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This is Part A of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A summarizes the general attractions of non-eliminative structuralism. Afterwards, it motivates an understanding of unlabeled graphs as structures sui generis and develops a corresponding ax...
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... "in giving the properties. . . of numbers you merely characterize an abstract structure" (from the quote in Section 1) matches the "interested only in the 'structure' or 'form' of a graph" in the Bender and Williamson quote from above. Similar to what was said in Section 1 about isomorphic systems and their structures, Mahadev and Peled hold that "two graphs G = (N, E) and H = (N, F ) are the same unlabeled graph when they are isomorphic" in the quotation above, and the same point is highlighted by Harary's depiction of isomorphic labeled graphs G 1 and G 2 , of which G 3 is the uniquely determined underlying unlabeled graph (see Figure 3). Finally, even when one should not mistake the graphical illustrations of unlabeled graphs for the depicted graphs themselves-as Bender and Williamson remind us in the quote above-figures such as Figures 2 and 3 convey a pretty clear graphical "Anschauung" of unlabeled graphs as structured entities. ...
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... 0 ', 'G 1 ', 'G 2 ' are in fact individual variables in the language of UGT, but for the sake of convenience I use the same symbols as individual constants in the metalanguage of UGT in order to refer to the three unlabeled graphs with less than three nodes. (Note that these names are used differently from what Harary took them to denote in the quotation in Figure 3 in Section 3.) Accordingly, one can determine the number of automorphisms for a given unlabeled graph (if restricted to the set of nodes of that graph): e.g., the unlabeled graph G 1 allows for precisely two automorphisms on its set of nodes, and the same holds for the unlabeled graph with three nodes depicted in the bottom-right corner of Figure 4. And one can prove cardinality facts about graphs: there exist precisely two unlabeled graphs with precisely two vertices (G 1 and G 2 ) 8 , four unlabeled graphs with precisely three vertices, and eleven unlabeled graphs with precisely four vertices. ...
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This is Part B of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A motivated an understanding of unlabeled graphs as structures sui generis and developed a corresponding axiomatic theory of unlabeled graphs. Part B turns to the philosophical interpretatio...
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... The unlabeled character of graphs freely generated by these two operations is described by axioms to be given presently. Following Leitgeb (2020a), these unlabeled graphs have undirected edges, at most one edge for any two distinct vertices, and no loop edges from a vertex to itself. The rules of inference are the same as for the P systems above. ...
Haskell Curry’s philosophy of mathematics is really a form of “structuralism” rather than “formalism” despite Curry’s own description of it as formalist (Seldin 2011). This paper explains Curry’s actual view by a formal analysis of a simple example. This analysis is extended to solve Keränen’s (2001) identity problem for structuralism, confirming Leitgeb’s (2020a, b) solution, and further clarifies structural ontology. Curry’s methods answer philosophical questions by employing a standard mathematical method, which is a virtue of the “methodological autonomy” emphasized by Curry (1951, 1963) and more recently with greater clarity by Maddy (1997, 2007).
... Compare, e.g.,Benacerraf (1965),Shapiro (1997) andLeitgeb (2020) on the significance of structural properties or structural relations in mathematics. 4 See, in particular,Linnebo (2008),Linnebo & Pettigrew (2014), andKorbmacher & Schiemer (2018) for present several different characterizations of the distinction between structural and non-structural properties, including the invariance account. ...
Mathematical structuralism can be understood as a theory of mathematical ontology, of the objects that mathematics is about. It can also be understood as a theory of the semantics for mathematical discourse, of how and to what mathematical terms refer. In this paper we propose an epistemological interpretation of mathematical structuralism. According to this interpretation, the main epistemological claim is that mathematical knowledge is purely structural in character; mathematical statements contain purely structural information. To make this more precise, we invoke a notion that is central to mathematical epistemology, the notion of (informal) proof. Appealing to the notion of proof, an epistemological version of the structuralist thesis can be formulated as: Every mathematical statement that is provable expresses purely structural information. We introduce a bi-modal framework that formalizes the notions of structural information and informal provability in order to draw connections between them and confirm that the epistemological structuralist thesis holds.
... Other systems of objects, like lattices, can be represented by a graph, their Hasse diagram. See alsoLeitgeb (2020aLeitgeb ( , 2020b.Content courtesy of Springer Nature, terms of use apply. Rights reserved. ...
We contribute to the ongoing discussion on mathematical structuralism by focusing on a question that has so far been neglected: when is a structure part of another structure? This paper is a first step towards answering the question. We will show that a certain conception of structures, abstractionism about structures, yields a natural definition of the parthood relation between structures. This answer has many interesting consequences; however, it conflicts with some standard mereological principles. We argue that the tension between abstractionism about structure and classical mereology is an interesting result and conclude that the mereology of abstract structures is a subject that deserves further exploration. We also point out some connections between our discussion of the mereology of structures and recent work on non-well-founded mereologies.
... But Bayesian nets, and graphs in general, are mathematical objects (e.g. Danks, 2014, p. 40;Leitgeb, 2020). They are defined as a finite set of nodes connected by a finite set of edges (Koski & Noble, 2009, p. 41) and sets, nodes and edges are mathematical objects. ...
Philosophers interested in the theoretical consequences of predictive processing often assume that predictive processing is an inferentialist and representationalist theory of cognition. More specifically, they assume that predictive processing revolves around approximated Bayesian inferences drawn by inverting a generative model. Generative models, in turn, are said to be structural representations: representational vehicles that represent their targets by being structurally similar to them. Here, I challenge this assumption, claiming that, at present, it lacks an adequate justification. I examine the only argument offered to establish that generative models are structural representations, and argue that it does not substantiate the desired conclusion. Having so done, I consider a number of alternative arguments aimed at showing that the relevant structural similarity obtains, and argue that all these arguments are unconvincing for a variety of reasons. I then conclude the paper by briefly highlighting three themes that might be relevant for further investigation on the matter.
According to the realist rendering of mathematical structuralism, mathematical structures are ontologically prior to individual mathematical objects such as numbers and sets. Mathematical objects are merely positions in structures: their nature entirely consists in having the properties arising from the structure to which they belong. In this paper, I offer a bundle‐theoretic account of this structuralist conception of mathematical objects: what we normally describe as an individual mathematical object is the mereological bundle of its structural properties. An emerging picture is a version of mereological essentialism: the structural properties of a mathematical object, as a bundle, are the mereological parts of the bundle, which are possessed by it essentially.
This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction instead simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations.
Рассматривается два варианта математического структурализма, один из которых строится на основании теории множеств, а второй – на основании модальной теории множеств. Показано, что теоретико-множественный структурализм сталкивается с рядом трудностей. В частности, он вступает в конфликт с принципом неопределенной расширяемости и с теоретико-множественным плюрализмом. В последней части статьи показывается, что конфликт с принципом неопределенной расширяемости решается при переходе к использованию модального теоретико-множественного подхода. При этом относительно адекватности решения трудностей с теоретико-множественным плюрализмом в контексте модального теоретико-множественного подхода остаются сомнения.
[Mathematical structuralism is characterized by the acceptance of the thesis that the subject matter of mathematics is structures. These structures can be understood in various ways. Structures, whatever they are, are “implemented” by specific collections of objects. One of the ways to classify different variants of mathematical structuralism relates to the possibility of constructing a theory of structures and how such a theory should be constructed. In other words, this method concerns whether it is possible to describe any mathematical structures using some unified mathematical theory, and what kind of mathematical theory it is. From the point of view of the set-theoretic version of mathematical structuralism, mathematics is the science of structures, which can be described using the apparatus of set theory. Specific mathematical theories “grasp” various kinds of set-theoretic structures, and the relation of realization that takes place between these structures and mathematical theories is supposed to be described by model-theoretic means. This version of structuralism presupposes a non-structural understanding of set theory, and the axioms of structure theory have a fixed interpretation. This version of set theory assumes that the entire universe of sets has already been built, that all stages of the cumulative hierarchy have already been realized, which conflicts with the very idea that every stage in the cumulative hierarchy is followed by the next stage. Another consequence of understanding the axioms of set theory as truths with respect to a fixed structure, as well as the fact that set theory as a basis receives a special status and is not interpreted in a structural way, is that such an approach does not imply any kind of set-theoretic pluralism. That is, only one set-theoretic structure is considered the only correct description of the universe of sets, while any alternative options proceeding from other axioms are discarded as false. In order to get rid of the conflict between the fact that the axioms of the theory of structures assume the universe of sets is already built, and all stages of the cumulative hierarchy of sets are already realized, and the principle of indefinite extendability, one can turn to modalities. Due to the fact that the axioms of set theory are understood in a modal way, it is possible to explain in a potentialist way the totality of all sets without the assumption that there is some universe of sets fixed once and for all. Accordingly, this is consistent with the principle of indefinite extendability. Set-theoretic pluralism still turns out to be significantly limited. That is, modal set-theoretic structuralism still does not provide a structuralist explanation for set theory.]
Relationism holds that objects entirely depend on relations or that they must be eliminated in favour of the latter. In this article, I raise a problem for relationism. I argue that relationism cannot account for the order in which non-symmetrical relations apply to their relata. In Section 1, I introduce some concepts in the ontology of relations and define relationism. In Section 2, I present the Problem of Order for non-symmetrical relations, after distinguishing it from the Problem of Differential Application. I also examine four main existing strategies to solve it. In Section 3, I develop my argument. The first step consists in arguing that—among those strategies—relationism can only accept directionalism. The second step consists in arguing that directionalism is affected by a serious problem: the Problem of Converses. I also show that relationists who embrace directionalism cannot solve this problem. In Section 4, I introduce and rebut several strategies on behalf of relationists to cope with my argument. In Section 5, I briefly draw some conclusions.
Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures.KeywordsSet theoryContinuum hypothesisHigher-order logicInformal rigour
