La objeción de Rieger y el horizonte de la ontología matemática

Abstract

Rieger has made an objection against the results obtained by Gödel in his first theorem (the incompleteness of formalized arithmetic). They can only be valid if we suppose that the formal system is interpreted in terms of the set of natural numbers and that there is a one-to-one correspondence between the figures of the system and the natural numbers. But we know that there are many models that satisfy Peano’s postulates (which Gödel uses as axioms) and which are not isomorphic with the classical model. Therefore the results might be different if a set were used, a non-classical model, for example, for intuitive arithmetic, and another, that of natural numbers, for example, to effect the one-to-one correspondence with the figures of the formal system. The result obtained by Gödel would thus be partial and not able to maintain his philosophical claims.
Gödel’s most important philosophical claim is the platonic position derived from his theorem of incompleteness. It follows from this theorem that while intuitive arithmetic goes beyond formalized arithmetic, the concept of number is not exhausted in the implicit definition established by the set of postulates: thus there cannot be a system of symbols which expresses completely the properties of natural numbers. But if the numbers transcend the symbolism which expresses them, they must belong to an objective world, ideal and independent of the consciousness that expresses them. Gödel came to the conclusion that not only the natural numbers but also the higher order’s sets belong to a world of ideal entities. Thus, Rieger’s objection is relevant to the fundamental problem of mathematical ontology: the objectiveness of the mathematical entity.
The principal point of Rieger’s disagreement is whether the existence of non-classical models makes the concept of a natural number relative. We do not think that we could reach such a conclusion. For, if it is demonstrated that Peano’s postulates can be satisfied by models which are different from the set of natural numbers, it only implies that: 1) no system can exhaust the concept of natural number; 2) the set of natural numbers is independent of the symbolic system which expresses them. If the relativistic thesis claims that the concept of natural number is equivalent to the concept of other entities which satisfy the same set of postulates, then those consequences are contrary to the relativistic thesis. In fact, in order to know whether two sets of entities are equivalent, they must be compared in all their characteristics. Therefore, if the symbolic system does not exhaust all the properties of the concept of the natural number, we have no criterion for knowing whether or not they are equivalent. On the contrary, the fact that they are two different systems demonstrates that they are not equivalent. There is, then, no relativity.
The relativity is derived, rather, from the categoricity. If formalized arithmetic were categorical, any set which satisfied it would be strictly equivalent. But in fact, the distinction between categorical and non-categorical theories has been abandoned. Any formal system sufficiently powerful to be interpreted by the set of natural numbers can be interpreted by other non-isomorphic sets. This does not imply any relativization. For instance, the fact that the group theory can be interpreted by such different sets as that of real numbers and that of the rotations of the square does not imply that the concept of both sets has been relativized. It only means that neither the concept of real number nor the concept of square can be fully defined by the concept of group. The same applies to the concept of natural number. The fact that the system of formalized arithmetic does not permit us to define the concept of the natural number has nothing to do with the relativization of that concept.
Thus, Rieger’s objection, instead of going against Gödel’s philosophical claim, actually works in his favour. Other results do, however, partially invalidate it. The main one is the Löwenheim-Skolem theorem; the demonstration that the set theory, if it were consistent, could be interpreted by means of a numerable set. Here we can certainly talk of relativization. The concept of a non-numerable set is relative to the symbolic system which is used. Cantor’s results depend upon the limitation of the symbolic theory that he chose to use. Consequently, the results of the set theory or the metatheoretical investigation do not compel one to think that transfinite sets exist in a platonic world. The character of “transcendentality” (non-numerability) depends on the structure of the symbolic system which describes the object. For this reason it does not seem to be objective.
On the other hand, it seems that the concept of natural number corresponds to something objective, for the fundamental reason that no interpretation allows that the properties of the natural numbers as they are expressed in Peano’s arithmetic can be relative to the symbolic system. By means of Peano’s arithmetic as well as the group theory we may know the properties which are common to an infinite number of sets of different entities. These properties have been sufficient up to now to make complete use of the natural numbers.
We come to the following conclusion. Classical Platonism has been, on the one hand, reinforced, since the results of the metatheoretical investigation lead one to think of the arithmetical facts as objective. On the other hand, it has been weakened, due to the relativization of the concept of a transfinite set. Not one of the classical positions on the ontology of the mathematical entity can explain these results. Platonism seems to be reduced to the region of the natural numbers, while there is another vague area that cannot be considered as truly objective. As for empiricism it has been demonstrated many a time that the mathematical entity cannot be empirical nor based on experience. The essays on neo-empiricism, like Kalmar’s, seem destined to failure. Nominalism is an ambiguous philosophy; its only unequivocal property is its reductionism. If the end of the reductions is empirical, then we are dealing with an empiricism, and it is therefore false. If the end of the reduction is not empirical, then we are falling into Platonism or other classical positions. As for formalism, we have already seen that the results of Gödel and of Löwenheim-Skolem do not allow for the reduction of the mathematical entity to the symbolism which expresses it.
Finally, there is intuitionism. The official thesis appears to be anti-platonic, for one cannot speak of a property when it has not been constructed. The concept of construction, however, sends us to the elements of construction. To construct is to arrive at a result after a finite number of operations with elements which existed beforehand. To construct a property is to operate with entities which have properties that have not been constructed. The contrary would lead to a regressus. The elements on which the arithmetical construction is based cannot be empirical, for if they were the statements upon the properties of numbers would only be valid for concrete objects, or if generalized, they would only have a probabilistical value. So, whatever we do, we cannot avoid the platonic horizon.
The result cannot be avoided. At the moment we do not see clearly what is the ontological status of the mathematical entity. The classical positions seem too ingenuous. One could say that the results of the investigation suggest a kind of limited Platonism, in which the region of the natural numbers is the only one which presents itself in an undeniable manner as totally independent of the subject. This does not imply that all mathematics can be reduced to arithmetic, nor that the propositions upon transfinite sets are untenable. It only implies that: 1) mathematical ontology is completely different from other ontologies; 2) between the mathematical entity and the language that expresses it there is a sui generis relation which is completely different from the relation between non-mathematical entities and the language which expresses them. The oddness of this relation is indicated in the Löwenheim-Skolem theorem, for its results both strengthen and weaken the conviction that the mathematical entity is independent of the language that describes it. While the specific relation between the mathematical entity and the language is not made quite clear, the problem of the mathematical entity will not be clear either. The results of the metatheoretical investigation compel us to look for a criterion in order to distinguish the mathematical entity which is truly independent from the language that expresses it, from that which depends either partially or totally upon the language.