Some Difficulties of Contemporary Structuralism
In general, the philosophical thesis expressing that mathematics is not concerned with individual objects, but with “systems” or properties of objects that share a common structure, is called structuralism.
A “system” is a domain of objects provided with certain functions and relationships fulfilling certain conditions. Structuralism is called non-eliminative, if it does not presuppose any ontology for particular objects. According to this kind of structuralism, mathematical objects are incomplete, because they lack an internal structure. A structure appears to be an abstract form or entity, and what the constants in mathematical propositions denote are not individuals, but just ‘positions’ or ‘places’ in this structure; they do not have an identity outside of the structure. The objects ‘denoted’ in mathematical propositions are structure-dependent. Thus, the ‘objectual’ character of Mathematics is a way of speaking. Structuralism is called eliminative, if it questions the existence of abstract mathematical structures and maintains that the nature of mathematical objects is exhausted by their positions in these structures. An example is the set-theoretic structuralism, where mathematics is considered as the study of different structures built up from sets that were thought to exist (Bourbaki). Both versions of structuralism involve two main tasks for the philosopher:(a) He must clarify the relationship between ‘structure’, ‘system’, ‘theory’, ‘subject’ and ‘position’ or ‘place’ and argue for or against their respective ontological significance.
(b) He should provide an explanation why mathematicians do not realize about what they actually speak, or if they realize it, that they do not expressly refer to it and continue to use singular terms.
Indeed, in eliminative structuralism, the ontological commitment of mathematical objects or at least sets, remains unexplained, while in non-eliminative structuralism, we are faced with two versions, each of which leads to its own problems: if we adopt a version in re, we consider that a ‘structure’ is all that can be instantiated by a system, but there is not an independent entity ‘structure’.
‘Positions’ in the structure are treated in terms of a function, that is to say, ‘3’, for example, refers to the object in the position ‘3’ in all systems instantiating the structure. The structure depends on systems that instantiate it and presupposes an ontological background. This background is the crucial point that cannot be itself included in a structural approach. If we adopt a version ante rem, we consider that the structure exists independently of the systems that instantiate it, and its ‘propositions’ express a generalization with respect to all systems that instantiate the structure (realistic interpretation of the structure). The ‘positions’ in the structure are then treated as ‘objects’ in a grammatical sense lacking any non-structural property. For example, the object ‘3’ is the successor of the object ‘2’. Do we have uniqueness of objects? In no way, unless we assume the uniqueness of the isomorphism between systems instantiating the structure. For example, let (E, the first level of the construction of geometric space. It is obtained by choosing the language of groups to serve as the tool of reasoning about representations of muscular sensations.[160] Similarly to Carnap’s Aufbau, the starting point (guided by experience) is for Poincare the definition of two two-place relations satisfying certain minimal empirical conditions: an external change a (with ‘x a y’ for ‘x changes in y without muscular sensation’) and an internal change S (with ‘x S y’ for ‘x changes in y accompanied by muscular sensations’). Further, he proceeds to a conventional classification of external changes: among external changes some can be compensated by an internal change, others cannot. If they can, experience teaches us only “that the compensation is approximately produced”; it gives the mind only “the occasion to accomplish this operation”, but “the classification is not a raw fact of experience” (Poincare 1898, 16). If compensation is possible, the changes are called changes of position, if not changes of state. In this way, he obtains the following result: modulo an identity condition with respect to the compensation by internal changes, Poincare defines the equivalence class of changes of position and calls it a displacement. Displacements form a group in the mathematical sense and it depends on the choice of its sub-groups whether the group corresponds to Euclidean or non Euclidean geometry.At first glance, Poincare’s approach seems just to be an abstraction process leading to a form of invariance. Nevertheless, in reality, the faculty to create the general concept of a group is the expression of a form of our understanding “existing in our mind”. The set of relations satisfying the group axioms resembles an ante rem structure, which is exemplified (in a Goodmanian sense) by the specific displacement-structure (= transformation group). In other words, the form in the mind leads to a special kind of epistemologically accessible universals without that one has “the possibility of deducing by purely logical means the particular form of the universal” (Agazzi 2014, 442). The formation of the group concept—a ‘universal’ or a second order form (concept) —
G[A1, A2, A3)
is suggested by a specific sensations system
g'(s1, s2, s3),
which is the material of the form (a vague part of the extension of the concept). The form is the conventional ‘abstraction’ of the various imagined laws of sensations corresponding to groups of axioms. Read as relational set (i.e. extensionally), the general group G is a model of the group axioms and these axioms are ‘only’ exemplified by the sensation system G’, which does not correspond exactly to the axioms. The elements of displacements groups are complete and independent entities with respect to the axioms of the group. But by analyzing the subgroups of the displacements groups (common to geometries with constant curvature), the variables of the axiom system are transformed into ‘places’ that lack ontological independence: they are depending on decisions that are taken concerning the property of distance, for which exists a choice between different possibilities — such a choice was not yet possible with respect to the general axioms of group!
Thus Poincare defends a structural point of view without completely disengaging the structure from an “ostensional” aspect and this allows him to dispense with a consistency proof.
What matters here is the general idea that the faculty of construction of the general concept of group pre-exists in our minds and led to a universal, and that this faculty is suggested by an imagined system of sensations. The concept obtained can be read as a model of an axiom system, in which originally the domain of quantification is composed by independently given elements, which lose progressively their independence and become incomplete. The genesis of the geometrical metric structure is neither seen as the creation of the concrete material from the universal (structure) nor as the creation of the universal from the concrete (sensations), but as the advent of relations linked to the concrete in a semiotic analysis of the universal. Contrary to in re structuralists, Poincare’s universal, i.e. the general group structure, is not ontologically but epistemically dependent on exemplifications. Nevertheless, Poincare doesn’t speak of this structure as such but uses it as a meta-mathematical tool for his psycho-physiological genesis of real actions with imagined sensations: the psycho-physiological model is the ratio cognoscendi of the existence of the faculty to build the general group structure in our mind.The exemplification of the structure by big varieties of systems (imagined sensations/mathematical and physical facts/objects) is not a logical but a semiotic operation. It’s only the structure that gives the common character to systems: the building up of the structure is a mastery occasioned by concrete systems (samples), which do instantiate and exemplify the structure.
The universal as mathematical structure of the first level is not underdetermined but, as relational property, indeterminate. If we stay within the framework of a Tarski-semantic, such vague ‘objects’ probably imply the move to a non-classical logic (Evans 1978, 208): it seems contradictory to say that there are objects and at the same time that it is indefinite whether they are identical or not.
Indeed, if an object is undetermined, it differs from another with respect to the property of being identical. Therefore, by contraposition of Leibniz-identity, the two objects are different. Contradiction! If it is found that logic led to some sorites with regard to vague concepts, we have at least two possibilities:(a) to solve the problem through a special logic appropriate to vague situations
(b) to find a better understanding of the relationship between a precise language and the linguistic practice in question.
We choose the second possibility. Based on the thought of Peirce, we explain vagueness as indeterminacy of meaning in terms of indeterminacy of an exemplification of a concept in a dialogue (Williamson 1994, 48, note 28). In this sense a vague ‘object’ or structure R is interpreted as a second order relation
R (Pi,..., Pn) « a (Pi,..., Pn) A ^'(Pi,..., Pn) A......
where ‘«’ signifies the exemplification of ‘R’ in a semiotic sense, by a system of axioms a’, p’.... The meaning of ‘R’ is then to develop according to the pragmatic maxim of Peirce:
“Consider what effects that might conceivably have practical bearing you conceive the object of your conception to have. Then your conception of those effects is the whole of your conception of the object” (C.P., 5.422; cf. Peirce 1878/79, p. 48 and C.P. 5.402).
The experimental perspective involved into the maxim is here performed by the exemplifications and limited by the constraints of definitions and conceptual formations of formalisms. The formalisms and proofs with respect to which the universal is a model in the usual model theoretic sense, take here a hypothetical form, ‘as if their variables referred to clearly identifiable objects instantiating the universal. The working out of universals as structures is still ‘growing’ and the extended structures still remain incomplete in principle, both with respect to their proper identity and the identity of their objects.
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