The Material Theory of Induction

John D Norton

Reviewed by William Peden

The Material Theory of Induction
John D Norton
Calgary: University of Calgary Press, 2021, £119.99/OA
ISBN 9781773852751

Even prior to its publication, John Norton’s book has stimulated debates about induction. Its publication will galvanize these discussions. Does it merit all this attention? Yes, and not just from philosophers of science. Practically all philosophers will find novel and thought-provoking ideas, with implications for their research.

It has been nearly two decades since Norton first advanced his material theory of induction. One reason for the attention it has drawn is that the theory retains some of the more attractive features of the predominant Bayesian approach, while discarding the more concerning. According to Bayesianism, inductive reasoning should consist in specifying one’s ‘priors’ (degrees of belief) in the relevant hypotheses, which should be consistent with the probability calculus. As one accumulates evidence, these priors are revised by a formal method called ‘conditionalization’. Bayesianism has the appealing feature that the priors of hypotheses can vary with local background knowledge. Similarly, learning via conditionalization differs from context-to-context. For instance, with the right probabilities, discovering that gravitational waves obey general relativity theory can increase our confidence that all phenomena conform to this theory, in the same way as the measurement of Robert Wadlow (who was just shy of 273cm in height) reduces our degree of belief that nobody will ever be more than 273cm tall. Thus, Bayesianism offers a formal theory of induction, but one that can be sensitive to different circumstances.

Norton agrees that context is important, but he comes from a very different starting point. An old story about induction is that its rationality depends on a background assumption of nature’s uniformity. This assumption should justify the expectation that our observed samples are at least roughly representative of the populations described in our conclusions. Thus, some sweeping claims about nature will entail that an inductive argument form like ‘all observed X are Y, therefore all X are Y’ tends to be reliable, and hence it is (defeasibly) rational to use this argument form.

Norton agrees that background knowledge of reliability is important, but rejects the traditional view that it is general assumptions that justify good inductions. No plausible assumption of this type will be sufficient to clinch reliability. Instead, inductions are justified by context-specific background knowledge. For example, the conformity of observed gas samples to Boyle’s law can be extrapolated to all samples of gas because of specific factual claims in continuum mechanics and other scientific discoveries about gases, not because of a general uniformity principle. Typically, this context-specific (or ‘local’) background knowledge will not provide deductive certainty, but it must warrant the hypothesis that our observed samples are likely to be representative of our inference’s target population. Norton thereby shifts attention away from justifying claims that general forms of inductive argumentation are reliable. Instead, the justification comes from knowing that an inductive argument’s subject matter is such that our observations are a reliable guide to our conclusions.

In this sense, Norton has a ‘material’ rather than ‘formal’ theory of induction. He rejects the latter because any inductive pattern can be systematically unjustified in some broad domain, relative to our background knowledge of the relevant local facts about the inductive argument’s subject matter. For example, using ‘all observed X are Y, therefore all X are Y’ is unreasonable when inferring from samples of some birds’ plumage to the plumage of that species in general, because we know that this pattern is unreliable in this context—partly due to the famous cases of black swans. Logics of induction have their place, but their boundaries are drawn by our background knowledge. Any formal system that picks out and endorses general patterns of inductive reasoning will sometimes be irrational to use, because we know that there are some (and perhaps many) broad domains where it will not be reliable. Overall, the material theory retains the sensitivity to local context in Bayesianism—one of its best features—but drops the Bayesian thesis that probabilities always provide the best tools for formalising induction.

The book covers a wide range of topics in the philosophy of science, with the material theory as an overarching idea. Two initial chapters introduce his theory and the basic arguments for it. In the next six chapters, Norton discusses a broad spectrum of issues in the philosophy of science. He argues that universal formal principles of induction are unnecessary and even pernicious. As some candidates for universal claims about induction, he covers replicability, analogy, epistemic virtues, simplicity, and inference to the best explanation. In the process, he develops fresh ideas of his own. The final seven chapters build on Norton’s existing criticisms of Bayesianism. His most persuasive critique is that Bayesianism cannot adequately handle states of ignorance. These situations occur when we cannot base our priors on background knowledge of physical probabilities (such as relative frequencies or propensities) in a satisfactory way. Norton argues, in contrast, that the material theory can do a much better job of accommodating reasoning under ignorance. In particular, it allows us to use a variety of formal models of belief that are specially adapted to different contexts of inductive inference, such as quantum mechanics.

A detailed exposition of such a voluminous book would be impossible, so I shall focus on a few ideas that indicate its relevance to a great variety of debates. For philosophers of science, it will probably be the arguments against Bayesianism that grab their attention. Norton is far from an uncompromising critic: he grants that there is a place for the Bayesian approach to induction. In his view, it is an appropriate method when our priors are derived from background knowledge about the relevant physical probabilities. For instance, we can play games of chance using joint distributions that approximate the real long-run relative frequencies of events involving fair gambling apparatuses. Chemical kinetics gives us the approximate radioactive half-lives of elements, and thereby can justify the expected value of a sample’s decay. Meteorological models provide probabilistic expectations for weather events. Insofar as Bayesian reasoning uses priors from such sources, Norton has no issue with it.

However, he details a range of realistic examples where no probabilities are warranted by our background knowledge. Moreover, he argues that trying to represent this ignorance using additive probabilities will be a Procrustean bed that either distorts or tacitly supplements our actual scientific knowledge in these situations. Perhaps the most interesting examples, detailed in Chapter 15, occur when our background knowledge consists of a seemingly deterministic theory, such as Newtonian mechanics. Norton explains how these theories not only fail to be deterministic, but also fail to justify any particular probability distribution that could provide a basis for Bayesian reasoning.

Some aspects of Norton’s criticisms are clearer in this book than in their previously published versions. He has been raising the problem of determining priors under ignorance for many years. Worrall ([2010], pp. 751–52) has noted how most Bayesians should not be troubled by this argument, because they believe that mere opinion is a perfectly adequate basis for a prior—they are ‘subjective’ Bayesians. In the book’s new discussions, we can see that Norton’s point goes deeper than simply rejecting subjective priors. He specifies a plausible adequacy condition for using an inductive rule like conditionalization: our background knowledge should support the rule’s reliability in the context that we are using it. When a Bayesian probability distribution is appropriately derived from background knowledge, it meets this adequacy condition. What Norton’s examples suggest is that this derivation is not always possible. Of course, subjective Bayesians can contest this adequacy condition. In doing so, they enter into the deep and torrid waters of general epistemology—Rinard ([2017]) is an example of a contemporary Bayesian skilfully swimming in these depths. Naturally, Norton’s arguments are not the last word on Bayesianism, but they do show that pelagic excursions like Rinard’s are necessary. More generally, they evince the important connections between the methodology of science and current issues in general epistemology. I hope that his book will entice many philosophers working on the latter to engage with the materialist–Bayesian debates.

Similarly, many metaphysicians would benefit from examining what Norton has to say about simplicity. More than a few philosophers think that science assumes that nature is ‘simple’ in some sense of that word. Like a growing number of contemporary philosophers of science, Norton thinks that simplicity is very domain-relative. Additionally, he provides a novel and attractive explanation for why it is nonetheless useful to retain ‘simplicity’ as a general term for describing some scientific theories. He argues that this term functions as a promissory note for more detailed descriptions of the evidential relations between a theory and our total evidence. These evidential relations are often extremely complex; only a brave soul would try to collate and explain even a small fraction of evidential support for and against general relativity.

Norton defends this account with a wealth of detailed and persuasive examples. It also fits with the image developed by philosophers like Cartwright ([2020]) of science as a ‘tangle’ of diverse types of output. True, science provides us with data and theories, but it also produces experimental designs, instruments, measuring techniques, and so on. It can often be impossible to briefly describe how such a ‘tangle’ supports a theory. Hence, shorthand phrases like ‘this theory is the simplest explanation of the data’ have a plausible function as promissory notes for richer accounts, which should be described in more detail if requested. Hence, science does not presuppose nature’s simplicity. Norton does not explore the implications of this idea, but an important consequence is that similar assumptions in metaphysics have no precedent in the methodology of science.

But why does it sometimes seem that science assumes simplicity? Norton offers one novel and intriguing explanation: good scientists do not infer the existence of objects or properties without warrant from the evidence. This epistemological principle can sometimes look like a presupposition of simplicity, since it constrains conjectures of complexity. Yet it is actually a methodological principle, not a substantive claim. His discussions also suggest some supplementary explanations. For instance, Norton notes how simplicity has pragmatic value: ceteris paribus, we prefer to have simpler theories and descriptions of the world, insofar they are easier to use. Perhaps the generality of this practical preference for simpler theories can sometimes seem like a general substantive assumption of nature’s simplicity, partly because our preference for simpler theories sometimes biases our reasoning.

Norton’s arguments about inference to the best explanation (IBE) should also interest and provoke a wide spectrum of philosophers. What do philosophers add to human knowledge in a scientific age? For decades, a popular answer has been that we provide ‘explanations’. If a philosophical explanation possesses sufficient theoretical virtues (simplicity, fruitfulness, coherence, and so on), then we can infer, provisionally, that the theory is true. There is nothing worrisome about our role, because scientists do the same thing. Some philosophers even regard their discipline—at least as they practice it—as a branch of science, distinguished mainly by its extreme generality. Against this ‘factive’ approach to IBE, some philosophers of science have proposed ‘pragmatic’ theories, which say that IBE can only justify judgements about usefulness, pursuit-worthiness, or some other pragmatic virtue, rather than inferring factual claims.

However, according to Norton, good use of IBE in science actually involves highly sophisticated eliminative induction,¹Norton does not rule out the reduction of some cases of IBE to other types of induction. Khalifa et al. ([2017]) discuss this possibility in detail. rather than any distinct type of reasoning. In his view, good IBE in science has several steps. First, we specify a set of plausible alternative explanations of a phenomenon. Second, we find that all but one has a severe problem with its evidential relations to our overall data. The exact details of these problems are very contextual, in accordance with Norton’s general account of induction. Third, we tentatively adopt the surviving theory, subject to various provisos. The inference is not deductive because our delineation of possible theories is always incomplete. As with simplicity, describing a theory as ‘the best explanation’ is a promissory note for the sophisticated and bespoke argumentation required for high-quality eliminative inductive reasoning. Norton’s theory is a fascinating alternative to both orthodox and pragmatic theories of IBE. Moreover, this model of IBE in science seems to have little in common with IBE as practiced in philosophy. Though he does not discuss this point, Norton’s account raises serious issues for metaphilosophy.

Thus, there is a very wide audience who can benefit from this book. Readers will not have to agree with everything Norton says in order to find useful ideas. This book is the most novel, thought-provoking, and stimulating work on induction in a generation.

William Peden
Erasmus University Rotterdam
peden@esphil.eur.nl

Notes

¹ Norton does not rule out the reduction of some cases of IBE to other types of induction. Khalifa et al. ([2017]) discuss this possibility in detail.

References

Cartwright, N. [2020]: ‘Why Trust Science? Reliability, Particularity, and the Tangle of Science’, Proceedings of the Aristotelian Society, 120, pp. 237–52.

Khalifa, K., Millson, J. A. and Risjord, M. [2017]: ‘Inference to the Best Explanation: Fundamentalism’s Failures’, in K. McCain and T. Poston (eds), Best Explanations: New Essays on Inference to the Best Explanation, Oxford: Oxford University Press, pp. 80–96.

Rinard, S. [2017]: ‘No Exception for Belief’, Philosophy and Phenomenological Research, 94, pp. 121–43.

Worrall, J. [2010]: ‘For Universal Rules, against Induction’, Philosophy of Science, 77, pp. 740–53.