Heinz von Foerster Festschrift

HvF Recursion … or was it Magic?

Skillful Good Fortune

A story is told of a lady riding a bus asking a young man if he could point out a certain stop where she needed to get off. "That's easy", he answered with a mischievous smile, "watch me and get off two stops before me"! That sounds just like the sort of referential trap a precocious young Heinz von Foerster might have sprung and to avoid I will start in the beginning.

It was May of 1959 when I was a young teaching assistant at the University of Illinois looking for a summer job. The prospects looked rather grim until serendipity lead me to the Biological Computer Laboratory BCL, at the Electrical Engineering Department of the University. A smashingly handsome man, Professor Heinz von Foerster, founder and grand maestro of BCL greeted me enthusiastically. His look was most compelling: athletic posture, strikingly well formed cranium, hairless except for a well coiffured halo-like ring of hair just above the ears, penetrating and playful eyes, and a command of circumstance that seemed absolute. With extraordinary flair and a charming smile he confided that he had been looking for a mathematician just like me and hired me on the spot. My timely arrival was attributed to skillful good fortune; a remark with which he would often "explain" favorable but otherwise mysterious happenings. I had entered the aura and spell of HvF never to be the same again.

Professor HvF reminded me of no one! He was and has remained unique. I have been told of some great men who are difficult to imagine young. I find it impossible to imagine HvF old! An anecdote was persistently told that until the age of 25 it was a toss-up whether the young Heinz would become a physicist or a stage magician. He never confirmed nor denied the story while brushing aside the vagaries of bad taxonomy.

In the early sixties Prof. HvF was invited by the Illinois Historical Society to lecture on the History of Magic on their gala annual meeting. The evening opened with a request for a volunteer to lend a dollar bill to the secretary of the society and who, in turn, sealed it in an envelope and guarded it prominently in his coat pocket. In the following hours the audience was spellbound as Prof. HvF progressively described and masterfully demonstrated the favorite magic tricks of various eras starting with the early Greeks. For each trick he computed the probability of its natural occurrence (like the probability of choosing a specific card out of a deck is 1/52) and plotted this number versus time on the horizontal axis. These probabilities markedly decreased as time progressed. Ergo, as people became more sophisticated more improbable events were needed in order to impress them. Only HvF could have portrayed the evolution of magic in this profound and delightful way and, in the process, relating some very spicy vignettes about the great magicians. The audience was totally captivated and mesmerized; these stolid historians would never be the same. At the end we were reminded of the dollar bill still in the secretary's possession. A pad of paper was passed successively to three randomly chosen members of the audience. Each wrote a number of their choice on the paper and when these numbers were added the sum, by skillful good fortune, turned out to be the same as the serial number on the dollar bill! Thunderous applause followed. No wonder it was always a problem finding a lecture hall large enough to accommodate all the people who wanted to hear and experience the one and only Professor Heinz lecturing.

Brain, Cybernetics and Learning

"I am convinced that a time will come when the physiologist, the poet, and the philosopher will all speak the same language and mutually understand each other." — Claude Bernard

Much has been said and written about specialized topics, Phaidros complained about two thousand five hundred years ago during a social gathering of Agathon's friends who came to celebrate his recent triumph as a poet. "Monographs on minute details in the lives of Hercules and Homer are swamping the market, but who discusses an interdisciplinary problem as, for instance, love?". Thus commenced in Agathon's house the immortal first interdisciplinary symposium which was attended by philosophers, statesmen, playwrights, poets, social scientist, linguists, medical doctors and students of various disciplines.' With this quote Prof. HvF convened the seminal symposium on the Principles of Self-Organization at Allerton, Illinois in June of 1960.

Indeed BCL was the truest continuous reincarnation of the symposium at Agathon's house, teaming with people of many disciplines speaking in strange tongues yet remarkable understanding each other! I entered it the day after I met HvF to receive my first assignment. Outside his office people waiting to see "Il Professore" were having animated discussions. I waited my turn wondering how I fit in this strange scheme of things. The answer was soon at hand: I had been chosen to work on the Ear. Startled, I reminded Prof. von Foerster that I was a mathematician. Yes, of course, he impatiently said that is why I want you to work on the Ear! He gave me the classic book of Wever and Lawrence on Physiological Acoustics, apologized for not having any office space for me, encouraged me to find an agreeable environment—such as the local mudhole known as the Lake of the Woods (it was actually weeds!)—to read the book and report back. This was my introduction to BCL's effort to explore and emulate Perception. It lead me to Mathematical Biology and a series of mathematical models of the cochlea of the inner ear modestly following the earlier work of Helmholtz. I became "hooked" and, to the chagrin of my professors in "pure" mathematics, stayed with BCL not just for the summer but until graduation and spiritually forever.

HvF nurtured and provided the "glue" for BCL where Cybernetics; perception, brain structure and function, and interdisciplinarity flourished. Ashby, Pask, Maturana, Gunther, Lofgren, Burckhart and others came, in turn, attracting lots of students. There were physiologists, psychologists, physicists, electrical engineers, philosophers, logicians, choreographers and mathematicians, linguists, artists, administrators and perpetrators, neurophilophers, spieltandlers and, of course, MAGICIANS; orchestrated and lead by the intrepid HvF. From Switzerland and England they came and China, India, Armenia, Hungary Germany, Chile, Sweden, Greece, Israel, Italy, India and everywhere in the USA. A veritable Babel of tongues and disciplines resonating in the BCL temple. But it was the charming and wise Mai von Foerster who really reigned supremely over BCL. She inspired us and together with the three special sons Johnny, Tommy and Andy rewarded us with magnificent gatherings at 4 Flora Court—the von Foester chateau in Champaign Illinois.

Some of the doings at BCL

The infamous "Doomsday" paper appeared in Science in the early 60's. From a population growth model, based on demographic data, it was deduced that in a few decades "we will all be squeezed to death" by overpopulation. Further scandalized by the titillating detail that the "crunch" will occur on a Friday the 13th (on Heinz's extrapolated birthday), applause, sensation and medieval criticism were heaped on HvF. The critics not only missed the subtle humor but also the point. Since then, the population growth has exceeded the doomsday scenario, as was shown in the graph with the yearly comparisons posted on HvF's office door. But not all work on population dynamics became "notorious". Mathematical models, done in collaboration with Dr. Hamilton of the Sloan-Kettering Institute, of white cell proliferation in blood provided insight on leukemia. Leukemic conditions were, in fact, emulated by allowing DNA components in the dead white cells to be reutilized by the maturing cells. I was sent to Dr. Hamilton's lab in New York City to better understand the physiology. A young lady Biophysicist there captured my interest but instead of impressing her, I managed to faint in front of her during an in-vivo experiment that I was observing; the embarrassment haunts me to this day. Upon my return, HvF delicately inquired if I may have been a bit indisposed.

The classic work of McCoullough, Pitts, Letvin (the Frogiest) \& Maturana on "What the Frog's eye tells the Frog's brain" steered our efforts to understand visual pattern-recognition and to implement it with plausible neural models and networks. The Numarete which counted convex objects was built—and it really worked—as well as edge, contour detecting and sharpening networks. The more general structure of linear "Neural Networks" was studied and equivalences between Action, Interaction and Cascaded networks were found. The ideas and results were later summarized in Heinz' paper in the J. of Theoretical Biology. And since then, these ideas and results have enjoyed remarkable and frequent "rediscovery". In the late 1970s (i.e. about 20 years later) a prominent member of that community gave a lecture at the IBM Scientific Center in Los Angeles, my "home" after BCL with the current results and ideas on edge and contour detection. Afterwards I gave her one of the early BCL reports containing the "recent" results and in fact more. She was furious and demanded to know why these had not been published? They had been … a bit ahead of their time.

Next to Heinz it was Ross Ashby who influenced me the most, as with his ideas on the "Topology of History" or explaining the behavior and structure of the society in India (where he served as a British soldier). "Life is a game we play with our environment" he posed. It is a game where we do not know the rules yet we manage to learn to win (i.e. survive). Ross constructed a gadget with "parts" (i.e. paper cut in various shapes) distributed inside an egg-carton. He proceeded to demonstrate that the gadget, playing one part in a two-party a game whose rules it did not yet "know", could reorganize itself until it won more often than it lost. Here was a "system" with sufficient pre-organization enabling it to self-organize dynamically so as to increase it's likelyhood of surviving. It was mind boggling. If someone recalls the details of the gadget's function I urge them to make them known. I have forgotten but would love to enjoy it over and over again.

Ashby among others kept admonishing us to cope with the non-linearity in our reality. As the resident mathematicians I felt especially "guilty" and in order to atone I did my graduate research on Nonlinear Systems. Classifying the world into Linear and Nonlinear systems seemed a bit like classifying the world into Bananas and Nonbananas. Just like there are many distinct "Nonbananas" it turned out that there are distinct classes of Nonlinear Systems which can be characterized by their specific "composition" rules. For example, the composition operation for Linear Systems is addition (and in this case it is usually called "superposition"). More generally then, certain nonlinear systems have an analogous composition rule (i.e. an operation other than addition). By way of an example consider a network composed of neurons. Knowing the function (i.e. input-output relation) of the neurons and their composition (i..e. interconnectivity) the network function can be computed directly from that of it's components via the composition operation. Some fringe benefits are that certain connections are possible but others not—due to incompatibilities between classes of nonlinear systems. I recall HvF, Ross and others like Michael Arbib liked this idea and urged me to continue it. After leaving BCL, IBM's business interests prevailed and gradually the compositions … decomposed.


"To iterate is human … to recurse divine"

Just prior to course registration one semester I asked Prof. HvF for suggestions on what courses to take; Learn how to Learn he answered unhesitatingly. I am still in awe with the profoundity of this advice and pass it on to my students though not with the HvF's flair—only he can do that. And how do we learn how to learn or even simply learn? In 1989 HvF gave one of the lectures of a distinguished series at the Univ. of Southern California in Los Angeles. It was vintage Heinz and he was roundly applauded, admired and celebrated by the local prominenti. I felt sorry for the next speaker since HvF is such a hard act to follow. Among other things he spoke about Epistemology or how do we know what we know.

I am continuously absorbing and learning from my experience at BCL and my association with Prof. Heinz. It is a life process … and perhaps beyond (we will not go into that). I had listened to the discussions on "Concept Formation" when I was at BCL often feeling that it was mostly over my head. At times I wondered not only about the sanity and well-being of the discussants but also the listeners including my own. Still they spoke with conviction so who was I to question.

So when Epistemology came up all these thoughts reawoke from my subconscious and I listened with rapt attention. May be this time I will either understand or convince myself of the other option. And I actually understood … at least partially. HvF spoke about learning by Recursion. The non-recursive part of the process (i.e. concept formation) consists of the initial examples that we learn. Continued use of the concept are (act as) the recursive steps sharpening and deepening the meaning. Well I thought I was finally getting it (modulo some details) and so did a number of the listeners as I discerned from the head-nods. Over 23 years after my departure from BCL I thought that I was finally graduating in the sense that I could understand some of the heavy-duty ideas in real-time — i.e. while HvF was expounding them. Well no such luck! Just as I was engulfed by these pleasant thoughts HvF hit me and others with a bombshell—the most powerful one yet. To the shock, astonishment, non-belief, daze, absolute bamboozlement of the listeners he denied Causality! Here I (and almost everybody else) had to part ways with him. I thought of my two Meta-Murphys1 namely:

  • Law of Attraction of Unfortunate Events "Once an unfortunate event occurs it tends to attract others of its kind", and it's close relative,
  • Law of Inopportune Timing "A fortunate event (if it occurs at all) tends to occur a bit too soon or a bit too late.

Our phenomenological existence described by Recursion and these Metas2 (which may also be recursively obtainable). But no Causality? This is like magic. I am still left wondering: Recursion … or was it magic? It is part of the mystery and legacy of HvF and I celebrate it everyday. Dear maestro Heinz, recursive epistemologist … and magician, I offer you my profound gratitude and admiration.


  1. The prefix Meta pertains to the fact that from each of them Murhpy's Law can be trivially deduced.}
  2. Instances of lack of skillful good fortune.

Alfred Inselberg
School of Mathematical Sciences Tel Aviv University Tel Aviv 61396, Israel, aiisreal@math.tau.ac.il
Senior Fellow San Diego SuperComputing Center, San Diego,
and Multidimensional Graphs Ltd, Raanana 43556, Israel

Heinz von Foerster Festschrift