Godel's Incompleteness Theorems

[Tim Gwinn <***>]

This is a correction to my original post. I inadvertently left out two prior paragraphs in the Rashevsky quote.

Tim

 

 

> -----Original Message-----
> From: ROSEN Forum [mailto:***]On Behalf Of James N
> Rose
> Sent: Thursday, March 25, 2004 11:47 PM
> To: ***
> Subject: Re: Comparing Rosennean Complexity
---snip--
>
> My other question about his thoughts would otherwise go to
> asking what he thought of Godel's incompleteness theorems. !?
>
> Can you cite any comments for me on that topic?
>

James,

 

In addition to Judith's quote, I think another one from Essays on Life Itself that speaks to your question is this:

    "Any question becomes unanswerable if we do not permit ourselves a universe large enough to deal with the question. Ax=B is generally unsolvable in a universe of positive integers. Likewise, generic angles become untrisectable, cubes unduplicatable, and so on, in a universe limited by rulers and compasses.

    I claim that Godelian noncomputability results are a symptom, arising within mathematics itself, indicating that we are trying to solve problems in too limited a universe of discourse. The limits in question are imposed in mathematics by an excess of "rigor," and in science by cognate limitations of "objectivity" and "context independence." In both cases, our universes are limited, not by the demands of problems that need to be solved but by extraneous standards of rigor. The result, in both cases, is a mind-set of reductionism, of looking only downward toward subsystems, and never upward and outward." [EL 2]

 

The possible impact of this for science, and biology in particular, is a few paragraphs later:

"I take seriously the possibility that there is no list, no algorithm, no decision procedure, that finds us the organisms in a presumptively larger universe of inorganic systems. The possibility is already a kind of noncomputability assertion, one that asserts that the world of lists and algorithms is too small to deal with the problem, too nongeneric.

    Indeed, the absence of lists or algorithms is a generally recurring theme in science and mathematics, one that reveals the nongenericity of the world of algorithms itself, a world too unstable (in a technical sense) to solve real problems. This was the upshot of the Godel results from the very beginning." [EL 3]

He ends the section with:

    "The main lesson from all this is that computability, in any sense, is not itself a law of either nature or mathematics. The noncomputability results, of which Godel's was perhaps the first and most celebrated, are indicative of the troubles that arise when we try to make it such." [EL 4]

 

Along the same lines as these remarks, is something that Rashevsky wrote in an article entitled "Physics, Biology, and Sociology: A Reappraisal" in the BMB [June 1966, Vol 28 No. 2]. Some of the wording sounds very much like Rosen's and I imagine there was an intermixing of ideas there between them. It is long, but seems worth quoting in full, especially since it is not readily available:

    "We perceive biological phenomena only through their physical manifestations. In effect we do perceive them as physical phenomena. Therefore it is logical to conclude that all these phenomena can be explained in terms of physics, even though possibly an extended physics. One can hardly quarrel with this. However, it must be strongly emphasized that explaining all biological phenomena in terms of physical models is not the same as deducing from the laws of physics the existence of biological phenomena. The latter is something quite different.

    That such a deduction may be impossible, in spite of the fact that all observable biological phenomena occur according to laws of physics, is especially suggested by Godel's theorem in mathematics. As Godel (1930, 1931) has shown there are mathematical statements which may be either correct or incorrect, compatible or incompatible with the basic postulates of a mathematical system, yet which can neither be deduced from that system of postulates, nor can they be proved false. If such a thing happens in pure mathematics, it is quite conceivable that a similar situation may occur in mathematical physics.

    It is true that Godel's theorem is proven for sets of postulates that are formulable in terms of the symbolics of "Principia Mathematica". Physics as a whole has not yet been put in that particular postulational form. The work of J.C. McKinsey, A.C. Sugar, and Patrick Suppes (1953) on the axiomatic foundations of classical mechanics; and the work of H. Rabin and P. Suppes (1954) on the axiomatization of relativistic mechanics, represent important steps in that direction. If all physics would  be axiomatized and put in the form of symbolic logic, then a direct application of Godel's theorem would lead to the conclusion that there are physical phenomena which cannot be deduced from the system of axioms. The proof of Godel's theorem amounts essentially (Kleene, 1952) to exhibiting a mathematical statement that cannot be proved to be correct or wrong. The theorem, however, does not tell us about every given statement whether it is a "deducible" one or not. A complete axiomatization of physics will lead us to a certainty that there are physical phenomena which are consistent with the axioms of physics, but which cannot be proved to be so. But even then we shall not know whether biological phenomena belong to that class or not.

    However, the revolution in mathematics, created by Godel's theorem, would justify the expectation of such a possibility. If biology does belong to the class of the "undecidable" statements of mathematical physics, then there seems to be only one way out of the difficulty. We must establish purely biological postulates and principles, without attempting to reduce them to physics. From a certain point of view we may consider this as another extension of physics. Physics then will "swallow" biology. Both the classical mechanists and the vitalists may argue that their respective points of view will then be vindicated by such a procedure. This perhaps proves the utter uselessness of the mechanist-vitalist controversy.  We shall not enter here into the question of nomenclature. We shall rather study the scientific implications of the above possibility.

    Applying consistently the analogy with Godel's theorem, we must conclude that the extended system of postulates of the "enlarged" physics will still remain incomplete. The system will be adequate to treat biological phenomena or at least a large portion of them. But other natural phenomena will always exist, which shall fall into the class of "undecidable" statements.

    We shall here make a surmise that the principles of biology are the "undecidable" statements of mathematical physics. We shall not wait for the proof, basing our surmise on the possible analogy with mathematics. Making such a surmise may seem pretty poor scientific methodology. We find ourselves, however, in good company. Oddly enough, some important statements in mathematics are now formulated in form of surmises, without proof. Church's Thesis (Kleene 1952) is one example. Markov's assertion (Markov 1961) that any algorithm is representable in the form of a normal algorithm is another one. Perhaps in natural sciences we should not attempt to be plus royaliste que le roi and therefore not to be more rigorous that the Queen of Sciences.

    Should the above surmise about the undecidability of biological principles within the framework of the postulates of physics prove to be wrong, less inconvenience will arise than if the above mentioned surmises in pure mathematics prove to be wrong. The new purely biological principles will simply be reduced to physical principles, just like the principles of classical optics were reduced by Maxwell to principles of electromagnetism. All the consequences of those biological principles will remain intact. This was the idea expressed by us when we tried to formulate the "principle of adequate design" and the principle of "biological epimorphism" (Rashevsky 1944, 1954, 1958, 1960). In view of the above we should now be prepared to consider them as possibly independent principles.

    What has been said about the relations between physics and biology holds, mutatis mutandis, about the relation between biology and sociology." [p. 288-289]

 

Regards,

Tim

 

P.S. - Apologies for the length and for the lack of umlauts. :)