Modal Structuralism. Thought of the Day 106.0

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Structuralism holds that mathematics is ultimately about the shared structures that may be instantiated by particular systems of objects. Eliminative structuralists, such as Geoffrey Hellman (Mathematics Without Numbers Towards a Modal-Structural Interpretation), try to develop this insight in a way that does not assume the existence of abstract structures over and above any instances. But since not all mathematical theories have concrete instances, this brings a modal element to this kind of structuralist view: mathematical theories are viewed as being concerned with what would be the case in any system of objects satisfying their axioms. In Hellman’s version of the view, this leads to a reinterpretation of ordinary mathematical utterances made within the context of a theory. A mathematical utterance of the sentence S, made against the context of a system of axioms expressed as a conjunction AX, becomes interpreted as the claim that the axioms are logically consistent and that they logically imply S (so that, were we to find an interpretation of those axioms, S would be true in that interpretation). Formally, an utterance of the sentence S becomes interpreted as the claim:

◊ AX & □ (AX ⊃ S)

Here, in order to preserve standard mathematics (and to avoid infinitary conjunctions of axioms), AX is usually a conjunction of second-order axioms for a theory. The operators “◊” and “□” are modal operators on sentences, interpreted as “it is logically consistent that”, and “it is logically necessary that”, respectively.

This view clearly shares aspects of the core of algebraic approaches to mathematics. According to modal structuralism what makes a mathematical theory good is that it is logically consistent. Pure mathematical activity becomes inquiry into the consistency of axioms, and into the consequences of axioms that are taken to be consistent. As a result, we need not view a theory as applying to any particular objects, so certainly not to one particular system of objects. Since mathematical utterances so construed do not refer to any objects, we do not get into difficulties with deciding on the unique referent for apparent singular terms in mathematics. The number 2 in mathematical contexts refers to no object, though if there were a system of objects satisfying the second-order Peano axioms, whatever mathematical theorems we have about the number 2 would apply to whatever the interpretation of 2 is in that system. And since our mathematical utterances are made true by modal facts, about what does and does not follow from consistent axioms, we no longer need to answer Benacerraf’s question of how we can have knowledge of a realm of abstract objects, but must instead consider how we know these (hopefully more accessible) facts about consistency and logical consequence.

Stewart Shapiro’s (Philosophy of Mathematics Structure and Ontology) non-eliminative version of structuralism, by contrast, accepts the existence of structures over and above systems of objects instantiating those structures. Specifically, according to Shapiro’s ante rem view, every logically consistent theory correctly describes a structure. Shapiro uses the terminology “coherent” rather than “logically consistent” in making this claim, as he reserves the term “consistent” for deductively consistent, a notion which, in the case of second-order theories, falls short of coherence (i.e., logical consistency), and wishes also to separate coherence from the model-theoretic notion of satisfiability, which, though plausibly coextensive with the notion of coherence, could not be used in his theory of structure existence on pain of circularity. Like Hellman, Shapiro thinks that many of our most interesting mathematical structures are described by second-order theories (first-order axiomatizations of sufficiently complex theories fail to pin down a unique structure up to isomorphism). Mathematical theories are then interpreted as bodies of truths about structures, which may be instantiated in many different systems of objects. Mathematical singular terms refer to the positions or offices in these structures, positions which may be occupied in instantiations of the structures by many different officeholders.

While this account provides a standard (referential) semantics for mathematical claims, the kinds of objects (offices, rather than officeholders) that mathematical singular terms are held to refer to are quite different from ordinary objects. Indeed, it is usually simply a category mistake to ask of the various possible officeholders that could fill the number 2 position in the natural number structure whether this or that officeholder is the number 2 (i.e., the office). Independent of any particular instantiation of a structure, the referent of the number 2 is the number 2 office or position. And this office/position is completely characterized by the axioms of the theory in question: if the axioms provide no answer to a question about the number 2 office, then within the context of the pure mathematical theory, this question simply has no answer.

Elements of the algebraic approach can be seen here in the emphasis on logical consistency as the criterion for the existence of a structure, and on the identification of the truths about the positions in a structure as being exhausted by what does and does not follow from a theory’s axioms. As such, this version of structuralism can also respond to Benacerraf’s problems. The question of which instantiation of a theoretical structure one is referring to when one utters a sentence in the context of a mathematical theory is dismissed as a category mistake. And, so long as the basic principle of structure-existence, according to which every logically consistent axiomatic theory truly describes a structure, is correct, we can explain our knowledge of mathematical truths simply by appeal to our knowledge of consistency.

Frege’s Ontological Correlates of Propositions

For Frege there were only two ontological correlates of propositions: the True and the False. All true propositions denote the True, and all false prepositions denote the False. From an ontological point of view, if all true propositions denote exactly one and the same entity, then the underlying philosophical position is the absolute monism of facts.

Lets disprove what Suszko called ‘Frege’s axiom’: namely the assumption that there exist only two referents for propositions.

Frege’s position on propositions was part of a more general view. Indeed, Frege adopted a principle of homogeneity (Perzanowski) according to which there are two fundamental categories of signs (Bedeutungen and truth-values) and two fundamental categories of senses (Sinn and Gedanken).

Both categories of signs (names and propositions) have sense and reference. The sense of a name is its Sinn, that way in which its referent is given, while the referent itself, the Bedeutung, is the object named by the name. As for propositions, their sense is the Gedanke, while their reference is their logical value.

Since the two semiotic triangles are entirely similar in structure, we need analyze only one of them: that relative to propositions.

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Here p is a proposition, s(p) is the sense of p, and r(p) is the referent of p. The functional composition states that s(p) is the way in which p yields r(p). The triangle has been drawn with the functions linking its vertexes explicitly shown. When the functions are composable, the triangle is said to commute, yielding

f(s(p)) = r(p), or f ° s(p) = r(p)

An interesting question now arises: is it possible to generalize the semiotic triangle? And if it is possible to do so, what is required? A first reorganization and generalization of the semiotic triangle therefore involves an explicit differentiation between the truth-value assigning function and the referent assigning function. We thus have the following double semiotic triangle:

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where r stands for the referent assigning function and t for the truth-value assigning function. Extending the original semiotic triangle by also considering utterances:

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Suszko uses the terms logical valuations for the procedures that assign truth-values, and algebraic valuations for those that assign referents. By arguing for the existence of only two referents, Frege ends up by collapsing logical and algebraic valuations together, thereby rendering them indistinguishable.

Having generalized the semiotic triangle into the double semiotic triangle, we must now address the following questions:

  1. when do two propositions have the same truth value?
  2. when do two propositions have the same referent?
  3. when do two propositions have the same sense?

Sameness of logical value will be denoted by (logical equivalence), while sameness of referent will be indicated with (not to be confused with the equiform to express indiscernibility) and sameness of sense (synonymy) by . Two propositions are synonymous when they have the same sense:

(p ≈ q) = 1 iff (s(p) = s(q)) = 1

Two propositions are identical when they have the same referent:

(p ≡ q) = 1 iff (r(p) = r(q)) = 1

Two propositions are equivalent when they have the same truth value:

(p ↔ q) = 1 iff (t(p) = t(q)) = 1

These various concepts are functionally connected as follows:

s(p) = s(q) implies r(p) = r(q), r(p) = r(q) implies t(p) = t(q)

In general, the constraints that we impose on referents correspond to the ontological assumptions that characterize the theory. The most general logic of all is the one that imposes no restriction at all on r valuations. Just as Fregean logic recognizes only two referents so the most general logic recognizes more than numerable set of them. Between these two extremes, of course, there are numerous intermediate cases. Pure non-Fregean logic is extremely weak, a chaos. If it is to yield something, it has to be strengthened.