Abstract
This paper discusses a conception of physics as a collection of theories that, from a logical point of view, is inconsistent. It is argued that this logical conception of the relations between physical theories is too crude. Mathematical subtleties allow for a much more nuanced and sophisticated understanding of the relations between different physical theories.
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Notes
Note, that structure is identified not only with “fundamental,” but also with what is “real,” “true,” and “physical.” Note, too, the date: These remarks were made 20 years before Anderson’s famous “More is Different” paper. Anderson (1972)
In light of the rest of this paper, I think Belot’s terminology is unfortunate. Such “emeritus” theories are often fundamental, once one realized that the conception of “fundamental” admits different, legitimate readings.
For example the velocity of the fluid near the breakup point becomes infinite.
See Eggers Eggers (1995).
The idea here is that near breakup the shape of the drop looks the same at various magnifications analogous to the self-similarity observed in fractal geometry. As a result, no parameters, in particular, no macro-parameters play any kind of role in the process of breakup.
In fact, the very idea of a well-defined drop with boundaries doesn’t apply anymore. Of course, this reflects the supposed inconsistency between the two theories. There is a singularity in the continuum theory, yet there isn’t one in the molecular theory.
The use of the Lennard-Jones potential is justified in investigations of this sort (interactions between closed-shell atoms) for the following reasons. It exhibits long-range van der Waals attraction, extremely strong short-range repulsion and has a potential well. Given these features, along with its relative ease of computational implementation, it is the potential of choice for investigations into generic properties of many molecular dynamical interactions.
It need not be the case that there are distinct theories involved. Even the “simple” case of understanding laminar continuum flow in a pipe requires the investigation of similar boundaries. This is the so-called Prandtl theory. See Darrigol (2005) for an excellent discussion. On the other hand, sometimes it is best to think of these situations as involving distinct theories as we will see in the next section.
The paraxial approximation allows one to identify the sine of an angle with the angle itself—a procedure that works for small angles.
See also Batterman (2005) for a discussion of controllable vs. uncontrollable idealizations.
A convergent series perfectly defines a function. The first term is an approximation to the function and the later terms are treated as the remainder. The series is convergent if the limit of the remainder as the number of terms goes to infinity is zero.
Similarly, one cannot derive the universal qualitative changes in states of fluids at their critical points from finite statistical mechanics—one needs divergences that only arise in the idealized thermodynamic limit (Kadanoff 2000, pp. 238–239)
Perhaps, one could set up the Physics hierarchy by bringing in quantum electrodynamics, but the classical theory is often discussed in nonquantum contexts.
This is to say that the total electromagnetic force on a particle, even when it is accelerating, is a function of the external fields only.
See below for a less naive continuum approach.
See Batterman (2002) for a discussion.
“\(m_0\)” is the bare mass, “\(\mathbf{v}\)” is the velocity, and “\(\mathbf{F}_{\mathrm{ext}}\)” is the Lorentz force.
Put another way, this is a principle that states the particle idealization is controllable—its effects are simply too small to be observed.
See Batterman (2002).
The latter expresses the conservation of the total (mass and field) stress-energy tensor.
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Acknowledgments
I would like to thank Nic Fillion, Chris Smeenk, and Peter Vickers for helpful discussions. This research was supported by a grant from the Social Sciences and Humanities Research Council of Canada.
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Batterman, R.W. The inconsistency of Physics (with a capital “P”). Synthese 191, 2973–2992 (2014). https://doi.org/10.1007/s11229-014-0468-4
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DOI: https://doi.org/10.1007/s11229-014-0468-4