On the Dialectics of Metamathematics
by Peter Várdy
The entirety of this article cannot be reproduced here. Its logical meat is missing: a detailed exposition of Bertrand Russell's attempts to banish the paradoxes of set theory through the theory of types and the prohibition of self-reference, as well as treatments of related problems by other logicians and mathematicians. Please consult the original article to connect these issues in mathemathical logic to the author's philosophical thesis.
I have long been fascinated by the question of the relationship between formalization and the extra-formal dimension of abstract thought. This article encapsulates the conceptualization I am after like no other. The philosophical highlights of the author's argument are here excerpted. R. Dumain
What on Earth does metamathematics have to do with dialectics? One may justifiably find the association of these concepts dubious. Metamathematics, the foundational science of mathematics, seeks to grasp the conditions under which a contradiction-free science of quantity is possible. The following essay would like to show, in taking up the foundational reflections proper to metamathematics, that the principal anchoring of consistent mathematical objectivations presupposes a ground which not only founds these, but at the same time surpasses them in principle. Such an immanent movement of the principal presuppositions of a spiritual construct is what Hegel understood by a dialectical development. Yet it is not my concern here to provide a Hegel-interpretation, but rather to demonstrate a development in the Hegelian sense with respect to a science. Only what is astonishing here is that it should be possible to show such a development precisely with respect to mathematics, a form of the human spirit extremely remote from dialectics. And yet this development also has to be understood dialectically: knowledge can neither avoid the quantitatively unambiguous nor be content with it.
‘God’, ‘freedom’, and ‘self-consciousness’ do not constitute an arbitrary selection of concepts from the history of occidental philosophy. Such is virtually impregnated with self-reference. Precisely its central concepts possess a reflexive structure. That means, however, that Russell’s prohibition has far-reaching consequences for the whole of philosophy. Were Russell’s insight that every self-reference robs the concept in question of any sense, to prove to be unconditionally true, then the central concepts of the philosophical tradition would be invalid. The reconstruction of a contradiction-free mathematics on the basis of a general prohibition of self-reference means that the theory of the foundations of mathematics, which is plagued by antinomies, can be rebuilt only on the ruins of metaphysics.
It is to Russell’s merit that he elevated this principle of non-reflexibility, of the objectivistic spirit in mathematical formalism, expressly to the rank of a principle. Here Russell’s principle radicalizes Cartesian dualism to the effect that now objectivity is not only encompassing with respect to subjectivity, but rather there is no longer any mediation between the mathematically graspable objectivity and self-conscious subjectivity; rather, a justifiable sense may not in general be attributed to any spiritual construct with a self-reflexive structure.
On the basis of the foregoing reflections, we may now draw the conclusion that the prohibition of circularity itself, in that it excludes every self-reference without exception, is self-referential. That means, however, that Russell’s insight into the senselessness of self-reference in fact does free the mathematical formalism of the antinomy by means of his prohibitive principle, but the price to be paid for this guarantee of freedom from contradiction is that the prohibition of circularity is equivalent to the antinomy.
In this metaphor, the modern principle of irreflexivity, the prohibition of circularity, takes the place of antiquity’s absolute foundation. Their relation is here simultaneously one of correspondence and extreme opposition: the "hypothesis" of the theory of types does not stand in a relation of deductive dependence on the prohibition of circularity—nevertheless the prohibition of circularity necessarily motivates the irreflexivity of the theory. Contrary to the self-reference of the ideal foundations of the ideas, the prohibition of circularity is a principle of irreflexivity; as such, however, it is at the same time necessarily self-referential.
The relation of principle and theory in modern science is radicalized in Russell’s foundation of science and at the same time transcended toward knowledge in the ancient sense.
The formal restriction of comprehension to irreflexive sets would thus introduce a source of difficulty into the deductive-axiomatic order. Precisely for this reason, a restriction of comprehension by Russell’s vicious-circle principle must remain outside of the formal-deductive order in a so-called motivating relation. The formal-deductive order is thereby reduced to a mere objectivation within mathematics, which expressly presupposes the informal mathematical intuition of the subject.
Put another way: formalized, the prohibition of circularity would be an antinomy.
Incidentally, it is noteworthy that a consistent formalization is by no means the only way in which a concept can be objectivated. For instance, the contemporary fine arts, by means of a realistic reflection, have also depicted reflexive structures. It is exactly at this point that Realism turns into Surrealism.
Magritte’s "La reproduction interdite" from 1937 shows a man standing before a mirror who sees reflected, instead of his face, his back. The work can be understood as an expression of the vicious-circle principle.
The Dutch graphic-artist M. C. Escher likewise worked on the problem of reflexivity ("Still Life with a Convex Mirror," 1934; "Hand with a Reflecting Sphere," 1935; "Still Life and Street," 1937; "Three Spheres," 1946). As far as I know, he was the only one in the fine arts who succeeded in comprehending the singular point of reflection in the identity of just one reflection, instead of in an infinitely regressive series of mirror-refractions: "Print Exhibition" from 1956 shows a gallery visitor standing before a print showing in turn a port town and in the town the same gallery with the same visitor standing before the same print. Now Escher’s mapping is most remarkable given the fact that the lithograph in front of the viewer in the picture is identical with the lithograph in front of us. In other words, a part of the depiction is identical with the whole, or the viewer views his viewing. Escher proceeded in this objective realization of self-reflection as follows: he depicted a view of a city (including the gallery and therein the views of the city, infinitely reflecting themselves in views of the city), originally drawn from the central perspective, subsequently with the help of a projection expanding cyclically clockwise, whereby the infinite series of depictions (including the gallery visitor) are telescoped in one another and thus are simultaneously identical and not identical with themselves. Noteworthy is that the network of the cyclical projection in Escher’s subsequent depiction seems to correspond to an elliptical geometry.
An objectivation of self-reference and therefore also of the prohibition of circularity is thus perfectly possible, whereas a mathematically objection-free, i.e., contradiction-free, formalization is hardly possible. Therefore the prohibition of circularity guarantees that the theory of types is contradiction-free only via an informal motivating relation to the latter’s formal-deductive order. Thus there is at least one proposition of mathematics whose principal relevance cannot be expressed as a formal proposition of the theory of types. This conclusion differs from the result maintained by Gödel as a corollary to his incompleteness theorem, insofar as the role of the consistency, which can be expressed formally only incompletely, is actually taken up in Russell by the vicious-circle principle: our faith that arithmetic can be represented without contradiction by the formally axiomatized theory is grounded on the vicious-circle principle.
The suggestion by Boudrie and Kluyt contains a consistent development of what had remained only implicit in the foregoing. As we saw, the prohibition of circularity is free of self-reference if and only if it is self-referential. That means: reflexivity may be eliminated only by reflexivity itself; to the extent that everything is irreflexive, self-reference immediately appears again. Irreflexivity entails reflexivity as its shadow, which it is unable to shake off. The principle of an irreflexive completeness itself yields a logically necessary self-reference. Were mathematical formalization the only way to objectivate anything that is possible, then mathematics would not be possible.
Yet this necessary self-reference of excluded self-reference, this negation of a negation, in an Hegelian parlance, is completely empty. It has no determination other than self-reference. It requires that exposition which does not only arise from the reductio ad absurdum of a complete irreflexivity. The formal necessity of self-reference retains for the moment the emptiness of the (contradiction-free) reversed paradox of the liar, who says: "This sentence is true." Yet it points to a void as the logical embedding space of metaphysical concepts.
Finally one could respond to Boudrie and Kluyt’s circular prohibition of circularity by saying that, beyond the credit due to the acknowledgment of self-reference as a necessary moment of non-self-reference, their prohibition not only suffers from the emptiness of its own reflexivity, but also from an abstractness with which it sets circularity and freedom from circularity in opposition to one another. For one not only has to give up abstract freedom of circularity, one also cannot hold fast to an abstract self-reference without it turning into its opposite. The antinomies of mathematical circularities were not the first to make this manifest, but rather Hegel’s critique of abstract self-consciousness already did so. Self-consciousness is possible only insofar as it, instead of ridding itself of its own negativity, refers its negativity to itself and posits it as self-sublating.
Russell’s vicious-circle principle actually achieves that. If we retain the prohibition of circularity, then we also retain its self-reference, which also includes the truth of its antinomic nature.
The Opposition between the Prohibition of Circularity and Formal Theory
Let us raise anew the question of how the prohibition of circularity as a principle of an antinomic, self-referential nature is to be grasped in relation to formal mathematical theory. The prohibition of circularity and formal theory are obviously opposed to one another, since the latter’s freedom from contradiction is enabled only by the (antinomic) exclusion of self-references. Now what kind of opposition is this?
It is not abstractly negative, if we understand thereby every kind of relation of two opposed terms which unambiguously exclude one another in an essential respect. In view of the essential definition of circularity, formal theory is in fact unambiguously circle-free; the prohibition of circularity itself, however, is not unambiguously circular, but just as circle-free. Via Aristotle’s primary determination of opposites, we fail to get an immediate hold of the said relation. I can show this here only very briefly for his four oppositions and on the assumption of the Aristotelian theory.
The prohibition of circularity and formal theory are not contradictorily opposed, since neither of the opposed terms can be asserted to be something positively determined in itself; nor can either of the two be asserted to be constantly true, while the other is false. Their opposition is just as little contrary, since the circularity and its negation cannot be understood as specifications, e.g., of self-referential or formalizable concepts. This is already clear in that the prohibition of circularity itself possesses the character of a concept. Thereby the prohibition of circularity does not fulfill the requirements of Aristotle’s theory of definition, since otherwise the genus could be predicated of the specific difference. The relation of the prohibition of circularity to the mathematical concept falls outside of a genus-species order. But neither can we understand the opposition privatively, for instance, as if the prohibition of circularity lacked freedom from circularity as something that would, according to its own nature, belong the former.
The question as to whether the relation of the prohibition of circularity to the mathematical concept may be understood as a relative opposition is even more difficult to answer. First, the terms of the relation, here thought as principle and as something principled, exist only in their constitutive relation. In other words, if we wish to think the relation of principle to principled as a relation, then we have to pass over the Aristotelian doctrine that a relation is always accidental and we may no longer understand the relation as a category in the usual sense. Furthermore, we have to consider whether the relata are so opposed in their relation that that determination attributed to one of the terms of the relation, is lacked by the other. This constitutive determination, the absence of circularity, is in fact unambiguously attributed to the mathematical concept; but neither circularity nor its absence are unambiguously attributed to the prohibition of circularity, but rather both simultaneously in the same respect.
This state of affairs calls for a negation that does not correspond to the traditional four opposites and that does not exclude what is negated, but rather negates in such a way that it simultaneously embraces what is negated. This preserving negation is Hegel’s concrete negation.
Perhaps this concretely dialectical negation corresponds to the relative opposition (as J. Hollak supposes).
The concept of such a "relative" opposition is then to be modified, however, precisely in the sense of the dialectical relation, namely: the opposed relation should no longer be thought to exist substantially outside of their relation.
Finally, the prohibition of circularity is non-formal in the sense that it in fact does not obey the prohibition of self-reference stated in it, but certainly founds here a region of extensive non-ambiguity. It thereby makes possible the region of the mathematical, in which the freedom from contradiction prevails. The precept of the freedom from contradiction and the prohibition of self-reference in fact prevail in this region, which is embedded in a non-formal region, in which certainly no precept of contradiction prevails. The prohibition is precisely sublated and not excluded, for the abstract negation would make itself known time and again only as a mathematical form of understanding, i.e., it would only reproduce the characteristic problematic of this form, instead of removing it.
It seems significant that just in this form of understanding—which understands itself as an absolute opposition to speculative reason— namely, in the mathematical, the necessity of Hegel’s concrete negation can be demonstrated. Its truth is more binding the less it was intended. Dialectical negation arises from mathematical objectivation itself, as a necessary condition of the latter’s own possibility.
This impossibility of an exhaustive formalization of several basic concepts of mathematical theory is familiar from the works of Gödel and Tarski. The informal sense of negation, which formal mathematics presupposes in its relation to the prohibition of circularity, was more precisely determined thereby.
In the foregoing we have found, in reference to the question of how the relation of the principle of excluded self-reference (the "prohibition of circularity") to the formal-deductive order of mathematical theories can be grasped, that the prohibition of circularity enables formal theory’s freedom from contradiction in that it simultaneously withdraws from this order: it is not a theorem of formal theory. It is self-referential, according to its structure antinomical, transgresses the distinction between syntactic and semantic references, and encompasses the formal-mathematical objectivation in a concrete-dialectical opposition to the latter.
It is an historically noteworthy fact that the difficulty of granting formal expression to Russell’s prohibition of circularity was never discussed. It remained simply unconscious, as if a state of affairs were repressed which could not be addressed until Gödel appeared and therefore was banished from mathematical consciousness. For would it not have been obvious to present Cantor’s axiom of comprehension now in an expressly qualified form? For instance: "Every set can be understood in turn as an element of sets insofar as it does not include itself as an element"? Such a proposition, however, would lead to difficulties and was consistently banned by Russell as nonsense. This according to a principle whose formal inexpressibility was subsequently accepted somehow without people becoming particularly thoughtful about this state of affairs.
Metamathematics seems to be confronted with the choice between an incomplete or contradictory self-foundation. This means, however: it has to opt either for a rejection of a thorough self-comprehension or for a self-relativization. In the proper sense, one is not at all free to opt for the first possibility, since its basis lies in its unfoundability. In other words, its possibility as a mere decision stands or falls with the deed, whereby its own grounds of justification are driven into the subconscious. One can omit the thorough self-foundation de facto; a mathematically acceptable foundation for this omission, however, would be self-destructive. By contrast, a decision in favor of self-relativization means that mathematics has sublated itself as a theory of everything possible and founded itself as one possibility of the human spirit.
The principle of circularity is, so to speak, a regulative principle for formal-mathematical objectivation also outside of type-hierarchies in which the prohibition of self-reference appears as such. In axiomatic set theory, there is in fact no prohibition of circularity, but if one asks a student of axiomatic set theory why he thinks that his system is free of the antinomies, he could answer with good reason that the operations which allow for the formation of certain sets are chosen such that, e.g., Russell’s class and other self-references can no longer be formed. Therein lies the reason for his confidence. At the same time, it is the reason for my assumption that a prohibition of self-references in the contemporary dialectical transformation of the ancient meaning of [Greek word] is a principle of mathematics, which, in that it in fact guarantees formal freedom from contradiction, simultaneously withdraws from mathematical objectivation and hints at a spiritual region in which, along with quantity, other categories, such as, e.g., ‘freedom’, ‘life’, justice’, and ‘person’, are also founded.
SOURCE: Várdy, Peter; translated by Marcus Brainard. "On the Dialectics of Metamathematics," Graduate Faculty Philosophy Journal , vol. 17, nos. 1-2, 1994, pp. 191-216. This text originally appeared as "Zur Dialektik der Metamathematik," in: Michael John Petry, ed., Hegel und die Naturwissenschaften (Stuttgart-Bad Cannstatt: Frommann-Holzboog, 1987), pp. 205-240.
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