The significance of the conceptual revolution in science derives less from the field models themselves than from their philosophical and epistemological implications. It is what they imply not only about the nature of the world, but about how one interacts with the world, that is important in understanding how the new view differs from the older, atomistic perspectives. One of the most important of these implications is that the Cartesian dichotomy between the res cognitans and the res extensa, the thinking mind and the physical object, is not absolute, but an arbitrary product of the human mind. Classical physics assumed that it was possible to make a rigorous separation between the observer and what she or he observes. Relativity theory and, in a different way, quantum mechanics require that the separation into an observer and a physical system be regarded as an arbitrary distinction entailing approximations that are not always negligible.

The breakdown of the Cartesian dichotomy also has methodological implications. When things are thought to exist “out there,” separate and distinct from the observer, the world has already been divided into two parts. The next step is to subdivide it further by regarding the exterior world also as a collection of parts. The parts, because they are intrinsically separate and individual, can then be analyzed sequentially as individual units; this is of course how Aristotelian logic proceeds. As long as the world is conceived atomistically, this approach is appropriate and, at least in theory, exact to any desired degree of accuracy. But the field concept has the effect of revealing limitations in sequential analysis. These limitations are especially likely to appear when the whole is (or can be considered as) a part of itself.

For example, consider a set φ = {a, b, c, d, . . .}. In this example there is no problem in regarding the set φ as the whole, and each of the elements a, b, c, d as parts of that whole. But now imagine a set a = {a, b, c, d, . . .}. From one perspective a is the whole itself, the entire set of elements enclosed within brackets. But from another perspective, a is a part of the whole, that is, one of the elements within the set. This problem is typical of paradoxes that arise from the field concept; it reveals an essential fallacy in the assumption that a whole can always be adequately defined as the sum of its parts. When classical, sequential analyses are applied to situations of this kind, paradoxes can become irresolvable antinomies.

I should like to turn now to more precise terminology and examine in some detail two examples in which the appearance of this kind of ambiguity proved to be decisive. In both cases, the paradoxes were revealed as a result of ambitious programs to extend the domain of classical analysis: in mathematics, the formalist program to prove that mathematics was free from contradiction; and in the philosophy of science, the positivist program to create an exact, objective language for science. These first examples are meant to convey a sense of how the generalizations I have been making about the field concept translate into specific examples from science. It is possible to see in them intimations of the complexities symbolized by the cosmic web.

In the early part of this century, the German mathematician David Hilbert suggested that it should be possible to prove that mathematics is free of contradictions by formalizing, one by one, the axiomatized theories of mathematics. Ernst Snapper, in a prize-winning article on the philosophical roots of mathematics,1 explains that to “formalize” an axiomatized theory T means (confining ourselves to first-order examples) to choose a first-order language L so that all of the undefined terms that appear in the axioms of T can be expressed through parameters of L. It is then possible to express in L all the axioms, definitions, and theorems of T, as well as all the axioms of classical logic. In this approach, one manipulates the symbols of L by means of exact syntactical rules, without necessarily being concerned about the content of the symbols. The advantage of creating the language L is that L can then be studied as a mathematical object in itself, independent of the content of T. Hilbert hoped that a theory T could be proved free of contradiction by demonstrating that all of the allowable syntactical combinations of L were free of contradiction.

At the heart of this formalist program is the attempt to create a vantage point from which one could talk about mathematics as an object in a language that would not be contaminated with what it was one wished to prove. The Hilbert program rested on the assumption that it is possible to make a rigorous separation between the theory and the theory-as-object.

The hope that this strategy would succeed was shattered in 1931 with the publication of Kurt Gödel’s paper, “Formally Undecidable Propositions in Principia Mathematica and Related Systems.”2 In this paper Gödel proved that for the mathematical system of the Principia, or more generally for any axiomatized theory with axioms strong enough so that arithmetic can be done in terms of them, the theory either will be inconsistent or will contain propositions whose truth cannot be demonstrated. Since inconsistencies are naturally to be avoided, mathematics finds itself impaled on the other horn of the dilemma; that is, it will contain propositions that cannot unambiguously be proven to be either true or false.

Formally undecidable propositions had long been known and formulated through various paradoxes. One classic illustration is as follows. On the first side of a piece of paper write the words “The statement on the other side is true.” Now turn the paper over and write “The statement on the other side is false.” Let us consider first Side 1 asserting that Side 2 is true. If Side 2 is true, however, then Side 1 is false. But if Side 1 is false, then Side 2 is not true, in which case Side 1 is true. One can pursue this line of reasoning forever without being able to reach a conclusive answer. The two statements together involve what Douglas Hofstadter calls a “Strange Loop,”3 a loop of reasoning that cannot be resolved because to accept either statement as true is to begin a loop which circles around to say that the same statement must be false. It is obvious such statements can be neither true nor false; they are inherently undecidable.

One way to analyze a Strange Loop is to consider it as a problem in self-reference. Each statement points to the other, and the other in turn points back, so that there is no independent vantage from which to evaluate either one. The Hilbert program had hoped to avoid this problem by separating the language L from the theory T. But this hope proved to be unfounded when Gödel demonstrated that it was possible to talk about number theory from within the theory itself. The problem of self-reference was thus revealed as unavoidable. Douglas Hofstadter explains:

What happens if one takes the statements one cannot prove and converts them to axioms? (Axioms, of course, are unproven statements.) In this case one has generated a new theory, because the set of axioms has changed; and in this new theory, new statements will arise that cannot be proven within that system. If these new statements are in turn converted into axioms, still other statements will arise elsewhere in the system that cannot be proven under those axioms. The process is interminable.

The implication of Gödel’s theorem, then, is that any theory that is not demonstrably false cannot be demonstrated to be completely true. Thus the program to prove all of mathematics true did not succeed. This does not necessarily mean that mathematics is false, of course— only that it cannot be proven true. The crux led Hermann Weyl to say that God must exist because mathematics is intuitively consistent, and the devil exists because it cannot be proven to be consistent. Whatever intuitive consistency one may grant mathematics, however, the inability to prove the truth of number theory is significant, for it reveals that even in mathematics, the most exact of the sciences, indeterminacy is inevitable.

Nor, it turns out, is this indeterminacy confined to axiomatic mathematics. It also appears in computation theory, in a problem that Martin Davis calls the Halting Problem.5 The question that the Halting Problem asks is whether it is possible to determine in advance if a computer will be able to find a definite answer—that is, come to a halt—for any given problem.6 The question has practical importance, for if it cannot be answered, one can suddenly find one’s computer involved in a Strange Loop of its own, which consumes expensive computer time and, in extreme cases (as in the infamous “page fault” error), renders the program useless. The answer to the Halting Problem, Davis explains, is no: there will be some computations which cannot be proven in advance either to have a solution or not to have a solution, in much the same way that the Incompleteness Theorem says that there are some statements within number theory which cannot be proven to be true or false. In fact, Davis shows how Gödel’s theorem (the Incompleteness Theorem) can be restated in terms of the Halting Problem, so that if the Halting Problem had a solution, the Incompleteness Theorem could not be true. Therefore, since the Incompleteness Theorem is true, the Halting Problem will not have a solution. The important point is that certain kinds of logical problems have no solution, not even using the most sophisticated computers imaginable. Davis makes this point explicitly: “Note that we are not saying simply that we don’t know how to solve the problem or that the solution is difficult. We are saying: there is no solution”7

What the Incompleteness Theorem does in mathematics, and what the Halting Problem does for the linear sequences of binary choices that comprise computer programs, is to imply that certain limitations in linear analysis are inescapable because of the problem of self-reference. It is because the tools for analysis are inseparable from what one wants to analyze that Strange Loops appear. In these examples, problems that cannot be solved through logical analyses appear as a result of considering both the tools for analysis, and the object to be analyzed, as part of the same “field.” They illustrate one way in which the emergence of...