Stephen Wolfram's Model of Information

In the 1940s, John von Neumann and Stanislaw Ulam began playing around with the idea of natural systems being extrapolated to initial conditions, then playing out as a sort of cellular automata. And I remember when I first read about cellular automata - James Gleick's book Chaos: Making A New Science had just come out - and it was filled with mind-spaghettifying ideas. Ideas like artificial life, the now-famous "Butterfly Effect," chaos mathematics and Benoit Mandelbrot and fractal geometry and fractal art, and it was  - much of it - way over my Generalist's head, but exciting. Cellular automata was in there. I had never heard of it.

                                                           Wolfram

Years later I picked up Stephen Wolfram's book after it came out in 2002: A New Kind of Science was about 1300 pages long, and was the manifesto of a guy who graduated with a PhD in particle physics from CalTech when he was 20, then received one of the first MacArthur "Genius" awards at age 21. This guy had a way to model just about everything: syntactic structures, social systems, particle physics.  Just about everything. It turns out he was a big-time guy in cellular automata, carrying on in the tradition of another "Martian," John von Neumann.

Wolfram's math was over my head, but books like this make me excited just to be in the presence of this sort of compendious mind. It's the kind of book I take off the shelf and open at random and read, hoping for some sort of inspiration. It usually works. Wolfram models information in our world upon his forays into cellular automata, in which you have a very basic system under initial conditions, and watch it evolve. He developed a taxonomy of the sorts of systems that arise, that he called "Class 1" "Class 2," and so on. These first two classes exhibit a low order of complexity; they tend to reach a level of constancy and repetition that's sorta boring. There's no surprises. They go on and on, ad nauseum, or die. A system like this? A clock.

His Class 3 level I think of as "noise." You can't predict anything. It's seemingly entirely random, like being bombarded by cosmic rays. If there's any structure at all, it's too complex. It seems akin to entropy. A system like this? Your TV tuned to a dead channel: all static and noise.

                                        cellular automata being simulated, played out. 

Wolfram's Class 4 is where the action is: these systems turn out lots of surprises. They're complex but there's structure; you can model from them and make a certain sense out of what's going on. Systems like this are intellectually exciting and basically describe any theory or "law" in the sciences. They're surfing the edge, almost falling into "noise" but never quite. It reminded me of Ilya Prigogine's ideas about complex adaptive systems and negative entropy, how life flourishes despite how "hot" it burns and uses resources. It creates information, structure, patterns, complexity. Indeed, Prigogine and Wolfram seem compatible enough to me...

                                 Shannon's basic equation for information theory:
                                 world-shattering stuff, turns out


My Other Information Systems
Probably because of my intellectual temperament - which includes not being particularly adept at math - I had always been very impressed with guys like Wolfram and what they were able to do with math, but I have also been suspicious that they're somehow operating from the conceit...or rather, flawed assumption that numbers can describe everything and that everything that's interesting to us is really just stuff that's interacting with the environment and doing computations. I thought these weird math geniuses had become hamstrung by the computer metaphor, and as I saw how different the human brain was from what they had asserted it was - a "biological computer" - I felt my suspicions confirmed.

I remember Timothy Leary giving a talk in Hollywood. He had been reading a recent book and was very enthusiastic about it. It was titled Three Scientists and Their Gods, by Robert Wright. So of course I had to read it. It's about Ed Fredkin, E.O. Wilson, and Kenneth Boulding. Leary seemed taken by Fredkin especially. This was an Everything Is Information Processing in a digital way stuff. Leary's psychedelic intellectual friend Robert Anton Wilson seemed interested in this view too, but never committed to it. RAW always seemed more committed to Claude Shannon's mathematical theory of communication - which is the gold standard for quantifying information - but Shannon's theory has information with no necessary semantic component; RAW made a heady brew from combining Shannon with Korzybski, who was all about semantics and our environment and how we make meanings.

Earlier, the originator of pragmatism, Charles Sanders Peirce, had developed a theory of semiotics that took into consideration the content of information using signs and a mind interacting with signs; he had begun to work out a system of defining, quantifying, and taking into account the evolution of a piece of information. This was the "pragmatic theory of information," but it hasn't gone all that far. Shannon's 1948 paper blew it off the map. But still, "information" had to have some sort of semantic component to it, or I had difficulty grasping it. Shannon's and Von Neumann's and Fredkin's and Wolfram's and Leary's ideas about "information" felt too disembodied to me; my intuition told me this couldn't be right. But I'm starting to come over to their side. Let me explain.


Modeling Natural Processes
Via cellular automata theory and the gobs of other stuff a Mind like Wolfram has, he said you can only get so far by modeling life as atoms, or genes or natural laws or as matter existing in curved space at the bottom of a gravity well. More fruitfully, we can model any natural process as computation. Big deal, right? Yea, but think of what this implies: Wolfram thought we can model a redwood tree as a human mind as a dripping faucet as a wild vine growing along a series of trees in a dense jungle thicket in the Amazon. Why? Because all of these systems were "Class 4" systems, and these are the only really interesting things going on. All of these systems exhibit the behavior of "universal computation" systems. (If this reminds you of fractals and art and Jackson Pollock, you're right: I see all of this stuff as a Piece. And so, apparently, does the math.)

Also: you cannot develop an algorithm that can jump ahead to predict where the system will be at Time X; this was proven by Alan Turing in 1936. You can't predict faster than the natural process itself. You had to wait to see what the system did; this blows to smithereens any Laplacian Demonic idea about knowing all the initial conditions and being able to predict everything. So guys like Ray Kurzweil - who has become more and more a sort of Prophet for quantifying the acceleration of information and making bold, even bombastic prediction about what will happen to our world, our society? Wolfram/Turing say no. There are no short cuts and our natural world is irreducible to anything close to Laplace's Demon. The system is too robust to reduce to even what Kurzweil seems to think it is. Robert Anton Wilson used the term "fundamentalist futurism" to criticize those groups of intellectuals in history that Karl Popper had called the enemies of the Open Society. I think the term may apply to Kurzweil too, but I'm not sure. Certainly it seems to apply to Hegelian historicism, most varieties of Marxism, Plato's Republic, and Leo Strauss and the Wolfowitz/Bush/Cheney Neo-Cons.

As I read Wolfram and Kurzweil, the latter seems to see our world as modeled within Wolfram's classificatory scheme as something like a Class 2 system: complex, but if you know enough about the algorithm that undergirds the whole schmeer: fairly predictable.

Arrogance? Aye, but human, all-too human, as Fred N wrote.

                                                    Drew Endy, now at Stanford

Synthetic Biology
Leary, with his penchant for neo-logizing, had in his 1970s book Info-Psychology, defined "contelligence" as "the conscious reception, integration and transmission of energy signals." There were eight fairly discrete levels of this reception--->integration----> transmission dynamic (modeled on the syntactic actions of the neuron). All well and good and trippy, but a team at Stanford led by Drew Endy has made a computer out of living cells.

Engineers at Stanford, MIT, and a bunch of other places have made biological computers. Do you know how a computer must be able to store lots of data? Well, it turns out storing data in DNA is insanely, wildly do-able and has more storage space than you can imagine. Perhaps you heard that some more of these everything-is-a-computer types stored all of Shakespeare's Sonnets in DNA. But that's small taters: it looks like we'll be able to store entire libraries, TV shows, movies, and CDs in DNA. Read THIS and see if you don't feel your mind getting a tad spaghettified.

So: a silicon chip uses transistors to control the flow of electrons along a path; Endy and his team at Stanford have developed a "transcriptor" to control the flow of proteins from one place to another, using Boolean Integrase Logic gates (or "BIL gates" so there's your geek humor for the day!). Endy says their biological computers are not going to replace the thing you're using to read this, but they will be able to get into a tiny, tight quarters and feedback info and manipulate data inside and between cells...something your Smart Phone cannot do.

Endy sees his biological computers as inhabiting a cell and telling us if a certain toxin is present there. It could also tell us how often that cell has divided, giving us early info on cancer, for example. It could also tell us how a drug is interacting with the cell, and make therapeutic drugs more individually tailor-made.

In a line that reminded me of dear old Crazy Uncle Tim, Endy told NPR that, "Any system that's receiving information, processing information, and then using that activity to control what happens next, you can think of as a computing system."

For more on bio-computing, see HERE and HERE.

I'm starting to swing more with Wolfram. But there are many other little snippets that are swaying me. I still like older forms of "information," more human-scaled and poetic and embodied.

But then there are the intelligent slime-molds, which I will leave you with. Grok in their fullness. Don't say I ain't never gave ya nuthin'!

How Brainless Slime Molds Redefine Intelligence.

Drippy Jackson Pollock Theories: Art and Mathematics!

Sometime in 1996, physicist Richard Taylor decided he'd also pursue a degree in Art History, so he's my kind of guy, obviously. He was on sabbatical and checking out Jackson Pollock's drip-art (1943-52). In 1949 Life magazine featured Pollock as the Hot New Guy Painter and then received tons of letters protesting this abstract expressionist's work, a common argument being something like, "My kid could paint that!" 50 years later, a major exhibit of Pollock at New York's MOMA had lines around the block, and a Pollock has sold for $40million. You explain it. I've got some fractal fish to fry.

                            A detail-section from Jackson Pollock's 1949 work Number 8

Taylor got the idea that Pollock, in his use of particular paints, his manipulations of the paints' viscosities, his bodily motions, his attacking the canvas by walking around it from every angle, using his famous drip technique, had produced fractals. Either Taylor intuited that Pollock had intuited the basic minute algorithmic structure of Nature, or Taylor thought it might be really interesting to pursue the idea and see where it took him, I don't know. But he began an interesting controversy that rattles and drips down to today.

Taylor pursued his hypothesis. As a physicist he was already familiar (an understatement) with Benoit Mandelbrot's book The Fractal Geometry of Nature. In order to test whether Pollock's paintings between 1943-52 contained fractals - whether they contained complex geometric patterns that could be detected at very small scales, medium scales, and larger scales, a hidden occult order that corresponded to well-defined parameters in fractal geometry - he scanned pictures of the drip paintings and put them in his computer. Then he divided the images into very small boxes that added up to five million drip patterns that were, when looked at by the naked eye, anywhere from four yards wide to less than 1/10 of an inch.

                                         Sabinoso canyons of New Mexico, aerial view
                             

If there were fractal patterns in Pollock, it didn't matter what the original size was that was placed into a tiny box in a computer image: fractals are "as above, so below": fractals at the tiniest levels are reflected in the largest levels. If we look at a tree, a twig fractally mimics the whole tree. Mandelbrot - whose work was sneered at when it first appeared (by the most eminent mathematicians of the day!), argued that he had discovered yet another non-Euclidean geometry, one that could be applied to the "roughness" of Nature. Previously, Euclid and other inventors of non-Euclidean geometry assumed smooth shapes basically governed "reality."

[Mandelbrot has won, and for further information - even though I assume nine of 10 readers of Overweening Generalist already know quite a lot about fractals - I've linked to a pretty good 50 minute Nova TV program, which includes Mandelbrot, Richard Taylor, Keith Devlin, Ralph Abraham and many others. If you're bored with my spew here and want to know more about fractals and Mandelbrot, skip to the end of this post. - OG]

                                          a computer-generated fractal image

Anyway, Taylor (1999 paper) found that Pollock's paintings were indeed filled with fractals. As Jennifer Ouellette wrote in Discover in their November 2001 issue about Taylor and Pollock, "Pollock was apparently testing the limits of what the human eye would find aesthetically pleasing." Let's be clear about why this might be so: if we contain fractal structures in our bodies, and Nature - say, a rain forest, mountains, trees, and clouds - are fractal, then we intuitively "understand" these very complex-looking images, because we evolved with them and they are in us. Studies have shown that people's stress levels can lower by as much as 50% by looking at fractal imagery.

This seems to be true at many levels. Imagine the sound of a baby pounding his fists on the keyboard of a piano. Then think of listening to someone playing major scales in every key on a piano with a metronome. Then imagine Beethoven playing variations on a simple theme.

Some evolutionary psychologists have surmised that, deep in our evolutionary ancestry, we became attuned to deviations in the visual environment that disrupted fractal patterns, and this aided in survival, thus making the argument that we are oddly soothed by looking at fractal patterns because they were primarily about survival, and predated aesthetic concerns. (Another Just-So story?)

What's weird is that Pollock did his drip paintings before Mandelbrot published his work on fractals. If we think Richard Taylor was basically correct and Pollock's drip paintings contain fractals, how did Pollock "know"? Almost every writer who's covered this story uses the words "intuition" and "instinct." Some link this to the fractal nature of our selves, but even if this wonderful idea is true, I still marvel at...how? My best guess, as of this date, is that Pollock somehow got himself into harmony with the way of Nature, what the ancient Chinese called Tao. 

"I can control the flow of paint. There is no accident." - Jackson Pollock

Taylor found, using the parameters of fractal geometry and his computer, that, as Pollock's drip technique evolved, the later, more complex paintings had even higher levels of fractal dimensionality.

The next step for Taylor was to build his "Pollockizer," which was a device that used fractal algorithms that allowed a machine to splatter or drip paint in ways that created paintings that were either fractally based, or, when he tweaked the numbers, produce just random, non-fractalized paintings. Then, to test his theory on the public, he displayed his machine-produced fractal paintings along with his non-fractal/ordinary "chaos" paintings. Of 120 people surveyed, 113 chose the fractal paintings as most pleasing.

Fractals are found in nature, but there are others that are man-made (like Pollock's dripworks), and there are still others that are computer-generated, and anyone can look on You Tube and find meta-psychedelic fractal videos that trippers during the Summer of Love could've never imagined. Taylor worked with perceptual psychologists and found that subjects preferred looking at objects that fell within the 1.3 to 1.5 level of dimensionality of fractals, irregardless whether the images were from nature, computers, or human-made. (This reminded me of my studies of why we find some people more beautiful than others.)

There's much more to Taylor's arguments about fractals in Pollock, but what he wanted to argue was that he had developed a way to quantitatively analyze the style of an abstract artist. As wild as Pollock's abstract paintings appear, Taylor said his method was objective and that he may have stumbled onto a way to authenticate and legitimate actual Pollocks, to detect the "fingerprints" of his style, and separate them from the (probably) hundreds of fakes out there. Conservators came running to Taylor, eager for him to apply his method.

The Method Gets Tested: Let the Drama Heat Up!
In 2003, Taylor analyzed 24 putative Pollocks and said they did not possess the fractal signature of Pollock.

In 2006, Alex Matter, the son of friends of Pollock, said he'd found 30 Pollocks that the famous painter had given to his parents. They were found in a storage bin in Long Island, New York. Taylor used his method and determined the paintings were not Pollocks.

Enter two physicists, Kate Jones-Smith and Harsh Mathur of Case Western Reserve University in Ohio. They argued that Taylor's methodology was flawed and that it should not be relied upon to authenticate paintings. In the journal Nature they said, "Several problems must be addressed before fractal analysis can be used to authenticate paintings." And one was the small-box counting technique Taylor used. Jones-Smith took her own simple childlike line drawing, Untitled 5:

...and subjected it to Taylor's method. She and her colleagues used the box-counting technique and found them to have the fractal dimensions of a Pollock! (?) Furthermore, Jones-Smith said Pollock's paintings lacked the range of scales needed to be considered as fractal, because the smallest speck of paint analyzed by Taylor was only 1000 times smaller than the entire canvas. Taylor responded in Nature that invoking this power law "would dismiss half the published investigations of fractals." Additionally, Jones-Smith found that, when she and colleagues analyzed Pollocks using Taylor's published methods, they found that two out of three Pollocks failed to satisfy Taylor's criteria. They charged that Taylor had kept some of his methodology unpublished. Also, Jones-Smith said Taylor only analyzed 17 of the drip paintings (other sources I've read give it as 20), which makes the sample size too small, as there were at least 180 drip paintings. This means that Matter's works may have really been Pollocks and that Taylor's idea about Pollock's stylistic fingerprints should not be used in authentication.

Taylor responded that Jones-Smith's Untitled 5 does not show fractal patterns. By late November, 2007, the idea of authentication by fractal signature seemed wildly unsettled.

Here's an interesting point for the OG: by the time physicists with the advanced ninja Mandelbrotian juju apply their numbers, I obviously lack the math chops to stick with the argument. It seems that when a story goes off in this direction - specialists clashing in a language I'm not conversant in - I retreat to the sidelines to watch the melee and see how it all plays out. And sometimes it takes years! Which is fine with me. I actually enjoy the drama of it all.

Enter: Four New Nerds
In early 2008 a paper appeared, "Multifractal Analysis and Authentication of Jackson Pollock Paintings," by Coddington of NY MOMA, Elton of Pegasus Imaging Corp, Rockmore of Dartmouth mathematics, and Wang of Michigan State mathematics. They looked at Taylor's methodology and how he'd determined that the 30 Alex Matter "Pollocks" were bogus. They extended Taylor's methods, adding the "entropy dimension," which they described as a quality related to the fractal dimension, and I'll just have to take their word for it. The four seem excited that geeks like themselves are making inroads into "Stylometry," and early in the paper cite its nascent use in literature, and this reminded me of a totally fascinating book I'd read a few years ago by Don Foster, Author Unknown.  Foster called what he did "literary forensics" (other times "forensic linguistics") and he used his computer and algorithms he'd developed about word choice, length of sentences, certain tendencies towards tropes, etc, to try to determine who some unknown author was. He was involved in the Unabomber case, he figured out who wrote Primary Colors, and showed that an incredibly literate and witty letter-writer to a Northern California newspaper, who signed as "Wanda Tinasky," was not Thomas Pynchon, as many suspected. It's a great read, but I've digressed. Again.

So yea, the four geeks. Stylometry would use advanced statistical analysis and improved digital representations to quantify works of art. They thought Taylor's box-counting method was legitimate. Hell, what do I know when they haul out their "entropy plots" and "slope statistics"? What did they conclude about Taylor's methods?

They thought Taylor was right: based on their entropy plotting, the Matter paintings were fakes. When they compared a known Pollock with one of Matter's "Pollock"s, they saw a "dramatic distinction between a secure Pollock and this drip painting found by Alex Matter."

Score one for Taylor.

2009: Jones-Smith and Mathur Team Up with Big-time Physicst Lawrence Krauss
They have done their best to debunk Richard Taylor. HERE is a short writeup on their doings.

2011: Fluid Dynamics Leaks In
Check out this article by Lisa Grossman from Wired. When I read the title I thought, okay, more of the math-physics-art geek wars and Richard Taylor, but noooo. What we get here is two geeks from Boston College and a Harvard mathematician claiming the first quantitative analysis of Jackson Pollock. The nerve! Grossman doesn't mention Taylor, but links to his 1999 correspondence to Nature about fractals in Pollock, co-written with Micolich and Jonas. There's no mention of Coddington, Elton, Rockmore and Wang. Why? Because they're not at Harvard or Boston College? What? If you're at Dartmouth it's a denigration?

Herczynski and Cernuschi of BC and Mahadevan of Harvard "believe they've done the first quantitative analysis of drip painting." Yea, hoo-kay.

Well, these fluid dynamics guys had just published an equation about how Pollock spread paint on a canvas, in Physics Today. So...why are they "the first?" Well, they're drawing on novel findings in the area of physics called fluid dynamics. How honey coils when you pour it onto a conveyor belt, the behavior of a dripping faucet, the movement of nanofibers and rope. Some whole other thing that ain't fractals. No, they've done a "quantitative analysis" of "inertial, gravitational, and viscous coiling regimes." Alright, but I think Taylor, Micolich and Jonas did a quantitative analysis in 1999, and the four nerds in 2008 did so also.

I will chalk all this up to the dimension of scientist's lives perhaps best shown in Carl Djerassi's books of what he calls "science-in-fiction," in which an anonymous author's short bio of Djerassi, we read, "illustrates, in the guise of realistic fiction, the human side of scientists and the personal conflicts faced by scientists in their quest for scientific knowledge, personal recognition, and financial rewards."

But just as Taylor thought Pollock was doing fractals before anyone knew about them - even himself - the fluid dynamics dudes say Pollock was doing experiments in fluid dynamics before any physicists had known about the way certain fluids of certain viscosities moved. (This all reminds me of a stunningly imaginative speculative work by a real Leonardo-level Generalist: Leonard Shlain's Art and Physics: Parallel Visions In Space, Time and Light. Here's a thick book on how, throughout history, artists seem to stumble upon "knowledge" about Nature before the physicists, astronomers, chemists, and biologists "discover" the same thing, and then quantify it. Shlain was a surgeon. Yet another digression. Sorry!)

Pollock was obsessive about finding new paints and pigments, and loved to alter their consistencies. Maybe the fluid dynamics guys have something here.

Meanwhile, Benoit Mandelbrot (who died in 2010) thought Taylor was right: Pollock drip paintings were fractal. He supported Taylor in a 2007 Science News article.

Harsh Mathur, Kate Jones-Smith's colleague and Richard Taylor's nemesis, is impressed by the fluid dynamics guys' work. But I find Mathur a tad glib when he says, "Either Taylor is wrong or Kate's drawings are worth $40 million. We'd be happy either way." Why? I'd still like to figure out whether Jones-Smith and Mathur had an ulterior motive for taking on Taylor.

As of 2011, Richard Taylor says that fractals are "just one key to authentication, and should be used with other methods. It's not a red-light green-light method." He's gone on to investigate neurobiology and fractals, using MRIs, EEGs, and skin conductance to measure stress levels of people looking at fractal imagery.

Tangentially, Frances Stonor Saunders, in The Cultural Cold War showed how the CIA covertly backed Western abstract expressionists like Pollock in order to win the war with the Soviets. The idea was that we were way ahead, because, well...just look how "advanced" someone like this Pollock character is! This is what a "free society" looks like!  A wonderful work of "hidden history" that almost reads like a byzantine spy novel, yet it's true. What a weird, weird world some of the CIA guys live(d) in.

Late in my investigation, I discovered Sarah Everts's article in Central Science. It serves as a recapitulation and extends my little ditty here. I like how she points out that fractals soothe, the debate rouses, the math and physics sooth, presumably due to their elegance.

"Technique is just a means of arriving at a statement." - Jackson Pollock

Here's a good overview documentary of fractals and their history, Mandelbrot, etc:

Other Works Consulted

• Chaos: Making A New Science, James Gleick, 1987
• Complexification: Explaining a Paradoxical World Through the Science of Surprise, John L. Casti, 1994
• Computers, Pattern, Chaos, and Beauty: Graphics From An Unseen World, Clifford A. Pickover, 2001
• Nature's Numbers: The Unreal Reality of Mathematics, Ian Stewart, 1995
• Patterns In Nature, Peter S. Stevens, 1974
• Trialogues At The Edge Of The West, Ralph Abraham, Terence McKenna, Rupert Sheldrake, 1992
• Jennifer Ouellette's home page
• Jackson Pollack site at Artsy

Imre Lakatos: Another Favored Hungarian

How does science work? In the long history of physics, from Aristotle to the string theorists of today, what precisely went on to bring about the change from one model or theory or mode of working to a newer way? How did similar things happen in math, chemistry, or other areas of knowledge that humans have sought to inquire into, to probe the inner workings of "reality"?

Perhaps the most influential High Culture text of the second part of the roaring 20th century was Thomas Kuhn's The Structure of Scientific Revolutions. Kuhn's creativity and rhetoric has proven very powerful. When your friends drop the word "paradigm," they may not know it, but they're probably borrowing from Kuhn at an nth removal. Kuhn said - of course this is far more complex than befits a single blog-post - but after a bunch of stabs in the dark, a new science gets going because some thinkers used metaphors that turned out to yield more satisfying corollaries. Eventually specialists working under the sway of a theory that seems to have won out cannot see any alternatives, or have been inculcated by their mentors in this theory that any rival theory or new idea must be wrong, or mistaken, or fraudulent. This is when a science is in its "normal" stage: scientists are working under such a powerful theory that it provides plenty of good problems to solve, and workers under this paradigm, while acknowledging that certain phenomena keep appearing that can't be accounted for in the theory, figure it's only a matter of time before someone working under this Great Theory will come along and figure it out. Meanwhile, anomalies keep piling up, and scientists keep putting them on the To Be Solved Later shelf. (Philosopher Ian Hacking's recent Q&A on the 50th anniversary of Kuhn's famous book, and how it's holding up.)

Eventually, a different way of thinking comes along. It is attacked by the dominant paradigm-workers; they have their whole lives invested in the reigning order. But the new paradigm starts to show that it can not only account for the older paradigm, but it starts to account satisfactorily for many of the anomalies the older paradigm couldn't solve; the older order and its adherents eventually die off, and the younger people with their new paradigm carry on, in their version of "normal science." A revolution has occurred.

Prior to Kuhn, possibly the most influential thinker in the philosophy of science was Karl Raimund Popper, who - and I'm leaving a woeful amount out here - said that a good theory must be "falsifiable" to be taken seriously. When tested, if falsified, great doubt is shed on the theory. If the theory is subjected to a thoroughgoing round of conjectures and is still found to fail, it should be abandoned for something better. (Popper's "Three Worlds" of knowledge idea.)

                                                    Imre Lakatos: One wily, funny,
                                                    rational cat!

Enter Imre Lakatos (say "EE moo ray LAK uh tosh"), who was born Imre Lipschitz, changed it to Imre Molnar to avoid Nazi persecution when they rolled into Hungary, then he later changed it to Lakatos (which means "locksmith" in Hungarian), possibly in honor of a Hungarian General who fought against Nazi rule in Hungary, but also maybe because his old shirts had "I.L." on them, so why not save money when it's scarce after WWII?

During WWII, Imre became a communist. He also attended the private seminars of Georgy Lukacs, the prime mover of the Frankfurt School. But eventually he had a difficult time dogmatically toeing the Communist party line (It's Imre Lakatos! Of course he couldn't help but question the Authorities!), and they threw him in prison for the crime of "revisionism." Imre spent 1950-53 in prison, then went back to academic life, radicalized and allied with at least one group that furthered the march toward the disastrous 1956 Hungarian Revolution.

And here we have that same old story, of political strife in Hungary driving its geniuses out of the country, where they usually flourished in England or especially Unistat. Imre found his home at the London School of Economics (LSE), where he met Karl Popper and a host of high-powered thinkers, became a very popular professor until Imre's untimely death in 1974, his lectures crowded with excited young intellectuals. Imre was never boring and told many jokes. As the philosopher Ernest Gellner said of Imre's talks: "He lectured on a difficult, abstract subject riddled with technicalities: the philosophy of mathematics and science; but he did so in a way that made it intelligible, fascinating, dramatic, and above all, conspicuously amusing even for the non-specialists."

He developed his method of doing the philosophy of science by first developing a method of dialogue surrounding a theorem in mathematics, showing that, if one questioned every attempted conjecture, a counterexample was almost always found. How did conjectures and counterexamples arise in the first place? One might be tempted to say they were arrived by well-educated thinkers using creativity on the problem, or even by something like Charles Saunders Peirce's "abduction" in logic.

Lakatos said a form of "thought experiment" was used, and, being the Hungarian sort of mathematical mind and seeming to fetishize "rationality" above all (more on this below), he admitted that thought experiments that led to tests of a theorem qualified as "quasi-empiricism." This mode of thought was very important and indeed, heuristic: it helped to generate auxiliary hypotheses, kernels of thought that, Hegelian-like (Lakatos was heavily influenced by Hegel) and with refutations and counterexamples...led to to the growth of knowledge. Moving on...

Lakatos wanted to find a middle way between Popper and Kuhn. At the LSE he became heavily influenced by Popper, but Kuhn's book (first edition published in 1962) had to be criticized and accounted for, and justified with Popper's falsifiability idea. Over the years, Lakatos often spoke of Popper1 and Popper2. The former was the Popper that is misread by his readers, or the Popper represented by bad readings of Popper; the latter represents the close, insightful readings of Popper, but many critics have said that Popper2 is really Lakatos himself.

Popper had said something along the lines that, scientists propose hypotheses, and these are tested. If the tests show the hypotheses did not work as thought, then this was Nature saying "No!" Think of something better. Here's Imre's rejoinder to that: "It is not that we propose a theory and Nature may shout 'No!'; rather we propose a maze of theories and nature may shout 'Inconsistent!'" - p.130, Criticism and the Growth of Knowledge. There's something telling here: in Imre's very complex epistemic formulations, we do not criticize individuals, we criticize the methodology being done within a certain research program. As his pioneering work in mathematics - in which he argued via imaginative reconstructions of actual different mathematicians' attempts to prove a conjecture, arriving at the idea that no theorem in informal math is "true" and "final," only that no counterexample had arrived yet - this was also how scientific knowledge should be worked upon. Highly mathematized, too rationalistic, too Idealistic? Maybe...

Imre thought that, because Kuhn rejected Popper's falsification and justification ideas, he'd fallen back on "mysticism" and, because Kuhn had not demonstrated a logic of scientific discovery other than something resembling a political revolution - or even plate tectonics - that his ideas were tantamount to irrationality and "mob rule." (See Imre in this passage from 1970's Criticism and the Growth of Knowledge) Speaking from my current state of confusion and ignorance, I think Imre wanted people to be as Hungarianly rationalistic as he was; however, they are not. I'm not sure they ever were. Rarely have groups of humans - even scientists! - behaved rationally in the way that Imre idealized. Scientists are humans too, with emotional stakes, cutthroat competitiveness, and adhering to party-line dogmas like Churchmen.

But now we're getting to why I think Imre Lakatos is such a fascinating thinker on the human stage. (He died of a brain hemorrhage suddenly at the age of 51.)

Eventually, Lakatos struck up a friendship with the anarchist-epistemologist and philosopher of science, Paul Feyerabend. Feyerabend, an Austrian, had fought for the Nazis, as he details in his wonderfully readable autobiography, Killing Time, eventually ending up with a professorship in Berkeley. Imre's mother and grandmother had died in Auschwitz. But Imre's and Paul's intense intellectual interests led to very heated and civil exchanges, and very many letters, all of Imre's to Paul lost, because of Feyerabend's carelessness. He said he tacked some to his walls in Berkeley to keep the rain out, if memory serves. Imre, meanwhile, dutifully filed every letter from Paul, and you can read Paul's letters to Imre in the delightful For and Against Method: Imre Lakatos/Paul Feyerabend, edited magnificently by Matteo Motterlini.

(John Kadvany's photos of Imre, Popper, Feyerabend, and Imre's Hungarian milieux.)

Despite Imre's intensely rationalistic mind, and his deep desire to take Popper's thought further, Feyerabend, a very close reader of Imre, Popper, Kuhn, and many others, repeatedly failed to see a difference between Imre's implied historical "logic" used in scientific discovery, and Feyerabend's own, "anything goes," best seen in the third edition of Feyerabend's, in my view vastly underrated Against Method. Throughout their correspondence, Feyerabend, playing Socrates tinged with Pyrrho, points up again and again that, when followed to its conclusion, Lakatos's "method" is virtually the same as his: anarchistic!

How could two seemingly polar temperaments be friends at all? First, both were deeply in love with intellectual thought, and especially dialogue. It was Imre who cornered Feyerabend at a party in 1970 and said, "Paul, you have such strange ideas. Why don't you write them down? I shall write a reply, we publish the whole thing, and I promise you - we shall have lots of fun." Paul's Against Method was his side; Imre died too soon to write his reply. Nevertheless, we can read the book of letters, and Imre's books and come to our own conjectures.

The second reason these two men were such good friends was humor. Imre's students knew that, no matter how difficult the subject of a lecture, some jokes would be forthcoming to retain levity. Feyerabend, a fascinating, irascible, complex character who I think qualifies as one of the great guerrilla ontologists of the 20th century, wrote in a letter to Imre dated 16 April, 1971:

"Criterion for being a lefty: you lose your sense of humour and become a self-righteous bastard (or bitch, as the case may be). In this, of course, you do homage to an age old American tradition: Puritanism. Now, I am un-American to the extent that I despise Puritanism, whether it comes from the right, left, or centre. 'But what about the truth?,' people ask. The truth, whatever it is, be damned. What we need is laughter. You have got the gift for laughter, even where your own position is concerned, so, as far as I am concerned, you are a good guy (and you are 'good' even theoretically, for your theory is equivalent to mine, as I have said above and in chapter 25 of my magnificent AM."

There is very much more to be said of Imre Lakatos, thinker extraordinaire, but I have gone on too long once again. I will leave by exhorting readers of Robert Anton Wilson, who are interested in his epistemology and his ontology, to maybe check out Lakatos and Feyerabend. Popper and Kuhn are far more well-known, but Imre and Paul should go a long way toward further mindcopulae, and I'm thinking especially of readers of RAW's The New Inquisition: Irrational Rationalism and the Citadel of Science. Also, as per Imre's inner working from within the logic of scientific discovery: when an idea within a research program is being tested via creative criticism and the dialectical sparks that occur therein, a progressive program will yield startling innovations. Nick Herbert talks about a couple instances of conjecture and refutation that led to innovation or new ideas that he was involved with, in quantum theory, here.

John von Neumann: Hungarian...Martian?

The world is filled with stories of alien spaceships sighted, and if one looks one can easily find that some people believe that the aliens - wherever on Earth they landed - got out of their ships and mixed in with the humans. They are among us, even now.

I have always loved the variations on those stories, never believing any of them.

During the time of the Manhattan Project, there was an ongoing joke that the Hungarians (Leo Szilard, Edward Teller and a few others) were so smart they must be "Martians" and they were speaking Hungarian as a cover. The smartest Hungarian of them all - maybe the smartest mathematical-scientific-freakish mind of the 20th century - was, for my money, John von Neumann. Even if you don't read the long Wikipedia link there, just have a glance at the right hand column, under "known for." See what I mean?

                                              Martian von Neumann, in 1945. Notice how his 
                                                              eyes seem to look straight into your own brain.
                                                              Can he read your mind, from wherever he is now?

Born in Budapest in 1903 to an upper-middle-class Jewish family, he was able to exchange jokes in classical Greek at age six. At that same age he could memorize telephone directories on sight and could divide two eight-digit numbers in his head, almost instantaneously, down to a decimal. He mastered calculus by age eight. His family, like most Jewish families in Budapest at that time, highly valued education and culture. When John's father bought an entire library from an estate they rebuilt a large room in their house to accommodate it, and John read the entire 44-volume universal history series in German and memorized it as a boy, and had retained total recall at the end of his life. He had a photographic memory, and anyone can do research on von Neumann and come upon some anecdote that is freakish, astonishing, and frankly, sorta "Martian." A colleague asked if John knew the first words of Dickens's A Tale of Two Cities. One wonders what the colleague was looking for, as JvN's memory was legendary. Most of us would reply, "It was the best of times; it was the worst of times," and be satisfied. John had to be stopped after 15 minutes.

A professor of history whose expertise was in the Byzantine empire, said that when he talked to JvN he realized JvN knew more about the subject than he did. How disconcerting...JvN tried to guard against this sort of thing - hurting people's feelings because he was so freakishly smart - but he was bound to fail every now and then.

The great Hungarian mathematician George Polya said, "Johnny was the only student I was ever afraid of. If in the course of the lecture I stated an unsolved problem, he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper." When JvN was teaching math at Princeton, he was not very good, because he'd talk so fast, demonstrate the math on the chalkboard, then quickly erase the equations before anyone had a chance to copy them. (He was probably bored? Absent-minded?)

The great physicist Hans Bethe, keeping with the Martian theme, said of von Neumann, "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man."

He earned a PhD in math from the University at Budapest at age 22, and there is very little evidence he ever studied. At the same time he earned a degree in chemistry from Zurich. He eventually made major contributions to mathematics in the fields of set theory, functional analysis, geometry/lattice theory, and there are now "von Neumann algebras." (Please don't ask for elaboration; I am no Martian!) JvN also helped revolutionize ergotic theory, underwrote the logic of quantum mechanics, and was a primary mover in the first atom bombs and other weapons for the US.

He made a signal contribution to economics: game theory, which has been used widely in other areas, including biology, logic (Prisoner's Dilemma), whether to cheat on your income tax, whether to donate to PBS if one watches it, the behavior of the stickleback fish, whether to believe the world is populated by cooperators or defectors, and how to conduct a Cold War under nuclear weapons. (Von Neumann, being persecuted as a child for his Jewishness and being exposed to authoritarianism, then seeing the rise of fascism in Germany, was a right-winger and favored a pre-emptive nuclear strike against the Soviets ASAP. How odd that his Game Theory would lead, as he later came to understand, Mutually Assured Destruction, or MAD, a term JvN allegedly coined.)

He also invented a computer that he called the "Mathematical Analyzer, Numerical Integrator and Computer," or MANIAC.

In the 1950s JvN told the US Senate he was violently anti-communist and much more militaristic than the norm.

Oh, and he designed the basic architecture for computers. I've heard academic philosophers to this day refer to any computing machine as a "von Neumann machine." In the 1940s he invented cellular automata theory, the theory of self-replicating machines. He analyzed the function of self-replication using pen and paper and seemed to prefigure the digital-mechanical structure of the DNA double helix, ten years before Watson and Crick. JvN theorized that the best way to mine an asteroid was to create self-replicating machines. I feel compelled to reiterate that he thought of this idea, and sketched out how it must work...in the 1940s!

Von Neumann loved the US. He saw a practicality of thought, a bulwark for a free world against authoritarian communism and Nazism and other fascisms, and the US as a country that would allow him to maximize his intellectual capabilities. In 1930, when the Institute of Advanced Studies opened near Princeton, the first three geniuses invited to be paid for life and think on anything they wished, were Einstein, Godel, and von Neumann. He and his wife owned the biggest house on the block in the neighborhood near Princeton, and loved to have long parties, sometimes two per week. He was a dapper dresser and once wore a three-piece pinstripe suit on a ride down into the Grand Canyon. Before emigrating to the US, he enjoyed Berlin's 1920s cabaret night life, and was a prodigious drinker of scotch...and could still do amazing arithmetical feats even when drunk. (Or he should have been drunk: he drank tumblers of scotch and never really appeared to be drunk!)  He loved off-color humor, Yiddish, and had what's been described as an Eastern European sense of humor. An example: A convict was playing cards with his jailers and when they found the convict was cheating, they kicked him out.

The virtuoso mathematician Stanislaw Ulam was friends with JvN, and they shared many in-jokes, my favorite being the one about asparagus. Asparagus was a delicacy in 1920s Berlin, and a man in a communal boarding house was eating more than his fair share. Finally, another boarder pointed out that others like asparagus too. Ulam and JvN turned this into a plan to write a twenty-volume treatise on "Asparagetics Through the Ages." What's so funny about this? Ulam and JvN had a code word, "asparagus," for anyone who tried "to obtain an unduly large share of credit for scientific work or any other accomplishment of a joint or group character." (p.221, The Martians of Science, Hargittai.) JvN was very open about his ideas, and never understood the rush to patent.

In Princeton circles, once more with the Martian theme, and according to William Poundstone in his book The Prisoner's Dilemma, von Neumann "was not human but a demigod who had made a detailed study of humans and could imitate them perfectly." A fellow Hungarian "Martian," Theodore von Karman, the father of supersonic flight and the US Air Force, claimed he was descended from Rabbi Loew of Prague, who created The Golem. Von Neumann made similar claims for himself.

This Martian died at the age of 53, in 1957, near Princeton, of cancer. As his mind began to falter, he was, by all accounts in terrifying agony, a screaming, uncontrollable terror. As he neared death, sedated, American guards watched his hospital room, wary that the Martian might babble high-level military secrets, but he began to speak in Hungarian, and the guards didn't understand it. His brother was at his side, and John von Neumann, going out of life on this planet, under heavy painkillers, recited Goethe's Faust from his photographic memory...

There is much, much more to be said about John von Neumann, the enigma. The Martian. But it seems unseemly for blog posts to go on too long...And besides, I'm due back on my own planet.

Reflections on Mathematics

In Eric Temple Bell's classic Men Of Mathematics, first published in 1937, a passage has long puzzled me into wonderment. Near the end of the book, on p.572, Bell writes, "At this point we may glance back over the whole history of mathematics [...] and note two modes of expression which recur constantly in nearly all mathematical exposition." Bell then gives two broad examples of phrases which different mathematicians have used. 

Here's one:

"We can choose a number less than n and greater than n - 2."

The second type of phrase is of this sort:

"There exists a number less than n and greater than n - 2."

Bell insists these two examples are not "merely stereotyped pedantry," and that mathematicians literally mean what they say when they use either formulation, and, even though they may seem different to an inconsequential degree, each phrase is pregnantly revelatory about the kind of philosophical disposition of the thinker. The differences are not "trivial."

Now I quote the section that fills me with a sort of giddy wonder in its implications:

"These two ways of speaking divide mathematicians into two types: the 'we can' men believe (perhaps subconsciously) that mathematics is a purely human invention; the 'there exists' men believe that mathematics has an 'extra-human' existence of its own, and that 'we' merely come upon the 'eternal truths' of mathematics in our journey through life, in much the same way that a man taking a walk in the city comes across a number of streets with whose planning he had nothing whatever to do." - p.573

Far be it for such a horribly rank amateur math enthusiast such as myself to take a side here (and yet just watch as I do take as side: typical? perhaps overweening?); however, the uncanny usefulness of math (I think I'm paraphrasing the great physicist of the quantum Paul Dirac there?) does seem to make one wonder...if, as most of the great mathematicians seemed to think, there exists some sort of Platonic Pure World of math, numbers, form...somewhere? this seems to a person of my own disposition as poetic, romantic, and frankly sort of crazy, math's incredible usefulness notwithstanding. 

What I'm saying is I'm with Bell's "we can"men. I have no proofs that math is a human invention, but I think cognitive scientists like George Lakoff and Rafael Nunez (co-authors of Where Mathematics Comes From) seem probably on the right track. We are incredibly agile symbol-making critters embodied with nervous systems who evolved systems of counting that became more and more diffuse and baroque and abstruse...and wonderful. And aye: useful. At this point in my life, I can't get behind the "there exists" people with their "extra-human existence"s. But I do find it odd in the extreme that so many profoundly great mathematicians have indeed believed in what mathematician-novelist John Casti called "the one true Platonic heaven."

I'd like to be so adept at math that my constructions...sorry!: my findings so enchanted myself that I believed in a Purer World somewhere else. Lord (and Pythagoras?) knows, I'd have great company...

It seems to me that, when I cash out the "we can" vision of how math works, I'm effectively saying what Robert Anton Wilson said of math: it is "pure fiction." 

In this sense, it seems a cousin to the Modernist poet Marianne Moore's definition of poetry: poetry was "imaginary gardens with real toads in them." Because if math is pure fiction, it has functioned as the language that undergirds/underwrites/instantiates all of science and technology, including the magickal gadget you're using to read this right now. 

I find it difficult to wrap my neurons around this weird problem (which is why I return to it in my private thoughts so often: I seem of the temperament of loving NOT being "sure" about anything). Other temperaments surely differ, and I may be too close to this wonderful problem to sort it out adequately, which is truly fine by me. I'm okay with the indeterminacy of it all. I have my guesses, but they may be wrong, or crucially inaccurate in some way. Possibly I lack "imagination" with regard to this vexing and passing strange conundrum (which seems not a conundrum at all for the hard core "we can" and "there exists" folk).

The present blogger-critter has found that any further probes into either basic orientation towards the ontological status of mathematics only yields more uncanny wonderment. Perhaps more on this some other day, but I should like to leave you with this quote from one of the giants of 20th century physics, Paul Dirac (ponder the implications!):


"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. This is only natural and to be expected. What however was not expected by the scientific workers of the last century was the particular form that the line of advancement of mathematics would take, namely it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation."
Paper on Magnetic Monopoles (1931)