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In pursuing philosophical enquiries in the foundations of physics, I have found myself operating with a concept of explanation which has a quite different logical syntax from the concept which Hempel has explicated. I suspect that the concept of explanation I operate with is not especially idiosyncratic, is not fully captured in the existing philosophical literature (though I am open to correction here), and that it may be the more appropriate concept if we are to consider explanation as the goal of physics. I therefore propose to give here a short account of the structure of this alternative concept of explanation.

. . In order to lay the foundation of such investigations on strict mechanical principles, I shall demonstrate the laws of motion of an indefinite number of small, hard, and perfectly elastic spheres’ acting on one another only during impact. If the properties of such a system of bodies are found to correspond to those of gases, an important physical analogy will be established, which may lead to more accurate knowledge of the properties of matter. If experiments on gases are inconsistent with the hypothesis of these propositions, then our theory, though consistent with itself, is proved to be incapable of explaining the phenomena of gases. In either case it is necessary to follow out the consequences of the hypothesis.

The general logical structure of Einstein's 1905 argument1 for the introduction of photons is shown (in a rather free rendering) in the diagram overleaf. (h is Planck's constant, k Boltzmann's constant, c the velocity of light, v frequency, T temperature, E energy, S entropy, p energy density, Sentropy density, V volume, and W probability.)2 It will be observed that the premises include both fairly low-level generalisations arising rather directly from experiments and fairly high-level generalisations arising from theory, and that the argument, apart from the last step, is of a deductive character. The aim of the present paper is to show that the argument by analogy which Einstein uses to mediate the last step is as a matter of fact redundant and that his conclusion already follows deductively from what he has already established. If my argument is accepted then Einstein's introduction of photons no longer takes the form of a plausible hypothesis suggested by analogy, but can be construed as a "deduction from the phenomena" in the sense of Newton and Ampere: i.e. as the rigorous deduction of a new explanatory theory from certain empirical generalisations taken as phenomena together with certain more general theoretical assumptions whose validity is taken for granted. I hope to show in some later articles that new hypotheses in physics are commonly introduced by "deduction from the phenomena" of this general kind and that there are no fundamental logical objections against this form of argument. I start with the assertion which Einstein has arrived at by an apparently deductive argument namely

It is argued in this paper that the valid argument forms coming under the general heading of Demonstrative Induction have played a highly significant role in the history of theoretical physics. This situation was thoroughly appreciated by several earlier philosophers of science and deserves to be more widely known and understood.

In his well known axiomatisation of classical particle mechanics (McKinsey, Sugar and Suppes [1953]; Suppes [I957]), Suppes was able to prove that masses and forces are independent primitive notions which cannot be defined on the basis of the other primitives. This result is normally taken to show that, contrary to Mach's positivist views, masses and forces are theoretical terms which cannot be eliminated in favour of observational terms. Taken at face value this conclusion seems also to provide strong support for hypothetico-deductivism as against inductivism. However it is difficult to accept Suppes's easily proved results at their philosophical face value. On the one hand (contrary to what most philosophers seem to suppose) mathematical physicists seem, in general, to have succeeded in eliminating theoretical terms in favour of more directly observational terms, in most other theories, whenever they have seriously set out to do so: for example everyone knows how to eliminate vector potentials from classical electrodynamics without loss; it is trickier in the case of quantum electrodynamics, but Mandelstam showed how to do it should we so choose; Wheeler and Feynman, following Tetrode and others, deliberately set out to eliminate even the fields from classical electrodynamics and in this they succeeded; the distinguished present-day positivist Robin Giles has published elegant formulations of classical and relativistic thermodynamics (Giles [1964]) and of quantum mechanics (Giles ['9701) in precisely defined observational languages in which the usual theoretical terms are later reintroduced by explicit definition. It would be surprising if classical mechanics proved to be an exception. On the other hand Suppes's choice of observational primitives is sufficiently idiosyncratic to invite suspicion. G. W. Mackey, in his [1963], presented in the first few pages an axiomatisation of classical particle mechanics which is very different from that of Suppes. Suppes took the position of each particle as given as a function of the time only. From this we can calculate the acceleration of each particle as a function of the time only. Mackey, on the other hand

Professor Popperx says that Lorentz and Fitzgerald 'do not say that a rod at rest in the ether would appear contracted if measured with the rods of the moving system. (Rather, one might expect, by implication, that it would appear expanded.)' Popper then goes on to emphasise the mutual character of the Einstein contraction, and stresses that it is this mutual character of the contraction which leads to the abandonment of the ether.

I believe that God1 is an extreme positivist, and presently I shall prove it. I also believe that we should try to emulate His system of beliefs. It follows that we should go in for extreme positivist critiques of existing theories in physics, and that this will yield a real prospect of improving them. But it also follows that we should not adopt an anthropocentric form2 of positivism. We need not, then, have positivistic doubts about the reality of the past, of physical objects, or of the microworld. I also happen to believe that the final correct theory of the universe will be almost unbelievably simple when axiomatized in an extreme positivist form.

Schrodinger's original interpretation of the Schrodinger equation had many attractive features lost in later interpretations of the quantum theory. But that interpretation runs into a number of formidable well-known and not so well-known objections. I argue, following the methodological precepts of Paul Feyerabend, that we need not regard any of these objections as fatal, provided we are prepared to opt for a number of bold and rather radical mathematical and theoretical conjectures. These would amount jointly to the conjecture that a fully time-symmetric consistently classically interpreted non-second-quantized analogue of existing quantum field theory would ( pace Jaynes, Tomonaga, Bell, and others) ultimately prove predictively equivalent to orthodox second-quantized theory. Introduction Schrodinger initially proposed his equation as a classical theory of matter waves directly analogous to Maxwell's theory of electromagnetic waves. |ψ| 2 represented a classical charge density functioning in the ordinary classical way as a source of electromagnetic fields, and acted on by these fields via the potential term in the matter–wave equation. This is a theory of coupled classical fields with no probabilities entering into its interpretation, and from a modern point of view it can be thought of in terms of the coupled Dirac and Maxwell fields, without second-quantization and interpreted in a purely classical manner.

This paper is directed at those who think that quntum mechanics really does contain a measurement paradox, and who want to understand how a theory could be both internally inconsistent in this way and at the same time empirically wholly successful. I think that I am now able to generate a whole class of theories which would share this peculiar feature. They would all be false in the universes they describe; they would contain genuine measurement paradoxes; and yet they would give correct predictions for all the results of measurements in those universes.

Clinical Presentation: A 4-week-old girl had difficulty breathing, reduced feeding, and upper limb weakness. She was born at 39 weeks’ gestation by forceps after a failed ventouse. At birth, minimal resuscitation was needed, with suction and facial oxygen. Results of subsequent examinations after birth were normal. At 2 weeks of age, she developed a dry cough, which gradually worsened, and on the day of admission, her breathing became increasingly noisy and she developed apneic episodes. There had been no exposure to cigarette smoke or pets. Dr Eason: The history of cough developing in the preceding 2 weeks and subsequent lower respiratory tract signs culminating in apnea is highly suggestive of respiratory syncytial virus bronchiolitis. Apnea is a common clinical finding in young infants with bronchiolitis. Noisy breathing could, however, be confined to the upper respiratory tract, and the change in breathing pattern suggests laryngomalacia. Laryngomalacia occurs when the larynx is soft and flexible; it is present from birth and exacerbated by upper respiratory tract infection. While it is a possibility in this case, the lack of stridor in the first few weeks of life makes it unlikely. Other causes of stridor include an enlarging subglottic hemangioma or a vascular ring; however, these would not give rise to neurologic signs. Bacterial infection by Haemophilus influenzae type B needs to be considered, but epiglottitis is a rare event now since immunization and uncommon at this age. Bacterial supraglottitis and tracheitis should also be considered. Physical Examination: Vital signs were all normal, with a temperature of 36.9°C, oxygen saturations of 98% in air by pulse oximetry, and heart rate and blood pressure within normal limits. An examination of the respiratory system revealed inspiratory and expiratory stridor, with tachypnea (respiratory rate, 62/min) and respiratory distress. Neurologic examination showed flaccidity of the upper limbs when both arms were held in a “waiter’s tip” position, indicative of an Erb palsy. There was normal grip and finger power. The lower limbs were normal except for a small paronychia on her right fifth toe. Dr Eason: The examination results suggest an upper airway obstruction despite some lower respiratory signs. The presence of bilateral shoulder muscle weakness (Erb palsy) is both confusing and of concern and will need further investigation. It implies the possibility of severe birth trauma, with damage of the phrenic nerve (cervical spinal nerve roots C3, C4, and C5) as well as the upper brachial plexus roots (cervical spinal nerve roots C5, C6), causing bilateral Erb palsy and vocal cord paralysis. In the absence of a history of shoulder dystocia, this seems very unlikely. Photographs taken in the first few days of life also rule out this diagnosis by showing the patient’s ability to lift her arms. Other possibilities include a type 2 Chiari malformation with downward displacement of the cerebellar vermis and the medulla to cause foramen magnum impaction syndrome. This can lead to abnormal brainstem and nerve root function. It is rare except as a complication of spina bifida and hydrocephalus. Neither a spinal lesion nor a large head circumference is present on examination. Bulbar palsies, due to abnormal function of lower motor neurons (eg, spinal muscular atrophy) may display similar signs if the vocal cords are involved. On examination, one would expect to see a lax open mouth with jaw muscle atonia, weakness, and wasting, particularly of the tongue, which would also show fasciculation. Infection with diphtheria or Epstein-Barr virus should be detectable on examination of the throat by the appearance of an adherent membrane or inflamed tonsils covered in exudate.