A Word from the Editor - James Swingle


Why Noneuclidean?

        Manhattan is a classical geometer's paradise—wide avenues traveling North and South in parallel, narrower streets traversing the island from East to West without ever meeting.  However, as in so many other ways, in this, too, Greenwich Village is a special place.  Walking west along 4th St., crossing 6th Avenue, you enter the West Village, where the simple, rational order of a Euclidean world is left behind. Streets suddenly bend like light shining through a prism.  You pass Sheridan Square, where some once seemingly un-crossable lines were permanently crossed quite powerfully in the Stonewall uprising.  And as you continue on, you reach the corner where the normally parallel West 4th and West 12th Streets meet.

        I've long walked by that spot and thought it would be the perfect location for a small eatery called Noneuclidean Cafe.  A place where people could sit down over some spicy food and a strong coffee and talk about art, discuss their personal journeys.  Unfortunately, the rent for such a space in Manhattan would leave me so deep in debt I wouldn't be able to afford a coffee in my own cafe.  And alas, I have no experience in the restaurant field.

        So, since rent is so much cheaper online than in Manhattan, I came here to create my cafe, my little place where, though everyone would have to bring his or her own coffee, they could nonetheless share their art, and discuss their personal journeys.  And I kept what I thought was a pretty cool name—Noneuclidean Cafe.  However, since it was no longer on the corner where West 4th St. met West 12th St., I did have a nagging feeling:  What the hell does it mean?  "Cafe" made sense—cafes have a long history of people sitting together to talk about what is most important to them, from art to revolution to ecstasy.  But why "Noneuclidean"?

        Please forgive me two quick paragraphs on the history of geometry.  I find it does to turn out to make for an interesting metaphor. 

    Euclidean geometry is based on five axioms1, or postulates.  The 5th axiom, the parallel postulate, states:

Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.2

Basically, this means that parallel lines never meet.  For centuries Euclidean geometry, in which parallel lines never met, was viewed as the only possible geometry.  However, many geometers were still bothered by this 5th postulate.  It just didn't seem quite as intuitive as, say, the 1st postulate:  "Any two points can be joined by a straight line."  Finally, in the 18th Century, it was shown that the 5th postulate could be changed, and that other, Noneuclidean, geometries were in fact possible:  hyperbolic geometry, which introduced ultra-parallel lines3, and elliptical geometry, in which parallel lines suddenly started to meet.  At this point, these discoveries showed that other geometries were logically consistent.  However, Euclidean geometry still seemed to be the way the world worked.

        And then Einstein showed up, and we found out that space was not in fact flat, but curved.  Well, Euclidean geometry describes a flat surface.  So suddenly, elliptical geometry, which describes a curved surface, was not just logically consistent, but a better way to describe the real world out there.

        I think the outline of the history of Euclidean geometry is something that gets played out again and again, on scales from grand historical narratives to our personal stories—those things we consider very common sense beliefs.  We start with a handful of ideas about what is the same as what, about what causes what—these are our premises about some situation or person or event.  And from these premises we build a structure of beliefs about how things work, or who someone is, or what something means.  And going back to our starting premises, we can find "proofs" for the various beliefs we construct along the way.  And as we do draw our conclusions, there is little doubt that we are describing reality with our beliefs, because they are all logical extensions of those starting premises.  Like Euclidean geometry for centuries, it is easy to fall prey to this tyranny of premises, of our unquestioned axioms.  Experiencing the world through the system of our beliefs, it's easy to forget that that filter of beliefs is there at all, that the world might not be an exact match to what we believe it is.  It's not that we are necessarily wrong—any more than Euclidean geometry is wrong—our beliefs provide an approximation of the world that allows us to act, to go about the business of living.  It's that it's easy to mistake our collection of beliefs for an exact match to reality4—just like many geometers mistook the logical system of Euclidean geometry for an exact match to reality.  So when I see the word "Noneuclidean," I think of it as a reminder to go back from time to time to test our premises. 

        For example, one premise upon which many of our institutions, as well as disciplines of study, are based is the premise that people act in rational self-interest.  That is the foundation of classical economics, and a central tenet of the free market philosophy.  However, does it describe reality?  Well, if one looks at evolutionary biology, the rational self-interested actor becomes a doubtful entity.  While all animals, including humans, certainly do act in self-interest at times, social animals like humans also act altruistically—it is in our genes' survival interest to do so5.  If one looks at empirical psychological experiments, the rational self-interested actor again doesn't seem to reflect reality.  Experiments show many situations under which people will not act in rational self-interest, but will be influenced by everything from the social environment6 to concerns for fairness7 to which words were surreptitiously inserted into a list they read at the beginning of the experiment8.  And yet, there are many who will tell you that classical economic theory is not just a system that approximates certain behaviors in a way that under some circumstances can be helpful, but a representation of the way the world is, the way people are.

        What happens to our world if we change that premise that people are rational, self-interested actors?  What if we try out the premise that people are sometimes rational, but often not, sometimes self-interested, and sometimes socially-oriented and altruistic?  It doesn't seem to me that this new premise is less reflective of reality than the classical economic one.  If one looks to evolutionary biology and experimental psychology, let alone our personal experience when people are often irrational, usually social, it seems to me the new premise is far closer to describing how the world works than the classical economic one.  And yet, large amounts of power are invested in institutions based on the belief that classical economics is true.

        On a personal level, what if we replace some of those premises about what we can't do, with premises that there are skills we don't currently have, but which we can learn?  What happens if, instead of holding premises like we "aren't good at public speaking," we change our premise to "public speaking is a skill that can be learned."  As someone who once stood with shaking voice and hands whenever I had to speak in public, who now loves teaching as one of my favorite activities, I'd certainly suggest the second premise, that public speaking can be learned, is both more empowering and more true.

        It seems to me that a lot of systemic change, on both a personal and a social level, starts with questioning some of our starting premises.  And so, I guess what I most of all want to suggest with the term Noneuclidean, is this testing of our premises.  And as we play with new premises, new starting points, we might find that many things we thought were forever separate from ourselves and our world will, like parallel lines in Noneuclidean geometry, in fact meet.

        So I created Noneuclidean Cafe to be a place where we can test some premises, and where people and things that don't always get a chance to meet can come together:  Where personal growth and the arts can sit down together; where unconscious and conscious are given voice to express themselves; where all change modalities find a chair; where traditional wisdom and Western science have a place at the table; where personal and social change are invited; and where all lines, whether straight or not, have the right to get married if they want.



 1  Euclid's 5 axioms are:

  1. Any two points can be joined by a straight line.

  2. Any straight line segment can be extended indefinitely in a straight line.

  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

  4. All right angles are congruent.

  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

2  Euclid worded the 5th postulate as stated in Note 1.  The wording used in the body of the article leads to the same geometry, and its relationship to parallel lines is clearer.

3  In Euclidean Geometry, all pairs of lines either intersect, or they are parallel.  In hyperbolic geometry, such pairs of lines, which neither intersect nor are parallel, do exist—they are called ultra-parallel lines.

4  I'm not at all making a generalized claim here that there is no truth or falsehood, that all views are relative, that all truths are subjective, or anything along that line of thinking.  In fact, I would argue that some beliefs are better approximations of reality than others.  I would, however, assert that our beliefs are always approximations of reality—containing generalizations, distortions, deletions—not the final word on how things really are.  And it is for that reason that I like to go back and check basic premises from time to time—for usefulness, for ethical consequences, and to see how well they do map to the world out there.

5  See, for example, Trivers, RL. “The evolution of reciprocal altruism”. The Quarterly Review of Biology. 1971. 46:35-55.  Collected in Natural Selection and Social Theory: Selected Papers of Robert L. Trivers.

6  For example, people's tendency to follow herd behavior without weighing the cost/benefit of such behavior to themselves.  See, for example, the chapter on Social Proof in Influence: The Psychology of Persuasion, by Robert Cialdini, pages 114 - 166.  He covers a wide range of cases where people's behavior is influenced non-rationally by what others are doing, from the prevalence of laugh tracks to the increase in suicides after a high profile suicide story in the news.

7  According to the rational self-interest posited by classical economic theory, individuals will act to maximize their return.  However, people will often act to punish a free rider, even at a high cost and no material benefit to themselves.  See, for example, Ernst Fehr and Simon Gächter, "Cooperation and Punishment in Public Goods Experiments", available online at http://ideas.repec.org/p/ces/ceswps/_183.html.

8  This is called "priming", where behavior is influenced by charged words added surreptitiously to a longer list of neutral words the subject reads before being put into an experimental situation.  In one experiment run by John Bargh, subjects who were primed with words like "aggressively", "bold", "rude", "bother", etc. interrupted a set-up conversation keeping the experimenter from talking to them, while subjects primed with words like "respect", "considerate", "appreciate", "patiently" waited without interrupting the conversation.  The experiment and the concept of priming is described in Blink, by Malcolm Gladwell, pages 53 - 58.


Mlodinow, Leonard, Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace,

    Free Press, 2001

Wikipedia, Euclidean Geometry, http://en.wikipedia.org/wiki/Euclidean_geometry

Wikipedia, Noneuclidean Geometry, http://en.wikipedia.org/wiki/NonEuclidean_geometry


By day, James Swingle does business training and personal growth workshops and coaching, which you can find out about at www.jamesswingle.com.  By night, he founded and edits Noneuclidean Cafe where he hangs out with writers and other dangerous types.


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