Just what is mathematics? This is a question that would seem very easy and turns out to be very hard. There are at present two major schools of thought to this: that mathematical objects are real things that exist outside of human thought and physical reality itself and are discovered rather than created. This school of thought has roots in Plato so is called Platonic or Neo-Platonic or simply Realist. Note that this concept of realism is quite different from what a physicist would think of realism in terms of quantum mechanics. The other side would say that mathematics is an evolving and contingent product of the human mind, an embodied product of the human mind, and merely that. Let’s call this group Human Only. For me, I’ve been rather unhappily in the Realism camp for some time until the last few years and am now firmly Human Only. But Realism can’t be completely vanquished as I hope to show in this essay. One question that keeps coming up is: if and when we contact another sentient being, will they recognize our mathematics. Carl Sagan certainly thought so when he had the Pythagorean Theorem and the Golden Section engraved on the gold tablet inside both Voyager spacecrafts. This would suggest that he would consider these mathematical objects just as real and universal as the physical gold.
The strange case of the Laws of Form
In the early 80s, there was a bar near the University on the south side of Speedway. A big open place with lots of tables, a nice place to waste a Saturday afternoon. The clientele there was eclectic and interesting, for example, members of the L-5 Society hung out there. Where else could you hear a drunken rant about the necessity of living in space? I was in this bar that I was talking to a guy who was recommending reading Laws of Form and telling me how it had changed his life. Smart guy, he was involved in systems admin and programming the first computer graphics mainframe in Tucson. Several years later I heard from a friend a few hints of the convoluted history of the purchase and attempt to start a computer graphics business. I never got the full story but it probably could be made into a mini-series at least as it involved local pot smugglers (and their money), Saudis (and their money), and lesbians. Oh My! I had heard about the Laws of Form before from glowing reviews in The Whole Earth Catalog and Nature, [4] but this recommendation pushed me to actually buy the book. Like most people, I barely got past the first 10 pages, put it in my library, and went on with my life. But from time to time I’d pull it out, wondering why it became so popular. In 2012 I moved out of my house and I had to downsize, shrinking my library by 2/3rds. I gave away a bunch of books from that era to someone who I thought might read them. For some reason, I kept Laws of Form. For this essay, I finally read the book cover to cover.
G. Spencer-Brown was educated at Trinity College, Cambridge, and taught at Christ Church, Oxford, University of London, and Cambridge University. He was a personal friend of both Bertrand Russell and Ludwig Wittgenstein. His studies moved from philosophy to probability theory to logic, especially the symbolic logic of Charles Sanders Pierce (1849-1914). During a brief stint working for the British National Railway, he started being interested in what today is called network theory. This interest resulted in the Laws of Form. Interestingly, he was a disciple of the Scottish psychologist R. D. Laing and Laing is referenced in his book. Spencer-Brown died in 2012 at 93 years of age.
G. Spencer-Brown’s Laws of Form was first published in April 1969 This is an attempt to provide an alternative and an expansion of Boolean Algebra. To do this he takes a basic geometric property, the ability to divide space into inside and outside, say drawing a circle on a white sheet of paper, which he calls a ‘distinction.’ From this simple premise he states some simple rules and turns this into a symbolic logic complete with a ‘calculus,’ a way to calculate. He then defines an alternating function that is equivalent to complex numbers. To this, he adds iteration over this function creating an analogy to time. He then shows a brief application to circuit theory. Spencer-Brown also claims that his new form of logic could provide clues to the four-color problem.
Here are his definitions and rules:
THE FORM
We take as given the idea of distinction and the idea of indication, and that we cannot make an indication without drawing a distinction. We take, therefore, the form of distinction for the form.
Definition
Distinction is perfect continence.
….
Axiom 1. The law of the calling
The value of the call made again is the calling.
….
Axiom 2. The law of the crossing
The value of the crossing made again is not the crossing.
…. [1]
So what to make of this little book with its William Blake quote and Chinese lettering (a koan perhaps, haven’t been able to find out.) An esoteric work linking deep spiritual knowledge to deeper mathematics? A one-off from an eccentric British polymath who maybe took one too many tabs of LSD?A brilliant work of mathematics? A nothing burger? All or none of the above? There was a conference about the book attended by Spencer-Brown at Esalen in Big Sur in 1973. [3] Supposedly Spencer-Brown got upset about something and left halfway through. In 2019 there was a 50th-anniversary conference at Liverpool University in the UK, there are videos of the conference.[10] A look into the mathematical literature shows that very little work has been done related to this book. In fact, it has generally been ignored. Most legitimate scholars who have studied and expanded Spencer-Brown’s work have been scientists, including social scientists.
One mathematician stands out though, Louis. H. Kauffmann at the University of Chicago. Kauffmann was an early researcher in Cybernetics, control through feedback loops. In the late 1990s, Cybernetics folded into the larger study of complex systems. Kauffmann is also a topologist studying knot theory. He has written several papers connecting Laws of Form to its roots in symbolic logic [7][9] and his 2019 talk makes a comparison of it to Lambda calculus. [10] Introduced by Alonzo Church in the 1930s, Lambda calculus is a formal theory of logic describing computation by abstracting the notion of a function. This is now called functional programming and is a way to use computers to solve proofs. Functional programming is central to the creation and current churn around digital currencies and decentralized finance. My point here is that Laws of Form and the Lambda calculus are legitimate mathematical objects exploring the roots of mathematics yet one has been ignored while the other has become foundational. It should be noted that in 1969 neural nets, a simplified model of living neurons was very popular in AI research. Like Laws of Form, they are also a logical structure built out of spatial geometry. Perceptrons by Marvin Minsky and Seymour Papert came out that same year. [6] In this book they proved that the then-current neural nets had a fatal flaw, they were not mathematically complete. This put back AI research for a decade until a neural net structure that is mathematically complete was found. That same year as Perceptrons was published, Laws of Form, a mathematically complete system based on spatial geometry, was also published.
Spencer-Brown does mention a connection between the Laws of Form and the 4 Color Problem and claims that both he and a D Spencer-Brown have done work on it. This seems to have been his brother although I can find no information on him. In the 1979 preface of his book, Spencer Brown claims to have a proof soon to be published. I don’t believe it ever was. Here is the question:
Given a map of countries, can every map be colored (using different colors for adjacent countries) in such a way so that you only use four colors? [11]
In 1852, Francis Guthrie, a student at University College London, was coloring a map of the counties of England when he noticed that at least four colors were required so that no two counties which shared a border had the same color. He conjectured that four colors would suffice to color any map, and this later became known as the Four Color Problem. During the next 120 years, various proofs were attempted and it was solved for higher-order objects like a torus. The original problem was first solved in 1974 By Kenneth Appel and Wolfgang Haken at the University of Illinois in 1974. The nature of this proof was very different than ordinary proofs in that they ran a then supercomputer for 1200 hours to “prove” the four-color problem. If the computer stopped then the proof worked. This was very controversial as it was the first-ever proof in pure mathematics that was done with a computer. Of course, it had to be checked and in 1975 a mistake was found in the algorithm, the recipe for writing the computer program. Several more were found. In 1994 Paul Seymour and his group at Princeton University wrote a new algorithm and solved it in much less computer time. This has stood as the final solution to the four-color problem.
Or not. One criticism is that the proof answered a yes/no question but gave little insight into why higher-order shapes, torii, etc., were so easy to solve by formal proof yet the ‘easier’ problem required computer assistance. A second question is how does this change the meaning of mathematics:
All mathematical theorems are known a priori.
Mathematics, as opposed to natural science, has no empirical content.
Mathematics, as opposed to natural science, relies only on proofs, whereas natural science makes use of experiments.
Mathematical theorems are certain to a degree that no theorem of natural science can match. [12]
The four-color proof is semi-empirical. Is mathematics now an empirical science? Important question. A modern computational mathematician would say that the algorithm is the proof, especially if written in the formalism of the Lambda calculus. Statistical proofs now exist. This changes the notion of mathematical realism in two ways. It removes this absolute esoteric Platonic notion of realism altogether. Embodied? Wait. If mathematics is now closer to natural science and natural science is the study of Reality, then what physical thing is being studied? Does this make mathematics real? Excuse me while my head explodes.
So why was Spencer-Brown so popular for a brief period? I believe he created an accomplished mathematical work that formed a complete logical structure out of simple Geometric truth and then published it oddly. This esoteric element turned off many mathematicians. In addition, at that time there was a group of people trying to reconcile science and religion and Spencer-Brown threw them a life raft. [2] Spencer-Brown’s version of mathematical realism might make some today uncomfortable:
One of the motives prompting the furtherance of the present work was the hope of bringing together the investigations of the inner structure of our knowledge of the universe, as expressed in the mathematical sciences, and the investigations of its outer structure, as expressed in the physical sciences.
The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance.
Although all forms, and thus all universes, are possible, and any particular form is mutable, it becomes evident that the laws relating to such forms are the same in any universe. It is this sameness, the idea that we can find a reality that is independent of how the universe actually appears, that lends such fascination to the study of mathematics. [1]
In addition, his pure mathematical structure has found no use, even in pure mathematics. This could change if one day someone ‘rediscovers’ his work. I also think the popularity is because something far deeper going on, something about how an individual, human or otherwise, is wired to the world. What is an individual but a distinction? I hope to explore this in future essays.
Spencer-Brown, G. Laws of Form. Julian Press, 1979.
Miller, Seth. “An Esoteric Guide to Spencer-Brown’s ‘Laws of Form,’” 2010, 52.
“Laws of Form: Spencer-Brown at Esalen, 1973” Accessed August 15, 2021. http://books.imprint.co.uk/book/?gcoi=71157100117690.
Beer, Stafford. “Maths Created.” Nature 223, no. 5213 (September 1969): 1392–93. https://doi.org/10.1038/2231392b0.
Kauffmann, Louis. “On the Map Theorem,” 2001. https://doi.org/10.1016/S0012-365X(00)00207-7.
Minsky, Marvin, and Seymour A. Papert. Perceptrons: An Introduction to Computational Geometry. Cambridge, MA, USA: MIT Press, 1969.
Knill, Oliver. “Coloring Graphs Using Topology.” Harvard University, 2014.
Robertson, Robin. “Something from Nothing: G. Spencer-Brown’s Laws of Form.” Accessed August 13, 2021. https://www.academia.edu/216970/Something_from_Nothing_G_Spencer_Browns_Laws_of_Form_cybernetic_version_of_article_.
Kauffman, Louis H. “The Mathematics of Charles Sanders Peirce,” 2001., 32.
westdenhaag.nl. “50th Anniversary Conference on Spencer Brown’s Laws of Form,” 2019. http://westdenhaag.nl/exhibitions/19_08_Alphabetum_3/more1.
Krantz, Steven G. “The Four-Color Problem: Concept and Solution,” 2007, 37.
Tymoczko, Thomas. “The Four-Color Problem and Its Philosophical Significance.” The Journal of Philosophy 76, no. 2 (February 1979): 57. https://doi.org/10.2307/2025976.
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