Confronting Science ’ s Logical Limits

To anyone infected with the idea that the human mind is unlimited in its capacity to answer questions, a tour of 20th-century mathematics must be rather disturbing. In 1931 Kurt Gödel set forth his incompleteness theorem, which established that no system of deductive inference can answer all questions about numbers. A few years later Alan M. Turing proved an equivalent assertion about computer programs, which states that there is no systematic way to determine whether a given program will ever halt when processing a set of data. More recently, Gregory J. Chaitin of IBM has found arithmetic propositions whose truth can never be established by following any deductive rules. These findings proscribe our ability to know in the world of mathematics and logic. Are there similar limits to our ability to answer questions about natural and human affairs? The first and perhaps most vexing task in confronting this issue is to settle what we mean by “scientific knowledge.” To cut through this philosophical Gordian knot, let me adopt the perhaps moderately controversial position that a scientific way of answering a question takes the form of a set of rules, or program. We simply feed the question into the rules as input, turn the crank of logical deduction and wait for the answer to appear. Thinking of scientific knowledge as being generated by what amounts to a computer program raises the issue of computational intractability. The difficulty of solving the celebrated traveling-salesman problem, which involves finding the shortest route connecting a large number of cities, is widely believed to increase exponentially as the number of destinations rises. For example, pinpointing the best itinerary for a salesman visiting 100 cities would require examining 100 × 99 × 98 × 97 × . . . × 1 possibilities—a task that would take even the fastest computer billions of years to complete. But such a computation is possible—at least in principle. Our focus is on questions for which there exists no program at all that can produce an answer. What would be needed for the world of physical phenomena to display the kind of logical unanswerability seen in mathematics? I contend that nature would have to be either inconsistent or incomplete, in the following senses. Consistency means that there are no true paraConfronting Science’s Logical Limits