Crystal Clear

In 1953, while working a hotel switchboard, a college graduate named Shea Zellweger began a journey of wonder and obsession that would eventually lead to the invention of a radically new notation for logic.

From a basement in Ohio, guided literally by his dreams and his innate love of pattern, Zellweger developed an extraordinary visual system—called the “Logic Alphabet”—in which a group of specially designed letter-shapes can be manipulated like puzzles to reveal the geometric patterns underpinning logic.

Indeed, Zellweger has built a series of physical models of his alphabet that recall the educational teaching toys, or “gifts,” of Friedrich Fröbel, the great nineteenth-century founder of the Kindergarten movement. Just as Fröbel was deeply influenced by the study of crystal structures, which he believed could serve as the foundation for an entire educational framework, so Zellweger’s Logic Alphabet is based on a crystal-like arrangement of its elements. Thus, where the traditional approach to logic is purely abstract, Zellweger’s is geometric, making it amenable to visual play.

Like his notation, Zellweger’s working methods are delightfully unconventional. While constituting a genuine research project in logic, his notebooks (made between 1953 and 1975) have remarkable visual appeal, passing through phases reminiscent of Russian Constructivism, outsider art, concrete poetry, and pop. These days we accept outsider artists, and are perhaps aware of outsider scientists, but Zellweger may be the first we could define as an outsider logician.

Although he has worked on the Logic Alphabet for over fifty years, his professional life has not been spent in departments of philosophy or mathematics, but in psychology, mostly at Mount Union College in Ohio, from which he retired in 1993. After half a century of obscurity, Zellweger’s work is starting to attract the attention of some mathematicians who believe it offers an exciting new perspective on logic. Christine Wertheim, co-director of the Institute For Figuring, has been studying Zellweger’s work, and talked with him about his unorthodox approach.

See also the Logical Garnet section in KAUFFMAN, L.H., 2001. The mathematics of Charles Sanders Peirce. Cybernetics & Human Knowing. 1 January 2001. Vol. 8, no. 1–2, p. 79–110. pdf

What’s different about the Arabic system? The Arabs (with help from India) settled on a consistent use of base ten, gave each number within this base its own symbol (the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and then imposed a macrostructure in which the units, tens, hundreds, and thousands were each given their own distinct columns or places. With this system you can more easily do arithmetic and uncover the patterns inherent in numbers. Is your Logic Alphabet analogous to the Arabic numerals? That’s right. The standard notations are analogous to the Roman numerals and mine is like the Arabic system, because there are sixteen logical connectives in all, but most notations only use three or four, cobbling the others together from combinations of these few. In my notation, all sixteen are given equal prominence, and each has its own symbol, just as each of the units in the base-ten Arabic system has its own symbol. And it’s because you use all sixteen that your notation shows the geometry and symmetries underpinning logic? Yes, you only see the symmetric patterns in the system when you look at the whole thing with all sixteen elements together. When you use only a few, you don’t see the beautiful crystalline structures. My notation is designed to highlight these wonderful patterns, not obscure them, as most notations do.

But why do you need to show these patterns? Both people and computers are doing fine using only three or four connective symbols, and not having a clue about the lovely patterns hidden below. Why do you think it’s so important to reveal these?

It’s a bit like cladding a house in glass so you can see how it’s actually built, rather than covering it all in plywood and stucco, which hide the underlying structure. To me, it’s important not to just be able to do something, like driving a car. I also want to know how the thing works. In my notation, you could say that the design puts what’s under the hood onto the dashboard, so you learn how the vehicle works while you’re actually driving it. This makes the whole act of driving much less passive, and ultimately much more satisfying. I also like the fact that one’s interaction with my notation is literally hands-on and physical, rather than just all in your head. Knowledge shouldn’t be disconnected from the body. The body should be used as much as possible as a part of the means through which we acquire and store knowledge. Why can’t logic be like that too?

Does this mean that your notation offers not just a new kind of logical notation, but a whole different mindset—a new perspective, really, on what logic is? It is a different mindset, but what’s really important is how this approach plays out in the teaching of logic. For most people, if they have encountered logic at all, it has probably seemed complicated, difficult, and full of abstract rules and symbols. My system is like a game whose simple and explicitly exposed structures enable you to easily understand the basic principles of logic, and even to begin doing some elementary constructions without learning any complicated rules. This is because the geometrical model on which it’s based can easily be embodied in objects you can manipulate with your hands and eyes. In fact, you can just play with the system as an interesting set of patterns in its own right. By flipping and rotating the elements, you can explore the symmetries embedded in the system without seeing it as a way of modeling logic. Your notation is specifically designed then to make these symmetry patterns tangible? The Logic Alphabet is a very special kind of writing that uses sixteen letter-shaped symbols whose forms have been carefully designed to mimic the geometrical patterns underpinning logic. In many ways, it’s more like a pictographic language than a phonetic or purely abstract one. However, the pictorial aspect doesn’t just lie in the shapes of the individual symbols, but also in the relations between them. What is important is the way you can turn one letter-shape into another by flipping and rotating, and that you can literally see this when you do it. Can you give me an example? For instance the Logic Alphabet includes the symbols “d,” “b,” “q,” and “p.” If you flip the d-letter to the right, it turns into a “b.” If you flip it upside-down it turns into a “q.” If you do both together, it rotates into a “p.” The other symbols in my Alphabet are also related by very specific symmetries which can be described by flips and rotations, and together all sixteen form a pattern you can have great fun with. But it’s easier to grasp if you look at a three-dimensional model first.

You built some models specifically for this purpose, didn’t you?

Yes. The Logic Alphabet is a script designed to be written on paper, so it’s only in two dimensions, but the overall patterns it captures are actually in four dimensions. Of course, we can’t build anything in four dimensions, but there is a three-dimensional version that neatly encapsulates the basic figures. In the 1970s, I built some models as teaching tools, which beginners can play with. In this way they become familiar with the symmetries and beautiful patterns, long before they are able to think about logic itself.

In this sense your whole approach is similar to the ideas developed by the great nineteenth- and early twentieth-century pedagogues Friedrich Fröbel and Maria Montessori, who founded the Kindergarten movement and the Montessori school system, respectively. Both believed that higher-level conceptual thought should be preceded by concrete hands-on play with geometric forms embodied in solid materials. That’s right. The whole way logic is taught today, as a system of purely abstract signs that only college-level students can understand, is completely backward. The Fröbel-Montessori method of acquainting children first with fundamental patterns, and then later putting letters and other symbols on these to show how the same forms are embodied in natural phenomena and geometric figures, is exactly right. I’ve just extended this methodology to logic. In fact, I think of myself as a Fröbelian logician rather than as an outsider, because I think of logical structures as figures of thought, figures that can be explored materially.