Workshop on Invariance, Objectivity, Symmetry

In order to register for the workshop, please contact office.hartmann@lrz.uni-muenchen.de

Program

Abstracts

Francesca Biagioli: “Mathematical Structuralism and the Relativization of the Kantian A Priori”

The idea of a relativization of the Kantian a priori to the presuppositions for the possibility of scientific inquiries – first articulated by Cassirer and others in connection with the revolutions of physics in the early twentieth century – has been enjoying a revival in more recent philosophy of science. However, it is being questioned whether this view provides sufficient resources to account for scientific change. Given Cassirer’s commitment to a structuralist account of mathematics in its internal development as well as in its scientific applications, it seems that the only strategy available to him and his followers is to establish an abstract relation of approximate inclusion between succeeding theories. This paper aims to explore a somewhat different strategy inspired by Cassirer’s functional model of concept formation in some key examples from his epistemological works, where the focus is on how structural procedures from nineteenth-century real analysis, analytic and projective geometry can find a fruitful expansion and a variety of applications to nonmathematical domains. My suggestion is that Cassirer’s account sheds light on the role of mathematics in the relativization of the Kantian a priori, and is of particular interest in the philosophical discussion of the epistemological implications of cases where there is a cross-fertilization between different branches of mathematics, as well as a co-evolution of mathematical and physical theories.

Elena Castellani: “Objects, Structures and Symmetries” (CANCELED)

The crucial role acquired in contemporary physics by symmetry principles and their exploitation through group theory techniques has motivated a group-theoretic approach to the question of characterising objectivity and, in particular, physical objects. In the philosophical literature, this approach – initially inspired by the seminal works of Hermann Weyl and Eugene Wigner and well-grounded in scientific practice – has often been taken to imply ontological commitments about the nature of physical reality (e.g., ontic structural realism). In this talk I will discuss the legitimacy of these kinds of inferences, especially in regard to such objects as the so-called elementary particles of fundamental physics.

John Dougherty: “Van Fraassen and McDowell on Empiricism and Objectivity”

Twentieth century criticisms of epistemic foundationalism pose a problem for traditional conceptions of empiricism, which claim that sense experience is the foundation of our knowledge. Two versions of empiricism that have been developed in response to these criticisms are those of Bas van Fraassen and John McDowell, who are each concerned to avoid what Wilfrid Sellars termed the “Myth of the Given”. In this talk, I suggest that McDowell's treatment of the objectivity of experience can help to answer objections to the effect that van Fraassen’s notion of “philosophical stance” is too relativistic.

Marta Esteves: “A Higher-Order Logic Theory of Sui Generis Structures: A Case Study and a Generalization”

Structuralism – a theory according to which mathematics is the study of structures – has received considerable attention in the last few decades (Chakravartty 2012; Doherty 2019; Schiemer and Wigglesworth 2019; Wigglesworth 2018). We subscribe to non-eliminative structuralism, a theory according to which structures exist independently of the systems that instantiate them, in that we search for a logical description of sui generis structures. We continue the work (started by Shapiro 1991, 1997) towards a higher-order logic theory of sui generis structures, based on the strategy used by Leitgeb (2020). Given that algebraic structures, albeit their importance in mathematical practice, have not received much attention in the structuralist literature, we will start by presenting the case study of groups, where we will axiomatize fundamental group theoretic notions using the language of higher-order logic. We will then extend this case study to a general theory of structures, by attempting to characterize the notion of a general sui generis structure, using an extension of the previous language to describe and adapt relevant category-theoretic notions. Finally, we will discuss the consequences of both the successes and the limitations of these axiomatizations for non-eliminative structuralism.

Johannes Korbmacher and Georg Schiemer: “Logical Concepts and Structural Properties”

Our starting point in this talk is the observation of a striking similarity between how philosophers of logic define logical concepts and how philosophers of mathematics define structural properties. To illustrate this observation, consider invariance based accounts. There is the so-called Tarski-Sher thesis, which states that a concept is logical if and only if its extension is invariant under arbitrary permutations of the domain of objects. Compare this to what we may call the Carnap thesis, which states that a property is structural if and only if its extension is invariant under taking isomorphic copies of objects. In both cases, invariance of extensions under a class of bijective functions is what matters. Moreover, both logical concepts and structural properties have been tied to definability. In the case of logical concepts, several results by McGee and Bonnay connect logicality under the Tarski-Sher thesis to definability in purely logical languages. Similarly, authors on mathematical structuralism have claimed that we can view structural properties as properties we can define from the basic properties and relations on structures. These similarities are not by accident. We claim that there is a close and systematic connection between logical concepts and structural properties. Based on a closer discussion of different invariance and definability criteria for both types of concepts, we first argue that logical concepts can be viewed as a kind of limit case of structural properties. The other way around, the claim is that structural properties are simply logical properties in the formal languages of mathematical structures. We discuss this claim in detail and show in what sense it vindicates a particularly structuralist account of logicism.

Hannes Leitgeb: “The Objectivity of Epistemic Reasons”

The philosophical literature on reasons usually speaks of a reason as being objective when some kind of factivity condition is satisfied. But one may also ask the following question, which concerns the objectivity of epistemic reasons in a different sense: do epistemic reasons only speak for propositions (with some strength) relative to a subject’s belief state, or does it make sense to say that epistemic reasons speak for certain propositions (with some strength) independently of a subject’s belief state? That is the question I am aiming to answer in my talk.

Christian List: “Consciousness and Objectivity”

Science aims to give us an objective picture of the world, a picture that does not depend on the perspective from which we look at the world. The inventory of facts as catalogued and explained by science is supposed to be invariant under changes in the perspective taken. Although this approach works well in most areas of science, it remains challenged when it comes to the explanation of consciousness. While some scholars think that conventional science will ultimately be able to explain consciousness too, others remain skeptical. In this talk, I will suggest that, insofar as conscious experience is irreducibly subjective, no objective picture of the world can do justice to it, and I will discuss an alternative picture, which recognizes subjective facts. To accommodate such facts, I will argue, we must embrace the idea that there is not just one objective world but several subjective ones. What we call “the objective world” is an abstraction and can be viewed as an equivalence class of all those subjective worlds that are equivalent with respect to all objective facts but may differ with respect to some subjective facts.

Alyssa Ney: “From Dynamical Invariances to Spacetime Structure”

One stumbling block for highly revisionary metaphysical proposals is the difficulty of showing how they can ground the ordinary macroscopic reality we know to be real. This challenge has been thought to be especially daunting for highly revisionary metaphysical interpretations of quantum physics that lack a fundamental spacetime framework. It is common to try to ground the existence of spacetime structure and consequently macroscopic objects in dynamical invariances of the fundamental quantum state. This paper examines two strategies along these lines in order to assess the extent to which they successfully address the explanatory challenges that arise when one eschews fundamental spacetime structure.

Alexander Reutlinger: “Articulating Invariantism. Revisiting the Counterfactual Independence Account of Scientific Objectivity”

Invariantism defines scientific objectivity via the notion of invariance. Inspired by Nozick’s work on objectivity, I will present a version of invariantism, according to which the key notion of invariance is spelled out more precisely as a specific sort of counterfactual independence (building on Reutlinger 2021). This invariantist view – the counterfactual independence account of objectivity – is intended to define a specific notion of objectivity: epistemic objectivity (aka objective evidence). Despite certain advantages, the counterfactual independence account of objectivity deserves to be articulated in a more nuanced manner. To do so, I will address the following questions concerning epistemological aspects of objectivity: What can be learned from Nozick’s examples of objectivity? How and why precisely does epistemic objectivity play a confirmatory role in science, given extant theories of confirmation? In what sense must objectivity be free of epistemic error? Can the counterfactual independence account be extended to capture the concept of objective belief?

Benjamin Zayton: “Two Generalisations of Bi-Interpretability”

Bi-interpretability is a well-understood and commonly accepted notion of equivalence for first-order theories. Although it is typically presented in a model-theoretic guise, it is amenable to a treatment in terms of syntactic translations between theories: Bi-interpretability is intertranslatability. A translation between two theories is a recursively extendable mapping from atomic formulas of the language of the first theory to defined formulas of the language of the second theory, which preserves theoremhood. In this talk, I will discuss two generalisations of the usual notion of interpretation and the induced generalisations of bi-interpretability. The first, bi-interpretability with parameters, is commonly used by model theorists, but has not yet been studied in terms of syntactic interpretations with parameters, which allow for the use of additional parameters in defining formulas. The second, piecewise interpretation, allows for definitions with multiple definientia – such definitions were considered by e.g. Carnap in his work on bilateral reduction sentences. It turns out that the resulting notion of bi-interpretability coincides with Morita equivalence, another recently developed notion of equivalence. Both of these notions have in common that they seem to outstrip the usual bounds of definability in translating formulas, and possible justifications for adopting these standards of equivalence will be put forward in the talk.