Interpreting mathematical breakdowns in physical theories

Landau and Lifshitz, Classical Theory of Fields, p. 102:

Since the occurrence of the physically meaningless infinite self-energy of the elementary particles is related to the fact that such a particle must [because of the impossibility of rigid bodies, according to special relativity] be considered as pointlike, we can conclude that electrodynamics as a logically closed physical system presents internal contradictions when we go to sufficiently small distances. We can pose the question as to the order of magnitude of such distances.

Ehrenfest, as quoted in Nickles:

Another objection may be raised against the whole [project of generalizing Planck's theory by reference to classical results]: there is no sense - it may be argued - in combining a thesis which is derived from the mechanical equations with the antimechanical hypothesis of energy quanta. Answer: Wien's law holds out the hope to us that results which may be derived from classical mechanics and electrodynamics by the consideration of macroscopic-adiabatic processes, will continue to be valid in the future mechanics of energy quanta.

Hartmann, S. (2001, June). Effective Field Theories, Reductionism and Scientific Explanation. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 32 (2), 267-304.

Soon after Dirac presented his first attempts towards a quantum theory of fields, Euler and Heisenberg applied this theory to the process of photon-photon scattering. The authors did not worry much about the fact that Dirac's theory had various conceptual problems at that time. Quite to the contrary, by working out interesting applications of the theory and by exploring its consequences, Euler and Heisenberg hoped to get a hint in which direction one has to look in order to find a satisfactory quantum theory of fields

Kramers and Holst, The Atom and the Bohr Theory of its Structure, London: Glydendal, 1923, pp. 109-110:

there thus appears to be an irremediable disagreement between the Rutherford theory of atomic structure and the fundamental electrodynamic assumptions of Lorentz's theory of electrons. As has been emphasized, however, Rutherford founded his atomic model on such a direct and clear-cut investigation that any other interpretation of his experiments is hardly possible. If the result which he attained could not be reconciled with the theory of electrodynamics, then, as has been said, this was so much the worse for the theory.

It could, however, hardly be expected that physicists in general would be very willing to give up the conceptions of electrodynamics, even if its basis was being seriously damaged by Rutherford's atomic projectiles. Surmounted by its crowning glory — the Lorentz electron theory — the classical electrodynamics stood at the beginning of the present century a structure both solid and spacious, uniting in its construction nearly all the physical knowledge accumulated during the centuries, optics as well as electricity, thermodynamics as well as mechanics. With the collapse of such a structure one might well feel that physics had suddenly become homeless.

Kramers and Holst, The Atom and the Bohr Theory of its Structure, London: Glydendal, 1923, p. 126:

That atomic processes on [Bohr's] theory took on an unreasonable character (compared with the classical theory) was nothing to worry about, for Bohr had come to the clear recognition that it was completely impossible to understand from known laws the Planck-Einstein "quantum radiation," or to deduce the properties of the spectrum from the Rutherford atom alone. He therefore saw that his theory was really not introducing new improbabilities, but was only causing the fundamental nature of the contradictions which had previously hindered development in this field to appear in a clearer light.

Kramers and Holst, The Atom and the Bohr Theory of its Structure, London: Glydendal, 1923, pp. 174-5:

If we imagine… that some slits are hit by light quanta from one atom and others from a second atom, it is pure chance if there are wave crests simultaneously in all slits, because the different atoms in a source of light emit light at different times, depending purely on chance. An understanding of the observed effect of a grating on light seems then out of question.

The theory of light quanta may thus be compared with medicine which will cause the disease to vanish but kill the patient. When Einstein, who has made so many essential contributions in the field of the quantum theory, advocated these remarkable representations about the propagation of radiation energy he was naturally not blind to the great difficulties just indicated. His apprehension of the mysterious light in which the phenomena of interference appear on his theory is shown in the fact that in his considerations he introduces something which he calls the "ghost" field of radiation to help to account for the observed facts. But he has evidently wished to follow the paradoxical in the phenomena of radiation out to the end in the hope of making some advance in our knowledge.

ter Haar, The Old Quantum Theory, Oxford: Pergamon Press, 1967. p. 44:

we must emphasize the inner paradox which is inherent to all these considerations. We are trying to find principles and rules to determine the stationary orbits, that is, those classical orbits which are quantum-mechanically allowed and which at the same time behave in a most unclassical manner by not changing their energy through the emission of electromagnetic radiation. One should not consider such rules as the adiabatic hypothesis (see below) or the correspondence principle as attempts to make classical and quantum theory compatible; rather, they were attempts to find a way towards a completely quantum-mechanical formulation, such as the one found by Schrödinger and Heisenberg.

Wilson, Mark. Wandering Significance : An Essay on Conceptual Behaviour. Oxford, , GBR: Oxford University Press, 2006. p 331:

Most of us have never been taught such precautions, however, and are only preserved from frequent error by the fact that the computations we attempt with complex roots generally prove quite local in character. Our shifting ‘‘weight’’ talk is quarantined through the same loose control.

Wilson, Mark. Wandering Significance : An Essay on Conceptual Behaviour. Oxford, , GBR: Oxford University Press, 2006. p 190:

the fact that a region can be descriptively avoided in this manner does not indicate that it is therefore unimportant: the condition at the shock front represents the most important physical event that occurs in our tube. It is merely that we can keep adequate track of its overall influence in a minimal descriptive shorthand… Indeed, the whole idea of variable reduction or descriptive shorthand is that we are able to locate some shock-like receptacle that can absorb complexities and allow us to treat its neighboring regions in a simplified fashion.

Wilson, Mark. Wandering Significance : An Essay on Conceptual Behaviour. Oxford, , GBR: Oxford University Press, 2006. p 201:

our gas tube case also shows that sometimes these very squeaks can be cleverly exploited to temporize on a need to shift patches radically. Heeding Riemann and Hugoniot's advice, we can declare, "Let's take this mathematically impossible blowup as an omen that a shock wave is forming there." This ploy allows us to frame an unexpected variety of in-between mathematical patch, where so-called weak solutions are now tolerated alongside our old formulae (the acceptance of the famous Dirac delta-function falls in place in here). Applied mathematics is full of procrastinating, halfway repairs of this ilk. Because the phraseology of the calculus can be reconfigured to encompass "weak solutions" fairly deftly, a casual observer can easily overlook their intrinsic oddities. A closer look reveals the delicate framework of controls that allow the Riemann/Hugoniot ploy to work.

Wilson, Mark. Wandering Significance : An Essay on Conceptual Behaviour. Oxford, , GBR: Oxford University Press, 2006. p 548:

Although it is customary to call these supplements "weak solutions" ("weak" because they relate to the original equations.. only in a roundabout way), they often signalize completely different states of affairs in their home locales. Thus, in Heaviside's circumstances the weak solutions signalize "internal dispositions of our circuit to act upon input currents according to certain patterns", rather than true "histories of current flow in a circuit" as the "regular solutions" do. In contrast, within many other forms of continuum mechanics context, the comparable weak solutions do represent "physical histories" just as ably as the regular solutions (for reasons mentioned in the fine print). In other applications yet, they convey a rather surprising warning content, "You can't handle this kind of case without bringing in more physics"… So decoded, it seems odd to designate a sentence that carries the message "I can't solve this kind of problem" as a solution, but such answers are invariably designated as "weak solutions" in mathematical practice, just as Heaviside's non-solution "solutions" are likewise so denominated.

Wilson, Mark. Wandering Significance : An Essay on Conceptual Behaviour. Oxford, , GBR: Oxford University Press, 2006. p 549:

Sometimes, as with shock waves, [the limit of a series of approximations] conveys the funny halfway meaning discussed in 4,vii: "Complicated physics that isn't really described by E becomes relevant along my shock front, but we can still stay within E's descriptive patch by adopting the Riemann-Hugoniot trick." But sometimes our weak solutions don't carry any clear meaning but trouble: we need to abandon our current approach in this case (in many parts of continuum mechanics we haven't been able to rule out the threat of these unwanted guests, which may spoil the utility of our theory in the very cases where it most matters).

Wilson, M. (1990). Law along the Frontier: Differential Equations and Their Boundary Conditions. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1990, p. 574:

the internal equations predict that the stresses at the ends of the cracks will be infinite. At first blush, this indicates that the material ought to break or flow. But if you integrate (in an "improper" way, of course) the stored stress energy over a finite region, one can obtain a finite number which can be used to predict how the crack will grow. In this context, one decides not to worry about the infinite stress embodied in the crack, but concentrates upon the ability of the region containing the crack to release energy to other parts of the material.

Wilson, M. (2009, January). Determinism and the Mystery of the Missing Physics. Br J Philos Sci, pp. 12-13:

Although from a modeling point of view we are inclined to object to the appearance of singularities where some density or velocity blows up to infinity, from a mathematical point of view we often greatly value these same breakdowns; for, as Riemann and Cauchy demonstrated long ago, the singularities of a problem commonly represent the precise features of the mathematical landscape we should seek in our efforts to understand how the qualitative mathematics of a set of equations unfolds. Insofar as the project of achieving mathematical understanding goes, singularities frequently prove our best friends, not our enemies. Accordingly, if we have already decided that our MP formulations overlook the missing physics pertinent to extended bodies anyway, why should we gussy up the mathematical formalism of MP with artificial assumptions concerning repulsion at close quarters? Such unwanted supplements may only camouflage the very singularities that we need to uncover in our attempts to understand how our system behaves when no danger of close contact looms

Wilson, M. (1989). Critical Notice: John Earman's A Primer on Determinism. Philosophy of Science 56 (3).

The differential equations with which we begin our modeling are initially treated as "purely formal". This means, surprisingly enough, that these equations are purposefully not assigned a precise mathematical meaning. Instead, following the lead of whatever physical clues are available, we search for special domains and special meanings for the mathematical operations in our "formal equations", so that, so interpreted, the precisified results would permit a uniqueness and existence proof. One then hopes that the special qualities assigned to the final construction will provide clues as to what sort of supplemental considerations need to be added to the physics of the problem. In short, we let the inadequacy of our original modeling direct us towards their means of correction

Wilson, M. (1989). Critical Notice: John Earman's A Primer on Determinism. Philosophy of Science 56 (3).

Although I have indicated reasons why a worker in, for example, celestial mechanics might wish to excise singularities from her theory, there are also reasons why they should not be totally despised either. Indeed, the presence of singularities often provides a most helpful clue to discerning how a theory behaves. Except for the singularities and critical points like equilibria, the behavior of one physical system may look mathematically very much like another. The locations of the singularities serve as landmarks in understanding the system. Accordingly, we may want to introduce singularities artificially into an otherwise untroubled modeling simply to figure out what's going on.