Abstract
Recent years have featured the existence of a variety of structuralisms, with an important partition between methodological versus philosophical structuralism. Inside philosophical structuralism, many trends can be identified, corresponding to various ontological stances. We argue here that another main partition has contributed to organize structuralism in the twentieth century, rooted in different technical and theoretical interests. This partition is largely transversal to the ones classically identified. Concretely, the paper will focus on possible differences between an arithmetical and logical notion of structure that can be traced back to the writings of Bertrand Russell and Rudolf Carnap, and a mathematical notion of structure, exemplified in the works by Bourbaki. This coexistence gives rise to a fundamental ambiguity that affects contemporary structuralism. Philosophically, in one case the attention is rather centered on a foundational and reductionist perspective, as featured by the Whitehead-Russell Principia and the Carnapian project of the Aufbau: the scientific construction of the world around the idea of structure. In the other, the focus is on epistemological and dynamical issues, as exemplified by two key issues in Bourbaki’s treatise: understanding the architecture of mathematics, offering a tool-kit to mathematicians. These two distinct meanings still coexist inside contemporary scientific practices and lead to different theoretical interests, as we will show thanks to various recent examples.
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Notes
- 1.
For a criticism of the assimilation between ante rem and Platonic structuralism see Folina (2020).
- 2.
See Bourbaki (1972), in particular Ch. III, section 3, on the transition from a Lie group to its Lie algebra and Ch. II, section 6, on the Hausdorff series.
- 3.
To which we will refer as “the Aufbau” from now on.
- 4.
On the Russellian origins of Carnap’s structuralism, see also (Schiemer 2020, 403 ff).
- 5.
In the same paragraph of the Aufbau, Carnap also mentions Cassirer’s theory of relational concepts (Cassirer 1910; 2nd ed. 1923, esp. 299). Cassirer’s structuralism refers to Dedekind and to Russell’s logic of relations. His contribution to (non-eliminative) structuralism is well documented, see in particular (Reck 2020).
- 6.
The idea of relation-numbers and of relation-arithmetic appears actually already in (Russell 1903, 321).
- 7.
(Russell 1919, II, 53 ff).
- 8.
(Russell 1919, 61 ff).
- 9.
We address here specifically the notion of structures as relation-numbers. For a general study of Russell’s structuralism and conflicts with structuralism, in particular Dedekind’s, and an overview of the recent debates on the subject, we refer to (Heis 2020). On complementary mathematical components of Russell’s project such as his theory of geometry and his doctrine of quantity, see Gandon (2012).
- 10.
In current terminology, total orders: “A relation is serial when it is asymmetrical, transitive, and connected. A series is the same thing as a serial relation” (Russell 1919, 34).
- 11.
“A relation is connected when, given any two different terms of its field, the relation holds between the first and the second or between the second and the first”. Russell does not exclude “the possibility that both may happen, though both cannot happen if the relation is asymmetrical” (Russel 1919, 33).
- 12.
About cardinal vs ordinal numbers in Russell, see (Heis 2020). Our forthcoming developments happen to be also relevant on these questions, as they show that the purely mathematical content of Russell’s relation-number arithmetic incorporates in a single natural framework cardinal and ordinal arithmetic.
- 13.
Recall that cardinal numbers can be viewed as relation-numbers associated to classes equipped with the trivial (or empty) relation. The definition we have just given extends ordinal arithmetics but not cardinal arithmetics as the process would equip the union of two sets A∪B with a non trivial relation. To extend cardinal arithmetic, given P and Q two (binary) relations defined respectively on A and B, one can define the (binary) relation T on A∪B whose restrictions to A and B are P, respectively Q, and such that aRb and bRa never hold for a in A and b in B, giving rise to another notion of sum. Mathematical insights on how these various constructions unravel in algebraic structures can be found for example in Foissy et al. (2016).
- 14.
As for sums, products of orders and relations can be defined in several ways. An example is provided by a simple generalization of the lexicographical ordering: assume that A and B are totally ordered sets. One can then equip the cartesian product A × B with the total order (a,b) < (a′, b′) ⇔ a < a′ or a = a′ and b < b′.
- 15.
(Carnap 1928, Preface to the second edition).
- 16.
Referred to as “the Aufbau” later on.
- 17.
- 18.
We do not discuss in detail these notions here. The reader may however grasp their meaning from the examples given below—this will be quite sufficient for our present purposes. A discussion of the theoretical background, the shortcomings of the article and more generally the work of Carnap on axiomatics can be found in (Schiemer 2000).
- 19.
“What Bourbaki considers important is communication between mathematicians; personal philosophical conceptions are not important to him” (Dieudonné 1982).
- 20.
Dieudonné has long been Bourbaki’s secretary. As long as he was active, he was the one putting the last hand to Bourbaki’s treatise volumes. He was also the main “public” figure of Bourbaki’s first generation of members: see, among others, (Dieudonné 1980, 1982). Dieudonné was also a popular writter of essays on mathematics and its history, he had a huge role in the reception of structuralism in France beyond the academia.
- 21.
- 22.
See Hersh (1979). The term “Platonist” here is of course much too strong and used simply as an indication of Bourbaki’s conviction of the robustness of mathematical contents and their relative independence on logical or empirical issues, the Introduction of (Bourbaki 2004) going without doubt in that overall direction.
- 23.
The combinatorial theory of orders which includes the arithmetic one, whose scope is more limited, is overall an important component of twentieth century combinatorics, see for example (Stanley 1986, Ch. 3, Partially Ordered Sets).
- 24.
- 25.
Other constructions study advanced algebraic structures on posets that include the generalization of the sum of two series as defined by Russell-Whitehead and the disjoint union of sets: that is, ordinal and cardinal sums. Recent examples of such constructions can be found in (Foissy et al. 2016), and (Foissy 2019).
- 26.
One might ask how the distinction between Russell and Carnap’s logical-arithmetic approach and Bourbaki’s mathematical-architectural approach articulates with other distinctions, such as that between set-theoretic and category-theoretic structuralism (Awodey 1996) or that between bottom-up and top-down approaches (Cole 2010). Each distinction originates from an attempt to solve a different problem: Awodey shows the inherently structural nature of the language of category theory, Cole underlines the ontological difference between an approach that starts from elements and an approach that starts from abstract systems. Our distinction emphasizes the difference between the logical effort to classify (the real, subobjects up to isomorphisms, and so on) and the mathematical goal of accounting for properties of objects as they occur in axiomatic theories, understanding the architecture of mathematics and offering a tool-kit to mathematicians.
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Cantù, P., Patras, F. (2023). Russell and Carnap or Bourbaki? Two Ways Towards Structures. In: Cantù, P., Schiemer, G. (eds) Logic, Epistemology, and Scientific Theories - From Peano to the Vienna Circle. Vienna Circle Institute Yearbook, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-031-42190-7_9
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