Equivalence and Reduction of Scientific Theories

(Affiliated Workshop, English)

**chedule.**

09:00--09:45 James Weatherall: *Theoretical Structure and Theoretical Equivalence*

09:45--10:30 Thomas Barrett: *Equivalent and inequivalent formulations of classical mechanics*

10:30--11:15 Sarita Rosenstock: *Categories and the Foundations of Yang-Mills Theory*

Coffee break

11:30--12:15 Laurenz Hudetz: *Definable Categorical Equivalence and Reduction*

12:15--13:00 Hajnal Andréka & István Németi: *Adventures in the
Network of Theories***Abstracts.****James Weatherall (Irvine): Theoretical Structure and Theoretical Equivalence**

Our physical theories often admit multiple formulations or variants. Although these variants are generally empirically indistinguishable, they nonetheless appear to represent the world as having different structures. In this talk, I will discuss some criteria for comparing empirically equivalent theories that may be used to identify (1) when one variant has more structure than another (i.e., when a formulation of a theory has ``excess structure'') and (2) when two variants are theoretically equivalent, even though they appear to represent the world differently. **Thomas Barrett (Princeton): Equivalent and Inequivalent Formulations of Classical Mechanics**

In this talk, I consider whether or not the Hamiltonian and Lagrangian formulations of classical mechanics are equivalent theories. I do so by applying a criterion for theoretical equivalence that was recently introduced into philosophy of science by Halvorson (2012, 2015) and Weatherall (2015). Discussion of this specific case yields two general philosophical payoffs. The first concerns the verdicts we make about equivalence of theories, and the second concerns how we might interpret what a physical theory ``says about the world.''**Sarita Rosenstock (Irvine): Categories and the Foundations of Yang-Mills Theory**

In this talk, I'll discuss the prospect of using a category theoretic notion of physical theory comparison, due to Weatherall and Halvorson, to evaluate the relationships between various formulations of classical Yang-Mills theory. I'll show how in this framework, a formulation of Yang-Mills theory using principal bundles is theoretically equivalent to one using holonomies, despite some claims by philosophers that the latter has less structure than the former. I'll also show that another formulation in terms of Wilson loops has strictly less structure than these, but contrary to the arguments of other scholars, this formulation does not preserve all of the gauge invariant content of the previous formations, which I argue is fully captured by their category theoretic structures.**Laurenz Hudetz (Salzburg): Definable Categorical Equivalence and Reduction**

Recently, categorical equivalence has been frequently considered as a fruitful criterion of theoretical equivalence (cf. Weatherall, 2015; Barrett, Rosenstock and Weatherall, 2015; Hudetz, 2015; Halvorson, 2016; Barrett and Halvorson, 2016; Weatherall, 2016; Halvorson and Tsementzis, 2016). Yet, the criterion of categorical equivalence is not free of problems. I show that categorial equivalence is too wide as a criterion of equivalence of theories. Then I propose a solution to this problem by specifying a strengthening of categorical equivalence, called `definable categorical equivalence'. This strengthened criterion employs the model-theoretic notion of definability in order to capture the idea that, if two theories are equivalent, it must be possible to reconstruct the models of one theory from the models of the other theory, and vice versa. I argue that the criterion of definable categorical equivalence constitutes an adequate explication. Finally, I show how to explicate reduction relations in terms of category-theoretic as well as model-theoretic notions along similar lines.**Hajnal Andréka & István Németi: Adventures in the
Network of Theories**

Network of theories is an efficient way of organizing scientific knowledge. Science can be thought of as a network of theories, where communication and division of labor between scientists can be achieved by various connections between theories. Team-work, and unity of science---dream of the Vienna Circle---can be realized by strenghtening the connections aspect of this network. Birth of new concepts, emergence and reduction of theories can be addressed in the context of this network.

In this talk, we will be concerned with theories written in first-order language (FOL) and interpretations between these theories. We understand FOL in a general way, e.g., many-sorted, modal, even Henkin-style higher order logics count as FOL. An interpretation is the act of refining the basic concepts of the language of a given theory by interpreting them as compound, derived concepts of another theory. We will use more general interpretations than is common today in logic. The novelty is that we can interpret both the universe of discourse---the kind of entities (objects) the theory talks about---, and the basic relations as derived, compound universes and derived compound relations, respectively. Thus, a kind of ``object-relation'' balance is restored and ontology gets refined, see e.g., Barrett and Halvorson 2016.

We want to illustrate that the above network is convenient to work with. We used it first in Madarász 2002 when we stated a precise logical equivalence between the observer-oriented and the observer-free approaches to special relativity. We had to face then that interpreting the universe of discourse is indispensable, and we elaborated tools for defining derived, compound universes (sorts, in logical terminology) in analogy with derived, compound relations. We will present two worked examples to some detail.

The first example is the equivalence of a 5-axiom FOL theory SpecRel of special relativity, and another 24-axiom theory SigTh for special relativity that uses meager resources as far as basic concepts are concerned. Both theories have advantages over the other. Interpreting SpecRel into SigTh gives, among others, an operational definition for building coordinate systems via using just signals. These investigations are being extended, in ongoing research, from special relativity to general relativistic space-times, e.g., to Schwarzschild black hole, de Sitter and anti-de-Sitter space-times. The second example is a research just started. It is an interpretation of the above 5-axiom SpecRel into a FOL-theory Maxwel of electrodynamics. This interpretation amounts to analysing the basic concept of a photon of SpecRel as electromagnetic wave (derived concept in Maxwel). To write up the axiom system Maxwel we use some ideas from Gömöri 2016.

-- Barrett, T. and Halvorson, H., From geometry to conceptual relativity. PhilSci Archive, 2016.

-- Madarász, J. X., Logic and Relativity (in the light of definability theory). PhD Dissertation, 2002. http://www.math-inst.hu/pub/algebraic-logic/diszi.pdf

-- Gömöri, M., The principle of relativity -- an empiricist analysis. PhD Dissertation, 2016. http://doktori.btk.elte.hu/phil/gomorimarton/diss.pdf

Chair: Laurenz Hudetz

Time: 09:00-13:00, 7 September 2016 (Wednesday)

Location: SR 1.005

James Weatherall

(UC Irvine, USA)

Professor of Logic and Philosophy of Science at the University of California, Irvine.

Thomas Barrett

(Princeton University, USA)

PhD student at Princeton University.

Sarita Rosenstock

(UC Irvine, USA)

PhD student at University of California, Irvine.

Laurenz Hudetz

(University of Salzburg, Austria)

PhD student at the University of Salzburg.

Hajnal Andréka & István Németi

(Alfred Renyi Institute of Mathematics, Budapest, Hungary)

Professors at the Alfred Renyi Institute of Mathematics, Budapest.

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