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| Mathematical Structuralism |
Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects.
Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general conditions—typically spelled out in axioms defining the structure or structures of interest—characteristic of the branch of mathematics in question.
Thus, in the basic case of arithmetic, the famous “axioms” of Richard Dedekind (taken over by Giuseppe Peano, as he acknowledged) were conditions in a definition of a “simply infinite system,” with an initial item, each item having a unique next one, no two with the same next one, and all items finitely many steps from the initial one. (The latter condition is guaranteed by the axiom of mathematical induction.)
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| Richard Dedekind |
All such systems are structurally identical, and, in a sense to be made more precise, the shared structure is what mathematics investigates. (In other cases, multiple structures are allowed, as in abstract algebra with its many groups, rings, fields, and so forth.)
This structuralist view of arithmetic thus contrasts with the absolutist view, associated with Gottlob Frege and Bertrand Russell, that natural numbers must in fact be certain definite objects, namely classes of equinumerous concepts or classes.
Historically, structuralism can be traced to nineteenth-century developments, including the rise of the axiomatic method and of non-Euclidean geometries leading to the recognition of multiple abstract spaces independent of physical space and of spatial intuition. David Hilbert, whose work in the foundations of geometry was especially influential in this regard, remarked that “points, lines, and planes” could be read as “tables, chairs, and beer mugs”.
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| David Hilbert |
In instructive correspondence with Frege, Hilbert championed the structuralist view of axioms in pure mathematics as defining structures of interest rather than as assertions whose terms must already be understood. In the twentieth century, the development of modern algebra and set theory informed the influential views of the Bourbaki, who explicitly espoused a set-theoretic version of structuralism.
Virtually any mathematical structure (or “space,” e.g. metric, topological, and so forth) can be conceived or modeled as a set of objects with certain distinguished relations and/or operations on the set, and set theory has the resources for describing a wealth of interrelationships among structures, vital to advanced mathematics. The branch of logic known as model theory develops these ideas systematically.
Despite the success of set-theoretic structuralism in providing a unified framework for all major branches of mathematics, as an articulation of structuralism, it confronts certain problems.
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| unified framework |
Notable among these is that it makes a major exception in its own case: despite the multiplicity of set theories (differing over axioms such as well-foundedness, choice, large cardinals, constructibility, and others), the axioms are standardly read as assertions of truths about “the real world of sets” rather than receiving a structuralist treatment. Questions then arise about this “fixed universe as background”: How does one know about this real-world structure, how rich it is at its various levels, and how far its levels extend?
The (putative) set-theoretic universe cannot be a set; yet as a totality of a different order, is it not indefinitely extendable, contrary to its purported universality? These and related questions have led some philosophers, logicians, and mathematicians to develop alternative ways of articulating structuralism.
Alternative Articulations to Structuralism
The main alternatives to set-theoretic structuralism to be described here are, first, the view of structures as patterns or sui generis universals, developed by Michael Resnik and Stewart Shapiro, respectively; second, an eliminative, nominalistic modal structuralism, traceable in part to Russell and Hilary Putnam and developed by Geoffrey Hellman; and, finally, a version based on category theory, as a universal framework for mathematics independent of set theory, suggested by Saunders Mac Lane and others.
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| Alternative Articulations |
THE VIEW OF STRUCTURES AS PATTERNS OR UNIVERSALS. On the view of structures as patterns or universals, apparent reference to special objects in mathematics is taken at face value. Moreover, the reason that such objects are typically identified only by reference to operations and relations within a structure is that in fact they are inherently incomplete.
They are to be thought of as positions or places in a pattern, on analogy with, say, the vertices of a triangle. For Resnik, identity and difference among positions make sense only in the context of a structure given by a theory. The number 2, say, is identified as the successor of 1, the predecessor of 3, and so on, but not intrinsically.
Indeed, whether the natural number 2 = the real number 2 is indeterminate, except relative to a subsuming structure specified by a broader theory; and then it would still be indeterminate whether the numbers of the new theory were the same as or different from the respective old ones.
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| Michael Resnik |
This theory-relativity of reference and identity—besides leading to complications in the account of the common mathematical practice of embedding structures of a prior theory in those of a later one, as well as in the account of applications of mathematics— reflects Resnik’s reluctance to think of patterns as an ontological foundation for mathematics.
Talk of patterns may be only analogical, helping free one from the grip of traditional Platonism. Thus, a mathematical theory of structures is not given, in part because its objects could not then be identified with those of existing mathematical theories, defeating its purpose.
In contrast, Shapiro takes ontology seriously and develops an axiom system governing the existence of ante rem structures, abstract archetypes with places as objects, answering to that which particular realizations have in common.
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| Geoffrey Hellman |
The background logic is second-order and the axioms resemble those of Zermelo-Fraenkel set theory but with an added Coherence Postulate guaranteeing an existing structure modeling any coherent second-order axiom system, where this new primitive is understood as analogous to the logical notion of satisfiability.
Knowledge of key instances of this postulate arises naturally, it is argued, from their learning how to use mathematical language together with certain axioms characterizing the structure of interest (e.g. the principle of continuity of the real number system).
Although this view circumvents some of the objections raised against the set-theoretic version, it confronts a number of objections of its own. One (due to Jukka Keränen and John Burgess) points out that, whereas objects in a structure should be distinguishable entirely in terms of internal structural relationships, this is possible only in cases admitting no nontrivial automorphisms (11 structure preserving maps from the class of places to itself other than the identity map).
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| John Burgess |
The natural numbers and the reals are “rigid” in this sense, but many nonrigid structures arise in mathematics (e.g. the complex numbers, permuting i and –i, or homogeneous Euclidean spaces under isometries, and so forth).
A further objection finds a circularity in the account of abstraction offered; the relevant structural relations can only be distinguished from others generated, say, from permutations of objects if those objects (the places) can be picked out independently, contrary to the idea of “structural objects.” (This revives a well-known argument of Paul Benacerraf against numbers as objects generally.)
Finally, although not committed to any maximal universe of sets, ante rem structuralism seems committed to a universe of all places in structures, contrary to the view that any such totality should be extendable.
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| modal-structuralism |
MODAL-STRUCTURALISM. Turning to modal-structuralism, this view dispenses with special structural objects and indeed even with structures as objects, recognizing instead the possibility that enough objects—of whatever sort one likes—could be interrelated in the right ways as demanded by axioms or conditions appropriate to the mathematical investigation at hand.
As suggested by Russell, the irrelevance of any intrinsic features of “mathematical objects” arises through generalization: statements “about numbers,” for instance, are not about special objects but about whatever objects there might be, collectively standing in the right sort of ordering.
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| special objects |
By speaking of wholes and parts and utilizing a logic of plurals—reasoning about many things at once without having to talk of sets or classes of them—such generalizations, even over functions and relations, can be framed in nominalistic terms.
The effect is to generalize over “structures there might be” without actually introducing structures as entities. Extendability is respected, as it makes no sense to collect “all structures, or items thereof, that there might be.” Assuming the logical possibility of countably infinitely many objects, one can recover full classical analysis and, with coding devices, modern functional analysis and more.
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| utilizing a logic |
The main price paid for all of this is the adoption of a primitive notion of possibility, something set theory explains in terms of the existence of models. The gain is a circumvention of problems of reference to abstracta and a natural way of respecting indefinite extendability of mathematical domains.
CATEGORY-THEORETIC STRUCTURALISM. The final approach considered here is based on category theory. Having arisen in mathematics proper to help solve problems in algebraic topology and geometry, it can also serve as a general framework for mathematics. Its basic concepts are mappings (morphisms or arrows) between objects, and their compositions.
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| Morphisms |
The objects are typically what the other approaches call structures, described in relation to other such objects via morphisms (“arrows only”), not internally via set membership. Morphisms typically preserve relevant structure (algebraic, topological, differentiable, and so forth). Topos es are families of objects and morphisms with richness comparable to models of Zermelo set theory; they can serve as universes of discourse for mathematics.
Generalizations of set-theoretic ideas are provided (such as Cartesian product, function classes, and logical operations, which generally obey intuitionistic laws, i.e. excluding the “law of excluded middle,” p or not p, for arbitrary p). In contrast to set theory with its fixed universe, topos theory promotes a pluralistic conception of “many worlds,” functionally interrelated (cf. Bell).
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| fixed universe |
It is clear that there are some interesting similarities between category-theoretic structuralism and modal-structuralism, and indeed the latter can be adapted to accommodate the former. Whether category-theoretic structuralism can stand on its own, however, is an open question that turns on such issues as whether its basic concepts are really intelligible without set theory, just what its background logic presupposes, and whether a theory of category of categories can serve as an autonomous framework.
In sum, structuralism has become a major arena for exploring central questions of ontology and epistemology of mathematics.
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| central questions |
