Godel’s Incompleteness Theorem-Reciprocal Movement

Between Syntax And Arithmetic-Godel’s Incompleteness Theorem-Reciprocal Movement (RM)

Foundational Crisis In Mathematics


Divergencies, convergencies and, at times, insurmountable paradox, has marked the history of mathematics. A major discontinuity in the history of mathematics occurred in the l880’s when Georg Cantor developed his theory of sets. Cantor developed a diagonal method from which he worked out the mathematics of infinities. The essence of this method is, according to Douglas Hofstadter, “the fact of using one integer in two different ways--or, one could say, using one integer on two different levels--thanks to which one can construct an item which is outside of some predetermined list.” [Douglas R. Hofstadter, Godel, Escher, Bach: an Eternal Golden Braid, 1979, p. 423]

The diagonal method developed by Cantor was not the problem, rather, the controversy sprang from what Cantor was able to prove with this method. For instance, in his investigations into non-denumerable sets Cantor was able to prove:

“Any line segment, no matter how small, contains as many points as an infinite straight line. Further, the segment, contains as many points as there are in an entire plane, or in the whole of space of n dimensions (where n is any integer greater than zero) or finally in a space of denumerable infinite number of dimensions.” [E.T. Bell, Men Of Mathematics, 1937, p571]

By bringing into question fundamental axioms such as: every magnitude is equal to itself; the part is less than the whole; two magnitudes, equal separately to a third, are equal to each other; etc., Cantor’s transfinite mathematics brought the very foundation of mathematics into question.

Three foundational schools of mathematics arose in response to Cantor’s transfinite mathematics. Each produced their own point of view concerning the nature of number. Bertrand Russell advocated that mathematics could be reduced to a system of logic while the school of thought representing intuitionalism accepted number as a “given”. Kronecker, a spokesperson for the intuitionalist school, is reported to have said, “The whole numbers were made by the good Lord, everything else is the work of man.” [A.S. Luchins and E.H. Luchins, Logical Foundations Of Mathematics For Behavioral Scientists, 1965, p.56] Not surprisingly the intuitionalists took exception to many of the concepts associated with the mathematics of infinitesimals.

In an attempt to end the controversy over number, the third foundational school, the formalists (Hilbert), set themselves the task of creating an axiomatic language (using methods acceptable to both logists and intuitionalists) that would define a formal number-theoretic system (a system inclusive of the natural numbers, analysis and calculus). If successful, this axiomatic language would have been both consistent and complete and it would have eliminated the paradoxes and antinomies that were found in set theory and classical mathematics. For better or worse, the formalist school of thought did not achieve their goal. What was achieved however was a 1931 paper by Kurt Godel which determined the goals of the formalist approach to mathematics to be unattainable. In his paper, entitled, On Formally Undecidable Propositions of TNT (typographical number theory) and Related Systems, Godel offered a proof which demonstrated that given any sufficiently powerful formal system, it is not possible for it to be complete and, while it may be consistent, its consistency is not provable within the system.


When The Foundation Of Mathematics Is Based On Indeterminate Numbers

Between Content And Form, Between Assimilation And Accommodation Are The Polar Phases Of The Total Life Process—And Reciprocal Movement (RM)


There are a number of important consequences that follow from Godel’s proof but the one that is most significant in respect to Piaget’s constructionist structuralism, it seems to me, is the one that implies the existence of supernatural numbers. The derivation of a true theorem in number theory, which is its own negation, may be interpreted as the requisite condition for the existence of supernatural numbers. These numbers are peculiar in that they have the property of being infinitely large while possessing no numeral representation. But, everything that can be proven for natural numbers can be proven for supernatural numbers, with the exception that natural numbers are determinate while supernatural numbers are indeterminate.

When the foundation of mathematics is considered in this light, Piaget becomes easier to comprehend. For instance, when Piaget describes the nature of knowledge as being like a pyramid of knowledge that “no longer rests on foundations but hangs by its vertex, an ideal point never reached and, more curious, constantly rising,” he is being very consistent with the latest developments in number theory. Piaget continues:

“In short, rather than envisaging human knowledge as a pyramid or building of some sort, we should think of it as a spiral the radius of whose turns increases as the spiral rises…This means, in effect, that the idea of structure as a system of transformations becomes continuous with that of construction as continual formation.” [Jean Piaget, Structuralism, 1970, p. 34]

The concept of “knowledge as structure” and structure as a “double movement” e.g., the “completeness proof” for natural numbers requiring supernatural numbers and the “consistency proof” for supernatural numbers requiring natural numbers, becomes generalized in the thought of Piaget as the relationship of interdependence that exists between content and form. It is in this interdependent relationship that we find the basis for Piaget’s constructionist theory. Piaget puts this conclusion in his own words:

“Since Godel,...the idea of a formal system of abstract structures is thereby transformed into that of the construction of a never completed whole, the limits of formalization constituting the grounds for incompleteness, or, as we put it earlier, incompleteness being a necessary consequence of the fact that there is no “terminal” or “absolute” form because any content is form relative to some inferior content and any form the content for some higher form.” [Piaget, Structuralism, 1970 p. 140]

With the knowledge of the interdependence of content and form firmly established, Piaget turns to the subject of Biology in order to give his theory a concrete elaboration.

Piaget was deeply influenced by Darwinian evolution. In Piaget’s understanding of processes and states, in terms of developmental stages, we see the depth of this influence. However, Piaget went further than Darwin in his elaboration on the effects of the environment on the development of the individual. In addition to natural selection, Piaget believed something more was going on in an organisms adaptation to its environment. For Piaget an active restructuring and accommodation of an organism to its environment also influenced the development of the organism. Piaget described this new level of accommodation in terms of the interplay of accommodation and assimilation that occurs in the normal life processes. Martindale explains:

“Basic to all Piaget’s explanations is a conception of the individual life process. This has two major aspects or phases: the assimilation of objects to individual activity, on the one hand, and the accommodation of activity to the object world on the other. The two processes of assimilation and accommodation are polar phases of the total life process. They are not always in equilibrium and may even operate in partial autonomy from one another. One of the fundamental facts of the life process is the establishment of an equilibrium between assimilation and accommodation.” [Don Martindale, 1981, p. 342]