Abstract
Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity.
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Notes
The first volume of Philosophy of Science had been published only the year before.
See also Mormann (2013, pp. 423, 426). Although Mormann emphasizes the philosophy of science, he evidently understands this broadly, including within it the philosophy of geometry and space. This is why there is no harm in his unqualified reference to “philosophy,” and we shall often follow this practice.
This is a common conception of topology among philosophers. For instance, Callender and Weingard write that “Topology is a kind of abstraction from metrical geometry” (1996, p. 21).
See also Mormann (2013, p. 431): “neither Russell nor any other philosophers of science ever took notice of.
the path-breaking work of the American mathematician Marshall H. Stone.” Russell may not be as culpable as Mormann states, since by the time Stone had announced and published his results in 1934–1937, Russell was no longer actively working on these topics.
Although this term finds its origin in the work of Leibiniz, by the time of Russell and Carnap’s writings, it had become synonymous with topology. Indeed, Henri Poincare’s influential early work in topology bore the title “Analysis Situs” (Poincaré, 1895). This work has been noted as an originator of modern topological concepts; mathematician and historian of mathematics Jean Dieudonné has claimed that before this work of Poincaré we can only speak of the pre-history of the discipline (Dieudonné, 2009, p. 15). Oswald Veblen’s textbook of topology was also titled Analysis Situs (Veblen, 1922). Additionally, it is clear from the context in Analysis of Matter that Russell took the terms to be synonymous; he first mentions analysis situs immediately prior to introducing Hausdorff’s definitions (Russell, 1927, pp. 295–96).
Key examples include An Essay on the Foundations of Geometry (Russell, 1897), which Russell wrote on fellowship shortly after completing his BA in mathematics (Wahl, 2018: pp. x–xiv), and his earlier work “The Logic of Geometry” (Russell, 1896), which concerns the nature of the Euclidean axiom of congruence. He argues (against Helmholtz) that this axiom is strictly a priori and as such it cannot be proven through experience of objects in space. As a result, he concludes that rejecting this axiom is absurd from both a logical and philosophical standpoint. In support of these results, Russell provides argumentation that is both philosophical and mathematical.
Here, Russell credits Whitehead with this approach to points without citing a specific work. But as noted by Mormann (2013, p. 430), Russell elsewhere credits a specific work by Whitehead in reference to his construction of points: the fourth volume of the Principia. This work never saw publication. As a result it is not possible to consult Whitehead’s point construction in his own words, at least not in that particular work. However, in his “On the Mathematical Concepts of the Material World” (Whitehead, 1906), we do find a strikingly similar construction to the one recited in OKEW. Whitehead’s construction of points (Whitehead, 1906, pp. 488–492) builds points out of a theory of interpoints, which are similar to the enclosures described by Russell. Whitehead’s techniques did not involve topological concepts. Incidentally, that work by Whitehead cites Veblen, who would later become well known for his work in topology. However, the citations are to Veblen’s dissertation which concerned geometry and predated his work as a topologist. There does not seem to be evidence that the topological notions employed by Russell later in The Analysis of Matter had their origin in Whitehead’s point construction.
Technically speaking, this definition captures what we would now call a Hausdorff neighborhood basis. Condition (D) is the Hausdorff condition, while a neighborhood system is generated by closing the basis under the superset relation.
Indeed, his most famous student was Kurt Gödel.
For the most focused critical discussion of Russell’s scholarship regarding the construction of points, see Mormann (2008). Note also that Russell’s attempted construction of points bears resemblance to later approaches to mereotopology using (ultra)filters, for which see footnote 22.
It is worthwhile to acknowledge that one of Carnap’s early philosophical influences came not from the analytic tradition as he explicitly invoked the Husserlian concept of Wesenserschauung in his characterization of intuitive space in “Der Raum” (Ryckman, 2007, p. 103; Friedman, 2019, p. 204). Nevertheless, historians also give good reason to understand the early Carnap as operating squarely in the early analytic tradition. Gottfried Gabriel points out that Carnap himself gave “greatest tribute” to Frege out of his influential teachers during his youthful time in Jena and Frieberg from 1910 to 1914 (Gabriel, 2007, pp. 65–66), among other nuances of the relationship between Carnap and the “grandfather of analytic philosophy.” Additionally, in his editorial notes that accompany a new translation of “Der Raum,” Michael Friedman cites the shadow cast by Russell and Whitehead’s Principia in Carnap’s own pointers to relevant literature in his account of formal space (Friedman, 2019, p. 174). If we take Russell to be a progenitor of analytic philosophy, then his influence on Carnap cited here is pertinent when attempting to categorize Carnap’s early work.
Space: A Contribution to the Theory of Science.
Foundations of Geometry.
Carnap, like his predecessors, does not clearly distinguish between a manifold’s topological and differentiable structure, the distinction between which would only come about a decade later (Veblen and Whitehead, 1931, 1932). Nevertheless, even though formal space has the standard differentiable structure associated with products of the real line, Carnap does focus his attention on its topological properties.
It is worth noting that Friedman argues that Carnap’s approach is substantially flawed. But, for our present purposes we need not weigh in on the efficacy of the structure of intuitive space in Der Raum (Friedman, 2019, pp. 187–188).
This work was later translated as The Axiomatization of the Theory of Relativity (Reichenbach, 1965).
For more on geometrical themes in Carnap’s early work, see Mormann (2007).
We agree with Mormann’s comments insofar as they apply to geometric topology specifically. In this sense, his assertions apply in the context of the historical figures of interest in this section (Russell and Carnap). However, we will see in Sect. 3 that this view of the relationship between geometry and topology is not conceptually required by the features of topology. Starting in the 1930s and more so in the 1940s, the connections between topology and algebra became pronounced.
Indeed, Poincaré’s famous namesake conjecture is a noteworthy early problem in topology.
It was also in that year that Whitehead moved to the philosophy department at Harvard, and as noted in the previous subsection, Whitehead influenced Russell’s engagement with topological ideas. It would be worth investigating further whether Whitehead had any influence on Franklin or Wiener.
For another account of topological regions that includes a study of maps between regions, see Roeper (1997).
Roeper also cites an earlier predecessor, by Arnold Johanson, that takes a similar approach using category theory (Johanson, 1981). Johanson cites Whitehead and Russell’s TAM, among others, as inspiration.
Stone himself had called them Boolean spaces for obvious reasons (Johnstone, 1982, p. xvi).
Alternatively, one can proceed from the Boolean algebra’s prime ideals, which are dual to the ultrafilters and more faithful to Stone’s original presentation (Johnstone, 1982, pp. xv–xvi).
For instance, they are exactly the spaces that are totally separated and compact. See Johnstone (1982, pp. 69–70) for a number of other, equivalent characterizations.
Grosholz (1985, Sec. 2; 2007, chap. 10.3) provides a comparison of Tarski and Stone’s work in this area and offers an account of the resulting significance of topology for logic as a branch of mathematics. In particular, she comments on the initial limitations of the analogy between topology and logic, which was initially restricted to quite limited logics and topological spaces, that had to be overcome for it to become widely fruitful.
What follows is not a comprehensive statement of the extensive publications by these authors and other scholars working on topological formal learning theory and epistemic logic. Rather, we intend merely to introduce the reader to one fruitful branch of intellectual development that has arisen from the introduction of topology into philosophy.
Again, this just scratches the surface of the application of topology in mathematical logic. See also, for instance, Grosholz (2007, chap. 10.5) for an accessible treatment of some connections between topology and model theory, and Aiello et al. (2007) for a variety of surveys of spatial logics that employ topology.
In this chapter on topology, Kelly also draws on theoretical resources found in the literature related to recursion and decidability (Gold, 1965; Kugel, 1977; Putnam, 1965). Although Kelly’s application of these ideas is interesting, our focus in this section remains on the use of topology as facilitated by the conceptual connections derived from Stone’s theorem. For more on the connections between recursion theory, topology, and the Borel hierarchy, see Grosholz (1985, Sec. 3; 2007, chap. 10.4).
Additionally, van Benthem gestured towards the future directions that might be taken by combining aspects of his project in Logical Dynamics of Information and the formal learning theory of LRI (van Benthem, 2011, p. 249).
Mormann also draws another contrast, that “20th century philosophy of science showed no interest in topology as an object of philosophical reflection. There has been no ‘philosophy of topology’ in analogy to disciplines such as ‘philosophy of physics’, ‘philosophy of biology’, or ‘philosophy of geometry’” (Mormann, 2013, p. 425). However, the analogy is not as strong as it might at first seem, since physics and biology are distinct sciences, while geometry historically has been a partly mathematical, partly empirical discipline. Topology, by contrast, is clearly a subdiscipline of mathematics.
Mormann (2020, Sec. 5) also shows how to formulate different versions of Leibnizian identity principles for objects in topological terms (i.e., using topological separation axioms). We also believe that the topological structure employed in such cases should be interpreted in terms of similarity (of the objects involved) instead of in terms of geometry.
We defer to Fletcher (forthcoming, Sec. 2) for the technical details.
Mormann (2020, p. 144) acknowledges this particular interpretation: “Every conceptual space is endowed with a metric that measures the similarity between objects.” But he does not seem to extend that interpretation to topology more generally.
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This article belongs to the topical collection “Metaphilosophy of Formal Methods”, edited by Samuel C. Fletcher.
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Fletcher, S.C., Lackey, N. The introduction of topology into analytic philosophy: two movements and a coda. Synthese 200, 197 (2022). https://doi.org/10.1007/s11229-022-03689-9
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DOI: https://doi.org/10.1007/s11229-022-03689-9