Abstract
The post-Newtonian view that evidence alone decides what theories are accepted in science fails to solve Hume’s problem of induction. But this Newtonian view is untenable in any case: in persistently accepting unified theories only, and ignoring endlessly many disunified rivals that fit available phenomena even better, physics thereby makes an implicit metaphysical assumption: the universe is such that all disunified theories are false. That refutes the post-Newtonian view of science. It is thus not surprising that the Newtonian view fails to solve Hume’s problem. But does aim-oriented empiricism do better? It does; it solves the problem. First, in acknowledging that physics makes a substantial, metaphysical assumption about the nature of the universe, aim-oriented empiricism is more rigorous intellectually than all versions of its post-Newtonian rival. Secondly, the hierarchical structure of aim-oriented empiricism provides the best means for developing metaphysical assumptions of science that represent the nature of the universe in increasingly accurate and truth-like ways. Third, it is shown that Hume’s argument that there cannot be necessary connections between successive states of affairs is false. What exists now may well determine necessarily what exists in the future. We are justified in accepting the results of science.
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Notes
- 1.
My Understanding Scientific Progress, Maxwell (2017b) especially chapter 9, gives what I like to think of as the definitive solution to the problem of induction. Aim-oriented empiricism solves Hume’s problem of induction. Here, I merely highlight a few crucial points of the solution.
- 2.
Some interpret the problem as arising not just in connection with science, but whenever we accept generalizations or laws on the basis of evidence. But if we can solve the problem in connection with science, that may be held to provide a basis for the solution to any more general problem.
- 3.
For my criticisms of proposals put forward by Jeffreys and Wrinch (1921), Popper (1959, pp. 62–70 and 126–145), Friedman (1974), Kitcher (1981, 1989) and Watkins (1984, pp. 203–213), see my (1998, pp. 56–68). For criticisms of more recent proposals, including those of Bartelborth (2002), McAllister (1996), Weber (1999) and Schurz (1999), see my (2004c). Many attempts have been made by philosophers to explicate the non-empirical criteria a physical theory must satisfy to be acceptable.
- 4.
Einstein (1949, pp. 21–25).
- 5.
What follows is a highly simplified version of the solution to the problem of what it means to say of a physical theory that it is unified. For refinements, and the full story, see Maxwell (2017b, ch. 5).
- 6.
In line with what I have said above, we need to restrict the problem, in the first instance, to the fundamental theories of physics. Non-empirical considerations influencing acceptability of theory in other sciences include results in more fundamental sciences—ultimately fundamental physics.
- 7.
Examples of theories disunified in this sense have been given in Chap. 4.
- 8.
If physicalism is true, there are infinitely many imprecise laws and theories that are true. For example, Kepler’s law that the planets travel in ellipses round the sun is false if asserted as a precise law, but becomes true if asserted as an appropriate imprecise statement: the planets travel in paths round the sun that approximate to ellipses to such and such an extent.
- 9.
This point was spelled out in Chap. 4.
- 10.
What does “physicalism” in this sense assert? It asserts that the universe is such that there is a yet-to-be-discovered physical “theory of everything” that predicts and explains all actual and possible physical phenomena, and is unified.
- 11.
See Maxwell (2021a) for a more detailed exposition of this last point.
- 12.
What does it mean to say that metaphysical thesis m1 accords better with thesis M than m2? It means that m1 can be “approximately derived” from M, and m2 can be approximately derived from m1, but m1 cannot be approximately derived from m2. Or that physical theories that accord with or exemplify m1 can be approximately derived from theories that exemplify M, and theories that accord with m2 can be approximately derived from theories that exemplify m1, but theories that exemplify m1 cannot be approximately derived from theories that exemplify m2. For an explication of what the crucial notion of “approximate derivation” means see Maxwell (2017b, pp. 90–91). Roughly, T1 is approximately derivable from T2 if, as a result of T2 being restricted in scope, being simplified as a result of quantities that are distinct from zero being set to zero, and being reinterpreted, a theory \( {\mathrm{T}}_1^{\ast } \) emerges that is empirically equivalent to T1. When making derivations, physicists in practice very often allow non-zero quantities to go to zero, and make other simplifications characteristic of “approximate derivations”. Thus, in practice, a derivation in physics tends to be an “approximate derivation”. The notion is not obscure. It is an entirely familiar feature of physics in practice.
- 13.
A metaphysical thesis is not empirically testable, but it can be empirically fruitful in being in accord with a sequence of theories of ever increasing empirical success: see Maxwell (2017b, pp. 80 and 131) for further details. An empirically fruitful metaphysical thesis, M, may be said to be one that “supports an empirically progressive research programme”—a programme of research that produces a sequence of testable theories, T1…Tn, each meeting with empirical success before being refuted and replaced by its successor that meets with even greater empirical success before being in turn refuted, this series of theories drawing ever closer to capturing M as a testable theory. See Maxwell (2005a, section 8, Imre Lakatos) for an account of the aim-oriented empiricist conception of research programme and how it differs from Lakatos’s (1970) conception.
- 14.
This positive feedback feature of aim-oriented empiricism may seem to be a fatal defect in the view, not a strength, for it seems to amount to vicious circularity, metaphysics being used to justify theory, and theory then being used to justify metaphysics. We shall see, however, that aim-oriented empiricism succeeds in solving this well-known problem.
- 15.
Standard empiricism fails to solve the problem of induction because it quite fundamentally lacks intellectual rigour. Highly influential and profoundly problematic metaphysical assumptions, instead of being acknowledged and thus thrown open to criticism and improvement, are repudiated and denied. A science that pretends to implement standard empiricism is riddled with hypocrisy and intellectual dishonesty.
- 16.
In what follows, universes of various kinds are considered: universes that are knowable, comprehensible, epistemologically malicious, or non-malicious. In every case, what is referred to is not just a kind of universe, but a kind of universe with respect to conscious beings of a certain type in the universe with a body of knowledge and specific capacities for improving knowledge—namely, ourselves. Our universe is perhaps comprehensible with respect to ourselves, but not comprehensible with respect to chimpanzees.
- 17.
We might discover that this level 7 thesis of partial knowability will, in the future, become false, perhaps because conditions on earth will no longer support life. Such a discovery requires, however, that we accept, as an item of knowledge, that the thesis of partial knowability is at present true—according to the argument being developed here.
- 18.
The term comes from Einstein. During a visit to Princeton University in 1921, Einstein remarked “The Lord God is subtle, but malicious he is not”—a remark later transcribed in German over the fireplace in the Einstein faculty lounge of the Princeton Department of Mathematics: “Raffiniert ist der Herr Gott, aber boshaft ist er nicht.” In declaring that the Lord God is not malicious, Einstein asserted that the universe is not structured in such a way as to lure us into developing apparently astonishingly successful theories which subsequently turn out to be hopelessly wrong. In Fig. 1.2, Chap. 1, the level 6 thesis is said to be that of meta-knowability, not non-maliciousness; however, below I show that non-maliciousness implies meta-knowability.
- 19.
A meta-knowable universe is not one in which the positive feedback procedure definitely exists, as it were, in the constitution of the universe, awaiting to be discovered. It may not exist at all. Rather, a meta-knowable universe is one in which, if the positive feedback procedure seems to be working and giving results, then it will be the case that if it breaks down, and turns out to deliver only the illusion of knowledge, this breakdown could have been discovered before it actually occurs. A meta-knowable universe, in other words, is not one that contains in its constitution the possibility of the positive feedback procedure appearing to give results which then break down, and this breakdown could not have been discovered even in principle before it occurs. There are no pockets of epistemologically malicious feedback procedures built into the constitution of the universe, as it were, as deceptive lures or traps, awaiting discovery and subsequent disaster.
- 20.
For further discussion, see Maxwell (2017b, pp. 103–120).
- 21.
This is interpreted here, remember, to be the thesis that asserts that the universe is physically comprehensible, the universe being such that there is a yet-to-be-discovered, unified physical “theory of everything” that, in principle, together with appropriate initial conditions, predicts and explains all physical phenomena.
- 22.
According to special relativity, the laws of nature have the same form relative to all inertial reference frames, however they move (with constant velocity) with respect to each other. But, as one goes from one reference frame to another with a different velocity, the electromagnetic field divides up differently into electric field and magnetic field—even though the synthesis of these, the electromagnetic field itself, is the same with respect to all these reference frames. Thus, given special relativity, it is the electromagnetic field, invariant with respect changes of reference frames, that enters into physical law, not its two, variable aspects, the electric field and the magnetic field. The electric and magnetic fields are two aspects of the one unified entity, the electromagnetic field.
- 23.
Minkowskian space-time is four dimensional, three of space and one of time. Given any point, P, in this space, there meet at P the apexes of two (four-dimensional) spherical cones, one spreading outwards into the past, the other spreading outwards into the future. Light in a vacuum travels along the surfaces of these cones. Any point inside the future cone is to the future of P; any point inside the past cone is to the past of P. Any point outside both cones could be in the past, present or future of P, depending on the choice of reference frame. Such a point will, in any case, be spatially separated from P.
- 24.
According to general relativity, bodies subject only to gravitation move along geodesics in curved space-time—a geodesic being the nearest thing to a straight line is curved spacetime—the shortest distance between two points. All bodies move along straight lines, even those subject to gravitational forces; it is just that gravitation curves space-time, and so curves the “straight line” that a body moves along.
- 25.
- 26.
For further discussion see Maxwell (1998, pp. 80–89, 131–140, 257–265), and additional works referred to therein.
- 27.
Furthermore, in Understanding Scientific Progress, I distinguish eight distinct versions of physicalism that become increasingly demanding and restrictive, as one goes from the first version to the eighth. The eighth asserts that the true physical “theory of everything” is unified to the extent that particles-and-forces on the one hand, and space and time on the other, are synthesized into one unified, fundamental physical entity. It is this version of physicalism, I argue, that we should accept at level 4.
- 28.
All fundamental, dynamical theories accepted so far in physics, from Newtonian theory (NT) to the standard model, can be formulated in terms of a Lagrangian and Hamilton’s principle of least action. In the case of NT, this takes the following form. Given any system, we can specify its kinetic energy, KE (energy of motion), and its potential energy, PE (energy of position due to forces), at each instant. This enables us to define the Lagrangian, L, equal at each instant to KE – PE. Hamilton’s principle states that, given two instants, t1 and t2, the system evolves in such a way that the sum of instantaneous values of KE – PE, for times between t1 and t2, is a minimum value (or, more accurately, a stationary value, so that it is unaffected to first order by infinitesimal variations in the way the system evolves). From the Lagrangian for NT (a function of the positions and momenta of particles) and Hamilton’s principle of least action, we can derive NT in the form familiar from elementary textbooks. It is this way of formulating NT, in terms of a Lagrangian, L, and Hamilton’s principle, that can be generalized to apply to all accepted fundamental theories in physics. Lagrangianism, then, asserts that the universe is such that all phenomena evolve in accordance with Hamilton’s principle of least action, formulated in terms of some unified Lagrangian (or Lagrangian density), L. We require, here, that L is not the sum of two or more distinct Lagrangians, with distinct physical interpretations and symmetries, for example one for the electroweak force, one for the strong force, and one for gravitation, as at present; L must have a single physical interpretation, and its symmetries must have an appropriate group structure. We require, in addition, that current quantum field theories and general relativity emerge when appropriate limits are taken. Even if the level 4 thesis of physicalism is true, it may nevertheless be the case that Lagrangianism is false. This is so if space-time is discontinuous in the very small. In fact developments in quantum gravity, having to do with “duality”, suggest that Lagrangianism may be false: see Isham (1997, pp. 194–195).
- 29.
- 30.
- 31.
Bayesianism, in this context, is the doctrine that scientific method can be understood in probabilistic terms. A hypothesis is put forward; scientists assign a probability of it being true prior to evidence in support of it. If the hypothesis is then verified by evidence, theorems of the probability calculus tell us how the “prior” probability is increased. A major difficulty is the basis, the rationale, for assigning prior probabilities to hypotheses greater than zero. If zero is assigned, the probability of the hypothesis remains zero, however empirically successful it may subsequently be. The fundamental problem is however that Bayesianism attempts the impossible: to make sense of science purely in terms of theory and evidence, and in the absence of any metaphysical or cosmological view about the nature of the universe.
- 32.
It is very odd that Roush should say that Bayesianism “can readily accommodate metaphysical assumptions in prior probabilities on hypotheses” for, in doing so, Bayesianism ensures that metaphysical assumptions influence what theories are accepted and rejected, just that which she sets out to criticize.
- 33.
In theoretical physics, a fundamental physical theory—such as Newtonian theory, Maxwellian electrodynamics, or Einstein’s theory of general relativity—once empirically verified and accepted, tends to stand unchallenged for decades, even centuries. No proposed rival theory comes near to providing a seriously competitor. No version of standard empiricism can do justice to this situation, whether it be a version of confirmation theory, inference-to-the-best-explanation, Popper’s falsificationism, or Bayesianism. Non-empirical requirements these views hold a theory must meet to be acceptable—such as theory “simplicity”, “beauty” or their equivalents—are left too ill-defined and open-ended for it to be possible to select just one physical theory as acceptable, without any serious competition. By contrast, aim-oriented empiricism does justice to this situation without difficulty. It is astonishingly difficult to satisfy the two demands that an acceptable theory must satisfy, namely (a) exemplifying physicalism in a sufficiently adequate way, and (b) predicting phenomena in a sufficiently accurate and extensive manner.
- 34.
Fred Muller has declared that this argument is similar to Pascal’s wager: see Muller (2004, p. 115). Pascal famously argued that we should live as if, and strive to believe that, God exists since if He does not we have little to lose, whereas if He does, we may have much to gain. But the comparison is not valid. In the case of non-maliciousness, entertaining the possibility that the universe might be malicious after all provides no basis whatsoever for action, as nothing can, in principle, provide a clue that a malicious episode is about to occur, and nothing could indicate what form such an episode might take. As far as Pascal’s wager is concerned, entertaining the possibility that God does not exist immediately opens up a whole vast domain of possible actions, indeed entirely new ways of life: living with integrity, discovering for oneself what kind of world this is, and what is ultimately of value in life, instead of acquiring such things from ancient texts and the sermons of priests. The very intellectual and moral integrity of one’s life may be said to be at stake. There is every reason to ignore Pascal’s wager, and no reason whatsoever to take maliciousness seriously.
- 35.
Hume (1959, pp. 163–164).
- 36.
See Maxwell (1968a). See also Maxwell (2019b, chs. 1 and 2) for a more readable account, and for criticism of the work of others that took up my idea but, in the process, degraded it by interpreting “necessity” to mean “metaphysical necessity” and not “logical necessity”, as in my original paper. Dretske (1977); Tooley (1977, 1987); Armstrong (1983); Swoyer (1982); Carroll (1994); Shoemaker (1980); Ellis (2001); and Bird (2007) all defended degraded versions of my 1968 paper on necessary connections, often without referring to my earlier work even though in at least some cases my work was known to the authors in question. The outcome of this body of subsequent work is that the thesis of my 1968 paper is almost entirely unknown to most philosophers today. Unknown too is the crucial connection this thesis has with the mind-brain problem, in establishing that physics is about only a highly selective aspect of everything, the causally efficacious aspect, the silence of physics about perceptual qualities thus in no way implying that these qualities do not exist in the world around us. Adopt the view that perceptual qualities do exist in the world around us, and it becomes possible to hold that what we directly perceive is the world around us, not our inner representation of it. That in turn makes it possible to say we do not ordinarily know enough about our inner perceptual experiences to know that they are not brain processes.
- 37.
For more details as to how this would work in practice, see Maxwell (1968a).
- 38.
I have ignored a number of important details spelled out in the original exposition of the argument, concerning such things as the nature of time, the relationship between theoretical and observational entities and properties: for these important details, see Maxwell (1968a). See also Maxwell (2019b, chs. 1 and 2), where I give a much more readable account of my 1968 argument establishing the possibility of necessary connections between successive states of affairs.
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Maxwell, N. (2024). Appendix 1: How to Solve Hume’s Problem of Induction. In: The Philosophy of Inquiry and Global Problems. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-031-49491-8_7
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