The following thoughts were set in motion by the prospect of my possible participation in upcoming workshops related to projective geometry. I found myself uncomfortable with providing (in response to invitations to participate) a simple listing of relevant themes. When I paid more attention to this feeling I became aware of a conceptual "context" that clarified my discomfort. At the end of this process I in fact arrived at a listing of relevant themes -- but the sense of this listing is rooted in this context, which I now attempt to sketch.

More and more people are awakening to the limitations of scientific thinking. Despite its impressive achievements in the inorganic realm, it is more and more perceived as the source of global problems such as climate change and social unrest. Its negative quality is often characterized as "materialistic", meaning that it looks to matter as the underlying source of all real phenomena. The initial naive view of matter as "stuff filling space" has had to be revised in the last hundred years since the discovery of quantum mechanics has effectively destroyed the initial sense-perceptible aspect of this definition. One could say, the "stuff" has evaporated into abstract mathematical formulas, but the underlying concept of space has remained stubbornly fixed. One reason for this is that no viable alternative has been proposed (curved space-time of relativity is not qualitatively different from euclidean space for the events of daily life).

According to an ancient tradition in medicine, illnesses are always accompanied by the requisite healing plant. The skill of the healer is to locate the healing plant that corresponds to the given illness. In the case of the "space" crisis referred to in the previous paragraph, it turns out that at the same time that the scientific revolution was being born (out of the work of Kepler, Galileo, Descartes, Newton, and others), a new geometry was discovered (by Desargues, a friend of Descartes) in reaction to the puzzling phenomena presented by perspective painting. This geometry, which received a second powerful impulse around 1820, is called projective geometry, and this report is based on the conviction that its destiny is to heal the spatial one-sidedness afflicting our modern consciousness. In a very literal sense, with projective geometry our concept of space can once again become "whole".

As with so many other domains of human life, it was Rudolf Steiner who first identified the healing role of projective geometry in this regard. His numerous references to it in his scientific lectures clearly showed the road leading to a renewal of scientific and, more importantly, of social thinking based on it. He laid particular emphasis in this context on the importance of developing the concept of "negative space" to balance the ordinary "cartesian" concept of space. Two examples: in the light course (the first natural science course, GA 320) he claimed that the phenomena of life required a new type of force, originating in the periphery of space and acting in a planar fashion from without. He called such forces universal to distinguish them from the "central" forces of classical physics, acting between point centers. A second example: in the warmth course (second natural science course, GA 321), in a discussion of the different states of matter, he indicates that the solid state corresponds to our familiar picture of objects filled with stuff, while the gas state has to be thought of as a “negative” space obtained by turning the standard space inside-out, a hollow space whose "interior" extends to the periphery; the fluid state is an intermediate one balanced between the solid and gaseous states. Finally, he also repeatedly emphasized how the expansion of natural science concepts in this direction was a prerequisite for a new social thinking.

These indications from Rudolf Steiner regarding the importance of an extended space concept for the renewal of science and society were taken up by a small group of students, notably George Adams and Olive Whicher, Louis Locher-Ernst, and Ernst Lehrs. (see references below). The work of these pioneers came to an end in the early 1960's. Since then, with some notable exceptions (e. g., Nick Thomas), progress in extending these results has been modest, despite the increasing severity of the unfortunate consequences of the one-sided spatial thinking.

In the center, a hexagonal pattern in ordinary space; surrounding that, the same pattern "translated" into negative space. |

In this context I consider it important to ask, what can be done to move forward in this important task? As I look back on my experience with the anthroposophical mathematical community, beginning with my first exposure in 1979 and 1981 through 3-week courses with Olive Whicher and Lawrence Edwards, resp., one aspect jumps out at me. I call it the education challenge. What do I mean?

Research is a lot like gardening. The most important ingredient for a successful garden is the quality of the soil. (In fact, one can say, there are no diseases of plants, there are only diseases of the soil.) In the same way the most important ingredient for successful collaboration is a thorough grounding in the fundamentals of the object of study, in this case, projective geometry. This shared heritage of theory and practice is the soil out of which scientific collaboration grows. In brief: we know too little about projective geometry! Although more and more people know something about projective geometry, very few people know enough to engage in research. For example, we all know the cookbook recipe for the polarity on a conic section: the polar line of a point outside the conic section can be obtained by joining the tangent points of the two tangents from the point to the conic section. But how many of us know the actual definition of this polar relationship, and can prove that it is unique? Without this deeper grounding in the fundamentals, one remains a spectator and cannot participate successfully. This is not in itself a problem, but it can become one if there are too few active participants, or if the presenters themselves lack a solid grounding in projective geometry. The latter has perhaps interest in an interdisciplinary undertaking; my remarks here are addressed however at strengthening collaboration within the mathematical community.

The situation is exacerbated by the fact that projective geometry is not typically taught in university so that most participants are self-taught. The result is inevitably a very uneven level of preparation. And, let's be frank, it's hard to learn thoroughly when there is no one there to answer your questions and to question your answers. To wrap up: the long-term health of our collaborative efforts depends on upgrading our qualifications in the fundamentals of projective geometry.

Before discussing possible solutions, I want to mention another symptom that make progress difficult. In the groups and seminars I have participated in, it is difficult to maintain a continuous thematic impulse or group participation. Themes come and go, as well as the participants. One response to this situation is to accept is as normal. Taken to the extreme, this results in a "self-organizing" format. Participants themselves suggest topics of interest they are willing to talk about; the job of the organizers is to fit the offerings with the available time slots. This can sometimes lead to successful serendipity; more often than not, however, my experience has been disappointing. Such meetings may be enjoyable/inspiring/interesting while they are happening but -- in the absence of shared goals and questions -- durable, productive collaborations rarely develop.

If my hypothesis with respect to the education challenge is correct, then one source of this second symptom can be seen in the poor quality of the soil: too much material is presented that lies beyond the skill level of most listeners, so synergetic interaction fails to materialize, or the material presented is so elementary that it is too far from research quality. So one might hope that improving the quality of the soil by an education initiative might also at the same time mitigate this second symptom.

Another ingredient of successful gardening is choosing a subset of the endless variety of plant life to plant and cultivate. Also in scientific research a certain pruning of themes is a necessary condition for abundant growth. These themes would have to be worked out together and not delegated to one or two "experts". Only then can obtain the long-term commitment for participants necessary for lasting results. These shared themes and questions can then serve as “guiding lights” when organizing meetings and workshops.

Now that I have sketched out the context in which I am working I will turn to a discussion of possible features of such a research program.

- Education. Here I think it would be good to strive to offer some regular course (at regular intervals during the year and directly made available as video on-line?) leading to the mastery at the level of, for example, the content of Locher-Ernst's "Projective Geometry". Integrating it into an on-line course platform would allow for remote learning (discussion forums, homework assignments, etc.)
- Counterspace
- What is counterspace? It would be good to establish an overview of the various versions of counterspace and where they are applicable. (I can think of 4 off the top of my head and there are certainly others). Such an overview is long overdue, as people become confused when they notice that the same word is being used to mean different things.
- Nick Thomas's work on counterspace, as far as I can tell, remains a closed book for most of the community. An overview of the contents of his book by someone who does understand it would be I think greatly appreciated.
- Path curves and path curve surfaces
- Given the primary role of counterspace in George Adams' plant work, it's natural to look for it also in the path curve systems. Nick Thomas wrote a short article to show that the collineation underlying a path curve system can be factored as the product of two null systems (correlations) (in an infinite number of ways) thereby bringing in a counter spatial element. And the pivot transformation that appears in the investigation of seed pods, etc., is also a correlation. It would be interesting to investigate further to see whether one can find other, deeper connection to counterspace.
- Fibonacci numbers and golden ratio in plant forms: is there a way to use the approach of "The Plant between Sun and Earth" to understand why the discrete patterns of leaf and flower formation very often reveal the Fibonacci sequence or the related golden ratio?
- At the end of this life George Adams investigated the use of path curve surfaces for purifying or enlivening moving water (as part of his involvement in the institute for flow research at Herreschried, Germany). This work (carried out with George Unger) was published briefly in the MPK but the work came to a stop with his premature death in 1963 and to my knowledge has not been revived (flow forms go in another direction). It would be good to pick it up again and see if it can be developed further.
- A purely mathematical question: The analytic/algebraic approach to path curves leads to the theory of Lie groups and Lie algebras (after all, Sophus Lie and Felix Klein discovered path curves). The "infinitesimal generator" of the path curve system is a traceless matrix A; the path curve orbit at time t is then given by the matrix g(t) = exp(At). (A is an element of the Lie algebra, and g(t) is in the Lie group.) To what extent can this relationship be expressed synthetically? Is there a way to represent the infinitesimal generator geometrically?
- Physics
- The polarity of kinematics and dynamics. This theme was emphasized repeatedly by RS as essential to a renewed, reality-based physics; he also related it (in the warmth course) to the polarity of mental picture and will in the human being. A related question is "What is a force?". As far as I can tell George Adams "Universal forces in mechanics" is the only subsequent publication to address this in a serious way. He shows how 19th century projective line geometry provides an elegant formulation of kinematics and rigid body motion where this polarity can be clearly delineated. Since my Ph. D. thesis builds on this approach, I could present a short account of these results. Also Adams' essay "Forces in space and counterspace" deserves to be included here.
- If there is interest I am also glad to present an introduction to using projective geometric algebra to represent rigid body mechanics (i. e., a reformulation of "Universal Forces in Mechanics" in modern terminology.)
- Long-term, I think it is an interesting hypothesis that the paradoxes of quantum mechanics may not be saying anything about physical reality, but are primarily the expression of attempting to model physical reality using a “single space”. I expect that when one integrates counterspace into the mathematical foundations of the theory, the paradoxes will either disappear or look very different. For example, the currently trending phenomenon of non-locality (as evidenced in the entangled particles used for quantum computing) might look quite different in a geometric framework where planes are primitive entities along with points.
- Astronomy/Cosmology
- Last year I gave a talk on the fact that the planetary orbits are in fact circles when considered in counterspace (a fact first noted by George Adams in “On etheric spaces”, 1933). I think that there are other astronomical themes that also might reveal new aspects by incorporating counterspace. For example, the search for dark matter whose gravitational pull is hypothesized to be behind the unexpected expansion of the universe might be more simply explained using the force of "levity" based in counterspace -- a force that naturally pulls towards the periphery of space.
- Metamathemical themes
- Goethe and mathematics. Despite George Adams' prolific work in applying counterspace to botany, physics, and other fields, his work has not been taken up by our colleagues in the natural science section (to say nothing of mainstream scientists). I have identified one possible ground for this: the belief that Goethe's scientific method excludes using mathematical terminology. I would be glad to give a talk on why this belief is false -- on the contrary, in projective geometry Goethe's method finds its ideal spatial vocabulary, promising a bridge between mainstream and phenomenological approaches.
- "Periphery". It is inevitable as counterspace becomes more well-known that the polarity of center and periphery will also attract more attention. For example, current literature discusses the "environment" of the plant as an entity that is more than the sum of the material surroundings. Or, one contrasts the centric nature of DNA with the peripheral nature of life processes themselves. Or, Rudolf Steiner (Bologna, 1913) indicated that the higher ego of the human being lives not in the bodily organization but in the "periphery". While not a strictly mathematical question, the nature of “peripheral” in these contexts is a fundamental one for applying the mathematics of counterspace in the real world. Hence it seems to me it is important that we as a group develop a concrete and differentiated sense of what "periphery" means in these sorts of examples if we are to engage credibly with scientists and thinkers not familiar with this usage.

Concluding thought: Perhaps the best way to develop an appreciation of “peripheral” in the sense of the previous paragraph is immediately available in the quality of own group process. That is, we can begin to understand it, to the extent that our work together begins to develop a true “peripheral” nature distinct from a simple sum of individual selves.

**Further reading:**

George Adams and Olive Whicher, “The Plant Between Sun and Earth”, Rudolf Steiner Press, London, 1980.

Louis Locher-Ernst, “Space and Counterspace: An Introduction to Modern Geometry”, AWSNA, 2003.

Ernst Lehrs, “Man or Matter”, Project Gutenberg (gutenberg.org), 2004.

Nick Thomas, “Space and Counterspace: A New Science of Gravity, Time, and Light”, Floris Books, 2008.