Continuity of Theory Structure: A Conceptual Spaces Approach

ABSTRACT

By understanding laws of nature as geometrical rather than linguistic entities, this paper addresses how to describe theory structures and how to evaluate their continuity. Relying on conceptual spaces as a modelling tool, we focus on the conceptual framework an empirical theory presupposes, thus obtain a geometrical representation of a theory’s structure. We stress the relevance of measurement procedures in separating conceptual from empirical structures. This lets our understanding of scientific laws come closer to scientific practice, and avoids a widely recognised deficit in current philosophy of science accounts, namely to risk a collapse of the physical into the mathematical.

Acknowledgements

We thank two anonymous reviewers as well as the journal’s editor for comments that helped to improve an earlier version of this manuscript. Peter Gärdenfors thanks the Vetenskapsrådet particularly for support to the Linneaus environment, Thinking In Time: Cognition, Communication and Learning. Frank Zenker acknowledges an Understanding China Fellowship from the Confucius Institute (HANBAN).

ORCID

Frank Zenker http://orcid.org/0000-0001-7173-7964

Notes

1 The relativism has mostly been ascribed, of course. But the ascription could arise because Structure did too little not to appear as endorsing a relativism Kuhn would later deny (Kuhn 1987, 2000; Larvor 2003).

2 Although Newton defined mass in terms of density and volume.

3 F refers to the force domain, m to mass, a to the second derivative of the spatial domain, x to one spatial dimension, P to pressure, V to volume, T to temperature, and so on.

4 Sneed–Stegmüller structuralism (Sneed 1971; Stegmüller 1976; Balzer, Moulines, and Sneed 1987, 2000) expresses this insight as: the partial potential models of CM, each of which features F = ma as a basic element, can be completed to full CM models; see Andreas and Zenker (2014).

5 To further support that scientific laws do not primarily concern propositions but relations between concepts, consider N. Goodman (1983) having distinguished law-like sentences from universal generalisations, here presuming a connection between ‘being law-like’ and ‘being confirmable by inductive inference’. Those who commit to a linguistic view of laws have since viewed induction as concerning inference, mostly analysing support relations between, again, sentences. P. Gärdenfors and A. Stephens (Forthcoming), by contrast, view induction as primarily concerning relations between properties and categories, that is, knowledge-what (rather than knowledge-how or knowledge-that), which cannot be reduced to knowledge-that.

6 Rather than treat the risk of collapsing this distinction, some have sought to embrace it, arguing that no account can be provided of what makes a structure physical rather than mathematical (e.g. Ladyman and Ross 2007).

7 Since a theory’s laws connect different domains, they also introduce constraints on combinations of measurement values. So provided measurement values for T-non-theoretical domains (e.g. in CM, time and position; see Sneed 1971) are obtained on entities the theory applies to (e.g. physical objects), the combination of these values should again satisfy specific laws (such as Newton’s second law for CM). So also combined values lie on the hypersurface representing T’s empirical content.

8 Masterton, Zenker, and Gärdenfors (2017) propose splitting this type into ‘change in the separability of dimensions’ and ‘change in the interference of dimensions’ to handle the non-classical properties of position and velocity in QM. The acceptability of this proposal depends on whether one interprets the non-commutative algebra (characteristic for QM) as distinct QM content formulated in an otherwise broadly ‘classical’ framework, F_CM, or whether one treats the non-commutative algebra as having changed F_CM itself, as F_CM developed into F_QM.

9 Friedman counts as a priori those principles situated on the second level, for instance, that express Newton’s integral 3D space with its Euclidian metric, and Newton’s separate 1D time, or Einstein’s integral 4D spacetime with its non-Euclidian metric.

Additional information

Funding

Both authors acknowledge funding from the Vetenskapsrådet (Swedish Research Council). Frank Zenker acknowledges funding through the European Union’s FP7 Framework Programme (grant number 1225/02/03) as well as the VolkswagenStiftung (grant number 90 531).