Confronting Science's Logical Limits; October 1996; Scientific American Magazine; by Casti; 4 Page(s)
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 G¿del 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.