(Joint work with Igor Douven.)

In the conceptual spaces framework, natural concepts are often modeled as convex regions within a similarity space—an assumption motivated by the principle of cognitive economy. Convexity is thought to enhance learnability, making such concepts easier to acquire than those that are represented by regions that satisfy topological criteria that are less stringent. In this talk, I critically examine this hypothesis by comparing the learnability of convex concepts to that of merely connected ones. I will present findings from both computational studies using neural networks that are supposed to approximate human concept learning and behavioral experiments with human participants. All studies were conducted within a shape-based similarity space designed to represent various types of containers

Most theories of concepts focus on first-order similarity—how individual instances share features or occupy nearby regions in multidimensional space. However, many conceptual phenomena hinge on detecting patterns of feature covariation rather than simple overlap (Richardson 2019; Solomon & Schapiro 2024). For example, two animal categories may differ in their average feature values yet share analogous internal covariation structures, supporting analogical reasoning and robust generalization. This talk explores the role of covariation in terms of higher-order similarity, arguing that structural similarity across covariational patterns is a crucial yet underappreciated dimension of conceptual organization. I argue that data analysis techniques such as principal component analysis can help formalize this kind of similarity. This perspective accounts for the formation of overhypotheses and structural phenomena such as consistent contrast and value systematicity (Billman & Davies 2005, Dewar & Xu 2010). By moving beyond first-order similarity to the structure of relations among features, we arrive at a two-tier model of conceptual knowledge: intra-concept coherence for local prediction and inter-concept covariational alignment for efficient generalization. This framework might help explain why some conceptual systems are easier to learn, transfer, and remember.

Johnson (1932) and Carnap (1950) independently derived formal versions of enumerative induction that allow for different initial priors and arbitrary sensitivity to new evidence. Though this achievement was monumental for inductive logic, the Johnson-Carnap system could not account for the effects of analogical influences on inductive inferences. The most notable criticism of the system with respect to analogical influences comes from Achinstein (1963). The contribution of my paper is twofold: first, I provide a conceptual mapping of the treatment of analogy in the inductive logic tradition, from Hosiasson-Lindenbaum (1941) and Carnap (1945; 1980) to modern commentators. In particular, I distinguish three kinds of extensions of Carnap’s inductive logic system with respect to analogies: axiomatic (e.g. Huttegger 2019), parametric (e.g. Romeijn 2006), and geometrical approaches (e.g. Sznajder 2021). The second contribution of this paper is to argue that the literature was operating on a productive but fundamental mistake: the extensions of the Johnson-Carnap system did not capture the analogical influence Achinstein had in mind. A recent paper by Huttegger (2019) captures this kind of influence but does not highlight it.

The contemporary debate on Carnapian conceptual engineering largely hinges on the late Carnap’s notion of explication. By contrast, the early Carnap’s notions of rational reconstruction and quasi-analysis have received comparatively little attention. Emphasis has been placed on the continuity (Dutilh-Novaes 2020) or discontinuity (Carus 2007) between rational reconstruction and explication. So far, the comparisons between rational reconstruction and explication have not taken into account the scientific revaluation of Carnap’s quasi-analysis, as developed by Mormann (2009) and Leitgeb (2007). In this contribution, I aim to bridge this gap by drawing a comparison between explication and rational reconstruction that takes into account the mathematical revaluation of Carnap’s quasi-analysis.

I will argue for a thesis of discontinuity between rational reconstruction and explication. Specifically, I contend that (1) rational reconstruction and explication are distinct kinds of conceptual constructions. Moreover, I contend that (2) rational reconstruction exhibits philosophical virtues in responding to objections commonly raised against Carnapian conceptual engineering. To demonstrate (1) and (2), I assume that explication is equivalent to Carnap’s (1950) performances of explication. I also assume that rational reconstruction is equivalent to Carnap’s performance of quasi-analysis in the Aufbau (1928).

Nelson Goodman formulated his general strictures against similarity as an explanatorily useful concept as a reaction to the usage of similarity as a fundament of cognitive development in empiricist philosophy and in Carnap’s Aufbau. His criticism influenced not only philosophers, but also cognitive scientists, who vividly debated on the role and nature of similarity since the 1970ies. One of the critical issues of the debate is the extent to which similarity itself is really fundamental or rather a result of reasoning, categorization and other cognitive processes. This talk aims to revisit the philosophical history of similarity as a foundational notion of explaining cognitive development. In this talk, I explore and cautiously defend the view that some variant of similarity should be considered as a fundamental notion, in line with Carnap’s original framework and the tradition of classical empiricism.

Theoretical virtues such as simplicity and explanatory power are valued by scientists because they enhance the credibility of scientific theories. Philosophers have long been interested in how these virtues attain their positive epistemic value (Douglas, 2013). Mathematicians also discuss virtues such as simplicity and explanatory power, but they often emphasize mathematical beauty as the primary theoretical virtue that drives their research (Engler, 1990). Although beauty is also valued by scientists (Ivanova, 2017), the ways in which mathematicians rely on beauty appear to be different. As Hardy (1940/2012, §10) put it, there is no permanent place in the world for ugly mathematics.

To explain its alleged epistemic value, mathematicians and philosophers have proposed various definitions of mathematical beauty. However, these definitions often conflict with each other and fail to accurately describe how mathematicians use terms such as “beautiful” or “elegant” (Inglis & Aberdein, 2014). This discrepancy creates a mismatch between the importance mathematicians place on beauty and the apparent failure of philosophical accounts to capture its positive epistemic value.

In this talk, I propose a solution to this mismatch. I argue that mathematicians, like scientists, rely on multiple theoretical virtues when judging proofs, theorems, and conjectures. Not all of these virtues have positive epistemic value: some are truth-conducive, while others may serve merely pragmatic purposes. Theoretical virtues in mathematics are therefore a diverse set, and beauty is only one of them.

To support this argument, I take a two-step approach. First, I’ll review prominent definitions of beauty found in the literature. I’ll argue that they fail to explain its epistemic value and fail to generalize. Second, I’ll argue that this failure can be explained by taking the components of the definitions at face value: they are all to be found in the diverse set of virtues that mathematicians rely on when judging proofs.

Sources

Douglas, H. (2013). The Value of Cognitive Values. Philosophy of Science, 80(5), 796–806. https://doi.org/10.1086/673716

Engler, G. (1990). Aesthetics in Science and in Art. The British Journal of Aesthetics, 30(1), 24–34. https://doi.org/10.1093/bjaesthetics/30.1.24

Hardy, G. H. (2012). A Mathematician’s Apology. Cambridge University Press. https://doi.org/10.1017/cbo9781107295599 (Original work published 1940)

Inglis, M., & Aberdein, A. (2014). Beauty Is Not Simplicity: An Analysisof Mathematicians’ Proof Appraisals. Philosophia Mathematica, 23(1), 87–109. https://doi.org/10.1093/philmat/nku014

Ivanova, M. (2017). Aesthetic values in science. Philosophy Compass, 12(10), e12433. https://doi.org/10.1111/phc3.12433

False dichotomies such as ‘proof vs. explanation’ or ‘argument vs. proof’ as described by Müller-Hill (2019) might result in encountering an impasse, when looking at practices in the mathematics classroom. Viewing mathematical practices as cultural practices reveals that explanation, argumentation and proving can all be observed and interpreted as situated practices in the context of proof and explanation in the mathematics classroom (Cobb & Bauersfeld, 1995; Krummheuer, 1995; Yackel & Cobb, 1996). Although conceptual clarity has merit, describing mathematical classroom processes and practices demands a more dialectical and integrative understanding of these concepts. The enactment of these practices by teachers and students varies considerably, reflecting cultural traditions, educational conventions, and the scripted interactions that characterize different classroom contexts (Zhuang, & Conner, 2022). These mathematical practices are thus fundamentally cultural practices, shaped by the social and institutional contexts in which they occur. The reconstruction and analysis of classroom practices offers an empirical basis for conceptual clarifications that enhance our understanding of both the practices and their theoretical underpinnings (Bredow & Knipping, 2023; Conner, Tabach, & Rasmussen, 2023; Knipping & Reid, 2019). Understanding mathematical practices such as ‘argumentation’, ‘proof’ and ‘explanation’ as cultural practices allows us to research classroom mathematical practices through an integrated educational and mathematical lens, treating these as practices that emerge from the intersection of mathematical content and social context.

References

Bredow, F., & Knipping, C. (2023). Teacher actions framing argumentation in the mathematics class. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi & E. Kónya (Eds.), Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13) (pp. 80-87). Alfréd Rényi Institute of Mathematics, Budapest, Hungary and ERME.

Cobb, P., & Bauersfeld, H. (1995). The emergence of mathematical meaning. Erlbaum.

Conner, A.M., Tabach, M. & Rasmussen, C. (2023). Collectively engaging with others’ reasoning: Building intuition through argumentation in a paradoxical situation. Triangle. International Journal of Research in Undergraduate Mathematics Education, (9), 666–693. https://doi.org/10.1007/s40753-022-00168-x

Knipping, C., & Reid, D. A. (2019). Argumentation Analysis for Early Career Researchers. In G. Kaiser & N. Presmeg (Eds.), Compendium for Early Career Researchers in Mathematics Education (pp. 3–31). Springer. https://doi.org/10.1007/978-3-030-15636-7_1

Krummheuer, G. (1995). The Ethnography of Argumentation. In P. Cobb & H. Bauersfeld (Eds.), The Emergence of Mathematical Meaning (pp. 229–269). Routledge.

Müller-Hill, E. (2019). Explanatoriness as a value in mathematics and mathematics teaching. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11) (pp. 276–283). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.

Toulmin, S. E. (1958). The Uses of Argument. Cambridge University Press.

Yackel, E., & Cobb, P. (1996). Sociomathematical Norms, Argumentation, and Autonomy in Mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. https://doi.org/10.2307/749877

Zhuang, Y., & Conner, A. (2022). Teachers’ use of rational questioning strategies to promote student participation in collective argumentation. Educational Studies in Mathematics, 111. https://doi.org/10.1007/s10649-022-10160-6

Wenn von einem ‛practical turn’ in der Philosophie der Mathematik gesprochen wird, so kommt damit normalerweise die (reale) Forschungspraxis der Mathematik in den Blick, nicht jedoch die Praktische Philosophie. Im Kontrast dazu soll im Vortrag in systematischer (und ggf. historischer) Perspektive explizit nach einem wechselseitigen Verhältnis von Ethik und Mathematik gefragt werden. Die politische Relevanz einer solchen Fragestellung wird in den letzten Jahren vermehrt gesehen und diskutiert (vgl. etwa Zuboff, O’Neil); immerhin ist die prägende Wirkung mathematischer Strukturen – insbesondere implementierter Algorithmen – für die Gestaltung moderner Gesellschaften mittlerweile kaum noch zu übersehen. Nach einer kurzen Sichtung der in dieser Hinsicht relevanten, konkreten Handlungsfelder soll darüber hinaus auf einer abstrakteren Ebene gefragt werden, in welcher Weise sich Ethik und Mathematik auf ihre jeweiligen (verwandten bzw. grundlegend verschiedenen) Gegenstände beziehen und welche Formen des (verwandten bzw. grundlegend verschiedenen) disziplinären Diskurses dazu entwickelt werden. Insofern Mathematik wie Ethik universelle (menschliche) Beobachtungs- und Beurteilungsmittel sind, wäre ein umfassender Vergleich von ähnlich universeller Reichweite; der Vortrag kann somit bestenfalls erste Überlegungen zu dieser Thematik diskutieren.

In the current literature on AI, at least two notions of “explainability” occur, one ‘extrinsic’ and one ‘intrinsic’: extrinsic explanations, on the one hand, occur when AI systems produce outputs which are taken by human agents as ‘explanations’ of certain states of affairs in a given sociotechnical environment (e.g. decision support systems in law or medicine). Intrinsic explanations, on the other hand, call for ‘causal explanations’ of certain outputs of a given system, taking into account the formal structure of the system as a technological artifact. In practical contexts, it turns out that the boundaries between both of these dimensions of explanation often seem to be blurred, while their underlying objectives vary significantly – this fact becomes especially problematic if ethical considerations are involved.

Based on a short review of recent philosophical literature on explanations, the talk provides an overview over the explainability debate both in ethics and mathematics/technology and places it within the broader context of an empirically informed philosophy of technology and mathematics. It then argues for an integrated account of AI explainability.