+ RECENT PUBLICATIONS
The Institute's second book is now available:
A Field Guide to Hyperbolic Space:
An Exploration of the Intersection of
Higher Geometry and Feminine Handicraft
by Margaret Wertheim
The Institute's first book, based on our Inaugural Lecture:
The Figure That Stands Behind Figures
by Robert Kaplan
The IFF and cabinet
The institute has on ongoing relationship with Cabinet magazine
to publish in each quarterly issue an interview with one of our
Things That Think:
An Interview with Computer Collector Nicholas Gessler
Cabinet issue 21
Where the Wild Things Are:
An Interview with Ken Millett
Cabinet issue 20
Evolving Out of the Virtual Mud:
An Interview with Ed Burton
Cabinet issue 19
Crystal Clear: An Interview with
Developing the Logic Alphabet
Cabinet issue 18
The Mathematics of Paper Folding:
An Interview with Robert Lang
Cabinet issue 17
Crocheting the Hyperbolic Plane:
An Interview with David Henderson and Daina Taimina
Cabinet issue 16
Clear: An Interview with Shea Zellweger
Developing the Logic Alphabet
by Christine Wertheim
This interview was first published in:
Issue 18, Summer 2005
| 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 geometrical 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 Froebel, the great nineteenth century founder of the
Kindergarten movement. Just as Froebel 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.
|All images curtsey of Shea Zellweger
history of logic is very interesting, and has undergone many phases.
In the Western world, it began about 2500 years ago when the Greeks
were developing a new form of civic structure in which debate and
argumentation replaced allegience to tradition as the major political
tools. Slowly philosophers realized that, if laws were to be based
on the outcome of arguments, an understanding of how valid arguments
are actually constructed was crucial. Formal logic began when thinkers
like Aristotle started using simple diagrams, like the famous Logic
Square, to study these structures. This was an amazing innovation
because it involved applying a mixture of algebra and geometry to
the study of language, that is, language in its role as the medium
of argumentation. During the middle ages, many thinkers dreamed
of being able to make a complete mathematical analysis of logic,
a complete formal notation for describing arguments and their components.
The Spanish nobleman Raymond Lull, a famous 11th century debaucher
who latter turned religious, was probably the first to have this
idea, and in that sense he is considered by some as the great-great-grandfather
of computing. However, the mathematization of logic didn’t
really get very far until the mid-nineteenth century when an Englishmen
named George Boole developed the first fully-fledged formal notation.
The study of logic radically changed at this point. It’s not
dissimilar to the way that in the sixteenth and seventeenth century
physicists revolutionized the study of motion by mathematizing it.
In what way does your work relate to this innovation?
SZ: Since Boole’s groundbreaking work most logicians have
looked to algebra for models of how to formalize logic, which is
why it’s now so abstract. But logic can also be modelled as
a geometric system, and when you do this you get quite a different
view because you see that it’s not just a set of abstract
symbols but is composed of a fascinating group of symmetric patterns.
These patterns are the basis for my Logic Alphabet.
CW: So that we can compare what you’ve done with the standard
notations, can you give a layman’s description of how logic
is thought of today?
SZ: Let me make an analogy with arithmetic. In arithmetic we construct
equations by combining numbers through operations such as addition,
subtraction, multiplication and division, symbolized by the signs
+, -, x and ÷. In logic you don’t combine numbers,
but propositions such as “cats have four legs” or “trees
have leaves” which are symbolized by letters such as A, B,
C and so on. Just as numbers can be manipulated through arithmetic
operations, so propositions can be manipulated by combining them
through logical operators called connectives. The most common connectives
are “and,” “or,” and “if…then…,”
which each have their own symbols. With these you can write such
logical expressions as: [ (If A then B) and A ]. Then, just as in
math, you can use the laws of arithmetic to find the solutions to
equations by reducing them to a simple value - for instance, the
solution to the equation [((2+7) x 8)) ÷ (3 x 2)] is 12 -
so in logic you can use laws to find solutions to logical expressions.
The ‘solution’ to the example cited above is ‘B’.
This might seem very trivial, but modern computers are built around
these kinds of operations. A computer is really just a great big
CW: But in standard logic there are symbols for only three
or four of the connectives. Yours has 16, why so many?
SZ: It’s much like the difference between the Arabic and Roman
numeral systems. The Romans had distinct symbols for 1, 5 and 10,
that is, ‘I’, ‘V’ and ‘X, and for
the higher numbers they added ‘L’ for 50, ‘C’
for 100, ‘D’ for 500, and ‘M’ for 1000.
All other numerals were constructed from combinations of this idiosyncratic
set. This means that the number ‘VIII’ can’t be
interpreted by the place-based method we’re familiar with
- which would give 5111. Instead you have to see it as a sort of
mini-equation that reads ‘5+1+1+1’, i.e., 8. In this
way Roman numerals were an ad hoc mixture of number-values and equations.
CW: What’s different about the Arabic system?
SZ: 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 macro-structure
in which the units, tens, hundreds, and thousands were each given
their own distinct collumns or places. With this system you can
more easily do arithmetic and uncover the patterns inherent in numbers.
CW: Is your Logic Alphabet analogous to the Arabic numerals?
SZ: That’s right. The standard notations are analogous to
the Roman numerals and mine is like the Arabic system, because there
are 16 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 16 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.
CW: And it’s because you use all 16 that your notation
shows the geometry and symmetries underpinning logic?
SZ: Yes, you only see the symmetric patterns in the system when
you look at the whole thing with all 16 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.
|CW:But why do you need to show
these patterns? Both people and computors 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?
SZ: 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 hides 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?
CW: 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.
SZ: 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.
CW:Your notation is
specifically designed then to make these symmetry patterns tangible?
SZ: The Logic Alphabet is a very special kind of writing that uses
16 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?
SZ: 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 16 form a pattern
you can have great fun with. But it’s easier to grasp if you
look at a three dimensional model first.
CW:You built some models specifically for this purpose didn’t
SZ: 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 70s I built
some models as teaching tools, which beginners can play with. In this
way they become familiar with symmetry and the 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
Froebel 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.
SZ: 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 Froebel-Montessori method
of acquainting children first with fundamental patterns, and then
later putting letters and other symbols on to 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 Froebelian logician rather than as
an outsider, because I think of logical structures as figures of thought,
figures that can be explored materially.
that the patterns underlying logic have a crystalline structure, and
that it is the symmetries of this crystal which your notation allows
us to see and play with. Can you explain this?
SZ: As I said when talking about the models, the Logic Alphabet is
like a two-dimensional presentation of a higher-dimensional figure.
The three-dimensional version of this figure is a Rhombic Dodecahedron,
which is formed by the interpenetration of a cube and an octahedron.
I call it the Logical Garnet because it looks like the gemstone Garnet.
The letter-shapes of the Logic Alphabet were specially designed to
make the symmetries embodied in this figure as easy to see as possible,
and as easy to manipulate.
don’t normally think of mathematical and logic notations as
works of ‘design’. We mostly assume that when a new
symbol is needed someone just makes it up without much thought about
its aesthetic qualities. At least, that’s what I thought until
I saw your work.
SZ: We take the Arabic idea of one numeral for each unit within
the base for granted, but it was a profoundly creative act. Many
other cultures had usable ways of symbolizing numbers, but none
of them were as powerful as the Arabic invention. Once you realize
that the logical connectives form a group of symmetry relations
you are almost inevitably led to the idea that these should be reflected
in the shapes of the symbols themselves. This is what’s missing
in the standard notations.z
CW:In the Logic Alphabet, all 16 connectives have their
own names, and each has a letter-shape as its symbol. I believe
you’ve devised a mnemonic to remember them by?
SZ: Yes, I call it the Olivia Story:
|SZ: It’s a
mini-drama with 16 letters and five acts about a man who keeps being
interrupted by his daughter’s noisy play. At first he tries
to persuade her to desist gently. When this fails he becomes increasingly
exasperated, and winds up ordering her out of the room altogether,
or leaving himself! I had to make up my own pronunciations for the
words beginning with the four nonstandard letters—–rAckett,
mIFty, rIFtee, yORders—–hence the increasing sense of
chaos and linguistic breakdown as the narrative unfolds.
CW: You have said that dreams played a part in your logical
research. Could you explain this?
SZ: I’ve worked on this project a long time, so my mind keeps
thinking about it, even when I’m doing other things. I never
know when an idea will resolve itself, and when it does it can be
at the oddest times, like in a faculty meeting. Once I woke up from
a dream with an image of the letter ‘c’ on a swing spinning
round and round. It gave me the solution to a problem I was occupied
with at the time
CW: You seem to have mostly been working alone...
SZ: I don’t feel that I’ve ever really been alone. I have
worked outside the fields of professional logic. I never taught it,
nor even enrolled formally in any classes. I’ve pursued this
thing in my spare time, and on the occasional sabbatical, but I was
always able to find ‘soft spots,’ generous mathematicians
who would from time to time listen to my ideas and give me help, especially
the great geometer H. S. M. Coxeter, and my faculty colleague Glenn
CW: In the 70s you took out patents on your ideas. Why did
you do that?
SZ: It was really an end run. Up to the mid-70s my papers always came
back with rejection slips. So when I started making the models I figured
that a record of my diagrams in the patent office would be at least
one place where my work would be anchored publicly. You can’t
patent ideas, such as logarithms, but you can get a patent on a slide
rule, which is a material embodiment of them. My first patent was
filed in October 1976, but not granted until June 1981. It was a long
and torturous process. You have no idea!
CW: Was it worth all the trouble?
SZ: Absolutely. The specification included in each patent notes that
my models are an “introduction to the crystallography of logic.”
For me this was the first time that this expression has been used
in print. This links logic to its deep roots in symmetry.
CW: I believe your work is finally receiving some attention
in the logical arena.
SZ: No. Logicians aren’t interested at all. Unless you speak
in their language they don’t want to know about your work. However,
there are a few mathematicians who have taken an interest. You can
find references to their papers on my web site at: www.logic-alphabet.net
CW: Given how entrenched the logic world already is in its
current notations, do you think your system, or something like it,
will ever take off?
SZ: Something like it, a notation of the same geometrical type as
the Logic Alphabet. Yes, absolutely. I think it is inevitable. This
is Froebel, Montessori, and Piaget all over again. Just give one generation
of children the hands-on opportunity to play with models like mine
in the early years of their cognitive development, and it could transform
the way we do logic.