The Unknown and The Unknowable

Joseph Traub [3.10.97]
Topic:

"Science is about understanding the universe and everything in it. Examples of scientific questions are: Will the universe expand forever, or will it collapse?; Will there be major global changes due to human activities, and what will be the effects on earth's ocean levels, and on agriculture and biodiversity? Note that there are, a priori, no mathematical models that accompany these questions. Science uses mathematics, but it is also very different from mathematics. Can we up the ante from mathematics and prove impossibility results in science?."

Introduction
By John Brockman

Starting in 1959, Joseph Traub pioneered research in what is now called "information-based complexity". Computational complexity theory studies the intrinsic difficulty of solving mathematically posed problems; it can be viewed as the thermodynamics of computation. "Information-based complexity" studies the computational complexity of problems with only partial or contaminated information. Such problems are common in the natural and social sciences and he is applying "information-based complexity" to a wide range of problems Other work ranges from new fast methods for pricing financial derivatives to investigating what is scientifically knowable.

—JB


 JB: JB: To what extent is your interest in the limits of scientific knowledge influenced by the work of Gšdel?

TRAUB: In 1931 a logician named Kurt Gšdel announced a result that astonished the scientific world. Gšdel said that there are statements about arithmetic that can never be proved or disproved. This impossibility result is about elementary arithmetic, not some arcane corner of mathematics. To the educated lay person, Gšdel's undecidability theorem may be the single most widely-known mathematical result of the 20th century.

Gšdel's theorem is just one of numerous impossibility results established in the last 60 years stating what cannot be done. Another famous negative result, due to the British genius, Alan Turing, states that you cannot tell in advance if a certain abstraction of a digital computer called a Turing machine will ever halt with the correct answer. Now, what all these impossibility results have in common is that they are about the manipulation of symbols, that is, they are about mathematics.

I've spent some of my time for most of a decade asking myself what does this tell us about the unknowable in science. Indeed, the first time that I spoke publicly about this subject was on February 1, 1989 at a panel discussion in memory of Heinz Pagels organized by John Brockman.

Science is about understanding the universe and everything in it. Examples of scientific questions are: Will the universe expand forever, or will it collapse?; Will there be major global changes due to human activities, and what will be the effects on earth's ocean levels, and on agriculture and biodiversity? Note that there are, a priori, no mathematical models that accompany these questions. Science uses mathematics, but it is also very different from mathematics. Can we up the ante from mathematics and prove impossibility results in science?

Ralph Gomory, the President of the Alfred P. Sloan Foundation, proposes a tripartite division of science: the known, the unknown, and the unknowable. The known is taught in the schools and universities and is exhibited in the science museums. But scientists are excited by the unknown. Parenthetically, artists go to art museums to learn; scientists do not go to science museums because those museums act as if it's all known and preordained. That may be changing; exemplars are the Exploratorium in San Francisco and the American Museum of Natural History.

Gomory's tripartite division proposes three distinct areas: the known, the unknown which may someday become known, and the unknowable, which will never be known. The unknown and the unknowable form the boundary of science. Here are examples of questions for which the answers are today unknown.

How do physical processes in the brain give rise to subjective experience? That is, explain consciousness. Can the healthy, active lives of humans be significantly prolonged by, say, a factor of two or three? How did life originate on earth? Will the universe expand forever, or will it collapse? Can we develop a grand unified theory of the fundamental physical laws? Why do fundamental constants, such as the speed of light, have their particular values? Is there life elsewhere in the universe? Is it intelligent? How do children acquire language?

For which of these are the answers unknowable? We cannot prove scientific unknowability. That can only be done in mathematics. This is sometimes not understood, even by professionals. I expressed my interest in the unknowable to a very senior European scientist. He immediately responded that this had been, of course, settled by Gšdel's theorem. Not so; Gšdel's theorem limits the power of mathematics and does not establish that certain scientific questions are unanswerable.

 

  What are some of the reasons why a scientific question might be unanswerable? I'll limit myself to just three here. The first is that insufficient data has survived. That can be a problem in ur-linguistics, archaeology, and history. The second is that contingent events, sometimes called frozen accidents, may limit our ability to explain certain phenomena. (On the other hand, as Stephen Jay Gould eloquently argues, historical explanations in science can be as convincing as those arising from general theories.) Finally, resources, such as energy, may simply not be available in our part of the universe to discriminate among contesting theories about the universe.

Of course we must be very careful in stating that something is impossible or unknowable. We're all familiar with some of the notorious announcements concerning impossibility, such as, there cannot be a heavier-than-air flying machine.

The unknowable has long been the province of philosophy and epistemology, with questions raised by giants such as Immanuel Kant and Ludwig Wittgenstein. My goal is to move the distinction between the unknown and the unknowable from philosophy to science and thereby enrich science.

What is the basis for my belief that we might succeed? The Zeitgeist seems right for tackling such questions. We have had great success in establishing impossibility results in mathematics and theoretical computer science. Although these ideas cannot be directly applied to science, I'm hopeful that the modes of thought might be transferable. Recent workshops at the Santa Fe Institute have brought together leading physicists, economists, cognitive scientists, biologists, computer scientists, and mathematicians who have strong interests in defining the unknowable in their own fields.

JB: What kind of predictive models will you use?

TRAUB: A central issue is the relation between reality and models of reality. I like to talk about this in terms of four worlds. There are two real worlds: the world of natural phenomena and the computer world, where simulations and calculations are performed. There are two model worlds: a mathematical model of a natural phenomenon and a model of computation. The mathematical model is an abstraction of the natural world while the model of computation is an abstraction of a physical computer. Incidentally, although most people are only aware of the Turing machine as the abstract model of computation, the real-number model is probably more appropriate for the continuous models of science, but that's a story for another day.

I'll give you one concrete example concerning reality and models of reality. I wrote about this in "On Reality and Models" which is a chapter in the recent book, "Boundaries and Barriers" edited by John Casti and Anders Karlquist, and which is also a Santa Fe Institute report.

All living matter is built out of proteins, and these proteins fold effortlessly. It takes nature less than a second. But even with the most powerful supercomputers, we cannot simulate protein folding, and theory suggests that the problem is what is technically called NP-complete, meaning it's conjectured to be computationally intractable. Why is there this dissonance between our models and reality? I'll confine myself here to just one possible explanation. Nature doesn't fold arbitrary molecules; the molecules that exist in nature have been selected by evolution for ease in folding. But in our theory we don't know how to model this selection.

Trying to understand the relation between reality and our perception of reality is an old issue. Some 200 years ago Immanuel Kant believed that three space dimensions and one time dimension has more to do with our brains than with "reality". Niels Bohr and Albert Einstein argued about this. Bohr said all I want from a mathematical model is the ability to predict and I don't know or care about "reality". Einstein felt that there is a reality which our theories describe. The debate continues today between Stephen Hawking and Roger Penrose with Hawking taking Bohr's view and Penrose taking Einstein's.  

  JB: Talk about the end of science versus the limits of science.

TRAUB: I assume that you're referring to John Horgan's book, "The Ends of Science". John sent me the manuscript last fall for my comments. I suggested some minor technical corrections and told him I totally disagreed with his thesis that science had made such extraordinary progress that its golden age was over and only mopping up was left. Incidentally, the manuscript was titled "The Ends of Science", which is an ambiguous and far more interesting title. Apparently the publisher changed that to "The End of Science", hoping to derive some advantage from the success of books titled The End of You Name It, starting with Francis Fukuyama's foolish "The End of History".

John writes very well indeed; he is a senior writer for "Scientific American", and his book features juicy anecdotes about many scientists who are household names. However, I never would have predicted the amount of media attention that the book has actually received. Its message is basically pessimistic. For example, a column in the New York Times stated that Horgan found in his interviews with some of today's leading scientists an atmosphere of anxiety and melancholy and little acknowledgment that the great era of scientific discovery is over.

Those are not the emotions of the scientists that I know. The ones with whom I'm in touch are vitally excited by their work. There's more to be done than ever, and we can't wait to get on with it.

I'm not saying that there aren't difficulties. Funding for research has leveled off and will probably decrease. Universities don't have tenure positions available for young scientists. The emphasis in some of the leading corporate laboratories has shifted away from basic research. The Federal laboratories are in turmoil due to budget cuts and re-direction. But such difficulties are to be expected after the period of unparalleled growth which followed the Second World War. Horgan is claiming it's all over because the fundamental discoveries have been made.

Earlier I mentioned a very partial list of big scientific questions. Let me repeat a couple of items from that list: How do physical processes in the brain give rise to subjective experience? That is, explain consciousness. Is there life elsewhere in the universe? Is it intelligent? Will the universe expand forever or will it collapse?

I don't find John Horgan's thesis, that all the important discoveries are behind us, very compelling. Furthermore, each major advance leads to important new questions. Reports of the death of science have been greatly exaggerated. Indeed, I believe they're just plain wrong.

JB: What are the great unknowables that you see worthy of study?

TRAUB: Remember that we have to distinguish between the unknown and the unknowable. There's a very big list of things we don't know. Which important questions might be unknowable? I'll mention just four of them here. Although what I'd said earlier about mathematical models is solidly grounded, what I'll now say is highly speculative. I hope the readers of this interview will be forgiving.

The first has to do with earth systems predictability. There are many interesting questions here. Some people believe earthquakes, like Per Bak's sandpile, are a self-organizing phenomenon and are intrinsically unpredictable. On the other hand, my colleague Lynn Sykes at Lamont Doherty Earth Observatory believes that though we probably cannot predict two to three earthquakes out, we can predict the next big one in a geographic area. Of course, warning a population that a big one is coming has major social consequences. Another example of a question with enormous implications for humans is whether there will be major global changes due to human activities. What will be the effects on earth's ocean levels, and on agriculture and biodiversity?

A second is the likelihood of other intelligent life in the universe. Stephen Jay Gould argues that it's extremely improbable because so many unlikely accidents have to happen. He believes if the tape were to be replayed, we would not have evolved. My Santa Fe Institute colleague Stuart Kauffman argues that order comes for free and that we are "at home in the universe". Other than by discovering intelligent life forms elsewhere, can this question be resolved?

A third is whether we can understand consciousness. Some believe this to be unknowable while others believe the answer can be found within science. I have a 16-month old grandson and as I was watching him while he was still pre-verbal I wondered about his cognitive learning processes. He sure was busy crawling, looking and manipulating but what was going on inside? And what did he learn in the fetal stage?

My final question is whether we can keep the U.S. economy, let alone the international economies with which we are so interdependent, on an even keel. Basically, things have been pretty good since the Great Depression. Right now, it looks good. But we have coupled nonlinear systems and we know that such systems often exhibit chaos. Can the system be kept going for a "long" period of time without dreadful economic disasters? How might this be achieved?

There are so many wonderful questions. I guess I've been very lucky. I got interested in computers over forty years ago and I keep expecting to run out of interesting questions. But I never have to strip-mine. I just walk along and pick up diamonds.