Abstract
There is a great deal of justified concern about continuity through scientific theory change. Our thesis is that, particularly in physics, such continuity can be appropriately captured at the level of conceptual frameworks (the level above the theories themselves) using conceptual space models. Indeed, we contend that the conceptual spaces of three of our most important physical theories—Classical Mechanics (CM), Special Relativity Theory (SRT), and Quantum Mechanics (QM)—have already been so modelled as phase-spaces. Working with their phase-space formulations, one can trace the conceptual changes and continuities in transitioning from CM to QM, and from CM to SRT. By offering a revised severity-ordering of changes that conceptual frameworks can undergo, we provide reasons to doubt the commonly held view that CM is conceptually closer to SRT than QM.
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Notes
Connectionism (Bechtel and Abrahamsen 2002) is a special case of associationism that models associations using artificial neuron networks.
Given a temporal dimension, the qualitative evolution of an object can be represented by a trajectory through the conceptual space.
Quality dimensions can be continuous or discrete, and discrete spaces can also have metrics allowing measures of similarity. For instance, if three objects, A, B, and C, differ along two discrete dimensions such that A differs from B only along one dimension, but A differs from C along both dimensions, then A is more similar to B than A is to C, which is to say that the distance from A to B in the (Euclidean) plane of these dimensions (taking the value 1) is less than the distance from A to C (taking the value \( \sqrt{2} \)) in that plane.
Holonomic system constraints can be expressed by a zero-valued function like \( f\left({x}_1,\;{x}_2,\;{x}_3, \dots, {x}_n,t\right)=0 \). It is characteristic of holonomic constraints, moreover, that they do not have velocities as arguments. For instance, the constraint \( {x}^2+{y}^2-{L}^2=0 \) on the position of a 2D pendulum of fixed length L is holonomic. A constraint that cannot be expressed in the above form is a non-holonomic constraint.
Those inclined to grapple with scientific realism, of course, take the question ‘Which dimensions are basic, which derivative?’ to be one of principle. When Isaac Newton introduced the force domain into mechanics, for instance, many natural philosophers advocated—on grounds of ontological suspicion—a reformulation that made force a defined domain. Similarly, those following Niels Bohr in claiming that all measurements are ultimately spatio-temporal determinations, from which the values along other dimensions are inferred, view space-time as the basic domain. Whether conventional or principled, the division itself is (with the possible exception of such “Babylonian” physicists as Richard Feynman) present, and important to observe.
We have previously argued that Kuhn’s account unduly assumes the primacy of the symbolic level of representation (Zenker and Gärdenfors 2015).
This version of the criterion is a strengthening of the criterion proposed in Gärdenfors and Zenker (2011, 2013) where only changes of scales; e.g., the change from Celsius to Kelvin, were considered. This strengthening corresponds to a more severe change of a theory, so we now put it after changes in the importance of dimensions.
Phase-spaces have been predominantly used to represent the degrees of freedom of complex systems and to model chaotic behavior. This use is related to their use in formulating physical theories in the tradition of Boltzmann (that we are interested in here) but often obscures that original purpose. For a historical introduction see Nolte (2010). For a brief introduction to classical and quantum phase-space see Tao (2007).
The phase space formulation of quantum mechanics is particularly difficult to work in relative to the other formulations of that theory.
There is a lesson for psychology, and other disciplines using conceptual spaces, here. In the literature on conceptual spaces one does not see the multiplying of dimensions with the multiplying of objects being located in the conceptual space, but if our thesis—that phase-spaces are just an example of a conceptual space—is correct, then this aspect of conceptual spaces should be recognized more widely.
Besides our own, for instance, we know of no other work on non-Euclidean conceptual spaces, which is obviously needed if one admits that the conceptual space of relativity theory is non-Euclidean.
One intriguing possibility that we cannot explore further here is to motivate a ranking—hopefully, but not necessarily, our ranking—of the severity of conceptual space changes by entailment relations, in a manner similar to how topological morphisms are ranked.
Perhaps the metric test can be reformulated to account for such examples in the following manner: Dimensions are treated as separable if the data supports the city-block metric as the best measure of “distance” in the space formed by those dimensions and are otherwise treated as integral.
The general schema is: \( T=\underset{\varepsilon \to 0}{ \lim }{T}^{*} \)
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Acknowledgments
Parts of this paper were presented at GAP.9, held September 14–17 September 2015 in Osnabrück, Germany, and at EPSA15, held 23–26 September 2015 in Düsseldorf, Germany. We would like to thank audiences at these events, two anonymous reviewers for this journal, Keizo Matsubara and Lars-Göran Johansson for comments and criticisms that helped to improve earlier versions of this manuscript. The authors acknowledge funding from the Swedish Research Council (G.M., F.Z., P.G.), the European Union’s FP7 program as well as the Volkswagen Foundation (F.Z.)
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Masterton, G., Zenker, F. & Gärdenfors, P. Using conceptual spaces to exhibit conceptual continuity through scientific theory change. Euro Jnl Phil Sci 7, 127–150 (2017). https://doi.org/10.1007/s13194-016-0149-x
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DOI: https://doi.org/10.1007/s13194-016-0149-x