the golden age of mathematical philosophy


The four decades from 1940 to 1980 were the Golden Age of Mathematical Philosophy. During this time, the greatest minds of the century brought together cutting edge theory from engineering, mathematics, and physics to speculate on the nature and meaning of mind, morality, society and metaphysics. Today I want to give tribute to some of the grand, sweeping philosophical magnum opera that were published during this era.

The intellectual environment of the 1940’s was one of frantic innovation and an overwhelming desire to find (or impose) order in the chaotic human realm. The decade began in the turmoil of World War II, and (as is often the case) a great destructive action generated an equal-and-opposite reaction: the most creative minds of the century came together to examine the problem of human behavior. It was also a time for generalists (as opposed to specialists), and the technical advances in physics, math and engineering at the turn of the century inspired theoreticians to turn their tools to the pressing problems of the War Era: society, morality, and behavior.

For example, the field of economics was pole-vaulted forward by the intensely mathematical “Theory of Games and Economic Behavior” (1944) by John von Neumann and economist Oskar Morgenstern. In this work we saw the beginnings of an in-depth mathematical analysis of micro-economic behavior, viewing decisions in terms of outcome prediction and expected utility. But the implications of the book were not purely economic. Instead, this work seemed to promise a greater goal: the possibility of using pure mathematics to understand the nature of human behavior itself. Almost at the same time (in 1943), two papers appeared that applied the engineering concepts of feedback and signal systems to understanding the more broad philosophical concepts of purpose and meaning: “Behavior, Purpose and Teleology” by Arturo Rosenblueth, Norbert Wiener, and Julian Bigelow, and “A Logical Calculus of the Ideas Immanent in Nervous Activity” by Warren McCulloch and Walter Pitts. Together, these papers took bold steps to apply a mathematical analysis of signals, logic and systems to the realm of thought, behavior, and purpose.

In the years from 1946 to 1953, the Josiah Macy Jr. Foundation sponsored a set of interdisciplinary meetings called the Macy Conferences, which brought together the most brilliant minds in engineering, mathematics, psychology, psychiatry, sociology and anthropology. The explicit goal of the conferences was the development of a general science of the workings of the mind. Interestingly, some of the biggest practical technological breakthroughs of the era came from ideas presented in these conferences: neural networks, information theory, systems theory, game theory. But everyone there also had a deep feeling of the broad philosophical implications of their work. The people who were there talk about the sense of awe and discovery in the room when Claude Shannon presented his ideas from “A Mathematical Theory of Communication” (1948), where he defined for the first time a mathematical definition of “information” that was so concrete and precise that he was able to graph progressive increase of information that each letter adds to a word, and each word adds to a phrase, when reading a sentence. Was it possible that there could be a mathematics of meaning itself?

At the same time, the engineering and code-breaking research of the war effort had led to amazing advances in the theory of computation and the mathematics of general problem solving. In addition to the practical advances in cryptography and computational theory, the power-house effort of the greatest minds of the day inevitably lead to philosophical speculation as well. Alan Turing, the mathematician who first envisioned the idea of a general all-purpose “programmable” machine, published “Computing machinery and intelligence” (1950) in which he speculates that the mind can be viewed as an abstract function, and therefore a computation that could in principle be carried out mechanically.

Any good book on the history of cybernetics and cognitive science can tell you (much better than I can) the historical stages of practical development that these interdisciplinary discoveries produced over the following three decades, from 1950 to 1980. But what is often overlooked is the truly fantastic, mind-bending speculations about mind, meaning, and the nature of the universe that came about at the same time. I couldn’t possibly do them justice in a simple blog post, hopefully I can whet your appetite to investigate some of these classic works on your own. These are some of the books that go beyond the practical, or even small-scale philosophical speculations. These are some of the books that let the imagination go free, and explore the possible connections between mathematics and purpose, morality, meaning, and the soul.

Laws of Form (1968) by G. Spencer-Brown

“Thus we cannot escape the fact that the world we know is constructed in order to see itself… In order to do so, evidently it must first cut itself up into at least one state which sees, and at least one other state which is seen. In this severed and mutilated condition, whatever sees is only partially itself. We may take it that the world undoubtedly is itself (i.e. is indistinct from itself), but, in any attempt to see itself as an object, it must equally undoubtedly, act so as to make itself distinct from, and therefore false to, itself. In this condition it will all this partially elude itself.

[…] We notice one side of a thing-boundary at the expense of paying less attention to the other side. We notice a dish to be washed up in the sink by paying scant attention to the not-dish universe that our definition of the dish-boundary equally defines. Were we to pay equal attention to both sides, we would have to attribute to them equal value, and then the dish boundary would disappear. The dish’s existence would cease, and there would be nothing to wash up…

Believe it or not (and despite what you may think from the quotations above), this is a book about logic. And by “logic” I mean the very basic kind you learned in school: you know, the “A implies B has the same truth table as NOT B implies NOT A:” kind of stuff. But what Spencer-Brown does is create a whole new fundamental calculus of logic that is based on subjectivity instead of objectivity. Instead of starting with basic concepts of “true” and “false” and “A” and “NOT A”, he starts with a different set of concepts: a distinction, and an indication. All of the entirety of mathematics is rooted in the action of an observer, claims Spencer-Brown. An observer that decides to draw a boundary in the universe (a distinction) and point to one side (an indication) and say, “THAT is what matters.”

The take-home message:If we could only imagine a world where basic LOGIC was rooted in subjective action rather than objective “truth values”, how much more tolerant might that make us all?


General Systems Theory (1969) by Ludwig von Bertalanffy

“Our civilization seems to be suffering a second curse of Babel: Just as the human race builds a tower of knowledge that reaches to the heavens, we are stricken by a malady in which we find ourselves attempting to communicate with each other in countless tongues of scientific specialization… The only goal of science appeared to be analytical, i.e., the splitting up of reality into ever smaller units and the isolation of individual causal trains…We may state as characteristic of modern science that this scheme of isolable units acting in one-way causality has proven to be insufficient. Hence the appearance, in all fields of science, of notions like wholeness, holistic, organismic, gestalt, etc., which all signify that, in the last resort, we must think in terms of systems of elements in mutual interaction…

“There is this hope, I cannot promise you whether or when it will be realized – that the mechanistic paradigm, with all its implications in science as well as in society and our own private life, will be replaced by an organismic or systems paradigm that will offer new pathways for our presently schizophrenic and self-destructive civilization.”

This is a massive book that literally tries to draw everything together under the unified idea of the “system”. Von Bertalanffy starts out talking about physics and entropy and ends up talking about purpose and life as an active, non-equilibrium system. Everything, von Bertalanffy says, can be understood as a system: a complex network of interaction between simple component elements, that gives rise to emergent dynamics.

The take-home message:Reductionist science will never understand nature, because when you view everything as a “system”, at every level of analysis the universe is more than the sum of its parts.


Steps to an Ecology of Mind (1972) by Gregory Bateson

“Consider a tree and a man and an ax. We observe that the ax flies through the air and makes certain sorts of gashes in a pre-existing cut in the side of the tree. If now we want to explain this set of phenomena, we shall be concerned with differences in the cut face of the tree, differences in the retina of the man, differences in this central nervous system, differences in his efferent neural messages, differences in the behavior of his muscles, differences in how the ax flies, to the differences which the ax then makes on the face of the tree. Our explanation (for certain purposes) will go round and round that circuit. In principle, if you want to explain or understand anything in human behavior, you are always dealing with total circuits, completed circuits. This is the elementary cybernetic thought.

“The elementary cybernetic system with its messages in circuit is, in fact, the simplest unit of mind; and the transform of a difference traveling in a circuit is the elementary idea. More complicated systems are perhaps more worthy to be called mental systems but essentially this is what we are talking about.

“But what about ‘me’? Suppose I am a blind man, and I use a stick. I go tap, tap, tap. Where do I start? Is my mental system bounded at the handle of the stick? Is it bounded by my skin? Does it start halfway up the stick? Does it start at the tip of the stick? But these are nonsense questions. The stick is a pathway along which transforms of difference are being transmitted. The way to delineate the system is to draw the limiting line in such a way that you do not cut any of these pathways in ways which leave things inexplicable. If what you are trying to explain is a given piece of behavior, such as the locomotion of the blind man, then, for this purpose, you will need the street, the stick, the man; the street, the stick, and so on, round and round. […]

“Under LSD, I have experienced, as have many others, the disappearance of the division between self and the music to which I was listening. The perceiver and the thing perceived become strangely united into a single entity. This state is surely more correct than the state in which it seems that ‘I hear music.’ The sound, after all, is Ding an sich, but my perception of it is part of my mind.”

This collection of essays weaves a complex tale in which Bateson presents a world view where everything is information, there are no objective “facts”, and the basic unit of mind is an elementary feedback circuit. He even uses a thermostat—the simplest mechanical feedback loop—as an example of the simplest possible mental unit, able to exhibit purposeful behavior.

The take-home message:the basic unit of mind is the feed-back loop, which means that the mind doesn’t stop at the edge of our bodies. Our minds necessarily and inherently involve every part of the environment that we interact with.


Tetrascroll (1975) by R. Buckminster Fuller

“Here is Goldilocks having a sky party with her three friends, the Polar Bear family. Goldy says the sky part is a ‘system’ because Goldy plus the three bears equals four entities (or star events), and it takes four events to produce a system. A system divides all the universe into six parts: the universe outside the system, the universe inside the system, and the four star events which do the dividing. […]

“Together the observer and the observed constitute two points differentiated against an omni-environment of nothingness with one inherent line of ‘awareness’ interrelationship running between these points. Euler’s generalized formula, which he named topology, says the number of points plus the number of areas will always equal the number of lines plus the number 2, which Goldy finds to be at minimum 2P + 1A = 1L + 2, which minimum set of awareness aspects of life adds to for, i.e. the observer, the observed, the line of the interrelationship, and the nothingness against which the somethingness is observed.”

This book deceptively bills itself as a re-telling of the Goldilocks tale, but is really a basic treatise on the way that geometry can be used to understand every aspect of the universe. Building on ideas described in some of the earlier books, above, Fuller describes everything as a system of signals and observations; but in this case, everything is couched in the language of geometry. The conversation ranges freely from the geometry of observation to the geometry of evolution to the geometry of morality and society. But beware, it can quickly evolve from the simple equation in the above quotation to things like, “Goldy next shows the bears how the three-face-bonded tetrahedra-arc in its initial neutral nontransmitting state becomes spirally extended positive or negatively to attain its information-transmitting state, only with the addition of one more face-bonded tetrahedron.”

So you probably shouldn’t try reading it to your small child.

The take-home message: The basic tools of geometry can be applied to understanding all relationships, whether they are relationships between ideas, people, or cultures.

Autopoiesis and Cognition: The Realization of the Living (1980) by Humberto Maturana and Francisco Varela.

“A cognitive system is a system whose organization defines a domain of interactions in which it can act with relevance to the maintenance of itself, and the process of cognition is the actual (inductive) acting or behaving in this domain. Living systems are cognitive systems, and living as a process is a process of cognition. This statement is valid for all organisms, with and without a nervous system.”

This book is based on work of evolutionary biologist during the entire 1970’s decade, and reprints a number of articles previously published in the 1970’s. You can actually read one of the core sections of this book in its entirety on the web: The Biology of Cognition. The theory of autopoiesis brings together a complete grand theory of life and cognition defined in terms of self-referential information processing in biological systems. The publication of this book effectively serves as a final “book-end” to close the golden era of mathematical philosophy, because although a great deal of work has been published since then extending autopoesis theory to social, moral and philosophical domains, all of this work has been intellectually “evolutionary”: a mere extrapolation of the initial insight. The truly profound synthesis of all of the ideas that came before it (information theory, feedback, signal systems, game theory, systems theory, and so on) into a ground-breaking ontology and semiotics rooted in a biological definition of cognition is summarized in this book.

The take-home message: When we understand the fundamental core of what life is, as a formal process, we realize that all living is a form of mental activity, and that reality itself is a product of that process. Thus the slogan: living is knowing, and knowing is bringing forth a world.

These are just a few of the gems of mathematical philosophy produced during the “Golden Age” of mathematical philosophy. I hope you will explore at least some of them. And whether you are convinced by their visions and speculations, or consider them to be speculative intellectual over-reach by a bunch of engineers who are eagerly over-applying new tools of discovery, reading these books will undoubtedly change your appreciation of philosophy—and possibly the nature of human-kind and reality—forever.