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Self-reference immediately brings to mind the self-referring paradoxes such as “this sentence is false,” or infinite regressions such as “the class of all classes.” However, not all self-reference is paradoxical and not all self-reference leads to infinite regress, and I will return to the topic of meaningful self-reference shortly. The puzzle of the logical paradoxes is vastly overrated. A paradox is no more complex than a simple doorbell buzzer or oscillator. An oscillator is a system which has the characteristic that being in one state causes a change to occur which brings about another state and vice versa, so that the system continues to alternate between the two states. A paradox is simply a logical oscillator, and there is no reason to try to legislate it out of existence simply because we find it annoying and cannot yet find a meaningful use for it within contemporary logic.
A stable oscillator is a system that uses the principle of negative feedback to maintain its own state within a certain range, and such negative feedback mechanisms are usefully embodied in all purposive living systems. The difficulty with paradox is that the range of its oscillation, truth to falsity, is so wide as to make it apparently meaningless. If the paradox's range of oscillation could be reduced from assertions of “truth” to probability of truth, it might become meaningful and no longer “paradoxical.” This seems to agree with Popper's (1962) statement that paradoxes are avoided when we realize our fundamental inability to assert that any statement is true or false:
We can avoid paradoxes without using such drastic measures ... [as strictly excluding all self-reference which] would exclude some very interesting uses of self-reference, especially Godel's method of constructing self-referring statements, a method which has most important applications in my own field of interest, the theory of numbers. They are drastic, moreover, because we have learned from Tarski that in any consistent language—let us call it “L”—the predicates “true in L” and “false in L” cannot occur (as opposed to “meaningful in L,” and “meaningless in L” which may occur), and that without predicates such as these, paradoxes of the type of the Epimenides, or of Grelling's paradox of the heterological adjectives, cannot be formulated. This hint turns out to be sufficient for the construction of formalized languages in which the paradoxes are avoided. (p. 310)
On this basis, “paradoxical” statements would be more analogous to real oscillators. In direct current circuitry, electrical oscillators are useful only for ringing doorbells, but they are tremendously useful in electronics, which is based on alternating (oscillating) current. Perhaps paradoxical self-referring statements will have a fundamental role in the development of a probabilistic logic which avoids the illegitimate postulates of absolute truth and falsity.
As I mentioned above, not all self-reference leads to paradox or infinite regress, and a simple meaningful example is the sentence “this statement is true.” Another meaningful example is the preceding sentence which refers to itself indirectly by referring to a previous sentence (marked with an asterisk * above) which in turn refers to it. The self-reference in these examples corroborates and reinforces the statement itself, but obviously does not prove it. Self-reference, usually implicit or indirect, is very common, and when this self-reference is not recognized, it is often the source of meaningless circular “proofs.” This is particularly true in a discussion of determinism, since in any meaningful discussion a number of rules are assumed, including not only logic, grammar, etc., but also at least a limited determinism. I assume that the words I use will retain the same meaning after I have written them, that they will evoke essentially similar responses in the reader, that the chain of logical arguments will determine the proper and “true” conclusion, etc. In order to avoid the trap of attempting to prove our assumptions, it is necessary to examine and keep in mind some of the assumptions and limitations of the tools that we use to gain knowledge: logic, the scientific method, and our ultimate and omnipresent tool, the human brain. Logic, mathematics, and the scientific method have been enthroned in the past as absolute principles, but the far-reaching mathematical theorems of Godel and others have considerably diminished this absolutism. The following extensive quotation is from Bronowski's (1966) excellent discussion of these explicitly self-referring theorems and their implications:
[Godel's] first theorem says that any logical system which is not excessively simple (that is, which at least includes ordinary arithmetic) can express true assertions which nevertheless cannot be deduced from its axioms. And the second theorem says that the axioms in such a system, with or without additional truths, cannot be shown in advance to be free from hidden contradictions. In short, a logical system which has any richness can never be complete, yet cannot be guaranteed to be consistent... A. M. Turing in England and Alonzo Church in America showed that no mechanical procedure can be devised which could test every assertion in a logical system and in a finite number of steps demonstrate it to be either true or false... Alfred Tarski in Poland proved an even deeper limitation of logic. Tarski showed that there can be no precise language which is universal; every formal language which is at least as rich as arithmetic contains meaningful sentences that cannot be asserted to be either true or false...
Such a system of axioms has always been thought to be the ideal model for which all science strives. Indeed, it could be said that theoretical science is the attempt to uncover an ultimate and comprehensive set of axioms (including mathematical rules) from which all the phenomena of the world could be shown to follow by deductive steps. But the results that I have quoted, and specifically the theorems of Godel and of Tarski, make it evident that this ideal is hopeless. For they show that every axiomatic system of any mathematical richness is subject to severe limitations, whose incidence cannot be foreseen and yet which cannot be circumvented. In the first place, not all sensible assertions in the language of the system can be deduced (or disproved) from the axioms: no set of axioms can be complete. And in the second place, an axiomatic system can never be guaranteed to be consistent: any day, some flagrant and irreconcilable contradiction may turn up in it. An axiomatic system cannot be made to generate a description of the world which matches it fully, point for point; at some points there will be holes which cannot be filled in by deduction, and at other points two opposite deductions may turn up.
I hold, therefore, that the logical theorems reach decisively into the systemization of empirical science. It follows in my view that the unwritten aim that the physical sciences have set themselves since Isaac Newton's time cannot be attained. The laws of nature cannot be formulated as an axiomatic, deductive, formal and unambiguous system which is also complete. And if at any stage in scientific discovery the laws of nature did seem to make a complete system, then we should have to conclude that we had not got them right. Nature cannot be represented in the form of what logicians now call a Turing machine-that is, a logical machine operating on a basic set of axioms by making formal deductions from them in an exact language. There is no perfect description conceivable, even in the abstract, in the form of an axiomatic and deductive system. Of course, we suppose nevertheless that nature does obey a set of laws of her own which are precise, complete and consistent. But if this is so, then their inner formulation must be of some kind quite different from any that we know; and at present, we have no idea how to conceive it. Any description in our present formalisms must be incomplete, not because of the obduracy of nature, but because of the limitation of language as we use it. And this limitation lies not in the human fallibility of language, but on the contrary in its logical insufficiency.
This is a cardinal point: it is the language that we use in describing nature that imposes (by its arrangement of definitions and axioms) both the form and the limitations of the laws that we find. For example, it may be held that if we can remove the arithmetic from physics, we may yet get an axiomatic system which is complete and consistent. I do not share this view, but it is arguable; yet it does not seem to me to bear in fact on our present formulation of the laws of nature. On present evidence, we must conclude (in my view) that the human mind is constrained to conceive physical laws in arithmetical language: the whole numbers are literally an integral part of its conceptual apparatus. If this is so, then the mind cannot extricate the laws of nature from its own language; and we are not at all, as Leibnitz and others have thought, in a “pre-established harmony” with the language of nature. (pp. 3-5)
There is massive evidence from joint efforts in neurophysiology, cybernetics, and mathematics (George, 1965; McCulloch, 1965; Miller, Galanter and Pribram, 1960) to corroborate the view that the functioning of the human brain can be characterized by the principles of logic and mathematics. On the psychological level the evidence is less complete, but also corroborative. Piaget's extensive work (1954, 1957) has shown that the child's sensory-motor and conceptual development can be understood using the mathematical concepts of sets, groups, and operations, and that a child's “irrational” behavior can be understood as the result of incomplete or inappropriate application of rational principles. In addition, much psychopathology can be understood as the unfortunate consequence of deductions based upon incorrect or contradictory assumptions which the patient will not revise, or as the consequence of entrapment in paradoxical situations. (Watzlawick, 1964, 1967)
If the human mind and brain do work according to the principles of logic and mathematics, then all the limitations discussed by Bronowski must apply to all human thought. Because of its logical insufficiency, thought can never be complete or consistent and it can never give us a complete or consistent picture of nature. If nature does obey a complete and consistent set of laws, it must be of a kind basically different from any we know, and fundamentally different from, and probably inaccessible to, human conceptual processes. “We have to remember that what we observe is not nature in itself, but nature exposed to our method of questioning.” (Heisenberg, 1958, p. 58)
I have said that the human nervous system obeys the principles of logic and mathematics, but it might be more legitimate to say the reverse, since the structures existed prior to the principles. This suggests that the fundamental principles and hypotheses of scientific knowledge are symbolic replicas of the functioning of the human nervous system. As such, they can be examined in perspective: not as absolute principles, but as adaptive strategies with a certain degree of success in dealing with nature. There is abundant evidence that the human brain is the product of a process of biological evolution in which genetic mechanisms determine structures which embody hypotheses about the organism's surroundings. Those hypotheses that are not appropriate are discarded, while those that have some validity are retained and refined.
Since the human brain is a successful outcome of over a billion years of biological hypothesis testing, we can reasonably expect that logic is a successful tool for knowing and dealing with reality. However, if we examine other organisms we find that while each one embodies hypotheses about its surroundings, every one of these hypotheses is approximate and has only a limited application to particular domains, under particular circumstances, and often with only a small statistical advantage over the hypotheses embodied in other organisms. Since logic is a product of this same process, it would be unlikely, though possible, that it alone did not share these limitations. The following quotation is from Popper (1962), discussing the growth of scientific knowledge. Note that if “conjectures” is replaced by “genetically determined structures,” and “refutations” is replaced by “natural selection,” the quotation serves as an excellent characterization of biological evolution:
The way in which knowledge progresses, and especially our scientific knowledge, is by unjustified (and unjustifiable) anticipations, by guesses, by tentative solutions to our problems, by conjectures. These conjectures are controlled by criticism; that is, by attempted refutations, which include severely critical tests. They may survive these tests, but they can never be positively justified: they can neither be established as certainly true nor even as “probable” (in the sense of the probability calculus). Criticism of our conjectures is of decisive importance: by bringing out our mistakes it makes us understand the difficulties of the problem which we are trying to solve. This is how we become better acquainted with our problem, and able to propose more mature solutions: the very refutation of a theory—that is, of any serious tentative solution to our problem—is always a step forward that takes us nearer to the truth. And this is how we can learn from our mistakes.
As we learn from our mistakes our knowledge grows, even though we may never know—that is, know for certain. Since our knowledge can grow, there can be no reason here for despair of reason. And since we can never know for certain, there can be no authority here for any claim to authority, for conceit over our knowledge, or for smugness.
Those among our theories which turn out to be highly resistant to criticism, and which appear to us at a certain moment of time to be better approximations to truth than other known theories, may be described, together with the reports of their tests, as “the science” of that time. Since none of them can be positively justified, it is essentially their critical and progressive character—the fact that we can argue about their claim to solve our problems better than their competitors—which constitutes the rationality of science. (p. vi)
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