Abstract
The principles which constituted the triumph of the preceding stages of the science, may appear to be subverted and ejected by the later discoveries, but in fact they are (so far as they were true) taken up into the subsequent doctrines and included in them. They continue to be an essential part of the science. The earlier truths are not expelled but absorbed, not contradicted but extended; and the history of each science, which may thus appear like a succession of revolutions, is in reality, a series of developments.
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Reference
For the whole burden of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of nature’ (Newton).
This part of philosophy of science is covered by that stock character, the logician with the dictionary, at best further equipped with a relative-simplicity sieve to compare existing theories.
Just as I do not believe that it is healthy to confine philosophy of science to linguistic analysis of statements about science, but do believe that philosophy of science should, amongst other things, present a truthful account of what actually goes on in science, so I believe it would be a pity to treat heuristics as a matter outside philosophy whereof we must be silent.
There are historical problems, both as to what causes the individual scientist to discover a new idea, and as to what causes the general acceptance of scientific ideas. The solution of these historical problems involves the individual psychology of thinking and the sociology of thought. None of these questions are our business here. What we are concerned with are the straight logical problems of the internal structure of scientific systems and of the roles played in such systems by the formal truths of logic and mathematics, and also the problems of inductive logic or epistemology concerned with the grounds for the reasonableness or otherwise of accepting well-established scientific systems’ (R. B. Braithwaite, Scientific Explanation (Cambridge, 1953), 20–1). Another issue customarily connected with the study of scientific method is the quest for “rules of induction”. Generally speaking, such rules would enable us to “infer”, from a given set of data, that hypothesis or generalization which accounts best for all the particular data, in the given set. Recent logical analyses have made it increasingly clear that this way of conceiving the problem involves a misconception: While the process of invention by which scientific discoveries are made is as a rule psychologically guided and stimulated by antecedent knowledge of specific facts, its results are not logically determined by them; the way in which scientific hypotheses or theories are discovered cannot be mirrored in a set of general rules of inductive inference…. ‘… there can be no general rules of induction in the above sense; the demand for them rests on a confusion of logical and psychological issues. What determines the soundness of a hypothesis is not the way it is arrived at (it may even have been suggested by a dream or a hallucination), but the way it stands up when tested’ (Carl G. Hempel, Mind, 54 (1945), 4 (c)). for the act of conceiving or inventing a theory, seems to me neither to call for logical analysis nor to be susceptible of it. The question how it happens that a new idea occurs to a man—whether it is a musical theme, a dramatic conflict, or a scientific theory—may be of great interest to empirical psychology; but it is irrelevant to the logical analysis of scientific knowledge’ (Karl R. Popper, The Logic of Scientific Discovery (London, 5959), 31). ‘We pointed out in the beginning of our enquiry (§l) the distinction between the context of discovery and the context of justification. We emphasized that epistemology cannot be concerned with the first but only with the latter; we showed that the analysis of science is not directed toward actual thinking processes but toward the rational reconstruction of knowledge. It is this determination of the task of epistemology which we must remember if we want to construct a theory of scientific research’ (H. Reichenbach, Experience and Prediction (Chicago, 1961 ), 381–2 ).
Thus I consider Hadamard’s The Psychology of Invention in the Mathematical Field (New York, 1954 ) relevant for the philosopher of science. One of the messages of this book is the importance of non-conscious processes in scientific work. I have experienced myself the phenomenon of failure at the conscious level to connect theories that subconsciously have been constructed jointly.
Cf. the role of parody in the development of the novel. On the other hand, different contemporaneous fields of physics developed distinct forms of treatment: in classical physics the picaresque novel of electromagnetism, the neo-classical poetry of thermodynamics and the dead language of mechanics.
Though the lack of cylindrical symmetry in certain experiments with beams of particles might rationally have been detected within the theoretic framework in which these experiments were conducted (and perhaps was: R. T. Cox, C. G. Mellraith and B. Kurrelmeyer, ‘Apparent Evidence of Polarization in a Beam of 13-rays’, Proc. US Nat. Acad. Sci., 14 (1928), 544), no conclusive test of cylindrical symmetry was in fact carried out till Lee and Yang suggested non-conservation of parity in their paper (to account for a flaw in elementary particle theory). The two-component theory of Weyl, proposed in a different context (2eitschrift für Physik, 56 (192g), 33o), was probably not influential in this development, though it was then revived.
An example of such a general result of observation is the fact of the occurrence of line spectra, calling for appropriate theories to account for them.
E.g., the identity of inertial and gravitational mass in Newton’s mechanics (see ‘footprint’, p. 225).
Again, L may be ‘deeper’ in the sense that the weaker S-theory of first quantization of atoms determines correctly the frequencies of the line spectrum emitted, but, unlike the stronger, deeper L-theory of quantum electrodynamic field theory, has no theory of the process of emission and fails to predict intensities.
This is a plausible requirement, which goes beyond the mere requirement that the new theory should agree with the old on factual data that are successfully explained by the old theory. It is the requirement of an explanation in terms of the new theory of the success of the explanation by the old theory. This requirement is invariably met by good theories, in which insight into why the old theory worked is never lost by the new. This almost trivial empirical historical point is so neglected in most discussions that I consider it worth emphasizing. It is of philosophic interest, like any non-tautological general truth.
A Study of Relations Between Scientific Theories: A Test of the General Correspondence Principle’ (Doctoral Thesis, London University, 1968 ).
Incidentally, I do not subscribe to the ‘volume’ view of science that sees a monotonically accelerating accumulation of science in recent times. The period 1900–35 (Planck to Yukawa) was particularly rich. I consider the subsequent stagnation (at least in the field of physics) to call for an explanation.
Thus classical electromagnetism, for instance, is unable to answer the simplest empirical question meaningful in its field of application, namely the question as to the motion of an electron in a uniform electrostatic field. On the other hand, botany has fair predictive power in its own field of application.
See also note 72, paragraph 2 following.
As is usually the case, the prevalent theory cannot be associated with any one man. It is the world of textbooks and secondary publications (review articles and reports on applications) that
I have come across two publications in this field, by Tisza, Reviews of Modern Physics, 35 (1963), no. 1, and Strauss (loc. cit.), respectively. They are written from points of view somewhat different from that of the present paper, and concentrate on inter-theory relations in present-day physics. While welcoming an implied point of Tisza’s paper, viz., the (within certain limits) arbitrary character of the choice of theories integrating the system, I disagree with several points of detailed application, in particular with Tisza’s choice of integrating system, which I attribute to his paying insufficient regard to criteria of simplicity. Philosophically, I would disagree with Tisza on many grounds, particularly with his instrumentalist view (theories agreeing within experimental error appear to be treated as identical). I have learned in conversation with Professor Tisza that one of his grounds of preference for a thermodynamic-type theory as integrating theory is precisely the paucity of the (phenomenological) parameters required for a complete description (cf. Einstein’s ‘Theories of Principle’). For his distinction of ‘dynamic’ and ‘chemical’, cf. Dürr, ‘Approximate Symmetries in Atomic and Elementary Particle Physics’, in Properties of Matter Under Unusual Conditions, ed. H. Mark and S. Fernbach (New York, 1969). Strauss’s ‘partial formal anticipation’ is not a ‘footprint’ in my sense, but rather one of a set of equivalent formulations of S, which is only singled out later from the standpoint of L. I am grateful to Professor Strauss for sending me an advance copy of his ‘Die Entwicklungsdialektik in der Physik and das Dilemma in der büergerlichen Philosophie’ (Deutsche Zeitschrift für Philosophie, 1971). It appears that extensionally his term ‘dialektische Aufhebung’ coincides with my term ‘inconsistent correspondence’. I flatly disagree with him when he says (ibid., p. 5) ‘Es zeigt sich nämlich, dass die dialektische Aufhebung einer Theorie T1 durch eine Theorie T2 in allen untersuchten Faellen mit einer Begriffsaufspaltung verbunden ist….’ The most interesting case discussed in the present paper is precisely L-degeneracy, which refutes Strauss’s statement.
The instruction to attempt to unify might be taken as a normative heuristic rule, which may fail: ‘What God has put asunder, let no man join together’ (Pauli).
F. H. Bradley, Appearance and Reality (Oxford, 1962 ), 457.
We are not, of course, referring to the fact that any theory that contains the arithmetic of natural numbers, is incomplete in the sense that there are statements in the language of the theory such that neither they, nor their negation, can be logically derived from the axioms. This Gödelian difficulty can be overcome, for instance, by joining to a physical theory all true statements of the mathematics used, i.e., usually at least those of arithmetic.
I was very pleased by Professor Dingle’s example of a school arithmetic problem allowing the solution ‘number of workmen = minus 3’, and his point that a modern physicist would conclude that there are three negative workmen. Whilst Professor Dingle intended this as a warning caricature, I welcome it as a fair example. An actual example is provided by Bonnor, who points out that Maxwell’s theory allows charges to move freely with the speed of light, and suggests a physical interpretation (W. B. Bonnor, ‘Solutions of Maxwell’s Equations for Charge Moving with the Speed of Light’, International 3ournal of Theoretical Physics, 2 (1969), 373—g). This amounts to giving physical significance to hitherto uninterpreted elements of the theory (cf. criterion 6). We should, however, distinguish sharply between the fruitful policy of tentatively extending interpretation to regions where a theory is genuinely open, and the heresy of misplaced concreteness when a special interpretation is given to certain approximations within a closed calculus (such as the ‘non-conservation of energy’ in second-order perturbation).
But see the amusing polemical remark in Nernst: Theoretische Chemie (i t th—t 5th edition, Stuttgart, 1926 ), 805.
Albert Einstein: Philosopher—Scientist, ed. P. A. Schilpp (New York, 1951 ), 53.
J. C. Maxwell, A Treatise on Electricity and Magnetism (reprinted New York, 1954 ), vol. 2, 274–66.
Zeitschrift für Physik, 35 (1g25), 765. This also has a footnote referring to statistics, making the major error of failing to apply his principle. Cf. Einstein’s failure to use ‘Bose—Einstein’ statistics in 1905, and Planck’s use of Bose—Einstein statistics in 5900.
A mechanically inexplicable invariant character…’ (Bohr, Annals of Physics (1923), 225–6); a totally ad hoc principle designed precisely to justify the theoretic treatment of electrons in a many-electron atom in the only way then available.
There are, of course, flaws which have not been removed (see also section IV). Indeed, some have been inherited by successive theories. The inability of classical electromagnetic theory to deal exactly with the motion of a charged particle in a field shows its incompleteness. This flaw has never been removed. In my view quantum mechanics is unsatisfactory because many such classical flaws are compounded in it. Incompleteness may be masked by the use of an undue number of independent axioms: the mere use of Lorentz’s electrodynamic equation of motion (however unsuccessful) as an axiom independent of the Maxwell field equations points to a non-linear L-theory in which the equation of motion is derivable from the field equations, as is the case for the gravitational field.
Though Hamilton applied the same formalism to both optics and mechanics, this did not lead to a revolutionary advance by itself. It was only after the wave theory of light had become accepted, and Fermat’s principle of least time rehabilitated, bringing out a sharp distinction between optics and mechanics, that de Broglie pointed out the relation between Fermat’s and Maupertuis’ principles that led to a revolutionary unification of wave and particle physics. ‘And though, among these problems of mathematical optics, it is not here intended to include investigations respecting the phaenomena of interference, yet it is easy to perceive, from the nature of the quantity which I have called the characteristic function, and which in the hypothesis of undulations is the time of propagation of light from one variable point to another, that the study of this function must be useful in such investigations also’ (J. Hamilton, ‘On a View of Mathematical Optics’, Brit. Assoc. Report (1831–2), 545–7).
Cf. Newton’s third rule of reasoning.
No pun is intended in the use of the word conservation, (i) in a context such as ‘conservation of energy’, and (ii) at a meta-theoretical level referring to the retaining of certain laws which may themselves be conservation laws.
This kind of argument is analogous to the dimensional argument, based on scale invariance, which again only allows us to assert that if there is a functional relationship between two quantities, it must be of a certain form.
Similarly, laws may be determined uniquely by the empirical determination of a finite number of parameters, provided we assume some general requirements to start with. Such ‘deduction from the facts’ is the empirical analogue to the mathematical procedure of ‘proof by a finite number of instances’; we may prove Pythagoras’ theorem by first establishing that the expression for the square of the hypotenuse is necessarily bi-linear in terms of the lengths of the other two sides, and then ‘determining’ the coefficients of the bi-linear terms by reference to a small number of particular instances, which may be particularly simple (in this case, the single instance of the isosceles triangle).
We may compare this with the discovery of Neptune on the basis of Newton’s law of gravitation. Less rigorous are such cases of ‘in-filling’ as Mendeleev’s postulation of the existence and properties of the ‘missing elements’ to conserve his periodic table, and the prediction of the S2 - particle in recent elementary particle theory.
See Koertge, op. cit., note 12, as against T. S. Kuhn, The Structure of Scientific Revolutions (Chicago, 1970), 107 and 48.
The polemic concerning the use of the term ‘element’ following Soddy’s discovery of isotopes is instructive. It was suggested by leading physical chemists (e.g., K. Fajans, ‘Radioaktivität and die neueste Entwicklung der Lehre von den chemischen Elementen’, 2nd edition (Braunschweig, 1920), 88–103) that the term element now had to be applied to each isotope separately. These chemists were, in fact, following Mendeleev’s (and Dalton’s) philosophy attaching primary significance to the combining mass or atomic weight of an element as a source of its individuation. If this policy of retaining Mendeleev’s ontology had been pursued radically it would have meant the break-up of the periodic system. It was fortunate for chemistry that the policy advocated by the other side in the polemic was adopted. This other side (e.g., F. Paneth, Naturwissenschaften, 43 (1g2o), 839–¢2) advocated changing the definition of the relevant term ‘element’ in an ad hoc fashion precisely to preserve the simple pattern of the periodic system. A change of definition was preferred to a change of structure. Ontology followed theory.
Kuhn, op. cit., 169: ‘new paradigms seldom or never possess all the capabilities of their predecessors’. Paradoxically, the lower-level (more specific) theories seem to provide a relatively more promising hunting-ground for Kuhn losses than the most general theories. The chronological sequence: Dulong–Petit theory of atomic heat applied to metals—theory of free electrons in metals—theory of free electrons plus Fermi statistics, appears to be an example of a temporary Kuhn loss at the intermediate stage, when the previous agreement between theory and specific heat data was lost, to be recaptured on the application of Fermi statistics. Here, the temporary failure was in a particular field requiring the application of a conjunction of theories.
See H. R. Post, ‘Simplicity in Scientific Theories’, Brit. J. Phil. Sci., r r (196o), 32, and H. R. Post, ‘A Criticism of Popper’s Theory of Simplicity’, ibid., 12 (1g62), 328.
We identify any theory with the collection of all theorems provable in it. (L 1 Q) is a sub-theory of L consisting of those theorems of L satisfying condition Q.
This case of ‘approximate correspondence’ may be subsumed under the general treatment of correspondence given here: a theory S, one of whose axioms may be ‘the mass of the electron is m’ includes amongst its theorems ‘the mass of the electron is m± 1%’. It is this latter statement which might be in S*. Of course, it is preferable to confine one’s discussion to the axioms rather than to the unmanageable richness of the set of all theorems of a theory. But this move, again, presents no hardship. We may stamp any publication ‘axiom system’, declaring every sentence (except possibly the expression of gratitude to the Office of Naval Research) an axiom. In the case of approximate correspondence we are well advised to confine ourselves to publications that are not frauds, in the sense that they honestly state conservative estimates of accuracy.
See note 38 above.
P. K. Feyerabend, ‘Problems of Empiricism’, in Beyond the Edge of Certainty, ed. R. G. Colodny (New Jersey, 1965), 175–6, makes a beautiful point by drawing attention to the case of Brownian motion ‘refuting’ thermodynamics. It is not just that this refutation is originally suggested by the L-theory. The refutation is dependent on L. The refutation establishes limits of acceptability on the concepts of the S-theory, provided we accept the interpretation in terms of the L-theory. Thus, in this case, the ‘refutation’ cannot be done within the resources of the S-theory at all. We need an L-theory for this criticism of S.
But this L-theory has to draw strength from somewhere else in order to be taken sufficiently seriously to refute the S-theory. This strength is not parasitic on S, but must be drawn from elsewhere, an established mechanics plus atomic theory (L). This L-theory has to be enriched to L’ by a set of factual requirements (see also note 53). It is the bridge theory L’—L which is at issue, and this bridge theory must achieve ‘(inconsistent) correspondence’ between L’ and S.
Historically, the L-theory (statistical mechanics) was produced by Boltzmann precisely to reduce thermodynamics to atomism. Boltzmann refrained from going beyond ‘relative atomism’ to ‘numerical atomism’ and from pushing the consequential aspect, fluctuations, which may be regarded as instances refuting thermodynamics (H. R. Post, ‘Atomism t 9oo’, Physics Ed., 3 (1968), note q.ç). In some views these instances are not even regarded as refutations of (a suitably formulated) thermodynamics. But the point I wish to make here is that historically this case is an example of the heuristic benefits of a reductionist programme. as An analogous example is Nernst’s third law explaining how the second law of thermodynamics degenerates to Thomson’s rule at low temperatures.
In certain cases the Q-relativized L cannot be ‘translated’ (in the intensional sense) into S*, not because of the richness of L, but on the contrary because L may fail to refer to restrictive conditions peculiar to S: SI- all animals have trunks; S*F all animals on this island have trunks; LI- not all animals have trunks; (LIQ.)I- all grey-skinned animals have trunks. We may, then, enrich L by a remark concerning extension, of the form ‘All animals on that island are grey-skinned’, and this enriched L-theory is then translatable, after applying Q-relativization, into S. This additional ‘law’, ‘animals on that island are grey-skinned’ (a fact not necessarily noted by the S-islanders), may in turn be explained by a further theory (e.g., climatic conditions). Actual examples of such ad hoc bridging theories being required include the modern account of the periodic system in terms of elements consisting of ‘contingent’ statistical proportions of isotopes, and the present attitude to the fact of prevalence of protons (rather than antiprotons) in our universe. In both cases, some cosmological theory would have to be invoked in turn to explain the low level bridging laws.
See G. Fay, ‘On the Correspondence Principle’, Acta Physica Hungaricae, 29 (1970), 2–3. One way out of the difficulty might be to postulate ad hoc particular initial conditions, somewhat analogously to Boltzmann’s way of accounting for the irreversibility of thermodynamics. In this case of quantum mechanics such initial conditions are, in fact, postulated (‘collapse of wave function’) as the result of measurement-observation. But no quantitative derivation of the validity of, say, Kepler’s laws within observational error has been achieved. It is possible that such quantitative agreement could be achieved by postulating ad hoc a suitable probability distribution consistent with (initial) observation which would not show appreciable dispersion within cosmic times. Even if this could be done, it would be an unsatisfactory explanation.
See also E. L. Hill, in Mind, Marter and Method, ed. P. K. Feyerabend and G. Maxwell (Minneapolis, 1966 ).
See H. R. Post, ‘Incompleteness of Quantum Mechanics’, in Quantum Theory and Beyond, ed. E. W. Bastin (Cambridge, 1971 ).
Thus, the flaw in classical statistical mechanics represented by Gibbs’ paradox points to a radical theory of non-individuality such as Bose’s (H. R. Post, ‘Individuality and Physics’, The Listener (to October 1963), 534–7). See also G. Weinreich, Nature, 184 (1959), 1825–6.
These charges against the General Correspondence Principle are dealt with at length by N. Koertge, loc. cit., note 12.
The laws of nature are regularities, that is, conditions between events, whereas the initial conditions are the unforseeable events’ (Hontappel, Van Dam and Wigner, ‘The Conceptual Basis and Use of the Geometric Invariance Principle’, Rev. Mod. Phys., 37 (1965), 596). Our remarks concerning the inferior status of boundary conditions apply also to so-called initial conditions. We are not concerned with the difference between these two classes of conditions which have to be added to the theory to yield definite predictions. Nor are we referring to such conditions as the vanishing of Schrödinger wave functions towards infinity, which are, in fact, laws, and only misnamed ‘boundary conditions’.
This may involve the aid of factual linkages (rather than coordinating definitions) which are added to L to form that theory L’ which is to supersede S. An example is provided by the relationship: mechanics (L)—statistical mechanics (L’)—thermodynamics (S). Cf. E. Nagel, The Structure of Science (London, 1961), 354–5, on connectability and derivability. Nagel seems to go further and to claim that ‘temperature’ is not reducible to ‘mean kinetic energy’, just because the meaning of these two terms is not identical. While admitting the dubious character of the various ‘disorder hypotheses’ (see also below, p. 246), I claim in this particular case that an operationally defined ‘temperature’ can be shown to be identical with the mean kinetic energy derived from statistical mechanics with the aid of such disorder hypotheses.
See note 46, and cf. note 53.
Interesting case studies are presented by the following apparent counterexamples: (i) Boerhave’s theory (which surpassed the present theory in simplicity, but lacked the fertile concept of heat capacity leading to a non-trivial concept of temperature). (ii) Arrhenius’ theory of weak electrolytes. It is not at all clear that the qualitatively successful predictions (variation of electrical conductivity with concentration) of this basically false theory (partial disassociation in the case of salts) can be accounted for by present computational methods (Debye-Hückel, etc.) available to deal with interaction between ions. (iii) First-order perturbation in the case of divergence in higher orders. In so far as first-order perturbation yields qualitatively correct results, the fact of divergence in higher orders must be regarded as a flaw, to put it mildly. This flaw would have to be removed in an L-theory which corresponds (probably inconsistently) to the first-order perturbation theory. (iv) Sommerfeld’s theory of the hydrogen atom. In certain respects any one-dimensional (‘orbit’) model can only be a poor approach to the wave-mechanical treatment. But Sommerfeld’s theory was successful in giving the same energy values as Dirac’s theory did later, though it took no account of either wave-mechanics or spin, and is therefore now abandoned.
Cf. Weyl, Raum, Zeit, Materie (Berlin, 191g), 263. ‘Maxwell’s theory and analytic geometry are practically interchangeable in their mathematical constitution…. Physics has no more significance for reality than formal logic for the realm of truth.’ Maxwell’s equations can be derived from the assumptions (i) that the electrostatic field is a vortex-free vector field; (ii) that its sources are conserved; (iii) Lorentz invariance.
W. Nernst, Theoretical Chemistry (London, 1911 ), 4–5.
As Einstein says: ‘It is similar to a man engaged in solving a well-designed word puzzle. He may, it’s true, propose any word as the solution; but there is only one word which really solves the puzzle in all its forms. It is an outcome of faith that nature—as she is perceptible to our five senses—takes the character of such a well formulated puzzle. The successes reaped up to now by science do, it is true, give a certain encouragement for this faith’ (Einstein, Out of My Later Tears (Westport, 1950), 64). Or again: ‘The historical development has shown that among the imaginable theoretical constructions there is invariably one that proves to be unquestionably superior to all others. Nobody who really goes into the matter will deny that the world of perceptions determines the theoretical system in a virtually unambiguous manner, although no logical way leads to the principles of the theory.’ (Address for Planck’s sixtieth birthday, quoted by Weyl, Philosophy of Mathematics and Natural Science (New York, 1963), 153.) Or Jean Rostand on the particular example of Mendel: ‘Mendelism has been established once and for all in biological thinking and nothing will oust it, for it expresses a considerable part of the real basis of life itself.’ While I think that this statement is too strong in claiming finality achieved in a finite number of steps, I do agree that a well-confirmed theory will never be scrapped entirely.
See Popper, Logic of Scientific Discovery (London, 1965), chapter 7.
Op. cit., note 25, 69.
See note 39 above.
See G. N. Lewis, ‘Ultimate Rational Units’, Philosophical Magazine, 49 (1925), 750.
See L. Tisza, ‘The Conceptual Structure of Physics’, Reviews of Modern Physics, 35 (1963).
In the absence of better rival theories, theories claiming absolute numerological simplicity cannot be rejected on grounds of relative simplicity. They can be rejected by showing that there is a high probability that randomly distributed data would fit formulae of the same form and simplicity (within the limits of experimental error, where applicable), in other words, by showing that agreement of the data with the theory does not pass the significance test as against a null hypothesis of randomness. Such tests might be used to reject such theories as Stoney’s or E. E. Witmer’s (Physical Review, 75 (1947), 125). They have been used in support of Prout’s (C. Marignac, 186o, reprinted in Alembic Club Reprints, 20 (Edinburgh, 1932), 56 f.) and Hudson’s hypotheses. ‘It has been seen that if the ring forms on the y-carbon there are twenty-four agreements, no disagreements and none in doubt. As the chances that an event which can happen in two equally probable ways will happen in exactly the same way twenty-four times out of twenty-four trials is only one in seventeen million it seems certain that the lactonic ring in these mono-basic sugar lactones forms on the y-carbon atom’ (C. S. Hudson, ‘A Relation between the Chemical Constitution and the Optical Rotatory Power of the Sugar Lactones’, Journal of the American Chemical Society, 32 (1910), 346). We see here that we can apply such significance tests even to theories covering an essentially finite set of data only, and without reference to experimental error.
M. Hesse, Models and Analogies in Science ( Indiana, 1966 ), Io. The policy of taking models seriously should also be implemented in replacing vague metaphors by definite models. What is the point of introducing terms such as ‘pressure’ into psychology? We should make clear just what we wish to commit ourselves to when using this term
See, for instance, H. R. Post on ‘Pragmatic Simplicity’ in ‘Simplicity in Scientific Theories’, loc. cit., note 39, 32.
A. Einstein, Out of My Later Years, op. cit., note 58, 54•
Certainly, in our own intellectual behaviour we rarely solve a tricky problem by a steady climb toward success. I doubt that in any one simple mechanism, e.g., hill-climbing, will we find the means to build an efficient and general problem-solving machine. Probably, an intelligent machine will require a variety of different mechanisms. These will be arranged in hierarchies, and in even more complex, perhaps recursive, structures. And perhaps what amounts to straightforward hill-climbing on one level may sometimes appear (on a lower level) as the sudden jumps of “insight”.’ M. Minsky, ‘Steps Toward Artificial Intelligence’, Proc. I.R.E., 49 (1961), Io.
Cf. Curie, ‘Symmétrie dans les phénomènes physique’,journal de Physique, 3 (1894), 393–415.
The use of mathematical desiderata by themselves in the construction of new theories is illustrated by such publications as: D. Atkinson, L. A. P. Balazs, F. Calogero, P. di Vecchia, A. Grillo and M. Lusignoli, ‘Toward A Unitarization of the Veneziano Model’, Physics Leters, 2gB (1969), 423; A. Salam, R. Delbourgo and J. Strathdee, ‘The Covariant Theory of Strong Interaction Symmetries’, Proc. Roy. Soc. London, Series A, 284 (1965), 146; M. L. Thiebaux, ‘Crossing-Symmetric Amplitudes with Arbitrary Regge Behaviour Based on Elementary Symmetric Functions’, Physical Review, 188 (196g), 2283. The implication seems to be that symmetry, unitarity and analyticity between them completely define the physical situation. For a still more optimistic programme anticipating uniqueness on the basis of analyticity and unitarity, see particularly G. F. Chew, The Analytic S Matrix (New York, 1966): ‘Historically, the recommendation of analyticity in the theory of sub-atomic particles developed as much from studies of field theory as from experimental observations.’ See also W. Heisenberg, Introduction to the Unified Field Theory (London, 1966), 20–1. I am not referring here to extreme a-prioristic theories, such as Eddington’s or Atkin’s (Quantum Theory and Beyond, op. cit. note 49, 191), which start from scratch with epistemological principles and attempt to construct a physical theory guided solely by mathematical considerations, though I see no binding philosophical reason why such an attempt could not one day prove successful. Physicists live from hand to mouth as far as mathematics is concerned. The timely discovery of Hilbert space, before the advent of quantum mechanics, is sometimes cited as a minor miracle performed by the Zeitgeist, or whatever. The problem may be turned round: perhaps Hilbert space was assigned its role in quantum mechanics partly because it was there, conveniently constructed in a university with exceptionally good liaison between mathematicians and physicists. (See also C. Reid, Hilbert (London, 1866), 183–280.) ‘One of the most remarkable discoveries in elementary particle physics has been that of the existence of the complex plane’ (Eden et al., The Analytic S-Matrix (Cambridge, 1866), vii).
H. H. Pattee, ‘The Recognition of Hereditary Order in Primitive Chemical Systems’ in The Origins of Prebiological Systems and of their Molecular Matrices, ed. S. W. Fox (New York, 1965), 385405. There does not seem to be sufficient evidence for the absence of ‘error’ at the detailed biochemical level: error at the biochemical level need not lead to ‘error’ at the macroscopic level, there is latitude in coding; moreover, there may be interesting biochemical repairing mechanisms at work.
Historically, there are not many theories in pure biology that are sufficiently structured to make structural analysis, let alone a comparative study of structures of pairs of theories, interesting. Possible candidates might be found in embryology, genetics and neurophysiology. The absence of highly structured theories in biology is itself worthy of philosophic consideration. The absence of highly structured theories in biology is itself worthy of philosophic consideration. ‘Diese Tatsache, dass die Interessen der Biologie sich nicht immer mit denjenigen der mathematisch-physikalischen Wissenschaften decken, hat man bisher unbeachtet gelassen; auch von den Biologen wurde ein GALILEI, ein DESCARTES als Begründer der neuen Auffassung des Lebens gepriesen, obwohl an dieselben keine beachtenswertere biologische Idee anzuknüpfen ist’ (E. Radl, Geschichte der biologischen Theorien in der Neuzeit (Leipzig, 1913; reprinted Hildesheim, 1970), vol. 1, viii). ‘Und die Biologen, die ja in der Regel erst dann etwas akzeptieren, wenn die Physiker es ihnen zurechtgedacht haben, werden diese neue Philosophie erst wieder aus der Physik zu sich herüberholen’ (A. Meyer-Abich, in his Foreword to Geschichte der biologischen Theorien in der Neuzeit, 1970 ).
See, for example, M. Sachs, ‘Positivism, Realism, and Existentialism in Mach’s Influence on Contemporary Physics’ in Philosophy and Phenomenological Research, 3o (r97o). A further complication is that ‘general relativity’ by itself as developed so far can hardly be considered to be more than a frame.
See, e.g., A. Grünbaum, Philosophical Problems of Space and Time (New York, 1964 ).
See section II. 4, p. 236.
This may be based on an interaction whose energy graph shows a minimum when plotted against the product of charge and particle number. Broadly speaking, we have a perennial conflict between trying to preserve continuity on the one hand and the atomistic programme (preserving linearity) on the other hand.
Concerning atomism, this is the paradigm case for philosophers of science going off the deep end about ontology: modern physics has, in my opinion, abandoned any naive notion of existence of elementary particles. We are dealing not with things, but with events.
Examples include the case of Bohr’s determination of the allowed transitions by the prior criterion of correspondence to classical physics for large quantum numbers; and Planck’s use of correspondence with Wien’s law for short wave length. While Planck might have used correspondence with Rayleigh’s formula for long wave length, he in fact seems to have been guided in that direction by little more than the empirical results of Pringsheim and of Rubens indicating the breakdown of Wien’s law. It might be fairer to credit Rayleigh (Philosophical Magazine, 49 (I goo), 539) with appreciation of correspondence in that part of the spectrum. Against the available experimental evidence he derived his expression on the basis of Boltzmann’s statistical mechanics, and suggested that, though it failed in general, equipartition might ‘apply to the graver modes’. He was sufficiently convinced of the correctness of some aspects of orthodox thermodynamics that he considered it an argument against Wien’s law that, according to it, the intensity of radiation of given wave length would reach a limit as the temperature is raised.
In many cases, the relations of physical phenomena in two different physical questions have a certain similarity, which enables us, when we have solved one of these questions, to make use of our solution in answering the other. The similarity which constitutes the analogy is not between the phenomena themselves, but between the relations of these phenomena’ (J. C. Maxwell, Elementary Treatises on Electricity (1881)). See also J. C. Maxwell, ‘On the Mathematical Classification of Physical Quantities’, 77w Scientific Papers of James Clerk Maxwell, ed. W. D. Niven (Dover, 1965), vol. 2, 257.
See note 65.
Thus, the frequency of occurrence of dislocations (whose existence had not previously been suspected) could be computed on thermodynamic grounds for any particular crystal structure.
This paper is based on a talk with the same title delivered at a meeting of the British Society for the Philosophy of Science in April 1965. I wish to thank Dr A. F. Chalmers, Dr Mary Hesse, Professor N. Koertge, Dr M. Machover and Mr P. M. Williams for helpful discussions on various points, though they are not to be blamed for any statements made in this paper.
J. Hamilton, The Theory of Elementary Particles (Oxford, 1959 ).
For a discussion of this clock paradox and its geometrical analogue, see H. Bondi, Assumption and Myth in Physical Theory (Cambridge, 1967), 51 f.
See ‘Explanation, Reduction and Empiricism’, in Minnesota Studies in the Philosophy of Science. ed. Feigl and Maxwell (Minnesota, 1962), vol. 3.
Ibid., 37, lines 6–10.
Op. cit., note 44, t 70.
W. V. O. Quine, From a Logical Point of View (Cambridge, Mass., 1964 ), i6.
Op. cit., note 44, 175; my italics.
Kuhn, op. cit., note 36, 17.
Ibid.
Ibid., 169.
Ibid., 2nd edition (197o), postscript, 205–6.
Ibid., 207.
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Post, H.R. (1993). Correspondence, Invariance and Heuristics: In Praise of Conservative Induction. In: French, S., Kamminga, H. (eds) Correspondence, Invariance and Heuristics. Boston Studies in the Philosophy of Science, vol 148. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1185-2_1
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