Stephen Barr’s “Experiments in Topology” (originally published in 1964, reprinted in 1989 by Dover) is extraordinary because it treats a sophisticated mathematical subject with accessible language that can be understood by motivated junior high students. It is extraordinary because its wonderful and copious figures are remarkably clear and elucidating. It is extraordinary because it captures the flavor, depth, and breadth of a very subtle subject with carefully written passages that boil down significant complications into understandable overviews. It is extraordinary because he gives a concrete enough treatment that the attentive reader can learn something substantial of the subject while the mysteries that are both alluded to and implicit may drive the curious reader to explore its nooks and crannies. It is extraordinary because its learning-by-doing (experimental / exploring) style is infectious and the reader may be emboldened to ask their own questions and attempt their own experiments. It is extraordinary because its effective survey of key ideas in the major branches of topology make it a useful reference. It is extraordinary because it can reward the casual reader with some basic guideposts for apprehending an advanced subject while the serious student who builds all its models and tries to understand their integrated significance can extract many deeper insights from the text.

I prefer this latter, in-depth, approach and organized, through Math Counts, a mathematics group in Philadelphia, 11 deep explorations on topics from the book each with a group of 6-10 mathematically-oriented colleagues. We thoroughly enjoyed the book and our explorations, although after 11 months we wanted to move on to other ideas.

The book uses homeomorphism as its first principle for exploring topology. Barr gives several definitions, but I found the characterization on page 5 to be the most helpful: “Any distortion is allowed provided the end result is connected in the same way as the original.” This exemplifies the kind of careful but informal style of the book.

In chapters 2-6 we are introduced to the topological surfaces that can be built with a sheet of paper, namely, the plane, cylinder, torus, Möbius band, Klein bottle, and projective plane. This might be called the piecewise linear topology of 2 dimensions and it provides an introduction to ideas from geometric topology and differential topology. Barr introduces algebraic topology with Euler’s formula in chapter 1 and Betti numbers in chapter 8. The treatment of graph theory in chapters 7-8 includes an effective overview of the four color theorem which one of our participants, Kurt Tekolste, observed follows some of the ideas of the actual proof (which was first announced in 1976, 12 years after the publication of the book). Barr includes a little bit of knot theory in chapters 6, 8, and 9. Chapter 9 on the eversion, or inside-outing, of the torus is a wonderful further introduction to differential topology. The basic ideas of point-set topology are introduced in chapters 10-11.

In short, the book gives a competent introduction to the material and the flavor of a broad swath of the main subfields of topology though Barr does not use the academic disciplinary names preferring an informal treatment. I would like to thank George Zipperlen for pointing out the breadth of coverage in the book and for providing us a lot of useful context about the various branches of topology.

For each of our monthly meetings, I prepared a set of exercises on a bite-sized part of the book (my questions were sometimes a bit off because I had never studied topology in any depth before). Working through those questions, both in preparation before a meeting and during each group session, took more time than I expected. We came to realize that beneath the surface of Barr’s elementary presentation live many subtleties which neither the book nor our mathematically experienced participants were able to easily explain to each other. We had doubts about the ideas and the language in the book. It took a lot of effort to integrate each of our partial insights into apprehension. In fact, it took so much time and effort that we usually failed to complete the prepared exercise set, so the next month’s frequently overlapped significantly. But in every instance we came to understand the posed questions and the book in a deeper way. And we got to enjoy the rewards of determined exploration: realization and discovery over and over again! This in-depth process convinced us that Barr’s writing is very good: once we understood what he was saying and what the often unstated caveats were, we were in awe at how effectively he spoke mathematical truth in language accessible to middle school students!

Although I agree with Bruce Trumbo’s assessment that much of Barr’s language will cause difficulties for the reader, I think that applies to all mathematics writing: unfamiliar language about unfamiliar subjects is always difficult to interpret especially when the precision of mathematical thinking is desired. It was the general assessment of our group that overall Barr’s writing is precise and nuanced despite its informal style. It is topology itself that is almost too complicated for such an elementary treatment! Barr should be commended for his brilliant success in introducing such a subtle subject to a broad population of readers while maintaining the integrity of mathematical precision even though it may take substantial effort for readers to understand many of its passages in any depth, let alone the integrated significance of them into the broad enterprise that is modern topology. However, it should be emphasized that the book rewards less intense study by giving a feeling for the subject even if you choose not to fully untangle the nuances in the guideposts provided by the text.

One delight in the book, which George Zipperlen observed, is how the chapter on the conical Möbius strip subtly motivates the cross-cap. Indeed, the conical Möbius strip, in the limit, is the cross-cap. At our June 2018 event, George said “Be patient with Barr, what seems like a side excursion will come back.” Indeed, I suspect that most of the juxtapositions and suggestions of the text are meaningful even though we did not have the wherewithal to explore very many of them: it is a rich book and a rich subject!

Not only that, but your own ideas can lead you to fascinating tangents that are not in the book. One of our participants, David Sternman, imagined cutting a torus along a Möbius strip which led Michael Reichner to post a link that led us to George Hart’s great demonstration video of how to cut a bagel for a better breakfast. And Jeannie Moberly impressed us with her wonderful construction of Chris Hilder’s Boy’s Surface model.

We noticed very few errors in the Dover edition. One of our participants, Bob Miller, helped us realize there is a typo on pages 47-48 where it says “bringing B and B’ together and joining BC to B’C’ ” it should say “bringing B and B’ together and joining B’C to BC’ ”. On page 91, the phrase “is not is no worse” should simply be “is no worse”. On page 138, “Laocoön statute” should be “Laocoön statue”. In Fig. 25 on page 144, the dashed line “y” should be external and not internal to the inside-out torus.

Here is a list of links to the event descriptions and associated exercise sets for each of the 11 events I organized on Barr’s book (several of the comments posted to these events are also interesting):

- Topological Surfaces from a Strip of Paper (feat. minimal length Möbius strip) (Oct 2017)
- Variations of the Möbius Band to Explore the Nature of Homeomorphism (Nov 2017)
- Exploring Homeomorphism through Experiments on the Möbius Band (Dec 2017)”
- Topological Experiments: The Conical Möbius Band & the Klein Bottle (Jan 2018)
- Experiments in Topology: Dissecting The Klein Bottle (Feb 2018)
- Exploring the Topology of the Projective Plane (Mar 2018)
- Map Coloring; Martin Gardner’s Projective Plane & variations (Apr 2018)
- The Symmetry of the Projective Plane (and the curious property of twist) (May 2018)
- Using Twist to Understand the Topology of the Projective Plane & its Symmetry (Jun 2018)
- Betti Numbers and the Symmetry of the Projective Plane (Jul 2018)
- 10 exercises in support of reading and discussing chapter 8 pages 123-128 (on Betti numbers) and chapter 6 pages 96-107 (on the projective plane)

- Deliberations in The Trial of the Punctured Torus (Aug 2018)
- Note: 13 exercises on chapter 9 are included in the event description instead of a separate PDF

In my research on Barr’s book, I found these additional resources:

- Dover’s product page for the book. Dover also provides a PDF of chapter 1 of the book.
- H. M. Cundy’s review in The Mathematical Gazette Vol. 50, No. 373 (Oct., 1966), pp. 323-324.
- Bruce Trumbo’s review in Mathematics Magazine Vol. 38, No. 1 (Jan., 1965), p. 50.
- Karen M. Daniels of the University of Massachusetts Lowell has 14 slides based on Barr’s book to introduce topology to her Graduate Geometric Modeling students.
- Maria E. Salcedo’s paper “Knotted Ribbons and Ribbon Length” includes a section on Barr’s treatment of the shortest Möbius strip.
- Matthew R. Francis has three blog posts on the book: Hearing Around Corners, Simply Connect (Further Adventures in Topology), and Time to Make the Paper Donuts.
- Alexander Bogomolny’s Cut-The-Knot site has a landing page with a brief review of the book and a copy of its table of contents and back cover.

I have found little biographical information about Stephen Barr. If you know anything about him or if you know of another good resource on the book or have some experience or question about it, please post a comment as I’d love to learn more.

Whoever Stephen Barr is or was, the book “Experiments in Topology” is an able tour guide to lead both casual and serious readers to apprehend some key ideas in topology, one of the most storied and important branches of modern mathematics.

]]>I subliminally remembered this quote at the end of my study of Modern and Contemporary American Poetry (ModPo) with Al Filreis of the University of Pennsylvania. On 18 November 2015, I attempted to explain the idea to the ModPo community.

But what did Bucky mean by “the subjective and objective always and only coexist”? Let me give my interpretation and suggest its profound significance for our lives and in characterizing the nature of Bucky’s notion of design science.

In Bucky’s *Synergetics* (and probably in his entire oeuvre), I think by “objective” he usually means voluntarily working to realize an objective, a goal, or a purpose whereas by “subjective” he means involuntarily subjected to happenings (which may be due to necessity or chance or circumstance). Bucky’s meanings for “objective” and “subjective” are logical variants of their root words “object” and “subject” even though they are not the most common in contemporary parlance.

Do you agree that “objective” and “subjective” can be used in this way?

Here is my evidence for Bucky’s usage: In 302.00 and 305.05, he explicitly identifies objective with voluntary and subjective with involuntary. In 986.032, he identifies objective with experimental and subjective with experiential. In 100.010, Bucky identifies objective with active/self and subjective with passive/otherness.

Do you agree with my interpretation of Bucky’s use of the words “objective” and “subjective”? Can you cite other Bucky passages that further clarify his thinking?

Does Universe relentlessly subjugate us to situations which we did not voluntarily choose? Simultaneously, are we not also the agents of ongoing genesis intentionally and objectively building our futures (to paraphrase Harold G. Nelson and Erik Stolterman in their profound 2012 book *The Design Way*)?

In 161.00, Bucky writes “When science discovers order subjectively, it is pure science. When the order discovered by science is objectively employed, it is called applied science.” In 326.03, we find “All the synergetic metaphysical consists of two phases (1) subjective information acquisition by pure science exploration, and (2) objectively employed information by applied science invention.”

Is the information of pure science about subjectivity? Is the information of applied science about objectivity? Is Bucky right about these distinctions?

I think so: pure science is about our abstract scientific knowledge. We don’t usually think of knowledge as subjective, but that is exactly what truths do: they make us subjects of the agreed wisdom. That is, truths imperially obligate us. For example, try as you will, the force of gravity will relentlessly pull you toward the center of Earth. All truths either subjugate us in this way or they are not truths.

This is the spirit in which I read Bucky’s assertions about “generalized principles” and “the cosmic intellectual integrity” (1056.11): we are all subjects of and must obey “the great design laws of eternally regenerative Universe”. According to Bucky, these laws may be seen as the cosmic intellectual integrity which can be called “God” (see 311.16). God (whether theistic or deistic) is a ruler to which we are subjects, to which we are subjugated.

Do you agree that truth is subjective in the sense that we are its subjugated subjects?

In addition, the notion of objectivity is about purpose which is involved in all applied science and design.

Do you now agree with Bucky that pure science or knowing is subjective and that applied science or design is objective? Why or why not?

We began with Bucky’s quote “the subjective and objective always and only coexist”. How can it be that pure science and applied science always and only coexist?

Albert Michelson’s 1881 interferometer which after further engineering with Edward Morley eventually led to a successful measuring of the speed of light. The example is typical: engineering breakthroughs are frequently prerequisite for scientific developments.

In 2015, I organized an exploration Are Science and Engineering One Inseparable, Essential Way of Knowing and Doing? based on Henry Petroski’s interesting 2010 book *The Essential Engineer: Why Science Alone Will Not Solve Our Global Problems*. Petroski effectively shows how good science involves good engineering and good engineering involves good science.

I think the argument can be made simply: in order to test a scientific hypothesis, each experiment needs to be exquisitely engineered. So there is necessarily doing and design in science. Conversely, each step of the engineering process is based on some (possibly imperfect) understanding of how the world will respond to an objectively employed instrument or action. There is some basic knowledge underlying each action to intentionally change or design our worlds.

Are you convinced that scientific knowledge (pure science) and design (applied science or engineering) always and only coexist? Can you identify additional evidence in support of this claim? Can you find any countervailing evidence?

Can we go further?

If science as knowledge is a way of passively knowing and the arts, in general, are a way of purposefully doing, do science and the arts always and only coexist?

Any knowing disconnected from the experiences of doing would be fantasy not science. And, conversely, in order to act in the world one needs to have enough understanding of how effects can be achieved that some nascent knowing must inform any purposeful action.

So does science always and only coexist with art?

Can we even conclude that the arts are sciences and that science is an art?

When Bucky says “the subjective and objective always and only coexist” does he mean that the pure (knowing) sciences and the applied (doing / creating) arts including engineering are mutually intertwined?

How does Bucky’s notion of design science fit in this schema?

In 165.00, Bucky writes “Generalized design-science exploration is concerned with discovery and use by human mind of complex aggregates of generalized principles in specific-longevity, special-case innovations designed to induce humanity’s consciously competent participation in local evolutionary transformation events invoking the conscious comprehension by ever-increasing proportions of humanity of the cosmically unique functioning of humans in the generalized design scheme of Universe. …[We function as the] subjective discoverer of local order and thereafter as objective design-science inventor of local Universe solutions…”. In 174.00 to conclude this passage, he extols the scientist-artists.

I interpret Bucky as advocating a design science that is an intentional, participatory, and competent initiative to induce change through design creation and invention informed, guided, and developed through a scientific approach to knowledge. It suggests to me that design science, as Bucky imagined it, may be the actualization of the always and only coexisting subjective and objective. Of pure science and applied science. Of the sciences and the arts.

Do you think the effective realization of our future through design science is an important part of what Bucky meant when he wrote “the subjective and objective always and only coexist”?

Is Bucky’s design science a way to imagine, articulate, and actualize this interrelationship between science and the arts mediated through competent design?

Are design scientists the scientist-artists who engage the interfused subjective intellectual knowing and the objective designing/doing through inquiry, exploration, and action to effectively create our futures? Are they the ones who help devise new ways of thinking, knowing, and doing that provide new ways to experience and co-create and shape our complex, participatory, and ever-evolving Universe?

What is your perspective? How do you think about the subjective, the objective, design science, and their interrelationships?

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As much as I love Kate Orff‘s work and her Living Breakwaters project, I regret using the word “best” in the context of the BFI (Buckminster Fuller Institute) Challenge Prize. I was also deeply moved by previous winners including John Todd’s Challenge of Appalachia, Allan Savory’s Operation Hope, Blue Ventures, and The Living Building Challenge: each of them ranks as “best” in my eyes. Moreover, the “best” aspect of the BFI Challenge is the large number of inspiring runners-up and also-rans who give me hope that the creativity of Humanity will overcome the profound challenges our civilization faces.

My mistake was lapsing into the conditioned reflexes of our special-case American culture with its fixation on casually dubbing a “best” or “favorite”: a cult of exceptionalism. Of course, we are all extraordinary! Exceptionalism and its undue attention on the so-called “best” ignores the nuances of the greatness and weakness inherent in each of our efforts and can foster distorting prejudices.

Are you working to curb your overuse of favoritism and exceptionalism?

The Bucky quote that Harold and I struggled to remember is

to make the world work for 100% of humanity in the shortest possible time, with spontaneous cooperation and without ecological damage or disadvantage of anyone. — R. Buckminster Fuller

The Synergetics Collaborative is a 501(c)(3) non-profit organization dedicated to discuss and build Synergetics as a subject through our scientific and educational programs. We are working to understand, develop, and consummate Bucky’s vision for a “science of synergy”.

The exploration instigated by my reading of *Synergetics* inspired me to think more comprehensively and more synergetically. However, *Synergetics* isn’t a how-to manual: to a large extent it is an exploration of fascinating models that together with hard work can lead to new ways of thinking. When viewed with curiosity and attention to its integrated significance, *Synergetics* is a tantalizing, not-quite-complete glimpse of the dynamical shape of dynamical thinking: a tool to strengthen one’s faculties for Comprehensive Anticipatory Design Science.

I think one of the most important aspects of Synergetics is its attention to modelability: “Comprehension of conceptual mathematics and the return to modelability in general are among the most critical factors governing humanity’s epochal transition from bumblebee-like self’s honey-seeking preoccupation into the realistic prospect of a spontaneously coordinate planetary society.” (Synergetics 216.03).

Some Resources on Synergetics:

- My essay synthesizing Scott E. Page’s “Model Thinking” with Synergetics to suggest a new kind of science
- My essay “Reading Synergetics: Some Tips” which recommends building the models and more
- The Synergetics Collaborative YouTube Channel
- The On-Line Edition of Synergetics

Inspired by Stuart Firestein, I see ignorance as a tuning in to the negative space, the questions, that structure knowledge. As Firestein notes, the facts are one study away from becoming reframed, emended, superseded, or rejected. The questions on the other hand organize our knowledge. So although facts and knowledge are important, it is the ignorance, the questions, that give us our framework for understanding. Bucky understood this with his great clarion call “Dare to be naïve” to express the pivotal importance of saying “I don’t know” and “could it be?” which are prerequisite to asking the right questions.

Unlearning is the process of integrating new knowledge and questions with facts already known. We learn that items previously thought significant are side-shows. Previously overlooked questions and ideas can profoundly reorient our thinking. Ignorance spearheads the unlearning and gives us the kind of deeper understanding needed to design better solutions.

My comment on unlearning some of our most cherished assumptions is inspired, in part, by this wonderful quote by mathematician Morris Kline “The lesson of history is that our firmest convictions are not to be asserted dogmatically; in fact they should be most suspect; they mark not our conquests but our limitations and our bounds.”

Do you covet ignorance and unlearning? Should you?

To me the expression “both/neither” suggests that at least in certain situations either/or binary logic (the so-called law of excluded middle) can be erroneous. Especially for concepts that are infused with ambiguity, contradiction and paradox. So tension/compression, ignorance/knowledge, conservative/liberal are all both/neither concepts: they always and only coexist, we can’t highlight one without unfairly slighting the other. Both are vital, yet neither can capture the synergetic whole of reality. I was awakened to the importance of ambiguity, contradiction and paradox from William Byers’ interesting book *How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics*.

Do you fear paradox and contradiction? Will you dare to explore the world of the Both/Neither?

I agree with Fuller and Channer that humanity now has the wherewithal to transcend scarcity as ontological reality. Although we live on a finite planet with finite time, energy, and resources, we now know enough that for most purposes we can all realize lives of greater comfort and capability than even the great Monarchs of 100 years ago.

I felt Harold’s emphasis sounded too much like talking to the world and collided with the values in the Bucky quote “You can’t better the world by simply talking to it. Philosophy to be effective must be mechanically applied.” To practically and mechanically develop this realization, I would suggest that each of us needs to understand the technological basis of our civilization and develop our design faculty to realize the abundance Fuller posited.

One way to realize your abundance is to work to better understand the complex of subjects that might be impeding your design initiatives. I prepared a guide outlining tools for such self-educational efforts in my slides for “Education Automation Now and in the Future”. These and other tools to learn how the world works can help you more effectively exercise your design faculty to achieve your rightful abundance in service to the World Game Ethic (in support of 100% of humanity and all that).

Of course, it isn’t easy and personally I have not yet fully transcended the scarcities that frustrate my ideas for better serving 100% of humanity. Have you?

Are the basic resources available for each of us to transcend scarcity as Bucky and Harold promise? Do you think our design capability can transcend scarcity? Can individuals practically apply this realization? How can we do so? Are you still suffering from the brutalness of scarcities? What will it take to realize your abundance?

In many of Bucky’s communications, he emphasized the idea of “utopia or oblivion” with humanity facing an imminent “final exam”. Bucky sometimes mentioned specific dates such as 1972 (see page 5 of the The World Design Science Decade Phase I Document 2: The Design Initiative) as representing the moment when the accelerating acceleration of trends reaches a critical point of qualitative transformation. I suggest in the video that Bucky really meant this in a timeless manner: so that each generation faces a “final exam” instead of a literal reading with a specific date followed, presumably, by utopia.

I think the reason Bucky’s communications sometimes focus on the specificity of “now” as a moment of transformation is to impart an urgent sense of responsibility to his audience. Dennis Dreher impressed this idea upon me at a Synergetics Collaborative event. I think Bucky’s point in making such dramatic statements is to confer a new and profound sense of responsibility upon Humanity. In my reading of Bucky, I weigh the ideas in these two Bucky quotes as more significant than the special-case, date-focused transformation passages:

- “We humans are manifestly here for problem-solving and, if we are any good at problem-solving, we don’t come to utopia, we come to more difficult problems to solve.”
- “My own working assumption [is] we … are here for all the local-Universe information-harvesting and cosmic-principle-discovering, objective tool-inventing, and local-environment-controlling as local Universe problem-solvers in support of the integrity of eternally regenerative Universe.”

I am inspired by this profound new kind of responsibility. Could it be that our responsibility to steward civilization, our home planet, Earth, our solar system, our galaxy and Universe itself is only just beginning? Stewarding Universe could be Humanity’s greatest project and greatest destiny: unless some generation abrogates their responsibility and fails its final exam, I expect we will face greater and greater stewardship challenges … eternally. I explore this idea further in this short science fantasy story about the purpose of humans in Universe.

So I agree with Harold that humanity is undergoing a profound qualitative transformation. But I see it as a transformation of dawning awareness of our comprehensive responsibility to steward Humanity aboard SpaceShip Earth and beyond. Indeed some of the challenges that future generations will face will probably make today’s problems look like elementary training exercises (Survivor-lite!). I embrace the challenge. All hands on deck!

Do you find my interpretation of Bucky’s vision to be apt or erroneous? Terrifying or enthralling? How are you going to help your generation pass its final exam?

In the video I mentioned the excellent free on-line course “Design” with Kart T. Ulrich of the University of Pennsylvania, the Stanford D School, and work in Design Thinking as three recent initiatives in design that should be compared and contrasted with Buckminster Fuller’s vision of Design Science, shorthand for Comprehensive Anticipatory Design Science.

Perhaps these four important elements of Bucky’s work may serve as a foundation for Design Science. First, Bucky’s World Game Ethic as an imperative for design is unique and may be one of his greatest legacies.

Second, Bucky’s invitation “dare to be naïve” emphasizes the importance of searching for the right questions, mindfulness to ignorance and unlearning, and a mistake-making mystique. Thereby the designer acquires access to the prodigious creativity in the a priori mystery of Universe.

Third, Synergetics as Bucky’s vision for a science for thinking comprehensively about how Nature works can help us design more comprehensively and considerately and so better meet the aspirations of the World Game Ethic in our design work. Synergetics, especially when buttressed with the vision from Model Thinking of using multiple models to enrich our understanding, can be a significant tool for design.

Finally, Bucky’s example of making visionary artifacts to help imagine and prototype ideas for better meeting our needs is a powerful way to expand the design frontier through STEAM (Science, Technology, Engineering, The Arts, and Mathematics).

This scientist-artist vision for Design Science may be better appreciated by these Bucky quotes:

- “It is not for me to change you. The question is, how can I be of service to you without diminishing your degrees of freedom?”
- “You can’t better the world by simply talking to it. Philosophy to be effective must be mechanically applied.”
- “You never change things by fighting the existing reality. To change something, build a new model that makes the existing model obsolete.”
- “A designer is an emerging synthesis of artist, inventor, mechanic, objective economist and evolutionary strategist.”

Are these the main elements of Design Science? What inspiration and vision do you derive from Bucky for the practice and development of Design Science? Which of these ideas are unique to Design Science? What great ideas from other design traditions ought Design Science incorporate into its conceptuality?

Resources on Design Science:

- The Bienniel Design Science Symposium at RISD
- BFI Resources for Design Science
- “Design Science: A Framework for Change” by Michael Ben-Eli
- “Comprehensive Anticipatory Design Science An Introduction” by Patrick G. Salsbury

In the video I paraphrased this Bucky quip “I learned very early and painfully that you have to decide at the outset whether you are trying to make money or to make sense, as they are mutually exclusive.” I think Bucky meant to critique how money-and-profit driven businesses often neglect important sense-making work due to an excessive concern for return on investment.

I worry that our overly profit-driven business culture is underinvesting in the vital initiatives necessary to prepare the next generation to pass its final exam because the return on investment is too distant and too uncertain. Is there money-making potential in protecting civilization from asteroid impacts? Can profit drive initiatives to restore damaged parts of our ecology to mitigate dangerous changes in the global nitrogen, water, and carbon/oxygen cycles? Probably not. Profit is dangerously myopic. I think that is the point of the Bucky quip I cited.

In the video I used Bucky’s quip to suggest we might refactor our economy without money. I don’t think that was Bucky’s point. Moreover, I don’t think it would work. I almost succeeded in making that point by explaining David Graeber’s perspective on debt: the moralities of economics bind us together as a community. That is, money, at least in the form of debt, is a deeply human form of social ethics. In addition, human beings are skilled at thinking in terms of games with their imposed rules (gamification is now a fast growing new field of study). Therefore, it is probably necessary to include money-honey gaming in designs for any improved socio-political-economic system. Do you think money is important for our socio-political-economic systems?

Although Bucky was usually assiduously apolitical, one of his last books “GRUNCH” was a scathing work on our socio-political-economic systems. These systems may require the attention of design scientists. What do you think? Do you think Design Science should stay assiduously apolitical and restrict ourselves to only work on artifacts? Is there a limited way in which Design Science should work toward improving our socio-political-economic systems? How? And how not?

Harold emphasized the loss of jobs as a consequence of the forces of social evolution that Bucky called ephemeralization (“doing more with less”). I countered with a question about how jobs are created.

In W. Brian Arthur’s book *The Nature of Technology: What it is and how it evolves*, he suggests that ever-evolving technology always creates new needs and opportunities in an economy. It could be that the amount of opportunity created by ephemeralization exceeds the job losses which Schumpeter called “gales of creative destruction”. Arthur explains that the very same new technological developments that incur this destruction also deliver “winds of opportunity creation”. Unfortunately, the skills to identify and develop these opportunities are often quite different from the skills needed in displaced industries. So Harold’s points about the problem of labor are a very significant problem. Arthur’s book provides deep insights on the nature and evolution of technology including ephemeralization (though he does not use Bucky’s word).

Finally, I will note that our tendency to emphasize the negative (job losses) instead of the opportunity in this discussion parallels our discussion on the value of openness in our culture and the discussion on the media (see below). Could it be that we over-emphasize the losses without fully appreciating the “winds of opportunity creation” that Arthur identifies?

In the video, I referenced James Boyle’s argument about our bias against openness which is eloquently presented in this short 17 minute video.

I also referenced Johanna Blakley’s 15 minute TED Talk which argues that because the fashion industry has almost no ownership rights on its designs, it is more creative and profitable than industries encumbered by intellectual property (she is at the University of Southern California). Another great Bucky quote applies “ownership is onerous”.

The exquisite video series “Everything is a Remix” by Kirby Ferguson explains the mechanics of openness and sharing in creativity: Part 1, Part 2, Part 3, Part 4, and Embrace The Remix, his 10 minute TED Talk.

Finally, here are two short videos on the subject: Nina Paley’s 3 minute video “All Creative Work Is Derivative” and Jason Silva’s exhilarating 3 minute video “Radical Openness”.

How do you understand the value of openness? And sharing? Are we biased against openness? Is the free culture movement and the free software movement a harbinger for a more open future? Will sharing, freedom, and openness empower our civilization to new heights? Or is it a mistaken pipe dream?

Yikes, I lapsed into the common but trite conditioned reflex that the media is too negative.

It is important to remind people of the need to be skeptical of media reports especially when they highlight negative news. Negative news always travels fast and the media reach all of us quickly and we can become transfixed on the negative. Another danger from the media is the ability to spread social contagions of all types. So we need to foster a healthy skepticism of all ideas but especially those coming from the media.

It would be helpful if our friends in the media would more diligently engage skeptical voices to help temper the social contagions that race through our culture. However, as Harold points out the media is also a powerful force of good. When those social contagions are movements of major positive reorientation of humanity’s consciousness, it is the media who help us realign so rapidly.

We also need to recognize that the media is an important part of the large scale conversation that steers our civilization. It is important to recognize biases toward negativity (which the media often overly accentuate, but do not cause). It is important to keep a balanced perspective on this powerful force in our society.

If you are interested in watching more Harold Channer interviews that discuss Bucky and his work, I can recommend these:

- Buckminster Fuller 1974
- Michael Ben-Eli 05-08-14
- Christopher Zelov 05-29-13
- Michael Ben-Eli 05-24-12
- Allega Fuller Snyder & Elizabeth Thompson 06-04-12
- David McConville 12-16-11
- Allegra Fuller Snyder & Jamie Snyder 07-10-08
- Michael Ben-Eli 10-04-07
- Michael Ben-Eli 02-20-07
- Thomas T.K. Zung 11-24-05
- Christopher Zelov Aug 1994
- Jay Baldwin Nov 1994

Let me know what you think of my conversation with Harold or these addenda. I’d love to see your perspective on any of the ideas above. I look forward to continuing the conversation!

]]>- Education Automation Now and in the Future. In this talk I recognize Buckminster Fuller as one of the conceptual founding fathers of the Open Educational Resources (OER) movement, detail six of his educational ideas, and give a brief review of several OER courses I’ve taken to indicate the kind of comprehensive education now possible using freely available on-line courses.
- Synergetics and Model Thinking. In this talk I synthesize Scott E. Page’s Model Thinking with Buckminster Fuller’s Synergetics. I introduce both subjects, then discuss the importance of model thinking. Then I sketch some ideas about how Model Thinking and Synergetics can inform a more incisive approach to science.

Please share any thoughts you might have about these presentations in the comments. I would value your feedback.

]]>For me the most enticing facet of *projective geometry* is the profound way in which it treats *duality*. Duality is the notion that certain fundamental distinctions have similar structure in their complementary forms. In comparing a form with its dual, the basic structure remains even though the roles of the forms reverse. Inside and outside. Convex and concave. Yin and yang. In 2D (two-dimensional) projective geometry, point is dual with line; in 3D point is dual with plane while lines are self-dual. The relationship of duality is so penetrating and pervasive in projective geometry, that we might consider it *the geometry of fundamental duality*. It provides a geometrical stage upon which duality can be studied in a pure form.

Another profound aspect of projective geometry is its elementary treatment of *incidence* where one considers the join (∨) and meet or intersection (∧) of two basic geometrical objects such as point, line, plane, and hyperplane. The most fundamental correspondence of geometrical forms associates points and lines in dual arrangement: the points on a line form a *range* and the lines through a point form a *pencil*. The correspondence between a pencil and a range is a basic *projection*. Next a *perspective* relation joins a pencil with two ranges or a range with two pencils; that is, by combining two elementary projections. Such a *perspectivity* maps points to points, or dually, lines to lines as shown in the figure.

In the essay “Design Strategy” in Buckminster Fuller’s book *Utopia or Oblivion*, he includes projective geometry in his list of recommendations for a curriculum of design science. The connection between projective geometry and design thinking is an area that deserves more attention.

The rest of this essay tersely describes a broad listing of some of the more basic models of projective geometry. *Models* are a powerful tool for learning and for understanding as explained in my essay about the Importance of Model Thinking (based on Scott E. Page’s course). The models included below should provide an introduction to and an overview of projective geometry for those new to the subject (Note: some of these models require background knowledge that is not explained here. They are indicated with a . I encourage you to skim or skip such models, but to read on as later models may be more tractable.) I hope the experts will find the succinct summary and references useful. Although this list is fairly comprehensive, there are many models that are necessarily omitted. If you have a favorite model, please post a comment about it.

- Configurations. A configuration (p
_{n},l_{m}) is a set of p points and l lines such that each of the points lies on n lines and each of the lines contain m points. So there are n lines through each of its p points and m points lie on each of its l lines. For example, (4_{3},6_{2}) is a complete quadrangle and its dual (6_{2},4_{3}) is a complete quadrilateral. If p = l and m = n, the configuration is self-dual. E.g., a triangle is (3_{2}). A finite projective plane of order n PG(2,n) is a self-dual configuration of the form (n^{2}+n+1)_{n+1}if such a configuration exists (which happens when n=p^{k}where p is prime and k is a positive integer). Most descriptions exclude the case where n=1 which would give a triangle (3_{2}). When n=2, we get the fano plane (7_{3}). This treatment emphasizes that there are many finite projective planes. Configurations are easy and fun especially since finite projective planes require non-straight lines to draw on paper. For more on configurations see [Coxeter2003, p. 233] and [Polster, pp. 27-38]; for more on finite projective planes see [Polster, pp. 5-7, 67-124 (includes many beautiful drawings)], [Coxeter1989, pp. 233-4] and [Coxeter2003, pp. 91-101].- Block Designs. A nice characterization of finite projective planes is provided by the subject of block designs (which originated as part of experimental design). Namely, the projective plane of order n PG(2,n) exists if and only if an (n
^{2}+n+1, n+1, 1)-design exists if and only if there is a complete orthogonal family of Latin squares of order n (I studied Latin squares with Thomas Zaslavsky in an undergraduate combinatorics course at Binghamton). See [Roberts, pp. 356-405].

- Block Designs. A nice characterization of finite projective planes is provided by the subject of block designs (which originated as part of experimental design). Namely, the projective plane of order n PG(2,n) exists if and only if an (n
- Axioms provide another approach to characterize projective geometry. This list defines a projective plane (see [Polster, p. 5]).
- Axiom of joining. Any two distinct points are incident with a unique line.
- Axiom of meeting. Two distinct lines intersect in a unique point.
- Axiom of a quadrangular set. There are four points of which no three are collinear (meaning no three lie on the same line). This axiom effectively defines a quadrangular set.

- Following [Polster, p. 67] and [Coxeter1989, p. 256], here is an axiomization for projective space (meaning 3-dimensional or 3D):
- Axiom of joining. Any two distinct points are incident with a unique line.
- Axiom of transversals to triangles. If a line does not include any of the vertices of a triangle but it intersects two of its sides, then it intersects the third side of the triangle.
- Axiom of minimal extent of lines. Each line contains at least three points.

- Central projection. [Hefferon] describes six models of the projective plane based on central or Gnomonic projection.
- In the first three models, P can represent a projector, a painter, or a pinhole camera which projects the source scene S to an image plane I. The geometry of the image plane I is projective.
PIS Central Projections. Adapted from [Hefferon].

- Hemisphere Model. In an attempt to simplify these three models, we can place P at the center of a hemisphere and map any point X in space to the point on the hemisphere on the line joining P and X (l=P∨X). Points on the equator get mapped to two points on the hemisphere, so they must be treated specially. Also see [Polster, p. 227-8].
- Sphere Model. We can do even better with a whole sphere and mapping any point X to the two antipodal points where the sphere meets the line joining X with P (l=P∨X). The model treats the two antipodal points as one “point” and a great circle (a circle on the sphere whose center is also the center of projection) as a “line” in the projective plane. Also see [Polster, p. 225].
- Extended cartesian plane. Continuing the analysis by choosing the unit sphere (meaning radius of 1) and the tangent plane z=1 so it touches the plane at the sphere’s North pole. The model now maps points X in space to points on the plane z=1 (where l=P∨X intersects z=1). There is a problem with points which map to the equatorial points on the sphere: they are parallel to the plane z=1! So the model imagines that in each direction around the equator there is an “ideal point” (sometimes called a “point at infinity”) in the plane z=1 which corresponds to the two equatorial points. Since the totality of these “ideal points” map to the great circle known as the equator, the collection of “ideal points” forms an “ideal line” (which is sometimes called the “line at infinity”). See [Wildberger] for a nice video introduction to projective geometry that explains this model.
The unit sphere tangent to the z=1 plane. Adapted from [Hefferon].

The sphere model gives a concrete finite perspective on abstract “infinity”. From this perspective “infinity” is just a poetic expression of a concrete reality. It could be that all uses of the word “infinity” in mathematics are like this. See [Zeilberger] for another ultrafinitist interpretation of infinity in mathematics.

The traditional description of this model starts with the traditional Euclidean plane (or more generally the affine plane) and affixes a single “ideal point” to each set of parallel lines. Each of these “ideal” points lies on an “ideal line”.

- In the first three models, P can represent a projector, a painter, or a pinhole camera which projects the source scene S to an image plane I. The geometry of the image plane I is projective.
- Bundle models of the projective plane. In 3D space, a bundle is the collection of all the lines and planes through a given point O (which we can think of as the origin or the center of the bundle). The modern algebraic treatment of projective spaces springs from this fruitful model.
- “Points” are represented as lines through O. “Lines” are represented by planes through O. The “line” joining two “points” is the plane perpendicular to them both. The intersection of two “lines” is the line perpendicular to their two normals (the normal of a plane is the line or direction or vector perpendicular to it).
- Perpendicular Vectors. In the bundle model we can represent points and lines as vectors anchored to the origin at O. A “point” is simply represented by a vector. A line is represented by the normal vector to a plane. A point is incident with a line if their dot product is 0 (that is, they are perpendicular). The line joining two points is given by the cross product of the vectors representing the points. The intersection of two lines is the vector given by the cross product of the vectors representing the lines (each normal to its plane). I find it remarkable and lovely that the cross product serves as both meet (∧) and join (∨) in this model: it emphasizes the duality at the vector operation level!
- Homogeneous coordinates model. This is effectively an algebraic representation of the perpendicular vectors model. Since our bundle is in 3D space, we can represent both points and lines as ordered triples since they are both represented by vectors. These are scale independent and so are homogeneous.

- Extended cartesian space. In ordinary Affine (or Euclidean) 3-space, we can add an ideal point for each set of ordinary parallel lines. In any given ordinary plane, the collection of its ideal points form an ideal line. All those ideal points and lines form an ideal projective plane, the so-called plane at infinity. See [Coxeter2003 pp. 103-109] and [Richter-Gebert pp. 209-11].
- A very abstract approach (see [Busemann and Kelly p. 1] and [Richter-Gebert p. 36]) is to remove special cases such as the distinction between intersecting or parallel lines from Euclidean (or Affine) geometry. This model sees projective geometry as a more general geometry with a minimum of special cases.
- Vector subspace model of projective spaces. Given any n+1 dimensional vector space over any arbitrary field F, the projective space of dimension n can be represented by its k dimensional subspaces, k=0,1,…,n. When k=0, the trivial vector subspace {0} determines the empty projective subspace. When k=1, the collection of 1-dimensional subspaces represent the “points” of our projective space. When k=2, the collection of 2-dimensional subspaces represent the “lines” of our projective space. When k=s, the collection of s-dimensional subspaces represent the s-planes of our projective space. If an s-plane is contained in an t-plane, we say that it lies on or is incident with it. This containment relation forms a lattice in our projective space. The lattice operations of join (∨) and meet (∧) give the incidence structure of the space. See [MacLane and Birkhoff, pp. 592-595].
- Abstract naming. This is just a notational convention to represent projective geometries, but names are models of a sort too. The projective space of dimension n over the algebraic field F is denoted PG(n,F).

As you can see there are many, many models for projective geometry. Since a complete enumeration of projective spaces has never been compiled, there are many more models to be imagined and constructed. If you have any questions about these models or know of any other projective models, please post a comment. I look forward to discussing them!

One of my many reasons for writing this essay is to provide background information for the Harmonic Perspective paper that Jeannie Moberly and I prepared for the 2012 Bridges Conference and Proceedings. I wrote a Harmonic Perspective page to collect information about our work on the subject.

- Herbert Busemann and Paul J. Kelly.
*Projective Geometry and Projective Metrics*. Dover, 2006. - H. S. M. Coxeter.
*Introduction to Geometry*. Wiley, second edition, 1989. - H. S. M. Coxeter.
*Projective Geometry*. Springer, second edition, 2003. - Jim Hefferon.
*Linear Algebra*see “Topic: Projective Geometry” in the chapter “Determinants”. On-line textbook with a free license. Accessed 10 July 2012. - Saunders MacLane and Garrett Birkhoff.
*Algebra*. AMS Chelsea Publishing, third edition, 1999. - Burkard Polster.
*A Geometrical Picture Book*. Springer, 1998. - Jürgen Richter-Gebert.
*Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry*. Springer, 2011. - Fred S. Roberts.
*Applied Combinatorics*. Prentice-Hall, 1984. - Norman Wildberger.
*MathHistory8: Projective geometry*(YouTube video). Accessed 22 July 2012. - Doron Zeilberger.
*“Real” Analysis is a Degenerate Case of Discrete Analysis*. Accessed 10 July 2012.

In addition to reviewing the course, I will also suggest that **model thinking** is a new more incisive kind of science. This approach and its nascent toolkit for understanding, decision-making, prediction, strategy, and design is vitally important for practitioners of all types. Model thinking may be just the type of tool humanity needs to solve some of its thorniest problems. As such its arrival into broader consciousness is not a moment too soon!

So if you want to be out there helping to change the world in useful ways, it’s really really helpful to have some understanding of models.

— Scott E. Page

There are many ways to model the world. One of the most popular is with proverbs or short pithy sayings (our modern media seem to particularly love this deeply flawed “sound bite” approach to knowledge). As Scott Page points out, there are opposite proverbs too. For instance, the opposite of “nothing ventured, nothing gained” is “better safe than sorry.” Proverbs and their more elaborate cousins, allegories, can model or represent the world with persuasive stories, but they provide little discerning power and little basis for deeper understanding. In contradistinction, *model thinking* with its greater concern for precision can help us more carefully distinguish a complex of important factors with their interrelationships and behaviors. Therein lies its power!

Is intuition sufficient? No! Philip Tetlock, Robyn Dawes and others have demonstrated that simple naive models outperform experts of all stripes. In 1979 Dawes wrote a seminal paper, The Robust Beauty of Improper Linear Models in Decision Making, which showed the effectiveness of even “improper” linear models in outperforming human prognostication. Tetlock has made the most ambitious and extensive study of experts to date and finds that crude extrapolation models outperform humans in every domain he has studied.

That is not to say that models are “right”. Page emphasizes that all models are “wrong” too! Which leads to his most profound insight in the course: you need **many types of models** to help think through the logic of any given situation. Each model can help check, validate, and build your understanding. This depth of understanding is essential to make better decisions or predictions or build more effective designs or develop more effective strategies to achieve your goals.

Is intuition important? Yes, absolutely! The *many model thinker* relies upon intuition to select and critically evaluate a battery of models or to construct new or modified models when appropriate. These models help test our intuition. Intuition helps tests the models! Writing out a model often identifies facets and elements of the situation which intuition misses. Intuition is essential to find the aspects of the models that are a bit off the mark — and all models are a bit off. Model thinking is not “flying on instruments” or turning control over to mathematical or computer models. Instead it is about evaluating and comparing diverse models to test, build, fortify, and correct our intuitions, decisions, predictions, designs, and strategies.

Page’s course is filled to the brim with fascinating models! One of the first models Page introduces is Thomas Schelling’s segregation model which represents people as agents on a checkerboard. We discover deep and unexpected insights about how people sort themselves into clusters where everyone looks alike, for example, the segregation of neighborhoods based on race, ethnicity, income, etc. It is the first of many agent-based models to be discussed.

Page’s lectures are filled with the wonder of discovery (another trait he shares with Buckminster Fuller). We learn that seemingly elementary topics like aggregation or simple addition hold unexpected surprises. However, I was disappointed that Page didn’t use Bucky Fuller’s definition of “synergy” as the “behavior of whole systems unpredicted by the behavior of their parts taken separately” to provide an explanation for the limits of aggregation. Page’s treatment is excellent. Perhaps I am a bit too fond of Bucky’s “*synergy*” especially considering the fact that it urgently needs to be rescued from its current hackneyed and misleading usage.

Another really important (and fun) model helps us distinguish between luck and skill. We are surprised to learn that results among the highly skilled are mostly determined by luck! Pages’ related videos on randomness are helpful in understanding the important notion of uncertainty which I’ve discussed before.

I was particularly interested in Page’s lecture on *tipping points* where a small change produces large effects. Page and his colleague PJ Lamberson recently wrote a paper on Tipping Points which models them as discontinuities. Page emphasizes that tipping points should be distinguished from exponential growth which is often erroneously imputed to have a tipping point.

Another fascinating topic on which Page has done pathbreaking research is the development of models to show the efficacy of diversity in problem-solving. I enjoyed finding and perusing the paper on diversity by Lu Hong and Scott E. Page. The presentation in the course was simpler. Page is a gifted teacher who can explain sophisticated cutting edge research with concrete clarity.

It intrigued me that Page’s research concerns such elementary yet pervasive models. Model thinking is a young discipline that is proving itself ripe for breakthroughs that can inform our basic understandings in profound ways. I found all of Page’s models thoroughly fascinating!

[S]cience is basically culture’s answer to the big problem of epistemology, which is how can we know anything at all.

—Stephen Stearns, Yale Biologist

The standard view of modern science is given by the hypothetico-deductive model which posits that scientists propose hypotheses which are tested by experiments as part of a contest of alternative ideas to choose a winner which is dubbed Scientific Truth. In my study of science and engineering I have been repeatedly surprised at the unreasonable efficacy and perspicacity of “inferior” theories. For example, even though Newtonian mechanics is replaced by Einstein and Quantum as Truth, physics still teaches Newton and engineers still use it. Even though we have known for more than 2000 years that Earth is spherical, carpenters continue to use Euclidean geometry instead of the theoretically better spherical trigonometry. Why are such “improper” models so profoundly useful?

Scott E. Page’s idea of the many model thinker provides the crucial insight: by exploring several models, critiquing them, finding each models’ limits, its strengths and weaknesses, we build and deepen our understanding. In this way model thinking may be a new kind of science which places the emphasis on development and comparison of models which explore a subject from many perspectives and many levels of abstraction. It could be that the center stage of science is a dynamic dance where many models express their unique insights while challenging us to see how and when to integrate them with other models to compose new dances and new insights.

It could be that each of the many models reveals parts of the truth. Perhaps, only an ever-expanding integration of multiple models can approximate truth. Ephemeral “truth” may simply be the best current synthesis of our collection of partially overlapping, partially contradictory, and profoundly interacting models.

The principle of multiple model thinking has an important historical precedent in T. C. Chamberlin’s **method of multiple working hypotheses** (see my essay on Scientific Understanding for a longer discussion of Chamberlin’s work).

In the use of the multiple method, the re-action of one hypothesis upon another tends to amplify the recognized scope of each, and their mutual conflicts whet the discriminative edge of each. The analytic process, the development and demonstration of criteria, and the sharpening of discrimination, receive powerful impulse from the co-ordinate working of several hypotheses.

T. C. Chamberlin, The Method of Multiple Working Hypotheses, 1890

The important thread of *modelability* in Buckminster Fuller’s Synergetics provides another deeply important role for a science of model thinking: to foster vital discussions with the whole of society. Science, technology, the arts, and culture are all changing our world in profound ways. Model thinking is a powerful way to engage the whole of humanity in a more incisive discussion to better understand how our complex world works. It provides a basis to make better decisions, project future trends, strategize and re-design our worlds to solve Humanity’s problems. As such Scott E. Page’s *Model Thinking* course is a beacon of light!

Comprehension of conceptual mathematics and the return to modelability in general are among the most critical factors governing humanity’s epochal transition from bumblebee-like self’s honey-seeking preoccupation into the realistic prospect of a spontaneously coordinate planetary society.

—R. Buckminster Fuller, Synergetics 216.03

Scott Page does something extraordinary in this course: he strives for and succeeds in providing a conceptual, clear, and concrete introduction to formal models. His gifted exposition, broad comprehensive perspective, and infectious sense of wonder inspired me to want more. Conceptual, clear, and concrete are also the values I took from Bucky’s *Synergetics*. In *Synergetics*, we often build geometrical models; in Page’s course we “run the numbers” and write out elementary arithmetical and algebraic relationships. Both can be appreciated conceptually without much math. But both require effort to master and appreciate in depth. Both reward the diligent student. Both are incomplete and challenge the attentive student to ask more questions and dig deeper.

Page is effective in developing one’s faculty to shift between the conceptual (broadly understandable) and the mathematical (where definitions and manipulations such as algebra predominate). Too often math is taught with too little concern for the concepts and it can become a drudgery of computing. Buckminster Fuller extolled conceptuality when he wrote “Synergetics makes possible the return to omniconceptual modeling”. It is a critical value necessary to help people get access to tools for understanding the complexity of our ever-changing worlds.

This is a profoundly revolutionary course: it makes model thinking accessible to a broad audience with a broad array of conceptual, clear, and concrete models!

The videos are dense which is typical for Coursera courses. To ensure my head didn’t explode, I watched only 2 or 3 every night. Since each “lecture” consisted of between 4 and 7 videos, it sometimes took me three days to watch a whole lecture. There were two lectures each week. Each day I would also rewatch the videos from a day or two before and take detailed notes. I took a total of 23 pages of handwritten notes. When I finished these preparations I took the weekly quiz. In this way I worked at a steady pace of only 1-2 hours a day (including the first watching, the note-taking, and the quiz-taking).

The quizzes often only took 20 minutes or so to complete (sometimes longer if I had to think about or research something). I insisted on getting 80% or better which meant that I often had to take a quiz twice (the second time usually went quicker). I love that Coursera encourages taking quizes and exams multiple times. At the beginning of a course, I am often “out of sync” with its style of posing questions. By retaking the quizzes I can work through any cobwebs in my thinking. In hindsight, I wish I had retaken each quiz until I achieved a 100% score. There were always a few questions that challenged me, but overall I found the quizzes and the course to be fairly easy (much easier than the Databases course that I took last fall).

With each lecture Page also provided additional reading material. I read it all and searched the Net for additional primary sources (for example, I found and skimmed a copy of Schelling’s 1971 paper). Admittedly this added another hour or so per day to my time investment. As it turns out, the quizzes only tested what was in the videos, so skipping the readings would probably not affect your score. The readings would benefit those who need to see the material again in different forms or those who want a little more depth. I loved them, but if you are pressed for time they can be skipped.

I used the discussion forums a bit. They are a valuable resource, but I found them too time consuming. I used them mainly to sample what others were talking about. I really appreciated the work of Timothy Riffe who put out some code for R and Jennifer Badham who contributed to a number of interesting threads and posted her excellent course notes.

My effort was rewarded with a Statement of Accomplishment for completing the Model Thinking course. My heartfelt thanks to Scott E. Page, the University of Michigan at Ann Arbor, and Coursera for their efforts in making this class available. It was exquisite!

I highly recommend this class. If you can spare about 10 hours a week for 10 weeks, this course will significantly improve your ability to use models to deepen your understanding and effectiveness. Click here to register for the **Model Thinking** course with Scott E. Page (next offering is in September 2012).

The pilgrim is Dante himself and his guide through most of the journey is the Roman poet Virgil. Open Yale Courses provides its own able guide in Giuseppe Mazzotta who presents a fascinating and deeply engaging course ITAL 310: Dante in Translation (videos at YouTube). Mazzotta places the *Commedia*, more commonly entitled *The Divine Comedy*, in the encyclopedic tradition (a circle of knowledge through the liberal arts) but he also calls it an epic, romantic, autobiographical, and visionary poem. Indeed by the end of the course, I had lost track of how many different angles on the poem Mazzotta had identified: prophetic, philosophical, historical, sublime, humanistic, theological, scientific, geometrical, musical, a poetry of hope, a poetry of the future, etc., etc.!

Dante makes large claims for poetry: poetry is a way of knowing.

— Giuseppe Mazzotta

We (Jeannie and I) read the text of each poem, *Vita Nuova* (Dante’s first book) and the *Commedia*, before watching the lecture covering that text. We did not read many of the notes. That allowed us to go faster, but it meant we missed a lot of background which, in hindsight, is necessary to more fully understand the poem and the lectures. The benefit was the reading had a continuity and we limited our time investment.

Mazzotta skips some cantos (major divisions in a long poem; song in Italian) in the poem, but we read both poems in their entirety. We read Mark Musa’s translation from our local library. We tried to watch a video every 4 days (that is our favorite pacing: just slightly slower than the actual Yale class), but sometimes the gaps were a bit longer. There are 24 videos most of which are 75 minutes long. In all there are nearly 27 hours of video. It took four months to complete (29 Jun–29 Oct 2011). In the videos Mazzotta’s gestures were full of meaning; but frankly, you could fully enjoy this course with the audio only mp3s. Only once or twice does he write something on the board and once he passes around a geometric model that might be the shape of Dante’s Universe.

After watching each lecture, I posted some thoughts to FaceBook as a way to reinforce the experience and share what I was getting out of the course with friends. In preparing those notes, I typically read the transcript of the lecture and looked up a lot of Greek & Roman mythology and the relevant historical figures, places, and events in Wikipedia. I often re-read parts of the poem. The lectures were so rich and interesting, that many of my comments became substantial. We generally watch video courses for our edutainment, so I commented only on the parts of interest to me (these were not student oriented CliffsNotes!). In my comments, I tried to reflect on the poem, share my frustrations with it, and wonder about it and its meaning.

During the course, I also used the translation provided by the Princeton Dante Project (they have have many additional resources on Dante). Another resource for learning more about the poem and the poet is Saylor’s ENGL409 course on Dante.

The experience of working through ITAL 310 was very challenging to me. I was unfamiliar with much of the historical knowledge that provides context for understanding the poem, its subjects, and its allusions. In addition, my training in logic, mathematics, and computers may have atrophied some of the skills needed to understand metaphor and poetics (the course was, in this sense, a very healthy exercise for me!). Since I aspire to be somewhat competent in all of humanity’s cultural traditions, I took the course as an opportunity to develop my poetic interpretation skills. But on first reading the poem was very “flat” to me. Mazzotta’s lectures exposed the richness and depth of the text. Bridging that gap made this the most challenging on-line video course I’ve ever taken (it was my 11th in four years). Jeannie on the other hand enjoyed the course as pure edutainment.

I was gratified to find that by the end of the course, I was starting to see some of the allusions and metaphors! I even hazarded a few interpretations on my first reading in some of the later cantos. My herculean effort to learn some of the background history and philosophy and my reflections on FaceBook (using hermaneutical techniques) together with Mazzotta’s exquisite guidance led to a small but perceptible expansion in my poetic skill. I was thrilled! But listening to Mazzotta’s final lectures and re-reading my notes to prepare this review, I must admit that my poetic prowess is still one of my weaker faculties.

Another purpose for watching this course was my personal quest through a circle (encyclopedia) of learning (namely, my search for interesting on-line video courses). ITAL 310 was a wonderful addition to that project: it broadened my understanding of poetics and western civilization. Which leads me to another big benefit of free on-line courses: you can experience a subject that is outside your strengths or even outside your comfort zone! You can confront fears about subjects like mathematics, engineering, science, or even poetry(!) by taking a class without risking a bad grade or worrying about the return on investment of tuition. You may find, as I did in this course, that even with a thick skull and years of neglect, one can still turn a new leaf and strengthen a weak faculty despite feelings of inadequacy induced by one’s past. I invite you to dare to experience a subject that you have long neglected or one in which you fear inadequacy: watch one of the many free on-line video courses and grow your mind with the wonders of new learning experiences!

Mazzotta discusses how Dante’s unfinished text “On the Vulgar Tongue” plays a central role. A key element is the concept of perspectivism which Mazzotta explains as “the presence of viewpoints, various viewpoints, which one somehow manages to control, or know, all viewpoints. … this is the way the whole of Inferno is written … Perspective means that the world that I see shifts, changes according to the position that I, the spectator, occupy in the field of vision. … [T]he perception of reality changes according to the position we occupy…. Dante uses this perspectivism, which I repeat, really means a way of assembling various points of view.” Mazzotta suggests that the “practice of perspective” dates to Dante’s time. He calls the *Commedia* “a perspectivist story.” I am impressed and fascinated.

Mazzotta’s claim that the *Commedia* is an “encyclopedia of learning” and that poetry is a way of knowing was amply demonstrated. The poem, especially when supplemented with Mazzotta’s commentary, provides glimpses to see, feel, and think about the interconnectedness and unity of knowledge. I was especially thrilled to find Dante substantiatively treating such unusual poetic subjects as mathematics, geometry and experimental science! The astute reader might notice some influence from these ideas in my thinking about the ways of knowing in my essay An Enquiry Concerning Scientific Understanding. I wonder, is it possible that Dante could help bridge the chasm some modern people wrongly imagine to exist between the humanities and the sciences?

In Canto XXIV of Paradiso, Dante says “faith is the substance of things hoped for, the evidence of things that are not seen.” Mazzotta’s interpretation: “He tries to make faith and reason co-extensive … belief, knowledge and faith really belong together, they implicate each other. They are not the same thing … it’s a way of acknowledging limitations of what one knows. … It really means, I think, at a deeper level, that faith itself is a mode of knowledge. … faith opens your eyes and it’s a way of showing you something about the world that the reason alone cannot do.” Interesting.

Mazzotta said, “The idea of knowledge is one that keeps changing … knowledge keeps always expanding and including voices that had been rejected”. One of my big takeaways from this poem was coming to appreciate that Dante is very deeply open minded!

To my mind, this “encyclopedia of learning” is perhaps the greatest quality of the *Commedia*. I found it inspiring. It is gratifying to find that this broad-minded perspective is evident in medieval Christian thought. It is not a new idea, but we need it now more urgently than ever. Our problems are more complex than ever before and demand the kind of spirited breadth of vision, perspective, and ways of knowing that Dante evokes!

I was deeply impressed by the vibrancy of the culture of the late Middle Ages. This period in history should not be called the “dark ages” which I now understand to be a disparaging term originating in the arrogance of the “enlightenment”. Dante, his contemporary Giotto, and many others demonstrate that the middle ages were full of intellectual and artistic vigor!

I will level one strong criticism against the *Commedia*. I found the horrific, unending stream of punishment throughout Inferno to offend my joy and wonder in mistake mystique. This was my biggest disappointment in the poem, but it is healthy to remember for how long humanity was ignorant of the wisdom of mistake mystique.

Mazzotta suggests that the modern notion of romantic love was invented during the middle ages by poets like Dante. I was fascinated to learn that romance may be a cultural artifact that is less than 1000 years old.

Mazzotta tells of “Dante’s universe of desire. We’re impelled by desire, and desire is really what moves us. It’s love that moves us; it’s desire that impels us to go one way or the other.”

As the pilgrim rises beyond Saturn, the seventh heaven after Earth, Mercury, Venus, Sun, Mars, and Jupiter, and nearly 700 years before NASA presented Humanity with its first picture of Earth from space, Dante wrote the following visionary passage which puts Humanity in cosmic perspective (Canto XXII of Paradiso): “And all the seven, in a single view, showed me their masses, their velocities, and all the distances between the spheres, as for the little threshing-floor that makes us so fierce all appeared to me from hills to river-mouths, while I was wheeling with the eternal Twins.” The Twins refers to Gemini, Dante’s astrological sign. I love the cosmic perspective! By ending here, perhaps, this image will help you ponder Dante’s cosmic perspective and his cosmic significance for a few moments longer ….

A big Thank You to Open Yale Courses and Giuseppe Mazzotta for a wonderful experience!

Acknowledgement. I’d like to thank Joshua Pang for making the astute observation that Mazzotta is a guide for the course, like Virgil was for the pilgrim, in his comments on my FaceBook.

]]>Many thinkers including Einstein, Buckminster Fuller, and D’Arcy Wentworth Thompson have argued in support of the traditional deterministic world view[1]. However, Quantum mechanics, machine learning, and behavioral economics are three prominent areas which have helped realign modern thinking to apprehend that randomness and uncertainty may be fundamental and pervasive. Leonard Mlodinow in a 2008 book goes further and argues that randomness rules our lives.

In preparing for and discussing randomness at a recent meetup of the Ben Franklin Thinking Society, I started to gravitate to the hypothesis that uncertainty and determinism may be like inside and outside or concave and convex. They may be both real, both partially right and partially wrong, both revelatory and misleading. It may be that each perspective is a “tuning in” to only part of a reality that is *both-neither*[2].

The principle of functions states that a function can always and only coexist with another function as demonstrated experimentally in all systems as the outside-inside, convex-concave, clockwise-counterclockwise, tension-compression couples.

— R. Buckminster Fuller, Synergetics 226.01

Here are several ways to see the dual and co-occurant qualities of the stochastic and deterministic models or world views.

In a deterministic model of the world, the fixed set of laws that govern everything apply to every quanta of energy or their constituents. So computing the state of the world requires applying these fixed laws to each such quanta from some initial state and iterating through each picosecond of time. Clearly, this is computationally infeasible except for the computer known as *Universe* itself. So any effective simulation or calculation will entail estimates and approximations, that is, randomness. Unwittingly, randomness imposes itself into the system!

Conversely, in a stochastic model the relationships between data are given by frequencies with respect to their sample space, the set of possible outcomes. What could be more deterministic than the elementary counting of frequencies? Indeed probability is basically a form of advanced counting in ratios. Deterministic indeed!

Now consider measurement. The basis of a scientific model involves measurable parameters. Data are measurements. Science has determined that all measurements involve uncertainty. MIT physicist Walter Lewin puts it emphatically: “any measurement that you make without any knowledge of the uncertainty is meaningless!” Measurement theory is built upon the law of error which is a principle of the science of randomness. Hard data acquires its validity and persuasiveness from the science of chance!

The key to understanding measurement is understanding the nature of the variation in data caused by random error.

— Leonard Mlodinow

On the other hand, the law of error is a central principle in statistics, the science of inferring probabilities from observed data. Such inference is the gold standard of scientific truth. The techniques of scientific inference are based on the mathematics of randomness. Like all mathematics, the theory is definite, rigorous, and repeatably verified by logic, proof and experiment. The sciences of probability and statistics are rigorous and deterministic like all mathematics!

Even in a fundamentally deterministic world, our understanding, decision-making, strategies, predictions, measurements, and designs are predicated upon uncertainty and randomness. To be effective we must be cognizant of these lingering unavoidable uncertainties.

Conversely, even in a fundamentally uncertain world ruled by randomness, pattern and order emerge and can be identified. To be effective we can and should seek the design and structure permeating through the apparent randomness.

From these considerations, I conclude that randomness and determinism always and only coexist. They are inseparable. Each provides a spectacular, incisive perspective on reality. The careful thinker or practitioner should be facile in using both types of models to get a more wholistic, more complete picture of the world in which we find ourselves. This is evidence that *both-neither* should be our guiding principle in seeking truth!

Do you find the argument compelling? Is it sound? Can you help me improve it? Do you see other ways in which these two models interpenetrate and interaccommodate? How do you see the interrelationship between determinism and randomness?

To better develop my understanding of a more complete set of models (beyond superficial determinism vs. stochasticity), I am excited about Scott E. Page‘s new and just started on-line video course on Model Thinking. I think we need many diverse models to sharpen our thinking and uncover subtleties in the complex systems and theories upon which our civilization is built. I am looking forward to wrapping my head around the 21 or so models in this course. You can register for the Model Thinking course by filling out the form at http://www.modelthinker-class.org/.

So if you want to be out there helping to change the world in useful ways, it’s really really helpful to have some understanding of models.

— Scott E. Page

Finally, here are three good audio-visual resources that explore issues of randomness further:

- Radiolab’s exquisite podcast on
*Stocasticity*. - World Science Festival: The Illusion of Certainty: Risk, Probability, and Chance.
- Persi Diaconis’ talk “On Coincidences” (scroll down near the bottom where the 1998 lectures live or search for
*Diaconis*).

[1] Click here to read my previous essay on randomness where arguments for determinism are discussed.

[2] Credit to Tom Miller for the wonderful expression *both-neither*.

It seems to me this Pickwickian sentiment is the thread that holds together the plot of *The Posthumous Papers of the Pickwick Club*, the title in the original publication. The sentiment emerges explicitly with Mr. Bolton’s clamour in chapter 1 and continues with the cab driver in chapter 2 who assaults Pickwick for taking notes on their conversation. Throughout there are countless incidents of turmoil and challenge which after at least some modicum of sound and fury end with wholesome benevolence and good will.

In the big picture of the novel, it is Pickwick’s relationship with Nathaniel Winkle which exhibits Pickwickianism in its most dramatically nurturing and good-hearted sense. It is Winkle, the sportsman, whose careless shooting wounds fellow Pickwickian, Tracey Tupman. Winkle’s disastrous outing on ice skates results in Pickwick’s scathing admonition “[You’re a] humbug, sir. I will speak plainer, if you wish it. An impostor, sir.” Then at Pickwick’s trial, it is Winkle who volunteers the “one instance of suspicious behavior towards females” which helps the jury decide against our good natured and innocent protagonist.

Through all these ills borne of Winkle’s youthful ineptness, our eponymous leader embodies Pickwickian good spirit and supports his protégé. He intervenes after the unfortunate incident with the Dowlers in Bath which sends Winkle in flight. The dénouement of the novel is Pickwick’s difficult mission to reconcile Winkle with his father. I was struck at Pickwick’s remarkable devotion to Winkle and how that relationship exudes the Pickwickian sentiment through and through. Is this the plot that GK Chesterton missed in critiquing the novel?

These good-hearted and good-humoured adventures in the large as well as the innumerable little scenes throughout the novel reinforce my view that the plot of *The Pickwick Papers* is a tale of adventure showcasing *the Pickwickian sentiment* which through numerous fun and funny tribulations end with good-humour, pride and exultation. A comedy indeed!

It may be that my exalted view of Pickwick is unjustified. For a more conventional perspective see Edward Pettit‘s write-up Dickens Literary Salon: Pickwick Papers with its sundry references.

How would you characterize *“Pickwickian”*? Does “the Pickwickian sentiment” constitute a plot?

When I discovered Buckminster Fuller it was his benevolence (“the planet’s friendly genius”) that attracted me. Could my enchantment with Pickwick have led me to another humanist like Bucky? I do not know. But both the great character and the great thinker share Pickwickian qualities. The one that seems most striking is Bucky’s Mistake Mystique — Pickwickian indeed!

Do you see connections between Bucky and Pickwick?

]]>Please let me know what you think of it.

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