Wednesday, November 24, 2010

Sense and self-indulgent nonsense

I recently referred - perhaps unfairly - to the propensity of many mathematicians and scientists to be naive and uncritical when thinking outside their areas of expertise. Of course, we are all inclined to be naive and uncritical at times, but the phenomenon is more striking in a person who exhibits high levels of critical thinking in a specialist area. My original observation was based in part on having read a lot of autobiographical and other material written by very gifted scientists and mathematicians and wondering why I found myself having to 'make allowances' for them.

The distinguished mathematician Gregory Chaitin is a case in point. Here are a few instances of the strange mix of sense and nonsense that flows from Chaitin's pen when he moves beyond the work which has made him famous. [All the quotations are from his book Meta Math! The quest for omega*.]

On medical care. "[I]n my grandmother's generation in the old country, women would have a dozen children, most of whom would die before puberty. So you were trying a dozen mixes of DNA subroutines from both parents. (In the Middle Ages, babies weren't even named til they were a year old, since so many of them would die the first year.) Now instead of trying to keep women pregnant all the time, we depend on massive amounts of medical care to keep alive one or two children, no matter how unhealthy they are. While such medical care is wonderful for the individual, the quality of the human gene pool inevitably deteriorates to match the amount of medical care that is available. The more medical care there is, the sicker people become! The massive amounts of medical care become part of the ecology, and people come to depend on it to survive ... "

On mathematical prehistory. "[F]undamental questions go back millennia and are never resolved. For example, the tension between the continuous and the discrete, or the tension between the world of ideas (math!) and the real world (physics, biology). You can find all this discussed in ancient Greece. And I suspect we could even trace it back to ancient Sumer, if more remained of Sumerian math than the scrap paper jottings on clay tablets that are all we have, jottings that give hints of surprisingly sophisticated methods and a love for calculation that seems to far outstrip any practical application."

Chaitin cannot resist a footnote: "Did Sumer inherit its mathematics from an even older civilization - one more advanced than the ancient Greeks - that was destroyed by the glaciers, or when the glaciers suddenly melted, or by some other natural catastrophe? There is no way for such sophisticated computational techniques to appear out of nowhere, without antecedents."

On ideas and creativity. "Let me describe what it feels like right now while I'm writing this book ... [T]he ideas I'm discussing seem very concrete, real and tangible to me. Sometimes they even feel more real than the people around me. They certainly feel more real than newspapers, shopping malls and TV programs ... In fact, I only really feel alive when I'm working on a new idea, when I'm making love to a woman (which is also working on a new idea, the child we might conceive), or when I'm going up a mountain! It's intense, very intense."

Chaitin elaborates and reiterates with references to beautiful women, art and food (" ... like an amazing ethnic cuisine I've never tasted before."). He continues:

"And I'm a great believer in the subconscious, in sleeping on it, in going to bed at 3 a.m. or 5 a.m. after working all night, and then getting up the next morning full of new ideas, ideas that come to you in waves while you're taking a bath, or having coffee. Or swimming laps. So mornings are very important to me, and I prefer to spend them at home. Routine typing and e-mail, I do in my office, not at home. And when I get too tired to stay in the office, then I print out the final version of the chapter I'm working on, bring it home - where there is no computer - and lie in bed for hours reading it, thinking about it, making corrections, adding stuff.

"Sometimes the best time is lying in bed in the dark with my eyes closed, in a half dreamy, half awake state that seems to make it easier for new ideas, or new combinations of ideas, to emerge. I think of the subconscious as a chemical soup that's constantly making new combinations, and interesting combinations of ideas stick together, and eventually percolate up into full consciousness. That's not too different from a biological population in which individuals ... combine to produce new individuals. My guess is that all this activity takes place at the molecular level - like DNA and information storage in the immune system - not at the cellular level. That's why the brain is so powerful, because that's where the real information processing is, at the molecular level. The cellular level, that's just the front end ...

"Yes, I believe in ideas, in the power of imagination and new ideas. And I don't believe in money or in majority views or the consensus. Even if all you are interested in is money, I think that new ideas are vital in the long run, which is why a commercial enterprise like IBM has a Research Division and has supported my work for so long. Thank you, IBM!"

It takes your breath away, does it not?

* Vintage Books, 2005

Tuesday, November 16, 2010

Through a crystal darkly

In previous remarks on randomness and computation, I mentioned the work of Gregory Chaitin, a mathematician and theorist who has written and spoken (he is a brilliant speaker) extensively for both specialist and general audiences. Chaitin's technical work is highly regarded, but his interpretations and extrapolations are sometimes a little idiosyncratic and he is inclined to sound a bit New Agey at times. (He is rumored to receive help in his thinking from a giant crystal!)

Paul Davies (a physicist and writer) is, by contrast, sober and restrained - even a little pedestrian by comparison - but he is a reliable guide within his areas of expertise. I recently came across a foreword by Davies to a book of Chaitin's essays* in which Davies gives his perspective on the significance of Chaitin's work and its implications for physics and our view of the world generally.

Chaitin (who had been obsessed from his childhood years with Kurt Gödel's incompleteness theorem) "greatly extended the scope of Gödel's basic insight," writes Davies, "and recast the notion of incompleteness in a way that brings it much closer to the real world of computers and physical processes. A key step in his work is the recognition of a basic link between mathematical undecidability and randomness. Something is random if it has no pattern, no abbreviated description, in which case there is no algorithm shorter than the thing itself which captures its content. And a random fact is true for no reason at all; it is true 'by accident' so to speak ... Chaitin was able to demonstrate that mathematics is shot-through with randomness ... Mathematics, supposedly the epitome of logical orderliness is exposed as harboring irreducible arbitrariness." (p. vi)

"[M]athematics contains randomness - or accidental, reasonless truths," Davies explains, "because a ... universal Turing machine [an idealized computer], may or may not halt in executing its program, and there is no systematic way to know in advance if a function is computable (i.e. the Turing machine will halt) or not." (p. viii)

But this limitation on what we can know or predict (known as Turing uncomputability) applies not just to mathematics and computers but also to scientific theories. On Chaitin's view, a scientific theory is like a computer program that predicts our observations (the experimental data).

Indeed, in the words of Paul Davies, " ... we may regard nature as an information processing system, and a law of physics as an algorithm that maps the input data (initial conditions) into output data (final state). Thus in some sense the universe is a gigantic computer, with the laws playing the role of universal software." (p. viii)

And if the laws of physics are computer algorithms, there will be randomness in the laws of physics stemming from Turing uncomputability. But, according to Davies, the randomness will, in reality, be "even more pronounced than that which flows from Turing uncomputability." (p. viii)

He points out that the real universe differs in a crucial respect from the concept of a Turing machine. "The latter is supposed to have infinite time at its disposal: there is no upper bound on the number of steps it may perform to execute its program. The only relevant issue is whether the program eventually halts or not, however long it takes. The machine is also permitted unlimited memory ... If these limitless resources are replaced by finite resources, however, an additional, fundamental, source of unknowability emerges. So if, following Chaitin, we treat the laws of physics as software running on the resource-limited hardware known as the observable universe, then these laws will embed a form of randomness, or uncertainty, or ambiguity, or fuzziness - call it what you will - arising from the finite informational processing capacity of the cosmos." (pp. viii-ix)

There are, it seems, different forms or levels or randomness. The 'mild' form which - as chaos theory shows - is implicit even in classical, deterministic physics; the pseudo-randomness which can be generated by simple computer algorithms; the well-known randomness inherent in quantum mechanics; and perhaps the deepest levels of all stemming from proven features of idealized computers (Turing machines) and from seeing the universe itself as a giant computer - one with specific limitations on its processing capacities.

These are difficult (and to some extent speculative) ideas. But I think they are worth pursuing and may even have profound implications for how we see ourselves and our world.

It is, of course, impossible to draw definitive political or metaphysical conclusions from them, but, if the ideas are sound, there will be such conclusions to draw.

Let me just mention two thoughts which come immediately to mind: Chaitin's and Davies' notions are utterly incompatible with any political ideology which attempts to predict, plan and control human affairs; and they also appear to undermine perspectives which incorporate notions of a providential force operating behind the scenes and impinging on natural processes, historical events and/or individual destinies. 

* Thinking about Gödel and Turing: essays on complexity, 1970-2007 (World Scientific, 2007).

Tuesday, November 9, 2010

Lessons of the masters

The French expression, maître à penser, has no English equivalent. A 'thinking master' is what I have always wanted and never found. Perhaps wisdom is unstable and only exists fleetingly in an action here or a thought there. A strange thing, the desire for discipleship (to be a disciple - not to have them). It may reveal deep psychological flaws, but I think not. In my case it reflects simply a combination of a desire to know and a certain laziness. (Why should I do all the work?)

Some years ago, George Steiner gave a series of lectures (which became a book*) on the topic of masters and disciples. Most of the relationships he describes end badly, by the way, not unlike love affairs.

The lessons I have learned from my hoped-for masters have pretty well all been negative, and the thinkers I have flirted with have all been seriously flawed in one way or another. Arts and humanities-oriented writers and scholars are too often ignorant or (worse) scornful of scientific knowledge. Scientists and mathematicians, on the other hand, are often amazingly uncritical in non-scientific areas, and especially in social and political contexts.

It is not surprising, perhaps, that politics works as it does, catering to the lowest common denominator, that legislators lack vision or that government debt is spiraling out of control in so many jurisdictions.

But I continue to be amazed when intellectuals - as happens all too often - align themselves with religions or discredited ideologies - the desire for discipleship trumping the desire for truth.

* Lessons of the masters (Harvard University Press, 2003).

Tuesday, November 2, 2010

No sense of place

I have recently been reading some Patricia Highsmith novels* from the 1950s and 60s. Three communication media - the old-fashioned letter, the (usually local) newspaper and the telephone - all play very significant roles in these stories. (There is also the occasional telegram - or cable - and books also appear.)

Highsmith's characters spend a large proportion of their allotted pages planning and writing letters, posting letters, organizing the material for writing more letters, waiting for letters and speculating as to why no letter has come or, more rarely, receiving a letter and analysing the contents. The local newspaper is good for keeping track of whether the body has been found or what stage the police have reached in their investigation. And the telephone looms as large as in the movies of the period.

In their way, each of these media enhances the sense of place and/or the sense of distance from other places. Even the telephone signals the sense of distance by the involvement of an operator.

Patricia Highsmith's world may not be the real world of the 1950s and 1960s - it is a slightly claustrophobic and chilling distillation of reality - but it reflects important truths about the crucial role communication technologies play in weaving a cultural milieu and defining a locality.

As traditional letters disappear from the communication landscape, as print is replaced by digital devices, and the telephone operator is remembered only in the "Operator! Operator!" of old films and TV, we are inexorably losing our sense of place.

*The blunderer, This sweet sickness, Those who walk away and The tremor of forgery.