the stay in Chicago, where, in addition to my usual impromptu maintenance and design challenges for my mother-in-law (and her mother, who turned 94 while we were there), I visited a sequence of bookstores and a Shinola Shop, all using the subway trains.

In the back of my mind (and occasionally in the front, as when recommending Revenge of Analog to bookstore owners), Revenge of Analog informed my search for books and magazines. Getting on at the Harlem stop, not far from O’Hare, it did not take long to get to Logan Square, where I visited CIty Lit Books, owned by Teresa Kirschbraun with whom I had a long discussion. After buying The Internet of Us by Lynch and another, The Book, by Houston, I moved on to  Uncharted Books, where I found Nick Disabato’s design publications, among other books. In Wicker Park, a few stops closer to downtown, I found Quimby’s, Myopic Books and Volumes. Over a few days I visited a few more including Ravenswood Used Books on Montrose and Unabridged on Broadway. I recommend both of these stores along with the previously mentioned stores, though I would have to say that the most engaging stores to shop were City Lit Books and Ravenswood Used Books. (In Wicker Park I also bought a notebook in the Shinola store.)


Somewhere in this summer and process of reorientation of focus and energy, I found myself realizing that I have to make changes in order to recapture the analog, face to face interactions that flow at their own pace. A maker space is one idea, as is a place to be, to connect, to converse, with little in the way of time constraints, perhaps some sort of updated version of the 18th and 19th century Salons. This is what I am finding the summer of avoiding email (checked only on Tuesdays and Fridays) and movies (we canceled Netflix and Youtube Red), and instead reading and thinking and walking and talking, has led me to. Yet another idea that is emerging is the recreation of a Bell Labs like environment, updated, but also very retro in its demand for time to think, with a focus on an organic interdisciplinarity that would have seemed natural to the innovators and thinkers in the 18th, 19th and very early 20th centuries.

I began the summer very burned out from interaction with the highly dysfunctional, ego-focused, post-student-focused academia (i.e. the new normal in academia), and have arrived at a point where I see what to do. Revenge of Analog was an important catalyst. In one way, it did not teach me too many new things, yet in another way, it was an absolutely critical inspiration, moving me towards understanding where I must go. But that is what good catalysts do — they take things you know or almost know and then push you to respond to the inspiration that emerges from your own unique experience and whatever new thoughts the catalyst might add to the mix.


If we are to have a healthy future, community focused activities and places to be together to talk and connect and explore and learn and create must be preserved and expanded. it seems fitting that I found David Sax’s brilliantly timed catalyst for this rethinking and renewal on the new book shelf, in the local public library.

While my interests have led me to pick a few projects in line with this vision, there are an enormous number of variations and innovations that promote and support connection and creative productivity. All of them depend on fundamentally analog, tangible, non-virtual experiences. As a part of my response to the book, I intend to encourage as many people as possible to read this book. In fact, I am considering starting a book club that would begin by reading Revenge of Analog.

Perhaps I can even convince the local library to add this book to their book club list so that they will have multiple copies on hand when we read the book together.

Metrics and Inequality

Metrics — measures of performance or value — drive what we do at every scale, from the small, individual scale to the massive global scales. When those metrics are founded on misconceptions of reality, they contort behavior in such a way as to appear to support those misconceptions. To get back to the natural order of things, away from the artificial reality created by those false beliefs, we must start by reseting our metrics.


I was reminded of this as I perused the Harvard Business Review (HBR) I had purchased for the purpose of inspiring thoughts and reactions. I do not peruse the Review very often, but when I do, I am usually turned off by a large amount of what I find. The price  of 16.95$ reeks of self-importance. And the articles overflow with much that I find distasteful in academia and in the broader, elitist culture — the same culture that is currently driving the world to the brink of destruction. But the metrics and implied metrics in the articles got me thinking about the influence of bad metrics, about the models of reality that implicitly encode inequality. Those models are everywhere.

Take the current focus in the news and social media on racism.

The real problem is that racism is an epiphenomena. Looking more deeply, we find the pervasive illusion of organic superiority/inferiority and the (negatively) powerful habits of ranking in all areas of life. These survive only because people can’t tell the difference between (1) powerful (negative) beliefs that become self-fulfilling prophecies and (2) fundamental truths. (While behavior does follow those unhealthy ideas, I am talking about potential here, not the reality created by those self-fulfilling prophecies.)

But to confront the fact that our brains are all pretty much equal, and what really matters is environment and opportunity, we have to face man’s inhumanity to man and our own moral degradation and greed.

And facing that fact is painful and difficult.

Once we begin to understand the effects of trauma of all sorts, of the massive power of emotions — actually, of our entire environment, we begin to understand the observed behavioral data differently. We begin to see that our beliefs in inequality combined with our inhuman treatment of others actually generate inequality. We begin to see that any solution to inequality that does not begin with the understanding that people are, actually, truly born equal is bound to fail.

Because we cannot fix inequality and believe in inequality at the same time.

Though it is a fact that there are organic differences, that there are a relatively small number of (very) basic groups of talents people are born into, any solution to inequality cannot succeed if it does not start with the understanding that these talents are not rankable, but are equally amenable to (even extreme) development.

When this position is taken, we see that inequality is pervasive, that the roots to racism are found in how we treat each other in every environment, including very white environments. In fact, if you were to restrict yourself to purely White Anglo-Saxon Protestant environments (though finding such environments is getting harder), one would find the fundamental disease that becomes racism in other environments.

When we begin building metrics based on the facts of equality, we begin to stand a chance of making a difference.

This brings me back to the HBR articles and their usual conformity to a traditional interpretation of behavioral data.


Of course the mistake the intelligent people who populate academia and the elitist cultures make, is the mistake that scientists often make, of not taking into account the effects of multiple time/context scales in their studies. It is sort of like the Chinese story of the man who lost his horse ( 塞翁失馬 — Sāi Wēng Shī Mǎ) in which what appears to be a good thing or bad thing depends on context that keeps expanding. Not taking all the different temporal/spatial/contextual scales into account, often leads to incorrect conclusions.

To many such observers, the data appears to confirm that (1) unfettered competition and greed are natural and probably  good and (2) inequality is organically based. (Note: I am not saying that all competition is bad, only that the current vision for competition is deeply unbalanced and actually unfair to many smaller entities that want to compete.) Of course, the more sophisticated the person, the more polished and palatable their presentation of these ideas.  But, as I observed above, the process by which we can see differently is uncomfortable for everyone and painful for most.

So instead, we pretend that the results of greed and inequality are some sort of natural law that we have no power over. And we end up missing the principle that enables us to find richness almost anywhere.

We do not realize that enough is a feast.


I am far from the first to observe that enough is a feast, that aiming for more than enough is wasteful, and that piling up great piles of wealth of all kinds (not just financial) and locking it away literally or figuratively is an obscene crime against humanity. It is just that even though it has been said before, by many others, it seems to be one of those things we need very frequent reminders of.

What I am interested in is a world in which taking time to think has priority over the rush of the over-achiever, where what my family and my dog thinks of me is more important than what my department or academia in general or the National Academy of Sciences thinks of me, where being a fundamentally independent thinker is more valued than status as a “thought leader”, where quiet generosity takes precedence over noisy philanthropy, and success is measured by whether or not I and those around me have enough, not if I have enough money or prestige to supply a small country.

In such a world, where “enough” becomes integral to our metrics, there is enough for everyone. And when this happens the enormous human potential that we have been obscenely wasting is unleashed.

When, as Bryan Stevenson makes a case for in Just Mercy, we understand that healing begins in seeing our own brokenness, we begin to understand why we strayed from “enough” in the first place. We then understand that everything good begins with healing, that, from the humility we gain in that process of healing,  every other good thing flows. Then we understand that humility is not so much the opposite of arrogance and the drive for status, as it is the opposite of spiritual blindness.

For blindness was the problem all along. What we needed, what we really wanted, was always at our fingertips. Only our inability to see the true order of things stood in our way.

Accepting this, we are set free to find healing and a rich abundance that has nothing to do with impoverishing others in any way.

Fun with simple analysis problems I

This last semester, I ran a fun, informal master class in problem solving. Actually, a graduate student of mine — Yunfeng Hu — who is an expert problem solver, produced all the problems from the immense library he built up over his undergraduate career in China.

I believe that the art and culture of problem solving is not as widely valued in the USA as it ought to be. Of course there are those that do pursue this obsession and we end up with people with high scores on the Olympiad and Putnam competitions. But many (most?) do not develop this skill to any great degree. While one can certainly argue that too much emphasis on problem solving along the lines of these well known competitions does not help very much in making real progress in current research, I would argue that many have fallen off the other side of the horse — many are sometimes hampered by their lack of experience in solving these simpler problems.


Here is a problem that arose in our Wednesday night session:

Suppose that

     |\frac{df}{dx}(x)| \leq \lambda |f(x)|                    (1)

for all x, that f is continuous and differentiable, and that f(0) = 0.

Prove that f(x) = 0 everywhere.

Perhaps you want to fiddle with this problem before looking at some solutions. If so, wait to read further.


Here are three solutions: in these first three solutions we are dealing with f:\Bbb{R}\rightarrow\Bbb{R} and so we will denote \frac{df}{dx} by f'.

(Solution 1) Consider all the solutions to g' = 2\lambda g and h' = -2\lambda h. These are all curves in \Bbb{R}^2 of the form y = g_{C}(x) = Ce^{2\lambda x} and y = h_{C}(x) = Ce^{2\lambda x}. We note that if y=f(x) is any function that satisfies equation (1), then everywhere its graph intersects a graph of a curve of the form y = g_{C}(x) = Ce^{2\lambda x} for some C\in\Bbb{R}, the graph of f must cross the graph of g_{C}. if we are moving form left to right the graph of f moves from above to below the graph of g_{C}. Likewise, f crosses any h_{C} from below to above, when moving from left to right in x. Now, supposing that f(x*) > 0 at some x*. Then simply choose the curves g_{\frac{f(x*)}{e^{2\lambda x*}}}(x) and h_{\frac{f(x*)}{e^{-2\lambda x*}}}(x) as fences that cannot be crossed by f(x) (one for x < x* and the other for x > x* to conclude that f(x) can never equal zero. (Exercise: Verify that this last statement is correct. Also note that assuming f(x*) > 0 is enough since, if instead f(x*) < 0 then -f also satisfies (1) and is positive at x*)

(Solution 2) This next solution is a sort of barehanded version of the first solution. We note that equation (1) is equivalent to

-\lambda |f(x)| \leq f'(x) \leq \lambda |f(x)|            (2)

and if we assume that f(x) > 0 on E\subset\Bbb{R}, then this of course turns into

-\lambda f(x) \leq f'(x) \leq \lambda f(x).            (3)

Assume that [x_0,x_1]\in E and divide by f(x)  to get  -\lambda \leq \frac{f'(x)}{f(x)} \leq \lambda. Integrating this, we have -\lambda (x_1 - x_0) \leq \ln(\frac{f(x_1)}{f(x_0)}) \leq \lambda (x_1 - x_0) or

e^{-\lambda (x_1 - x_0)} \leq \frac{f(x_1)}{f(x_0)} \leq e^{\lambda (x_1 - x_0)}.            (4)

Now assume that f(x*) > 0. Define u = \sup \{ w | f(x) > 0 \text{ for all } x* < x < w\} and l = \inf \{ w | f(x) > 0 \text{ for all }x* > x > w \}. Note that l \neq -\infty implies that f(l)  = 0 and u \neq \infty implies that f(u)  = 0. Use equation (4) together with \{\text{a sequence of }x_0\text{'s} \downarrow l \text{ and }x_1 = x*\} or \{x_0 = x* \text{ and a sequence of }x_1\text{'s} \uparrow u\},  to get a contradiction if either l \neq -\infty or u \neq \infty.

(Solution 3) In this approach, we use the mean value theorem to get what we want. Suppose that f(x_0) = 0. We will prove that f(x) = 0 on the interval I = [x_0 - \frac{1}{2\lambda}, x_0 + \frac{1}{2\lambda}].

(exercise) Prove that this shows that \{f(x) = 0\text{ for some } x\} \Rightarrow \{f = 0\text{ for all } x\in\Bbb{R}\}. (Of course, all this assumes Equation (1) is true.)

Assume that x\in I. Then the mean value theorem says that

|f(x) - f(x_0)| \leq |f'(y_1)| |x - x_0| \leq |f'(y_1)| \frac{1}{2\lambda}               (5)

for some y_1\in I. But using equation (1) and the fact that f(x_0) = 0, this turns into |f(x)| \leq \frac{1}{2}f(y_1). By the same reasoning, we get that  |f(y_1)| \leq \frac{1}{2}f(y_2) for some y_2 \in I, and we can conclude that |f(x)| \leq \frac{1}{2^{2}}f(y_2). Repeating this argument, we have

|f(x)| \leq \frac{1}{2^{n}}f(y_n)                  (6)

for some y_n \in I, for any positive integer n. Because f is continuous, we know that there is an M < \infty such that f(x) < M \text{ for all } x\in I. Using this fact together with Equation (6), we get

|f(x)| \leq \frac{M}{2^{n}} \text{ for all positive integers } n                (7)

which of course implies that f(x) = 0


Now we could stop there, with three different solutions to the problem, but there is more we can find from where are now.


Notice that one way of looking at the result we have shown is that if

(1) f is differentiable,

(2) f(x_0)=0 and

(3) for some \delta > 0, we have that f(x) \neq 0 when x \neq x_0 and x\in [x_0 - \delta, x_0 + \delta],

then

\limsup_{x\rightarrow x_0}A_{f}(x) \equiv \left|\frac{f'(x)}{f(x)}\right|\rightarrow\infty              (8)

Note also that if we define

a(f) \equiv \sup_{x\in\Bbb{R}} A_{f}(x)                  (9)

we find that

a(f) = a(\alpha f) \text{ for all }\alpha\neq 0.               (10)

Let C^{1}(\Bbb{R},\Bbb{R}) denote the continuously differentiable functions from \Bbb{R}\text{ to }\Bbb{R}. If we define C_{\lambda} = \{f | a(f) \leq \lambda\} we find that not only is \bigcup_{n\in\Bbb{Z}^{+}} C_n not all of C^{1}(\Bbb{R},\Bbb{R}), we also have functions satisfying 0 < b \leq f(x) \leq B < \infty whose a(f) = \infty. So we will restrict the class of functions a bit more. The space of continuously differentiable functions from K\subset \Bbb{R} to \Bbb{R}, C^{1}(K,\Bbb{R}), where K = [-R,R] (compact!), is closer to what we want. Now, C_{\infty} \setminus\bigcup_{n\in\Bbb{Z}^{+}} C_n contains only those functions which have a root in K.

We will call the functions in C_{\lambda} \subset C^{1}(K,\Bbb{R}) functions with maximal growth rate \lambda. This is a natural moduli for functions when we are studying stuff whose (maximal) grow rate depends linearly on the current amount of stuff. Of course populations of living things fall in the class of things for which this is true. from the proofs above, we know that if f\in C_\lambda, then it’s graph lives in the cone defined by exponentials. More precisely

If a(f) = \lambda then for x < x_0 ,   \frac{f(x_0)}{e^{\lambda x_0)}}e^{\lambda x}  \leq f(x) \leq  \frac{f(x_0)}{e^{-\lambda x_0)}}e^{-\lambda x}   and for x > x_0 we have \frac{f(x_0)}{e^{-\lambda x_0)}}e^{-\lambda x}  \leq f(x) \leq  \frac{f(x_0)}{e^{\lambda x_0)}}e^{\lambda x}.

(Exercise) Prove this. Hint: use the first proof where instead of 2\lambda you use \alpha\lambda and let \alpha\downarrow 1.

(Remark) Notice that Equation (10) and \lambda < \infty implies that scaling a function in C_\lambda by any non-zero scalar yields another function in C_\lambda. As a result, we might choose to consider only

F \equiv f\in c_\lambda\text{ such that }f(0) = 1

or

F \equiv \{\text{ functions whose minimum value on }K\text{ is }1\}.

In both cases we end up with subsets that generate C_\lambda when we take all multiples of those functions by nonzero real numbers.

(Exercise) If we move to high dimensional domains, how wild can the compact set K be and still get these results? It must clearly be connected, so in \Bbb{R}^1 we are already completely general with our K above.


Moving back to Equation (1), we can look for generalizations: for example, will this result hold when f:\Bbb{R}^{n} \rightarrow \Bbb{R}^{m}? How about when f maps from one Banach space to another? How about the case in which f is merely Lipschitz?

Lets begin with f:\Bbb{R}^{n} \rightarrow \Bbb{R}^{m}.

In this case, the appropriate version of Equation (1) is

||Df(x)|| \leq \lambda ||f(x)||                    (11)

where ||Df(x)|| denotes the operator norm of the derivative Df(x) and ||f(x)|| is the euclidean norm of f(x) in \Bbb{R}^m.

Notice that

D\ln(||f(x)||) = \frac{1}{||f(x)||}\left(\frac{f(x)}{||f(x)||}\right)^{t}Df(x)                    (12)

where \left(\frac{f(x)}{||f(x)||}\right)^{t} is an m dimensional row vector and Df(x) is an n\text{ by }m dimensional matrix. (Thus the gradient vector is the transpose of the resulting n dimensional row vector.)  Now we can use this to get the result.

Let \gamma(s) be the arclength parameterized line segment that starts at x_0 and ends at x_1 the The above equation tells us that

\int_{\gamma} D\ln(||f(x(s))||) ds =  \int_{\gamma} \frac{1}{||f(x)||}\left(\frac{f(x)}{||f(x)||}\right)^{t}Df(x)  \leq \int_{\gamma} \frac{||Df(x(s))||}{||f(x))||} ds.        (13)

Thus, we can conclude that

\ln(||f(x_1)||) - \ln(||f(x_0)||) \leq \lambda ||x_1 - x_0||

which implies that

-\lambda ||x_1 - x_2|| \leq \ln\left(\frac{||f(x_1)||}{||f(x_0)||}\right) \leq \lambda ||x_1 - x_0||

and we can proceed as we did in the second proof of the problem in the case that f:\Bbb{R}\rightarrow \Bbb{R}. We end up with the following result

If ||Df(x)|| \leq \lambda ||f(x)||  and ||f(x)|| \neq 0 \text{ for all } x\in B(x*,r)\subset\Bbb{R}^n, then

e^{-\lambda ||x - x*||} \leq \frac{||f(x)||}{||f(x*)||} \leq e^{\lambda ||x - x*||}

for all x\in B(x*,r).

(Exercise) Show that this result implies that if f(x) = 0 anywhere, it equals 0 everywhere.

(Exercise) Show that this is implies the one dimensional result we proved above (the first theorem we proved above).

(Exercise) Our proof of the result for the case f:\Bbb{R}^n\rightarrow\Bbb{R}^m can be carried over to the case of f:B_1 \rightarrow B_2 where B_1\text{ and }B_2 are Banach Spaces — carry out those steps!


We come now to the question of what we can say when we are less restrictive with the constraints on differentiability.  We consider the case in which f:\Bbb{R}^n\rightarrow\Bbb{R}^m is Lipschitz. The complication here is that while we know that f is differentiable almost everywhere, it might not be differentiable anywhere on the line segment from x_0 to x_1.

Consider a cylinder C_{x_0}^{x_1}(1), with radius 1 and axis equal to the segment from x_0\text{ to }x_1. Let E = C_{x_0}^{x_1}(1) \cap \{x| Df(x)\text{ exists }\}. Since f is differentiable almost everywhere, we have that \mathcal{L}^n( C_{x_0}^{x_1}(1)\setminus E) = 0. Therefore almost every segment L generated by the intersection of a line parallel to the cylinder axis and the cylinder, intersects E in a set of length ||x_1 - x_0||. We can therefore choose a sequence of such segments converging to [x_0,x_1].

lip-cylinder

Since Df exists \mathcal{H}^1 almost everywhere on the segments [x_0^k, x_1^k]  and f is continuous everywhere, we can integrate the derivatives to get:

-\lambda ||x_1^k - x_0^k|| \leq \ln\left(\frac{||f(x_1^k)||}{||f(x_0^k)||}\right) \leq \lambda ||x_1^k - x_0^k||.

And because f is continuous we get that

-\lambda ||x_1 - x_0|| \leq \ln\left(\frac{||f(x_1)||}{||f(x_0)||}\right) \leq \lambda ||x_1- x_0||.

so that we end up with the same result that we had for differentiable functions.


There are other directions to take this.

From the perspective of geometric objects, the ratio \frac{||Df||}{||f||} is a bit funky — for example, if f(x) = volume of a set E(x)\subset \Bbb{R}^n  = \mathcal{L}^n(E(x)), where x can be thought of as the center of the set, we have that Df will be a vectorfield \eta times \mathcal{H}^{n-1} restricted to the \partial E(x). Thus, ||Df|| will be an n-1-dimensional quantity and f a n-dimensional quantity. We would usually expect there to be exponents, as in the case of the Poincare ineqaulity,  making the ratio non-dimensional.

On the other hand, one can see this ratio as a sort of measure of reciprocal length of the objects we are dealing with. From the perspective, this result seems to say that no matter what you do, you cannot get to objects with no volume from objects with non-zero volume without getting small (i.e. without the reciprocal length diverging). This is not profound. On the other hand, that ratio is precisely what is important for certain physical/biolgical processes. So this quantity being bounded has consequences in those contexts.

This does not lead to a new theorem: as long as the set evolution is smooth, the f and Df are just a special case where f:\Bbb{R}^n\rightarrow\Bbb{R}^1 and even though actually computing everything from the geometric perspective can be interesting, the result stays the same.

in order to move into truly new territory, we need to consider alternative definitions, other measures of change, other types of spaces. An example might be the following:

Suppose that X is a metric space and f:X\rightarrow \Bbb{R}. Suppose that \gamma:\Bbb{R}\rightarrow X is continuous and is a geodesic in the sense that for any three points in \Bbb{R}, s_1 < s_2 < s_3, we have that \rho(\gamma(s_1),\gamma(s_3)) = \rho(\gamma(s_1),\gamma(s_2)) + \rho(\gamma(s_2),\gamma(s_3)).

If:

(1) for any two points in the metric space there is a gamma containing both points and

(2) for all such \gamma, g_{\gamma} \equiv f\circ\gamma is differentiable

(3) and \frac{|g_{\gamma}(s)|}{|f(\gamma(s))|} \leq \lambda

then, we have that

-\lambda \rho(x_1, x_0) \leq \ln\left(\frac{|f(x_1)|}{|f(x_0)|}\right) \leq \lambda \rho(x_1,x_0).                       (14)

And, again we get the same type of result for this case as we got in the Euclidean cases above.

(Exercise)  Prove Equation (14).

(Remark) We start with any metric space and consider curves \gamma:[a,b]\subset\Bbb{R}\rightarrow X for which

l(\gamma)\equiv\sup_{\{\{s_i\}_{i=1}^{n}| a = s_1 \leq s_2 \leq ... \leq s_n = b\}} \sum_{i=1}^{n-1} \rho(\gamma(s_{i}),\gamma(s_{i+1})) \leq \infty.

We call such curves rectifiable. We can always reparameterize such curves by arclength, so that \gamma(s) = \gamma(s(t)), t\in[0,l(\gamma)] and l([\gamma(s(d)),\gamma(s(c))] ) = d-c. We will assume that all curves have been reparameterized by arclength. Now define a new metric

\tilde{\rho}(x,y) = \inf_{\{\gamma | \gamma(a) = x\text{ and }\gamma(b) = y\}} l(\gamma).

You can check that this will not change the length of any curve. Define an upper gradient of f:X\rightarrow \Bbb{R} be any non-negative function \eta_f:X\rightarrow \Bbb{R} such that |f(y) - f(x)| \leq \int_{\gamma} \eta_f(\gamma(t)) dt.

Now, if \frac{|\eta_f(x)|}{|f(x)|} \leq \lambda, we again get the same sort of bounds that we got in equation (14) if we replace \rho with \tilde{\rho}. To read more about upper gradients, see Juha Heinonen’s book Lectures on Analysis in Metric Spaces.


While there are other directions we could push, what we have looked at so far demonstrates that productive exploration can start from almost anywhere. While we encounter no big surprises in this exploration, the exercise illuminates exactly why the result is what it is and this solidifies that understanding in our minds.

Generalization is not an empty exercise — it allows us to probe the exact meaning of a result. And that insight facilitates a more robust, more useful grasp of the result. While some get lost in their explorations and would benefit from touching down to the earth more often, it seems to me that in this day and age of no time to think, we most often suffer from the opposite problem of never taking the time to explore and observe and see where something can take us.

Finding Quietness

Rereading parts of Glynne Robinson Betts’ 1981 book, Writers in Residence, recalled simpler, deeper times, when finding places of quietness and taking time to think was part of the routine many people used in order to hear themselves and others. In fact, reading this again prompted me to expand the time I spend without Internet interruptions. Steps as simple as ignoring email for extended periods or as comprehensive as turning the computer off for the entire weekend, are emerging as a necessary part of reclaiming quietness and time to think.

There is nothing profound in these decisions to disconnect — whatever is profound happens as a result of taking that time to see and listen and think.

When I do slow down, every pause, every quietness, every moment taken to see, to listen, to think, rewards with a rich, living connectedness and depth that cannot be exhausted. The fabrics of the past and future join with the present, without seams, without a sense that I am working to recall, to see, to feel. Time opens up, I enter, to travel my own path, to sit or stand or walk … stopping time, finding passage to places beyond space and time.

To the strictly modern intellect, what I have just said probably seems like non-sense. Reason, based on easily observable facts, will find little irrefutable evidence that a skeptic would find compelling.

I therefore offer no argument to convince the skeptic. Instead I say, “Come and see”.

When we begin to let go of dogma, the regard of peers, and the comfort of the in-group, room for discovery is created. Launching into quiet spaces, where fear is replaced by stillness, a boundless infinity surprises. We find flow.  In this personal place without limits, I find an overflowing garden, teeming with life. On the living path, everything is illuminated.

Yet this is something I cannot really transmit. It is only something I can hint at in what I write, faintly, incompletely. The experience of discovery, of knowing, of traveling to those places that are here and beyond at the same time, cannot be captured in words.

To see, you must see though your own eyes. To see, you must choose to slow down, find quietness, and dwell there.

I believe that most – possibly all – human beings have, at one time or another, experienced immersion in flow and a connection to the place without limits. There is a resonance emerging from any such experience, no matter how brief, that enables those with that experience to hear each other.  But life often seems to conspire to crush those memories, to remove our ability to hear and see. In the quiet, we can be moved to remember, to see, to hear. In the quiet we remember the place without limits.

In writing something of what I see and hear, there is a chance that faint recollections will be stirred in those that read, in the way Writers in Residence stirred my memories, my recollections of a time when quietness and time to think was plentiful.

The thought of this possibility brings a subtle sense of connection, of silent conversation, with those as yet undiscovered friends. Lingering in rediscovered quietness, we move against the flow of noise and commotion and modern distraction, encouraging all those in our circle of influence to rediscover for themselves their own place without limits.

Connection vs Attention

At our fingertips, in the present, in the place we find quietness, we may find boundless inspiration for a rich, creative life. When this is focused and refined under the influence of our own uniqueness, our own particular genius, we find illumination and a deeply satisfying flow.

Far too often we surrender who we are in an effort to gather attention, when what we are seeking is connection with others, connection with a richly creative life. Human society has almost completely abandoned the cultivation of truly individual genius for the pursuit of attention. As a result we are infinitely poorer.

Yet this choice is completely under our control — we may refuse this lopsided bargain. We may instead choose an abundance that more than makes up for whatever loss of fame or fortune our choice entails. Those that turn away from that obsession, towards quietness and life, find a healing, restoring force, gently coaxing them back to playfulness, to a place of freedom, to originality and creativity.

Immersion in nature, connection to the life that surrounds us, communion with quietness that speaks and engages us with Infinity — here we find sustenance for a life that never loses freshness or originality.

Boldly choosing connection rather than attention, such a life does not sacrifice its own brilliant originality to the temptations of fame or fortune, nor does it hide from the face of fear. Instead, that life enriches everyone and everything it touches, and in so doing finds connection.


In a present quietness cultivated, we find inspiration for a rich, creative life.

P1060219-1600-wide

 

Doing Mathematics

I have come to question a significant portion of the culture in academia, even while I have developed a deeper connection with other parts of that same culture or at least the culture that we could have. While I am deeply committed to mathematics as a creative occupation, and to teaching and mentoring in mathematics, my experience in academia after re-entering it seven years ago has strengthened my rejection of the many parts of that culture because they hinder the best research and teaching.

There are many aspects I could discuss, but here I am singling out four: the question of what makes a mathematical result or paper worthy of recognition together with the place of exposition in mathematics,  the value of awards and recognitions in mathematics, and the effects of federal funding on mathematics and academia.

As opposed to trying to do some sort of statistical study — a study which would only be meaningful if there were sufficient numbers of people following the ideas I propose, and there is not! — I will invoke common sense and intuitions that are commonly agreed on, but usually discarded as a guide for actions because of the economic realities of higher education; the institutions that pay us expect and reward the defective model and very few actively step outside those bounds.


We start with a relatively innocuous idea that papers that answer questions completely, are best.

What comes from the idea that results are best if they are definitive? Frankly speaking, I believe this idea is part of a cluster of ideas that impoverishes mathematics and mathematical culture.

I first thought about this when reading Bill Thurston’s 1994 article On Proof and Progress in Mathematics. In this article he contrasted how he approached his first work on foliations (resolve all questions, definitively!) versus his later work in geometry and the huge difference a more generous approach made in creating a rich, open, inspiring environment that many others got involved in, rather than the pinnacle of achievement that was admired from a distance.

Instead of maintaining a museum of monuments, we should propagate a countryside filled with rich, diverse gardens of ideas and a zoo of people tending and changing and expanding and creating new gardens.  While the first model leaves a trail of impressive facts, fit for admiration and worship, the second model is defined by engagement and inspiration for widespread creativity.

When Henry Helson visited Poland after the war, he was struck by the purity and simplicity of the mathematical culture that was also very generous. As he relates in his 1997 Notices article, Mathematics in Poland after the War, he was struck by the combination of generosity and fun that pervaded a culture that was serious about mathematics, but happy to publish things that did not aim to grab and own whole swaths of mathematical territory. Rather they published relatively short papers, each of which presented one new idea very clearly.

That exposition has been neglected, in spite of all the lip service to the contrary, can be seen in the response to the astrobites.org site, which has gained a lot of attention in the astrophysics community because of the large contrast between the high quality exposition that astrobites.org offers and the usual difficulty that non-experts have in reading scholarly papers.

I am now convinced that the high art of exposition should be valued as highly as the construction of brand new theorems, that publishing in such a way as to leave much to others is better than cleaning up an area and creating a monument: that what gets considered valuable mathematics ought to be greatly broadened. If anyone finds value — maybe because of explanations that require original thought, maybe because it brings the ideas to new audiences, maybe because it helps students see something clearly, maybe because it brings the understanding to the general public, and yes, possibly because it is completely original and surprising in construction — then it is valuable mathematics, worthy of the deepest respect. In this new model, the quality of the writing becomes very important. (I suspect that some will take issue with that statement saying that this is not a new model, but I will disagree and point to the enormous quantity of poorly written articles and books, some of which are also very valuable, even though they are not written very well. Of course, there are papers and books that are very, very well written. But it seems that this is considered a cherry on top, rather than something that should always, before anything else, be there.)

I am not urging that there be an effort to police exposition, but rather that this be given a great deal more attention at every level of education and practice. If we must have awards, let them go to those that have explained things well, have written things well. Better yet, train students to pursue the intrinsic rewards of doing anything well, from explaining derivatives to a confused calculus student to proving some new, highly technical theorem.

To encourage such changes, we would need to revisit how we reward and support the mathematical enterprise. This brings us to the consideration of the last two cultural components I said I was going to discuss: awards and federal funding.

Why do mathematics? For me, it is another form of art and at the same time, an exploration of the universe we live in. Knowing and understanding and explaining and inspiring others to do the same, exercises deep creativity and generosity; this is an occupation worthy of human beings that value themselves and others. Of course, there are an enormous number of occupations that can beneficially occupy the human mind and spirit. And each one can be as satisfying and beautiful and useful in its pursuit. By useful, I mean useful as an occupation, not useful as a tool to bend the world to my will. It is the occupation itself that is valuable. What happens to us and those we teach and share with, when we occupy ourselves (in a healthy environment!) is the greatest justification for any occupation.

From this position it becomes clear that awards and honors that many aspire to are actually a distraction. The reward is in the occupation itself. There are of course honors that have more to do with real appreciation rather than ranking and fame, and for such honors there is a place in a healthy culture. But the greed that masquerades in all of us as something more beautiful, seeks fame and fortune as a substitute for love and respect, whose lack actually gives room to that greed in the first place.

When the American Mathematical Society proposed the status of Fellow of the society, the negative side effects of such a program were pointed out rather eloquently by multiple individuals. In particular, I remember that Frank Morgan’s argument against the establishment of the program, and Neal Koblitz’ refusal of the offer of the status of Fellow. Of course, there is also the curious case of Perelman who refused the Fields Medal, the mathematical equivalent of the Nobel prize, whose recipients are given a demi-god status. For an interesting telling of the story and more, see Sylvia Nasar and David Gruber’s article Manifold Destiny in the August 28, 2006 issue of the New Yorker. (In the story, they quote Gromov, another prominent mathematician. Even though I very much doubt Gromov’s explanation of Perelmans refusal as a result of some great purity on Perelmans part, it is a story worth reading and thinking about.)

The influence of federal funding in mathematics, while it has enabled a great expansion of the enterprise, has led to a degradation of the culture, and not only in mathematics. It is well known that federal funding has turned academia into a serious addict, willing to do anything for the next fix of federal funds. That, combined with, spurred on by, the neglect of higher education in the public sector, has led to the very bad state of affairs in which grant money reigns supreme, fame (which can be turned into money!) comes second and teaching, for all the lip service it is given, occupies the lowest realms of academia. Proof of this diagnosis is not needed by anyone in academia (other than administrators who profit from illusions proposing some other reality), but if proof is needed, one need not look any further than the way adjuncts and instructors, who do a great deal of the teaching, are treated. Both in terms of the dismal pay and the insecurity of their jobs, we are saying that teaching is not what a university is really about — it is just what we have to do to keep up the charade.

But this is also where the tragedy lies; it lies in the immense impoverishment that results when teaching is not given top priority. It is a law of nature that real greatness, true stature, is proportional to the service to others that an entity or person actually provides. You may prefer to see this as my definition of greatness and stature. Either way, assuming this to be true, we have traded real nobility for a meager, greedy existence when we accept the perverted system of values that we currently have at research universities — and even, in some ways at teaching universities.

While small liberal arts college do in fact value teaching, they still take advantage of the situation generated by research universities and often pay their adjuncts obscenely low wages. It is tragic and funny at the same time that such colleges are usually full of people who think that businesses ought to raise the minimum wage, provide health care and longer paid vacations, and all sorts of other good ideas, but when it comes to the situation they have power over, they turn a curiously blind eye. But there is also this idolization of research universities, of elite institutions and this admiration pulls in some of the poison that they could otherwise easily avoid.

But, as I wrote in the previous post in this blog,  Learning to Think and to Act, research is a critical piece in education. It inspires and illuminates and brings a freshness and vitality that should be insisted on. On the other hand, research without teaching becomes selfish and elitist and aimed at goals that can at times be silly and irrelevant in their isolation.


 

What then, can we do? If the system is so far astray, what can be done?

In my opinion, the most powerful thing you can do is inspire change in your own sphere of influence by a focus on the place of freedom you actually have. Having your principles and philosophy aligned with life and love, and consistently acting in accordance with them, has always been the most powerful thing anyone could do.

Creative exploration and teaching, with a deep sensitivity for those that struggle; the pursuit of both pure and applied research, with generosity, and an acute sense for which applications are morally admirable; a discipline of simplicity, eliminating the pursuit of rank or awards or status or recognition — these are still the fundamental components of a culture worth immersing myself in, worth spreading to others. Taken together, they create a deeply rewarding occupation, an occupation that quietly, powerfully, moves us forward, and higher.

Learning to think and to act

I found William Deresiewicz’ book in a roundabout way. After reading an article in The Nation he had written, I read The Disadvantages of an Elite Education also by Deresiewicz and this led me to his book  Excellent Sheep: The Miseducation of the American Elite and the Way to a Meaningful Life.

The book is written empathetically, with a soul in plain sight. Whether you are applauding or arguing, you are engaged. During this personal conversation with the book, that turned into an email back and forth with the author, I decided to write something in response.

I also decided that I will now have all my students read this book. You might wonder why. After all Washington State University is not an elite university. Though we do have students who are brilliant and professors that are as creative and as interesting as those at any university, our university is not top ranked and is unlikely to be so anytime soon. It is true that those dedicated to innovative research and a deep, thorough education, can find agreeable environments here and there at this university. But that is by no means an across-the-board phenomena.

So why should I require my students to read William’s book? After all, it is aimed at students at, or thinking about being at, the most exclusive universities in the world.

The reason is very simple.

The elite universities that Excellent Sheep takes apart have a hold over the imaginations of just about everybody involved in higher education. Why? Because, even if you are not at one of these universities, you are strongly inclined to using those universities as a standard, a measuring stick. As a result, the elite universities end up infecting everybody else.

Tragically, the infection has been chosen by those lower ranked schools … it is a self-inflicted ailment, inflicted because of a lack of imagination and courage.

The freedom and breathing room that the lower ranked schools have — nobody is fighting them for their lower status — could be used to innovate and set a new standard of excellence. There are so many defects with what is considered elite that a faculty with imagination and vision and a disregard for status and tradition, could create something that would actually out-rank the elites by any natural, organic measure focused on real quality.


A Vivid Diagnosis

Excellent Sheep begins with 4 chapters in which the problems with elite education are outlined with frankness and clarity. The mad rush for students to become super-students, driven by parents that believe the Ivy League hype and encouraged by universities that have sold their souls to money, status and the illusion of greatness, has created a class of elite students that are maxed out,  stressed out, with little capacity for truly independent thought and little moral fiber. The vast majority of them have no real idea of who they are or what their own passions are. They console themselves with high paying jobs which in the end have little capacity for supplying them with purpose and the satisfaction that comes with following your own muse.

And they make terrible leaders: visionless, risk averse, conceited, and entitled. They are ill equipped to the jobs to which they aspire, that the world hands to them because they are “the best and the brightest”, a term that the author reminds us was invented to describe the technocrats who led us into the quagmire in Viet Nam.

After his rousing diagnosis and illumination of the multitude of problems with the elite schools, he transitions to his vision of what college should do for you and how he sees the humanities playing a big role in the re-imagination, the re-vitalization of education. These 6 chapters in parts 2 and 3 provoked the most thought on my part.

One part of his prescription for education centers around the idea that the humanities, taught correctly, teach students how to think, how to be skeptical and doubt the ideas and opinions they have accepted without critique. He explains how great books, with greatness defined organically and broadly, prompt thought and discovery and exploration leading to deeper self-discovery.  While he is not in any way claiming that this is new, his message that this is not happening at the Ivy league schools is something that is not well known.

It is here that I occasionally diverge, but not because I disagree with the general outlines of what he is saying. Rather it is in a few of the details and the extent to which he caries things. He simply does not go far enough sometimes. (Though I dare say that he goes further than almost anyone seems willing to take things.)


The Heart of the Matter

As will become clear in my own story, told later on, I believe in God.

Of course, exactly what that means is a long discussion. In fact it seems that saying you believe in God or don’t believe in God is almost a statement without information, at least if you think about what you believe.

Where this becomes important to this essay is in Deresiewicz’s acceptance of Gould’s idea that the arts and humanities on the one hand, and the sciences on the other, are separate magisteria. I believe this is wrong, that in fact the spiritual realm ties everything together and a God that creates is the beginning of wisdom in the search for an explanation of the ultimate unity of everything.

Of course enormous damage has been done to the conversation that should happen here, both by the believers and the unbelievers. In fact, it is hard to overstate the extent of this damage.

But if one can find quiet spaces in which to discuss and examine these questions, the questions can begin to be seen as an attempt to draw out an understanding that allows both the believer in a God that creates, and a believer in a universe without God, to benefit from each other’s insights.

The quietness and respect and time to think and observe that this enterprise takes, is founded on emotional health. This is where the real problems often lay. Because of the enormous damage that dogmatic ideologies and religions have done or threaten to do to us, we often find it very hard or even impossible to enter discussions with the patience and quietness necessary for such conversations to deepen and enlighten.

But where those conversations can happen, the effect is very powerful.

And it is precisely this environment that we should find in college — an environment where true diversity is respected and encouraged and challenged and supported. Free and thoughtful discussions that illuminate the mind and soul do not need, and in fact are damaged by, the force of dogma, ridicule, combative attitudes and the inability to listen because you have found the truth. Trusting this and boldly engaging in such an enterprise enables us to learn from each other, not just shout at each other. The blunt instrument that science and scholarship devolves into in an adversarial environment, would show itself to be a subtle revealer of mysteries in the environment characterized by love and respect. For love is the only thing that truly moves us to a place of progress.

Love does not imply agreement — our experiences in our families can teach us this. And it is not something that gets in the way of freedom, though twisted conceptions of love could tempt you to believe otherwise.

The freedom that such an environment gives and inspires, begs to be filled up with a rich curriculum covering thought and action in a broad way. In addition to classes and seminars, there would be maker spaces in every subject, jobs for students that range over a widest possible directions and a culture that made working and serving, alongside rather than from above, the norm.

Making and maker spaces, though they are in vogue in some corners of many universities, predominately in engineering departments, have yet to become truly integral anywhere, and that includes engineering departments. Yet, turning thought into tangible action is very valuable for students, if for no other reason than the intellectual and spiritual benefits of the manual crafts, as pointed out in Matthew Crawford’s book Shop Class as Soul Craft.

There are many reasons for teaching all students both academic expertise and a manual trade. To begin with, their way in the world would be much more sustainable, much less fraught with economic peril. Yet even this immediate effect would give them a sense of freedom in their academic pursuit since they would never need to compromise with a job that was nominally aligned with their expertise while actually being a betrayal of their muse or their morals or both.

Yet there are deeper reasons for pursuing manual training in parallel with the more apparently intellectual pursuits. Exercising a creative manual skill is the perfect counterpoint to intellectual pursuits even if only for the rest and deep satisfaction of producing tangible, visible results that are also useful. Yet there is more. The exercise of the faculties used to do practical work also broadens the mind and strengthens key abilities which in turn give us a much more robust approach to problem solving in the more overtly intellectual arenas. In my own experience, that I cover in a bit more detail below, building and tinkering played a significant in teaching me how to solve problems that cross disciplinary boundaries, as most real problems do.

Of course, the variety of things that one can do along the lines of skilled trades is very large and certainly not confined to things one does in a shop. But all of them give you an ability to be less dependent on others for your economic security, both in terms of what you can do for money and what you can do without so much money. If this is combined with a choice to live simply, on less rather than more, to avoid student loans (or in the very least refuse loans that cannot be nullified with a bankruptcy), students gain even more freedom. In not surrendering their own freedom, they are equipped to encourage others to live an equally simple, inspired life.

Along the lines of simplicity and inspiration, there is the matter of having and taking the time to think, as well as the related issue of the overloads that colleges encourage.

I consistently advise my early doctoral students to take at most two classes per semester and fill the rest of the required hours with research credits which are designed to deepen the studies in the classes they are taking. I do that because I can do that. But undergraduates often take 4-6 courses of which 3-4 might each be worthy of their full attention during the semester, and there is little I can do about that. Needless to say, they skim the surface and do not begin to master the subject. Of course, sometimes a whirlwind tour is sufficient, but whenever real thought and effort are merited, the overload takes it toll and mastery or even a touch of depth eludes them.

If this were to be addressed in a meaningful way, in parallel with real efforts to help the students find their muse, we would need to reduce the number of courses significantly and deepen the courses they did take by a significant amount. The result would be revolutionary.

If we moved away from grades to evaluations and portfolios, so that someone wanting to know something about a student would have to look at the students work, not just some set of grades or even worse, a single measurement like the GPA, we would encourage real depth and mastery.  There would be real incentives to think about what they were doing and act on the inspiration that followed.

An example of the kind of problem we are up against can be illustrated by the case of the elementary undergraduate linear algebra course at WSU. The course is a 2 hour course because the engineers did not want to waste precious credit hours, precious thanks to the accreditation requirements to which they are beholden. But the organic reality is that for anybody doing anything computational, linear algebra is arguably the most important mathematics class they will take. To do it justice, in line with what advantage mastery of the subject will give them, they should be taking a 6 to 8 hour course. Almost every applied calculation ends up requiring some linear algebra, with many problems requiring a lot of linear algebra. Yet politics between mathematics and engineering combined with the shackles imposed by accreditation generated a 2 hour class.   And as a result, most of the students that have taken linear algebra have little to no mastery of what may be the most important mathematics they take.

This is not unusual. Instead of doing what makes sense, we do what some set of people have decided is important, even though they are far from the facts and realities.

Of course, some of this is simply in the air — it is the spirit of the age to give yourself no time to think, to fill all your space with sound and action and tweets and email and messaging with facebook or instagram. Because of the way this swallows up personal, quiet spaces, it should be the first task in college to teach the students to repossess their own minds and souls. They have to take back time to think and see, and hear what the quiet has to say to them. If their could be one thing that you would ask your students to give up, it would probably be their mobile devices. As unrealistic as this might seem, an honest assessment of the situation would make it clear that these devices are robbing many students of the ability to think and focus deeply.

What I have described above boils down to a gentle, bare-handed exploration of ourselves and the universe, unmediated by electronic devices, unnarrated by our culture, unaccompanied by music through our earbuds, in an environment rich in quietness and time to think, broadened and deepened by experience with skilled trades and frequent, face to face interactions with other human beings who know how to listen. Such an environment would produce educational results of a very different nature than the ones we currently see.


Inspired Learning

If colleges understood research and teaching to be something that is far broader than is now imagined and practiced, then it would be discovered that research and teaching are not at odds, that each can enrich the other and that the stranglehold federal funds have on academia need not continue.

Currently what is valued is papers and external funding and if you had to choose one on which to bank your hopes of getting tenure, it would be external funding. At schools where teaching is the focus, this is not the case. But those schools usually look to research universities with admiration, so that to the extent there is change at those schools, the small steps here and there that can be seen reflect this misplaced admiration.

What would a balanced, sustainable attitude towards teaching and research be like?

To begin with, exposition of well known results and new research results would be highly valued. Ideally, every result would be accompanied by three expositions. There would be one which was a careful, well written record that other researchers could read and understand. Then there would be something that advanced undergraduates and graduate students could read and gain most or all of the picture from. Finally there would be something that was designed for the complete non-expert who is nonetheless inquisitive and motivated to know something about the area.

Right now we often have only the most inaccessible of the three expositions, the research paper. And that is often poorly written and primarily intended stake out territory and give the authors credit for having written a paper. Another goal is to have others cite the paper, to give the authors work “impact”. Supposedly the impact that everyone wants is about dissemination and scholarly, or even societal, benefit.

Yet, if real dissemination and wide benefit were the goal, careful, highly accessible expositions would be considered critical.

Some would argue that talks at conferences serve the purpose of part or all of the exposition I am advocating. But anyone acquainted with conferences and conference talks would know that these talks rarely transmit knowledge to anyone that doesn’t already know almost everything in the talk!

An example of the power of taking the time to explain can be seen in the blog, started by graduate students in astrophysics, focused on explaining the papers that appear in the astrophysics section of the archive at Cornell, arXiv.org. The blog, astrobites.org is a beautiful example that deserves to be imitated in all areas of scholarship.

It is widely understood that research in the widest sense (forget now about publishing or impact) is part of what makes a teacher a truly inspiring teacher. The attitude that explores, that innovates and creates playfully, that asks questions and tries to answer them is critical for the best teaching. Valuing this broader definition of exploration and research would go a long way towards bringing truly high quality teaching and research together.

Some professors are very good at mentoring whole crowds of students, others are very good at explaining very subtle, advanced ideas in classes, yet others are good at running hands on explorations of known and new ideas and environments. And there are many other ways in which professors contribute deep value, if value is measured in a natural, common sense fashion. But the current reward system rewards very little diversity, instead trying to force everything into narrowly defined research or teaching boxes. The result is that the system we have is biased against teaching and towards an insular, largely irrelevant, industry of research. And when it is not irrelevant, it is often beholden to some corporate or defense funded interest.

The low status that teaching has at many schools can also be seen in the way adjuncts are treated. The pay is criminally low, with little to no job security. As many others have noted, this reality makes a mockery of the claim that teaching is a top priority.

Approaching the integrated teaching and research mission more imaginatively, we might consider a system in which all of the teaching staff were tenured faculty, but where the roles they played were as varied as the individuals that made up the faculty. This would be easier with an administrative structure that was grass roots and not top down, but that would be an advantage and not a drawback. If adminstration were tasked with support, and not supervision, this would go a long ways towards eliminating the unhealthy feedback that has strengthened the current damaging definitions of research and teaching. Such a reconfigured support system could easily be directed by an inspired faculty to support a rich, new vision for integrated research and teaching.

And that would be something to get excited about.


Courage to Innovate

And, coming back to the pernicious influence of the elite schools on everybody else, if schools cared little for reputation or accreditation or status that depended on them bowing down and abdicating their own ability to innovate and imagine an effective path to education, we would have a huge diversity and richness in choice when deciding how and where to pursue the education that fits us. For every style of thinking and learning and doing, there would be a place where we could go to learn to think and act, a place where we could become more rather than less.

When I returned to a university setting full time after ten years at a national lab, I was surprised to find how little imagination many professors were willing to exercise to improve their situation and the situation of their students. I also found professors that actively promoted the idea that their university was inferior and that good students should go elsewhere.

I now realize that much of this is the result of a fatalistic acceptance of an environment in which innovation and common sense changes that are within reach, are obstructed by completely visionless, top-down administrative structures. Such systems are presided over by administrators that get to where they are not through the exercise of vision, but through risk aversion accompanied by a conservative point of view that stifles creativity and innovation.

(Are there administrators who do not fit this unflattering characterization? Of course. But they are a very small minority and are effectively neutralized by the effect of the rest of the system.)

As mentioned above, I think a part of the solution is a complete revision of the role of administration, from supervision and direction, to simple support with no supervision. The faculty, thus empowered could innovate and make the changes needed to reward the naturally occurring diversity that would keep things thoroughly inspiring and alive.

One could imagine a structure that was lean enough in non-teaching, non-research expenses that a tuition of 15-20K$ per year, if supplemented with donations that simply supported the infrastructure and equipment, would be sufficient to fund the operation of the school. Such a school would depend on the community for housing and small industries to capitalize in students learning trades, as well as student labor to keep the college running.To keep things focused on the right priorities, federal funds would not be allowed and students would not be allowed to use loans whose repayment was immune to bankruptcy. Instead, private donations for infrastructure and equipment and supplies would be sought, and innovative strategies for student funding would be pursued and supported. Choosing the moral high ground, some avenues of research would be avoided, but the freedom that this brought would be worth the price. (Such a funding model would also eliminate some of the mostly costly areas of research. This would be an acceptable price, especially since much of the most costly research is of questionable societal value anyway.)

By reducing the number of classes the students took, deepening the ones that remained and offering a rich profusion of enrichment experiences giving students exposure to ideas and activities outside their areas, students would experience depth and breadth in healthy balance. The enrichment activities could range from talks by visiting scholars and faculty to hands on activities that brought students into intimate contact with skilled trades. Using graduate students to help teach and mentor, one could have an environment that encouraged teaching and research that were integral, even inseparable.

Of course, the idea that one could operate a college with a much smaller overhead than is usually the case depends on greatly reduced services that have little or nothing to do with education. While there would be no reason why physical activities would need to curtailed, traditional sports would be absent. The legal structure of the school might be one of a cooperative that was supervised by the faculty and supported by a very small set of support staff and a larger contingent of students. But there would be many other options for the organization.

Graduate students, given free tuition and a small stipend to live on, would be expected to be full participants in the integrated teaching and research mission of the university. The suggested undergraduate tuition levels above would support two to four graduate student positions per professor which would be about right in that, this would translate into about one graduate student graduating every year or two for every professor, assuming an average time to graduate of 4 years.

A truly novel feature would be the presence of students in the skilled trades who would be full members of the community, along with those that were teaching those crafts. This would lead to opportunities and advantages that would enrich the university in many ways.

Where would the faculty to staff such a place come from? How would those who had been trained by such a dysfunctional system be able to guide and power such a unconventional approach to education? While it is true that many professors have let themselves be stunted by the system and robbed of their vision and idealism, most have small sparks that could be nurtured back into the enthusiasm that once motivated them. And there are always professors in the system who have never surrendered their vision, who would welcome the chance to be a part of something creatively alive and imaginative. Even if there are only 1% of the professors that are out there that would opt to be a part of something like this, that would be more than enough to get a movement going. If you count the graduates who have left academia because they cannot abide the state of academia, you have many more qualified candidates for something new and different.

Likewise, how would we find students interested in committing to a small, unconventional university? Where would you find these individuals that did not care about accreditation or reputation or status, who had the maturity to recognize that those concerns are separate, and often diametrically opposed to, real excellence and depth? I believe the answer is the same; There would be a large absolute number of students willing to make the commitment even though the relative proportion of all students might be very small.

This minimalistic description gives an idea of the kind of thing that could be created if there were a small group of people that shared the vision. Of course how it all worked out would be a function of exactly who the founders were, but letting go of the dysfunctional form that academia has evolved into and embracing a revitalized, integrated vision for teaching and research, it is certain that the students and faculty of the resulting organization would not lack a sense of purpose and accomplishment. Happiness would follow, as it always does when basic human needs for face-to-face connection and creative productivity are met.

 


Ready For College?

If everyone took 2-3 years to work and travel after high school, doing work that was useful to someone and travel that taught them independence and benefited others as well, students would arrive at college with a maturity level that enabled them to get a far greater amount out of the experience. This would also go far in counteracting the enormous waste of time and money that happens when students come to college to party and get a job certificate.

When my son graduated from high school we strongly supported his decision to take a break and it is something that I recommend to everyone that will listen.

My experience with returning students is that overall, they are all better off for the break. They return with a sense of purpose and determination to spend their time and money well, to get maximum value from their college experience. They are also much more inclined to heed advice or at least listen carefully before finding their own version of the right thing to do.


Making it personal

My own story is one of extensive meanderings that I now recognize were critical pieces of my education. It began with homeschool and music and parents with very broad interests and experience, combined with the southern New Mexico environment filled with eccentrics, who, contrary to apparent belief today, did not actually encourage me to believe in perpetual motion or the hollow earth theory (both of which were believed by people I knew). Instead, that environment gave me freedom to explore and theorize in my attempts to understand the universe, God and how everything operates. Even though he was a vocal artist and teacher, my father was also able to do almost anything with his hands and because of this my brother and I grew up building and tinkering, which both of us have continued to do in our own substantial shops.

Pausing for a moment on this point, it is important to note that in addition to choosing not to have television, my parents provided all the resources for us to create and invent and fabricate things that we imagined. Eventually we learned trades — I learned piano tuning and my brother learned automotive technology. Though it has been 35 years since I tuned a piano, the habit of working with my hands, of creating things in my shop, has stayed with me to this day. The same goes for my brother who now has other people work on his cars, and instead spends his spare time creating impressive works of art from wood and metal in his shop.

We moved to Eastern Washington to attend Walla Walla University, a high quality parochial school, filled with students and professors that could have been at places with higher reputations. After college, in an attempt to deal with the trauma of losing both parents, there was more wandering that led through graduate schools, divorce, oceanography research, life in a little cabin in the hills above the Santiam River, a research seed farm, construction, consulting, 7th and 8th grade teaching, research in a medical school, remarriage and graduate school in Portland, Oregon.  This led to Los Alamos where I started my career as a mathematician, in an environment that was ideally suited to someone with ability and energy who had, so to speak, come out of the woodwork. (This is the biggest tragedy in the decline of the national labs — they were places where good ideas and ability got you somewhere independently of where you were from. They were also places with an almost unique ability to nurture very high quality, inter-disciplinary work.)

Between the wandering and the graduate school in Portland, there was a conversation with a cousin. I had just remarried and was running a small research lab in the medical school in Portland. We visited his small farm in Oregon and he noticed that I was emotionally out of sorts, not peaceful about something. He remarked that when I had lived down in his area (that little cabin above the river) he had noticed that every time I went on walkabouts in the hills and mountains, I could leave in a state of anxiety or under some emotional cloud, but I would return with peace and clarity. He said, “You should do this every day!”

I took his advice to heart and upon returning to Portland started taking long walks in the woods and forests where I lived. On those walks I discovered who I was and what my passion was. I had found my muse. In conversations with God (my atheist friends have alternate, though sympathetic descriptions for what I experienced), I began to see things in a different light, finding that there was a living path into whatever you wanted to do, one that was as different from the usual career trajectory as a rich, living garden was from a herbarium with its dried and described plants.

I returned to graduate school with an entirely new perspective (and a new baby son).

Since then there has been evolution in thought and perspectives, but the experience in the woods and forests remains pivotal. I believe that the experience explains why I have often approached situations with a different viewpoint, believing that there are many ways to solve problems, that there is usually something good to be preserved and yet that is no reason to insist on keeping the bathwater with the baby.

The experience is also the reason that I view my job as a professor and mentor very broadly. i see it as my duty to encourage students to find their own muse, to listen to talks like David Levy’s No Time to Think, to read books like David Shenk’s The Genius in all of us and Buscaglia’s Love,  to get some of their news from places like Truthdig and the Real News Network, to publish open access papers and think carefully before giving up copyright and think deeply about what it implies before you accept money from places like the defense industry or the security industry. In addition to teaching geometric measure theory  and nonlinear analysis and other fascinating subjects, in addition to guiding dissertations and projects and interactions with industry, it is my duty to prompt them to think, to live examined lives and settle for nothing less than wholeness and emotional health. In that way, and only in that way, will I have helped set them free to travel and thrive on a sustainable, living path.

The conclusions they arrive at may be very different from mine, but then, thinking in unison is never a good goal. What I do know is that they will have the tools, not only to adapt and thrive, but also to correct and restore and recover from the mistakes that they will inevitably make.


Inspired and Provocative

I found a variety of reviews of Excellent Sheep when I was reading the book and not surprisingly, some people loved the book, other hated it. One that I enjoyed quite a bit was James McWilliams’ review, Why Did ‘Excellent Sheep’ Alienate So Many Readers?, which appeared in the October 2014 Pacific Standard. Like James, I believe that it is probable that those who disliked the book are those for whom the book hits too close to home. Though a thoughtful reading of the book will often inspire vigorous discussion, it seems to me that such a reading will also recognize that the book is long overdue, that the author has the experience to make the book worth reading and that attacks on the book reveal more about the attackers than the book.

But even those attacks on the book are useful, for they remind us that the emotions, acknowledged or not, can easily overwhelm everything else, irregardless of how much sophistication and skill one uses to try to disguise the fear or pain that drives our responses. And if we read those other reviews sympathetically, they will remind us that we are all susceptible to these reactions and complicit in a society that lets fear rob us of deeper insight and deeper lives.

To rob fear of its prey, to turn around the slide into the illusions of our modern age, we must first understand the guiding delusions and then direct our energies towards inspired, counteracting  goals. As an antidote to the current delusions and an inspiration for change in higher education, I know of no better book to begin with than this book by William Deresiewicz.

 

Heresy and Freedom

Reading the words of Albert Schweitzer and bits of the life of Roger Williams is both inspiring and motivating: inspiring because they were both independent thinkers and motivating because their writing and lives ask for reflection and response.

I especially like the Epilogue in Out of My Life and Thoughts, Schweitzer’s autobiography; It seems more important today that it could have seemed in 1931 —

I am in complete disagreement with the spirit of our age, because it is filled with contempt for thought … The organized political, social and religious associations of our time are at work convincing the individual not to develop his convictions through his own thinking but to assimilate the ideas they present to him. Any man who thinks for himself is to them inconvenient and even ominous. He does not offer sufficient guarantee that he will merge into the organization. Corporate bodies do not look for their strength in ideas and in the values of the people for whom they are responsible. They try to achieve the greatest possible uniformity. They believe that in this way they hold the greatest power, offensive as well as defensive.

Yet I disagree with Schweitzer’s theology on significant points: I believe in Jesus the Messiah, that he was in fact God and Man, the His death did provide a way through annihilation for all of creation and a path to eternal life, that he was raised on the third day, that he is coming again. Further, I believe in the literal creation of the earth by the actual word of God and I believe in a struggle between God and Satan/evil centered on the existence of free will and God’s claim that free will is harmonious with good and life. And I believe that only in that context — of a cosmic struggle, that history begins to make sense. But I do not retreat into magic for explanation (the first part of this paragraph not-withstanding), neither do I believe that Jesus came to establish a religion, nor do I condemn in any way those that fail to believe as I do.

All this clearly begs for a much longer discussion, though more immediately it most likely triggers strong responses in many readers.

The facts that I have so many differences with what has become traditional Christianity — to the point that I find most of the dogma of Christianity to be unhelpful,  and that I am clearly at deep odds with the religion of science (and that is precisely what science has become — a religion) , often lead to me feeling isolated and politely ignored.

Which brings me to Roger Williams (1603-1683), one of the early proponents of true freedom in the new world. His life was one of innovation with concrete results that continue to this day. In addition to insisting on true freedom of conscience, and as a result being exiled to Rhode Island that he later obtained a charter for from the King of England, he was a friend to all those that did not fit in with the dogma and narrow way of life prescribed by the Puritans. Yet his defense of freedom to choose what we believe was not founded upon an idea that truth was somehow elusive. Neither was that freedom relegated to only those areas in which he had no strong opinion. Rather, it was his position that liberty of conscience was fundamental and God given. As a result, it was something that men could not withhold without grave consequence. And it is this that made his ideas so powerful.

Moving forward to the present, I find it disconcerting that the liberals and progressives with whom I agree on many fundamental principles, have such trouble applying what they believe when it comes to acceptance and tolerance as a practice rather than a theory. It reminds me of the same sort of dissonance between the theory and practice of Christianity.  There are of course many kind and respectful (and therefore tolerant) Christians and many similar liberals (some of whom are Christian) — in fact they might even be the majorities. But the minorities in these cases are loud and, sadly, often unchecked by their more reasonable brethren. And in both cases, underneath, separate from the individuals, there is that institutional desire to bend the wayward ones towards the “truth”. To me, this is the hallmark of a formal religion — the desire to bend the will of others to my way of thinking, the fear that somehow if others do not believe as I do, very bad things will happen to my tribe, my organization, or even me personally.

Ridicule that those who feel superior heap upon the “unenlightened” follows quickly. The stupid creationists and the unsaved, deceived evolutionists, the godless atheists and the ignorant bible-thumpers. And so on and so forth. Resorting to labels instead of real conversations based on a deep respect for life, freedom and a shared humanity, we retreat into our own smaller tribes. We view others through a montage of the worst of the loud proponents of the “others”. And because of the fear we gather and cultivate, we do not seek to find grounds for conversation.

Of course, there are terrible ideas and histories that beckon us to the side of fear: the atheist Stalin generating extreme suffering, the (unbiblical) doctrine of hell and the horrendous implications of what that would mean about God, the deep loss of freedom that results for various people on the fringes when socialists get the upper hand, the equally deep loss of freedom for the poor and the economic heretics when unfettered capitalism holds sway, the demonic inhumanity of the inquisition and the horrors of the Nazi experiment. These are indeed strong inducements to side with fear.

Yet there are also examples of real freedom and the alternative power of discourse based upon deep respect, not the least of which includes the life of Roger Williams. The path to  connections which illuminate and defeat fear all involve listening and talking and writing and reading and thinking and more listening and talking and listening … conversations of the right type are at the center of this model of positive change.

While I do believe that some conversations only work in very small settings due to the complexity of the subject matter and the fact that these conversations trigger deep fears that can shut down thinking and spiritual insight, there is still room for writing things down, for conversations that are slow and deliberate and open. Even though it is unwise to “cast your pearls before swine” — meaning that communication without the preparation of a history of respect and real connection can be worse than pointless — one can and should opt for the hope that there are those out there that do in fact seek real conversations, that are seeking to explore and will appreciate co-travelers, co-explorers. As long as we do not try to speak to each other from on high, from the perspective of one who is enlightened, but rather from a position alongside others, ready to listen and see, especially when in disagreement — so long as this is true, we will find ample opportunity for connection and progress.

To disagree is the extent to which my freedom, when combined with respect and love, permits me to fall out with others. And in this version of falling out, there is no reason that the conversation has to stop. While some ideas and ideologies must be restrained from crossing the line and actually hurting others, that line is not something that we are usually dealing with when focused on the present, local circumstances, whatever they are.

But to disagree and ridicule and belittle, to disagree and use force of any kind — spiritual, intellectual, physical — to try to bend others to my will, is a serious abuse of freedom. In fact, I believe that this abuse is the root of many problems in the present and the past, but that is another discussion.

So while I disagree with the theological conclusions of Albert Schweitzer, I also find deep inspiration in his writings. I find that in this falling out, nothing is lost and much is gained.

I think that Roger Williams would approve.

 

Using Photography

I am building a website for the Analysis + Data Group that I am helping establish at WSU. I am trying something new, in order to communicate to potential student recruits much more than a usual mathematics website communicates. I want students who visit the website to begin to get a feel for how we think, who we are, even what it is like to think with us and learn with us. To do this I am partly using non-standard (for mathematics) photography: no mugshots allowed!

(A little about the group: there are five principal members — myself, Bala Krishnamoorthy, Haijun Li, Charles Moore, and Alex Panchenko — and about 12 Graduate Fellows in it — that number will firm up this fall. We will focus both on pure analysis and applications of insights from analysis to data problems.)

Here are a few of the photos I have already taken (or my son Levi has taken of me), that I will be using in the new website.


20140612_144412-wide-short

Alex Panchenko, who combines insights from nonlinear analysis and statistical physics in his work in analysis/applied analysis.


20140612_173724-cropped-1

Chuck Moore, who works on problems in PDE and Harmonic Analysis.


20140613_180757-cropped-1

KRV, about to explain 4.2.25 to Levi (and Obi).


This project has revived my interest in photography. As a result, I have also been doing some macro work, on my walks with Obi.

20140629_124341

20140629_125157

The last picture, of the aphid farm being tended by the ants, pushes the limits of the Samsung S-4 phone I have been using to get these pictures. So it has prompted me to get some new equipment. In that process I discovered Mathieu and Heather’s wonderful blog on  mirrorless cameras and photography MiirrorLessons. They introduced me to the electronic photography magazine Inspired Eye  which I also recommend without reservation. The two founding editors of the magazine — Olivier Duong and Don Springer — have precisely the right attitude/philosophy. That philosophy results in an environment that is rich and generous, a fact that is made abundantly clear once you read and  experience the resulting publication. They get art, in a fundamental way. That might seem like a funny statement since the magazine is about photography, and mostly street photography at that. But I stand by what I said — they get art in a way that very few do.

And life is about art, be it creative work in mathematics, or the way one makes food, or the way we (can) relate to others, or how we think and write. While it doesn’t upset me any more, I still protest when people express the idea that mathematics is somehow a non-creative non-art. Those kinds of statements are my cue for very gentle, non-forceful illumination.


I will post a link to the new group website when it is released in a few weeks.

Beginning Again

I am beginning to see tiny, yet brilliant slivers
of something that happened at the Cross that was so
enormous, so full of wonder, so full of illumination,
so powerful that it brings a deep stillness to everything
it touches.

A defeat of death and sin and all that is evil, a defeat
so complete that time held its breath, so awesome that
everything changed in that instant.

The tearing of the veil was the beginning of a tearing,
of a cracking, a disintegration of all the separates us
from God.

We were at that moment in a completely new era.

Time had begun its transformation, back to where it
began.

Face to face, in the silence that sings and heals, with
words that speak without words, nothing is withheld.

Nothing is withheld.

---

KRV

An Invitation to Geometric Measure Theory: Part 1

While there are a variety of article-length introductions to geometric measure theory, ranging from Federer’s rather dry AMS Colloquium Talks to Fred Almgren’s engaging Questions and Answers to Alberti’s Article for the Encyclopedia of Mathematical Physics, I will take a different approach than has been taken in any of these and introduce geometric measure theory through the vehicle of the derivative.

The Derivative, Geometrically

The derivative that is encountered for the first time in calculus is defined as the limit of a ratio of the “rise” over “run” of the graph of a function. For y = f(x), this becomes

\frac{df}{dx}(a)=\lim_{x\rightarrow a}\frac{f(x) - f(a)}{x-a}.

This is visualized as the slope of the secant lines approaching a limit – the slope of the tangent line – as the free ends of those lines approach (a,f(a)). This is illustrated in the first figure.

secant

The derivative as \hat{L}_a, the optimal linear approximation to f at a, is another, very useful way to think about the derivative. Here, we focus on the fact that the tangent line at (a,f(a)) approximates the graph of f(x) at (a,f(a)) as we zoom in on the graph. More precisely, writing x = h+a,

f(x) = f(h+a) = f(a) + \hat{L}_a(h) + g(h)h,

where \hat{L}_a is linear in h, g(h)\rightarrow 0 as h \rightarrow 0, and the tangent line L is the graph of the function y = f(a) + \hat{L}_a(x-a) .

Exercise: use the facts that (1) linear \hat{L}_a:\Bbb{R} \rightarrow \Bbb{R} have the form h\rightarrow sh, s a scalar, and (2) g(h) \rightarrow 0 as h \rightarrow 0, to rearrange this last equation for f(x) into the original definition of a derivative.

Using the equation above to get

\left|f(x) - (f(a) + \hat{L}_a(x-a))\right| \leq (\sup_{|s|\in[0,\epsilon]} |g(s)|)|h| for  h\in[-\epsilon,\epsilon],

we are able — after some work (see the exercise below) — to get this nice geometric interpretation:

tangent-geometry-1

The figure illustrates the fact that the graph of f(x) lies in cones centered on L, whose angular widths go to zero as we restrict ourselves to smaller and smaller \epsilon-balls centered on (a,f(a)). Inside the \epsilon_1-ball, the graph stays in the wider cone, while in the smaller, \epsilon_2-ball the graph stays in the narrower cone.

Let’s restate this. Defining

  1. p \equiv (a,f(a)),
  2. B(\epsilon) to be the ball of radius \epsilon centered on p,
  3. F\equiv\{ (x,y) | y= f(x) \},
  4. C_L(p,\epsilon) to be the smallest closed cone, symmetrically centered on L, with vertex at p such that F\cap B(\epsilon) \subset C_L(p,\epsilon), and
  5. \theta (\epsilon) to be the angular width of C_L(p,\epsilon),

we have that

f is differentiable at a \Leftrightarrow \theta(\epsilon) \rightarrow 0\text{ as }\epsilon \rightarrow 0

Here is a figure illustrating this:

in-cone

Exercise: provide the missing details taking us from the above inequality bounding the deviation from linearity to the above statement that {f is differentiable at a \Leftrightarrow \theta(\epsilon) \rightarrow 0\text{ as }\epsilon \rightarrow 0} using the facts that (1) the above inequality defines cones that are almost symmetric about L and (2) the \epsilon-ball centered at p is contained in the vertical strip (x-a,y-f(a)) \in [-\epsilon,\epsilon] \times(-\infty,\infty)

With this shift to a geometric perspective, we are now in a position to take a step in the direction of geometric measure theory.

Note that in our definition the cones contain all of the graph as they narrow down and we zoom in. But what if all we know is that a larger and larger fraction of the graph is in a narrower and narrower cone as we zoom into p? That is precisely the idea that approximate tangent lines capture. We will introduce two different versions of the concept.

Densities as a path to an approximate tangent line

Tangent Cones

The tangent line discussed above is also the tangent cone. The tangent cone of a set in \Bbb{R}^n can have any dimension from 1 to n. For nicely behaved k-dimensional sets, the tangent cone will also be k-dimensional. In the case of the usual derivative of functions from \Bbb{R} to \Bbb{R}, we are working in the graph space \Bbb{R}^2 with 1-dimensional sets. Moving to tangent cones, we can approximate one dimensional sets which are not graphs or, more generally, arbitrary subsets of \Bbb{R}^n.

We now define the tangent cone of F\subset\Bbb{R}^n at p.

To obtain the tangent cone, begin by translating F by -p. (This moves p to 0.) Define F(\epsilon) \equiv (F\cap B(\epsilon))\setminus p. Use a projection center at 0 to project the translated F(\epsilon) onto the sphere of radius \epsilon centered on 0. Take the closure of the resulting subset of the \epsilon-sphere. Finally take the cone over this set. Call this set T_p^\epsilon(F). That is,

T_p^\epsilon(F)=\{\Bbb{R}\geq 0\}(\text{Closure}(\cup_{x\in F(\epsilon)}\frac{x-p}{|x-p|})).

Now define the tangent cone of F at p to be the intersection of T_p^\epsilon(F) at any sequence of \epsilon_i‘s going to zero; \epsilon_i = \frac{1}{i} will do. Thus the tangent cone of F at p, T_p(F) is given by:

T_p(F)=\bigcap_i T_p^\frac{1}{i}(F).

Here is a figure illustrating the key idea:

tangent-cone-1

Note: the tangent cone is centered on the origin, 0, but I will be plotting it as though it were centered on p. Similarly, the tangent lines will sometimes be thought of as linear subspaces (i.e. centered on the origin 0, and other times as the shift of that linear subspace to p.

In the case of a differentiable function f:\Bbb{R}\rightarrow\Bbb{R}, this tangent cone is the usual 1-dimensional tangent line.

Densities

Now we need \theta^k(\mu,F), the k-dimensional density of F at p.

Define \omega(k) such that it agrees with the volume of the unit ball in \Bbb{R}^k when k is an integer (there is a standard way to do this using \Gamma functions). Let \mu measure k-dimensional volume. Typically this will be k-dimensional  Hausdorff measure, \mathcal{H}^k. Whatever intuitive idea you have of k-dimensional measure is good enough for our purposes. (At the end of this post I also define Hausdorff measures more carefully.)

Now, \theta^k(\mu,F) is given by

\theta^k(\mu,F)=\lim_{\epsilon\rightarrow 0}\frac{\mu(F\cap B(\epsilon))}{\omega(k)\epsilon^k}

when this limit exists. When the limit does not exist, we work with the limsup and liminf of the right hand side which are called upper and lower densities of F at p and are denoted by \theta^{*k}(\mu,F) and \theta^k_*(\mu,F) respectively.

Approximate Tangent Cones

We now define the approximate tangent cone at p to be the intersection of closed cones whose complements intersected with F have density zero at p:

\tilde{T}_p(F)=\bigcap\{\text{closed cones }C\text{ with vertex }p|\theta^k(\mu,(\Bbb{R}^n\setminus C)\cap F)=0\}

Originally (in this section), we were aiming at having a definition of approximate tangent line that was invariant to (small) pieces of the set F outside the sequence of cones, provided those pieces got small enough, quick enough. Now we can make that more precise. We want a definition of approximate tangent line that ignores such excursions of F provided these excursions have density zero at p. Rather anti-climatically then, here is the definition we have been waiting for (though you might have already guessed it!)

A 1-dimensional set has an approximate tangent line at p when the approximate tangent cone is equal to a line through p.

When the curve is an embedded differentiable curve, the tangent line and the approximate tangent line are the same.

Remark: in general, when we are dealing with k-dimensional sets in \Bbb{R}^n, we will get approximate tangent k-planes.

Exercise: can you create examples of one dimensional sets which have a (density based) approximate tangent line at p but not the usual tangent line at p?

Exercise: prove that a tangent line to a continuous curve is also the (density based) approximate tangent line at p.

Integration as a path to an approximate tangent line

There is different version of approximate tangent k-plane based on integration. (The one dimensional version is of course an approximate tangent line.)

We start with the fact that we can integrate functions defined on \Bbb{R}^n over k-dimensional sets using k-dimensional measures \mu (typically \mathcal{H}^k). We zoom in on the point p, through dilation of the set F:

F_\rho(p) = \{x\in\Bbb{R}^n | \;\;x=\frac{y-p}{\rho}+\text{ p for some }y\in F\}.

We will say that the set F has an approximate tangent k-plane L at p if the dilation of F_\rho(p), converges weakly to L: i.e. if

\int_{F_\rho} \phi d\mu\rightarrow_{\rho \rightarrow 0} \;\; \int_L \phi d\mu

for all continuously differentiable, compactly supported \phi:\Bbb{R}^n \rightarrow \Bbb{R}.

In the next two figures, we illustrate this for the case of 1-planes – i.e.lines: in the first figure, L is the weak limit of the dilations of F, while in the second it is not.

Note: solid green lines are the level sets of \phi while the dashed green line indicates the boundary of the support of \phi. Note also that the \rho‘s of 0.4, 0.1, and 0.02 are approximate.

approximate-2

approximate-2-wrong

Exercise: can you create an example of a one dimensional curve which has the usual tangent line at p but not an (integration based) approximate tangent line at p?

Closing Note On Hausdorff Measure

We would like a notion of k-dimensional volume or k-dimensional measure. In many cases, the right notion turns out to be k-dimensional Hausdorff measure. We already know what 1,2, and 3-dimensional measure is as long as the objects we are measuring are regular enough, like subsets of lines, rectangles, and cubes. It does not seem too much of a stretch to think that we can extend these measures to things that are somewhat wiggly. That is, we can still easily imagine measuring the length of a subset of a smoothly turning curve, or the area of a piece of a surface that undulates slowly. Hausdorff measure permits us to measure not only such smooth sets (giving the same result as any reasonable extension of the usual Lebesgue measures to the nice cases), but also to measure very wild sets (like fractals).

How to compute the k-dimensional Hausdorff measure of A\subset \Bbb{R}^n:

  1. Cover A with a collection of sets  \mathcal{E}= \{E_i\}_{i=1}^\infty, where diam(E_i) \leq d \;\; \forall i. Here, diam(E_i) is the diameter of E_i.
  2. Compute the k-dimensional measure of that cover: \mathcal{V}_\mathcal{E}^k(A) = \sum_i\omega(k) (\frac{diam(E_i)}{2})^k
  3. Define \mathcal{H}_d^k(A)=\inf_{\mathcal{E}} \mathcal{V}_\mathcal{E}^k(A) where the infimum is taken over all covers whose elements with maximal diameter d.
  4. Finally, we define: \mathcal{H}^k(A)=\lim_{d\downarrow 0}\mathcal{H}_d^k(A).

Remark: Suppose that for any  \epsilon > 0, there is a cover \{E_i\}_1^\infty of A, such that \sum_i diam(E_i) < \epsilon. Then for k \geq 1, \mathcal{H}^k(A) = 0.

Here is a figure illustrating Hausdorff measures:

hausdorff

Clearly, this can be difficult to compute. It turns out though that in \Bbb{R}^k, \mathcal{H}^k = \Bbb{R}^k. And by use of mappings, this can take us quite a ways in computing \mathcal{H}^k(A) for integral k and rather general A.

Exercise: Show that if 0 < \mathcal{H}^\gamma(A) < \infty then \mathcal{H}^\alpha(A) = \infty and \mathcal{H}^\beta(A) = 0 for \alpha < \gamma < \beta.

Thoughts on receiving a negative review

This is a slightly edited version of something I wrote in 2009, not long after arriving at WSU from Los Alamos. It remains as pertinent now as it was then. Coincidentally, Gaza is again in the midst of increased mayhem.

Today I received a copy of a review of a paper I am an author on. Needless to say, the reason I am writing about it here is that the review was negative in a way that was not helpful. While the reviewer did make some good points, and we will address those points, it was done in an unfriendly way.

Have I seen worse reviews? Of course. So why write about this review? I suppose because it comes at a time when I am being reflective and when I am thinking about such things more carefully. The error that reviewer made was in not reading the paper carefully enough. Of course we can improve the paper and make it less susceptible to misinterpretation, and we will, but I think that the acceptance of this status quo of negativity and a cultivated attitude that looks for errors and ignores insights, ends up robbing our society of a great deal of original, creative productivity.

In my new position in the mathematics department at Washington State University, as I look around and get the intuitive sense for this university and put that in context of what I have observed at other universities, I see a pattern. And that pattern is tradition and conservatism and narrowness that has its roots in narrow self interest. It inhibits interdisciplinary work. It makes people far more apt to see the mistakes in other work, rather than finding the insights and innovations.  It makes people timid and afraid of adventure, of risk.

Do I like it in academia? Yes. There is still a decent amount of good and potential for a great deal more. There is freedom to develop truly new initiatives. And there are some students and colleagues who are inspired and inspiring.

But the threads I am disturbed about are simply local expressions of global states of human consciousness that we all observe in their horrific consequences: Gaza, the economic crisis, epidemics in Africa, etc.  Underlying everything are multiple threads, but the one that I see everywhere is an unconsciousness, a blindness that is deeply disturbing.

In this state, humans think there is no connection between their personal negativity and selfishness and the atrocities in Gaza. It is acceptable or even good to inflict inhuman atrocities on your enemy, but evil for those “terrorists” to strike back in the ways they can. The unconscious see a great gulf between them and the “terrorists”. They believe that some people are intrinsically good and some intrinsically bad. And the end justifies the means. The work of Chris Hedges — see for example, “I don’t Believe in Atheists” or “War is the Force that Gives us Meaning” or his columns in truthdig.com — feels and proclaims aloud the absurdity of these inconsistencies.

So what can my response be — be it to the reviewer, or the critical, narrow nature of some in academia, or the unmotivated, narrow minds of some students, or the unthinking, unconscious state of some people I run into in my daily life? Certainly, becoming negative and critical is not the answer.

It seems to me that the only thing I can do is to spend all my energy creating a personal atmosphere of rich, creative productivity and connection based on love. Generating happy beauty and a vibrant, living atmosphere, beckoning to those in the sphere of my influence to cooperate in creating little bits of heaven on earth, even if only locally, is the only real evidence there is for the existence of love or heaven … or God.

Geometric Measure Theory by the Book

There are an armful of texts that I have used to learn and teach geometric measure theory. In this note, I will give a review of these texts, which are:

  1. Herbert Federer’s Geometric Measure Theory
  2. Frank Morgan’s Geometric Measure Theory: A Beginner’s Guide
  3. Krantz and Parks Geometric Integration Theory
  4. Lin and Yang Geometric Measure Theory – an Introduction
  5. Leon Simon’s Lectures on Geometric Measure Theory
  6. Pertti Mattila’s The Geometry of Sets and Measures in Euclidean Spaces
  7. Evans and Gariepy’s Measure Theory and Fine Properties of Functions
  8. Ambrosio, Fusco, and Pallara’s Functions of Bounded Variation and Free Discontinuity Problems
  9. Enrico Giusti’s Minimal Surfaces and Functions of Bounded Variation

My two favorites are Leon Simon’s Lectures on Geometric Measure Theory and Evans and Gariepy’s Measure Theory and Fine Properties of Functions. Before I dive into comments on each of the books, here is a bit of history concerning my path into the subject.

The Backstory

I was first turned on to geometric measure theory by David Caraballo, the last student to finish with Fred Almgren before Fred died. (Fred who was famous for his deep results in geometric measure theory, was a student of Federer and a professor at Princeton.) I met David at the Nonlinear Control Theory Short Course, organized by Hector Sussmann and Kevin Grasse, at the 1999 Joint AMS-MAA meetings in San Antonio. David and I became instant friends and I was soon swept away by David’s passion for geometric measure theory, realizing that this field was also particularly well matched with my mathematical muses.

In particular, David talked up Evans and Gariepy’s text, so the first thing I did when I got back to Los Alamos was hibernate in my office and immerse myself in that text. It was beautiful and exhilarating — and I was hooked.

At the time I was working on inverse problems and dynamical systems. The inverse problems involved images and it was the image analysis that drew me further into geometric measure theory. The biggest influence in this migration was of course the rising prevalence of total variation regularization in image analysis methods. The first papers I read were David Strong’s. These inspired me to introduce total variation regularization into the sparse tomographic reconstructions methods we were working on. (There were other people dabbling with these methods at the same time at Los Alamos, but Tom Asaki and I took these methods and ran with them. Abel inversions were the tomographic workhorse at Los Alamos, so this was one of the first targets of our work. The first papers can be found here: Abel inversion using total-variation regularization and Abel inversion using total variation regularization: applications)

Through my work in image analysis, I met Andrea Bertozzi, and while visiting Duke at her invitation early in 2003, I met Bill Allard. Bill and I started a close collaboration on data analysis work at Los Alamos, and I began picking up pieces of geometric measure theory from him. Bill was a student of Fleming’s at Brown, but he had also very carefully read and commented on Federer’s entire text as Federer was writing it. (If you know Federer’s book, you can’t help but be impressed by this.) After graduating from Brown, Bill moved to Princeton where he did his seminal work on varifolds.

So why does image analysis lead rather naturally to geometric measure theory? For the simple reason that edges are a big deal in images, while functions of bounded variation (BV) are a very natural class of functions to use when representing functions with discontinuities (edges). And functions in BV are particularly nice, because they are wild, but not too wild — we can still make sense of derivatives (they are nice measures), and all sorts of other nice properties are still at our disposal. For example, sets, whose characteristic functions are in BV, still have usable, generalized outer normals and as a result, we still have the divergence theorem in such regions. Analysis is still nice, or at least possible. (De Giorgi used this class of functions to solve the minimal surface problem in one of the three papers in 1960 that solved this problem. Another was written by Federer and Fleming, the third by Reifenberg. All three used different methods.)

Comments on the Books

I have studied, referenced or taught out all the above monographs. I now give more detailed comments on those texts. But I am not going to introduce the subject of geometric measure theory here — that will be the subject of other posts (coming soon), nor will I simply outline the contents of the texts, since I don’t think that adds much value. Instead, I will add the things you can’t get from a perusal of the table of contents.

Here are the comments:

  1. Federer’s 1969 Geometric Measure Theory: To a very large degree, this is still the ultimate go-to reference for the contents of the first 4 (of 5) chapters. This is not to say that that content has not evolved, but rather that it is still the foundation for current work. (For example, Solomon and White have enabled us to avoid the difficult structure theorem in getting existence for minimal surfaces, but that structure theorem is still very important.) This text is also rather notorious for it’s density and difficulty. Some of that difficulty is due to the (sometimes understandable) impatience that readers bring to their reading, but it also seems that the lack of pictures in this particular book is a rather eloquent statement about its accessibility. I have learned that it is much easier to read when you translate nearly everything into pictures (because you can!), though its terseness encourages me to use it as a reference, not a text. But it is a completely indispensable reference! Every students should own a copy and read pieces as needed.
  2. Frank Morgan’s Geometric Measure Theory: A Beginner’s Guide: Frank wrote his highly successful text as a path into, and an inspiration for the study of, Federer’s book. In contrast to Federer, Frank draws lots of pictures, many of them very enlightening. Almost everybody in the younger generations — people who could have used Frank’s book first — have first read Frank’s book before using other texts. This is as good a place as any to explain that there have been two main branches of effort in GMT, one focused on variational problems like the minimal surface problem, and the second focused on the geometry of sets and measures, with a particular focus on harmonic measures. Morgan and Federer come out of the first branch, while Mattila’s text, which is commented on below, comes from the second branch. (I will comment below on the two new branches of GMT — GMT in metric spaces and GMT with a view to data analysis.) I always use Frank’s book as an illuminating reference for newcomers. I recommend it very highly as a first exposure and an evangelical tool.
  3. Krantz and Parks’ Geometric Integration Theory: At first I was skeptical because I am into nice figures (I am an xfig devotee) and some of Krantz and Parks’ figures were very bad. But then I used the text as a reference when proving the deformation theorem for simplicial complexes (see Simplicial Flat Norm with Scale for the paper) and I was impressed with the exposition. Last year I used it as a text for a graduate class. While there are aspects of the text I still don’t like, I do recommend it as a reference that every student should own and consult. (By the way, Federer originally wanted to name his book Geometric Integration Theory, but didn’t because Whitney had already written a book with the same name. Whitney’s book is relevant for those interested in geometric measure theory, and it is now available from Dover books!)
  4. Lin and Yang’s Geometric Measure Theory – an Introduction: I have not used this as a text. I was discouraged from using it by the typos, which it is important to note were also very irritating to Fanghua Lin because he had tried to get the publisher to correct them! But it does contain a very nice selection of topics, maybe the broadest selection in the above texts.
  5. Leon Simon’s Lectures on Geometric Measure Theory: As I said above, this is one of my two favorites. I am currently (Fall 2012) teaching a graduate class from this text. The book is not so easy to get, but if you are willing to persist, you can get it from the Centre for Mathematical Analysis at Australian National University. (Though the last time I ordered a bunch of copies, they all had the first page of the index missing and they were clearly photocopy’s of the original print run of the book.) I used Leon’s text one other time, when I taught a short course on GMT at UCLA in the spring of 2007. Why do I like the text so much? There are flaws, like typos and a typewriter type font that takes getting used to and things I would change here and there. But Leon’s selection of topics (not too many!), his versions of theorems, the way in which he gives enough details in proofs, but not too many (leaving many implicit exercises and problems for the reader), and the way in which he puts everything together has generated a book that I really like to study and teach from. I recommend it very highly as a primary text, after or alongside Frank’s book. As a side note, Leon is working on a second edition of this text. It should be very good!
  6. Pertti Mattila’s The Geometry of Sets and Measures in Euclidean Spaces: As mentioned above, this text comes from the harmonic analysis branch of the subject. As a result, it does not deal with currents, which were developed for their use in GMT (on minimal surfaces) by Federer and Fleming. Mattila’s book is well written and challenging, but not so challenging that students don’t like it. It has explicit problems, which many students like. (I always preferred implicit, fill-in-the-details or what-if-I-weaken-this-assumption type problems.) It covers lots of material, including Marstrand and Priess’ results (about which I would recommend De Lellis’ notes on Rectifiable Sets, Densities and Tangent Measures ), fractals and connections to singular integrals. It contains things that none of the other books I am commenting on have, and it is the only representative of the harmonic analysis branch of GMT in my selection of books. And it does have a different flavor, as one might expect. I consider it something that every student of GMT should own.
  7. Evans and Gariepy’s Measure Theory and Fine Properties of Functions: As noted in my story above, this was the first book I saw on the subject. It deals with the same subjects that the first part of Federer and the first part of Simon deal with. It does not delve into currents. The writing is very clear, the proofs are complete, and the amount of filling in, in the proofs, is consistently small enough to make it fairly fast to study, but often enough to keep you very engaged. I found it inspiring when I first read it and it is still usually the first book I have my students buy. The chapters cover measure theory and integration, Hausdorff measure, Radon measures, area and co-area formulas, Sobolev spaces, BV functions (including detailed development of the structure theorem for sets of finite perimeter), and a final chapter on things like Radamacher’s theoorem and extension theorems like Whitney’s. CRC even lowered the price from 180$ to 90$ (it has crept back up a bit) in response to our complaints about the price! As far as prerequisites are concerned, most students find this more accessible after a first course in graduate analysis, but some might be happy with it as an introduction to analysis, as long as some other text, like Royden or Folland is also on hand. I recommend it very highly as a text and reference.
  8. Ambrosio, Fusco, and Pallara’s Functions of Bounded Variation and Free Discontinuity Problems: David Caraballo was also the first to tell me about this book. It prepares the reader to deal with the existence results for the Mumford-Shah functional, which is an image analysis functional for used for image segmentation. Ambrosio and De Giorgi proposed the space of special functions of bounded variation (SBV) for use with free boundary problems in 1987. In a 1989 paper by De Giorgi, Carriero, and Leaci, these ideas were used to prove the existence of minimizers for the Mumford-Shah functional. Later Ambrosio developed the theory of SBV more fully and this book is a logical follow-on to these works. I have not used this book, though I have read a small bit here and there. I think it is well liked by students, but it is unreasonably expensive at 250.00 (list price). Shame on Oxford University Press! They should not be following the example of the Dutch profiteers! This is an object lesson in why, if you are a mathematician, you should publish your own book and not give it to some publisher to exploit. OK. Done with my soapbox. If you can afford it, get this book! If you can’t, write to Oxford and complain bitterly about their crazy prices and the fact that they are limiting access to an excellent and fascinating book!
  9. Enrico Giusti’s Minimal Surfaces and Functions of Bounded Variation: I read much of Giusti’s book and liked it a great deal. I recommend it as very a good source for the subjects it covers. This book is inspired by De Giorgi’s path to a solution for the minimal surface problem, though it contains more material since it was written over 20 years after that work by De Giorgi. Here is review of Giusti’s book by Fred Almgren. It contains a very detailed account of the contents of the book and some nice history as well. Again, this book is overpriced at 183.00 for the paperback but luckily, Springer (who owns the book’s publisher Birkhauser) sells it for 91.50! So buy it directly from Springer. And I do recommend buying it. You will enjoy studying it if you have any interest in this subject.

Finally, as promised above, a few words about the other two branches of GMT. As mentioned above they are GMT in metric spaces and GMT with a view to data.

As far as I know, GMT in metric spaces had its genesis with Ambrosio and Kirchheim’s paper, Currents in Metric Spaces. It is a very active area of research, including for example, analysis on the Heisenberg group, where paths are allowed tangents in proper subspaces of the tangent space instead allowing the path to have arbitrary directions in the tangent space. (These are called sub-Riemannian spaces.) As far as I can tell, this branch was inspired by the growing area of analysis in metric spaces.

By GMT with a view to data, I mean both GMT applied to data and new GMT inspired by data. That the flow of ideas goes in both direction is very important and the reason why this is a very exciting, productive, high-potential place to work. This is where I am working. It includes the work at the intersection of image analysis and BV/SBV functions, like the work of Rudin, Osher and Fatemi which introduced TV regularization to image analysis and the segmentation functional of Mumford and Shah. Other examples include the work of Jones, Lerman, Schul and Okikiolu on Jones’ beta numbers, the multiscale flat norm work I am doing with collaborators (inspired by the connection between the L1TV functional and the flat norm), as well as the applications of curvature measures (and things like curvature measures) to data. Examples of this last item include the work of Adler, Taylor and Worsley at the intersection of statistics and integral geometry and the work of Chazal, Cohen-Steiner, and Merigot on boundary measures. Many more examples exist, but these examples give a flavor for the kinds of things happening at the intersection of GMT and data.

Actually, the GMT data branch has the other three as subbranches for the simple reason that all three have very useful insights to offer data and because data suggests problems leading to new ideas in each of the three areas.

Afterword

As mentioned above, I like figures. But, while not having figures in a geometric measure theory text doesn’t make so much sense to me, it is the case that students should be drawing their own figures anyway. This is part of the work involved in making the subject yours, in internalizing the ideas and techniques. One might even argue that books without figures are better for students who must then draw figures for themselves in order to grasp what is going on more fully. But I would not go quite that far. I think that drawing pictures, as many and as often as you can, should be the part of the GMT culture. Figures should be common, and they should also appear in books. I do believe that authors should not try to draw pictures for everything, but should draw just enough to get the students going themselves, help students avoid pitfalls and inspire them as they struggle to master the ideas. At the very least, this discussion helps you see why I can still like Leon Simon’s book so much even though there are no figures in it!

It should also be noted that even though I talk about the two older and the two newer branches of GMT, it would be silly to insist on a bold demarcation of the boundaries between branches and a subsequent classification of everything and everyone. Part of this is because the intersections between branches are very large. Another part of this is because the two newer branches are by their nature agnostic, caring only about generalizable (in the case of metric spaces) or useful (in the case of data analysis) ideas or developments in GMT. Using Federer to supply another example, even though Federer’s most famous paper (the one with Fleming that solved Plateau’s Problem) was focused on calculus of variations, he also established other significant pieces of the foundation for the entire field. It would therefore make no sense at all to try to assign him to a single branch. So thinking about the field as characterized by branches is useful only if you do not take it very seriously, or worse yet, turn the branches into fences impeding travel!

Higher Education: the real problem is not the cost

Even though there is a lot of talk about money and cost, the real problems in higher education are the mistaken ideas that have gained respectability through the motion of the cultural herd we all live in. Here are four such ideas:

(Mistaken Idea 1) Everybody should go to college after high school because most jobs actually justify requiring a college degree as a qualification.

I do believe that everybody should have the opportunity, when they have real desire, to learn more deeply, to go to college and interact with mentors, etc. But the current way in which college is a knee jerk path to jobs is just silly. Giving people something they really don’t want in response to ill-founded ideas on what is useful to them is a recipe for very unhappy situations.

(Mistaken Idea 2) Adding computers and the Internet to the educational enterprise makes it better.

In fact, the way most people use the Internet changes the way they think, making it very hard for them to focus, think independently or deeply about anything.See for example the research cited by Nick Carr in his (possibly too) provocative book “The Shallows”.

This constant tweeting, emailing, texting, surfing, video-gaming that college kids are immersed in is seriously degrading their ability to focus deeply. I come to that conclusion through a combination of direct observation of the lower level classes I have taught — differential equations, business calculus and linear algebra — as well as the results reported in Carr’s book, as mentioned above. Another very interesting talk to listen to is David Levy’s Google Tech talk “No Time To Think” which can be found on youtube here.

 The illusion of greater knowledge hides the fact that less is being internalized, reasoning powers are weaker, and that the capacity for depth has been seriously degraded.  While it is true that a disciplined use of the Internet can be very helpful in research, the ways it is used by the vast majority has little to do with these positive uses.

(Mistaken Idea 3) Grant money for research has improved the overall quality of education.

In fact, the addiction that universities have to grant money has driven them to pay lip service to education while in reality, by any organic measure, they have relegated education to a least important role.

Grants have evolved from a luxury to a necessity in the sciences (and many other areas of academia). Universities are now completely dependent on that funding, and the drive to get that funding takes precedence over everything else. In particular, while professing a great focus on teaching, a close look at status/promotion/pay/etc at institutions of higher education (other than the teaching colleges) will show that teaching is most definitely not a top priority. Of course, an important part of the blame for this is the decreasing revenue from states, which necessitates an even greater emphasis on grant getting and an increase in class size. This last trend – bigger class sizes – has a seriously negative impact on teaching quality. Teaching is inherently a one-on-one or one-on-few exercise. Mentoring and training on the large scales that is being adopted to deal with the lack in funding is very difficult unless one agrees to the (large) drift in the definition of education that accommodates the growth in class size.

(Mistaken Idea 4) Education is what happens in a lecture class, by reading a book, or by searching the Internet.

Education is about helping students discover and follow their passion, in collaborations with teacher/mentors. It is also about training the students to develop and exercise their moral muscles. Independence of thought and action and the exercise of compassion and generosity results in an education that is robust and versatile, forming a foundation for long term creativity and happiness. It leads to a sustainable society.

So, books and lectures and Google Scholar can all be tremendously valuable to the process of education, but education is very far from the simple mastery of some subject of study. Even if the lectures are very inspiring, the books extremely well written and all the papers that are needed can be found and downloaded, these are merely a small to medium sized piece of an education.

Summary

Education — which should be about helping students discover and follow their passion, in collaborations with teachers/mentors — is severely hampered by the mistaken ideas outlined above. The acceptance of those ideas greatly disadvantages students and professors, and in the long term, our entire society. Independent and creative (yet disciplined) thought is a critically important ingredient of a happy society. When there is a shortage, as there currently is, everything from economics and technology to moral strength and community suffer. But because the time scale at which this happens is longer rather than shorter, it is invisible to politics as we know it. As a result, positive progress will come from the bottom up, from the grass roots.

The main task at hand, is the fueling of a grass roots movement by helping people realize they are operating on mistaken assumptions, that the real crisis in higher education has nothing to do with the cost and debt, but rather with the nature and substance of education.

Connection

Loneliness, 
moving through me,
flies away, having drawn me to a deeper stillness. 

A vision, 
flowing like music  
compels me to continue on. 

Expression, 
fresh in its originality, 
luminous and living in action and influence, 
brings release.

Seeking 
connection, 
    I find God. 

The pain 
of 
loneliness 
is transmuted 
into the awe of companionship 
with Him in whom I live and move and have my being.

The Power of solitude … and Social Connection

Awhile ago, Eric Blauer blogged this:

“Of this there is no doubt, our age and Protestantism 
 in general may need the monastery again, or wish it 
 were there. ‘The Monastery’ is an essential dialectical 
 element in Christianity. We therefore need it out there 
 like a navigation buoy at sea in order to see where we 
 are, even though I myself would not enter it. But if 
 there really is true Christianity in every generation, 
 there must also be individuals who have this need. […]”
  —Kierkegaard’s Papers and Journals:  A Selection,
 translated and edited by Alastair Hannay, 
 47 VIII I A 403, pg. 275

in response to something I had written him. In turn, that prompted me to write the following.


The monastery in its essence has always been there.  At least in its original, unpolluted version of time in solitude with God, it has always been accessible. The solitude of walks with God in nature, the quiet seclusion in which we hear and see, is closer than most think.

In fact, we are invited to find it by waiting:

“But they that wait upon the Lord shall renew [their] strength; they shall
mount up with wings as eagles; they shall run, and not be weary; [and]
they shall walk, and not faint” Isaiah 40:31

“… in quietness and in confidence
shall be your strength …” excerpted from Isaiah 30:15

Which my walk has combined to:

“They that wait upon the Lord shall renew their quietness and confidence”

The monastery, as an ideal, is flawed. In pursuit of this ideal, a culture is robbed. For it is fundamentally wrong to view communion and union with a mate, interaction with the world, and social flow as distractions from a deeper walk with God. Acted upon as a model for spiritual depth, such views lead to an impoverished life, an impoverished culture.

Yet the simple, solitary walk with God is a powerful experience leading to deep insights and fresh originality. Spiritually, creatively, we are drawn to the greatest depths by an existence constantly moving between a walk with God and a walk with others.

The monastery impulse, stripped down, reduced to its essence of deep communion with God, is a powerfully transformative impulse. Enlarged by communion with others, it grows generous. Freed from the burden and unnatural restrictions of tradition, it becomes the source of such a rich profusion of creativity and connection that observers are constrained to recognize that something extraordinary is at work.

In such an atmosphere, where love and depth, generosity and creativity flow freely, no arguments are needed to persuade others that we have good news, for it is self evident.

Who we are becomes the only argument we ever need.

Rage

   I rage with a lonely rage

   against isolation, blindness and a smiling callousness,
   against the denial of our nakedness, our need

   against the illusion of goodness

--

sing to me connection, sing to me life, flowing,
quietly moving me to vision

sing to me a fountain of companionship

--

   what will these gods do for you? 

   ... these gods of all false comfort, 
   taking credit for gifts not of their making, 
   these gods who rob us, yet remain barren ...

   ... what will they do for you?

   I rage with a white hot rage

--

Sing to me songs of comfort
songs from silence
silence ... singing

--

   I rage

   against cleverness,
   cleverness masquerading as depth,
   against sophistication, imprisoning the wounded soul

   against a proud intellect, withering the spirit

--

sing to me