Modern mathematical sciences are hard to imagine without appeal to efficient computational algorithms. We address several conceptual problems arising from this interaction by outlining rival but complementary perspectives on mathematical tractability. More specifically, we articulate three alternative characterizations of the complexity hierarchy of mathematical problems that are themselves based on different understandings of computational constraints. These distinctions resolve the tension between epistemic contexts in which exact solutions can be found and the ones in which they cannot; however, contrary to a persistent myth, we conclude that having an exact solution is not generally more epistemologically beneficial than lacking one.

Some ecologists have argued that theoretical model building in population and community ecology has gone evidentially unconstrained. In the essay, I argue that “bottle experiments” offer ecological model building evidential constraints and illustrate this by considering work on chaotic models tested by the dynamics of flour beetles. Critics reply that these experiments are importantly unlike nonmanipulated natural systems and thus do not constitute genuine tests of the models. I conclude by considering two responses to this worry and a suggestion on how to move forward on this debate.

A common view among statisticians is that convergence (which statisticians call consistency) is a necessary property of an inference rule or estimator. In this paper, this view is challenged by appeal to an example in which a rule of inference has a likelihood rationale but is not convergent. The example helps clarify the significance of the likelihood concept in statistical inference.

In his new book What is a Law of Nature?, David Armstrong gives an account of functional laws on the basis of the theory, originally proposed independently by Armstrong himself, Dretske, and Tooley, and further developed in this work, which asserts that laws are relations of necessitation between properties. On the theory, properties and relations are universals, and so a law is a relation between universals and is itself a universal. There are two reasons why Armstrong's account of functional laws is worth discussing. First, any theory that purports to be about laws that are investigated by science–I assume that we are only interested in such theories–must cover functional laws and not merely be adequate for generalizations about ravens or grue. Second, Armstrong claims that the main rival to his theory, the regularity theory of laws, cannot give a satisfactory account of functional laws (p. 22 and p. 37, all references to What is a Law of Nature? unless otherwise stated). If this is true, and if Armstrong's theory can provide a satisfactory account of functional laws, then this may incline us towards his point of view.

This article addresses the question of whether we should conceive of mechanisms as productive of change in a regular way. I argue that, if mechanisms are characterized as fully regular, on the one hand, then not enough processes will count as mechanisms for them to be interesting or useful. If no appeal to regularity is made at all in their characterization, on the other hand, then mechanisms can no longer be useful for grounding prediction and supporting intervention strategies. I conclude that, if the New Mechanistic Philosophy is to be successful, a stochastic characterization of mechanisms must be adopted.

Following an account of the incommensurability argument, an objection, based on assumptions concerning rival theories, is examined and rejected. This rejection leads to an alternative direction of criticism of incommensurability, a direction that involves the articulation of comparative standards of theory evaluation that are independent of meaning invariance.

Recently, Nathan (1986) criticized Bar-Hillel and Falk's (1982) analysis of some classical probability puzzles on the grounds that they wrongheadedly applied mathematics to the solving of problems suffering from “ambiguous informalities”. Nathan's prescription for solving such problems boils down to assuring in advance that they are uniquely and formally soluble—though he says little about how this is to be done. Unfortunately, in real life problems seldom show concern as to whether their naturally occurring formulation is or is not ambiguous, does or does not allow for unique formalization, etc. One step towards dealing with such problems intelligently is to recognize certain common cognitive pitfalls to which solvers seem vulnerable. This is discussed in the context of some examples, along with some empirical results.

This article introduces Harvey Brown and Oliver Pooley’s ‘dynamical approach’ to special relativity, and argues that it may be construed as a relationalist form of Einstein’s ‘practical geometry’. This construal of the dynamical approach is shown to be compatible with related chapters of Brown’s text and also with recent descriptions of the dynamical approach by Pooley and others.

Precaution is a relevant and much-invoked value in environmental risk analysis, as witnessed by the ongoing vivid discussion about the precautionary principle (PP). This article argues (i) against purely decision-theoretic explications of PP; (ii) that the construction, evaluation, and use of scientific models falls under the scope of PP; and (iii) that epistemic and decision-theoretic robustness are essential for precautionary policy making. These claims are elaborated and defended by means of case studies from climate science and conservation biology.

In this article I argue that structure in chemistry is a creature of abstraction: attending selectively to structural similarities, we neglect differences. There are different ways to abstract, so abstraction is interest dependent. So is structure. First, there are two different and mutually irreducible notions of structure in chemistry: bond structure and geometrical structure. Second, structure is relative to scale (of energy, time, and length): the same substance has different structures at different scales, and relationships of structural sameness and difference vary across the scales. However, these facts have no tendency to undermine structure’s claim to reality, or its metaphysical seriousness.

The permutation symmetry of quantum mechanics is widely thought to imply a sort of metaphysical underdetermination about the identity of particles. Despite claims to the contrary, this implication does not hold in the more fundamental quantum field theory, where an ontology of particles is not generally available. Although permutations are often defined as acting on particles, a more general account of permutation symmetry can be formulated using superselection theory. As a result, permutation symmetry applies even in field theories with no particle interpretation. The quantum mechanical account of permutations acting on particles is recovered as a special case.

There is renewed interest in the logic of discovery as well as in the position that there is no reason for philosophers to bother with it. This essay shows that the traditional, philosophical arguments for the latter position are bankrupt. Moreover, no interesting defense of the philosophical irrelevance or impossibility of the logic of discovery can be formulated or defended in isolation from computation-theoretic considerations.

This paper defends the Causal Theory of Reference against the recent criticism that it imposes a priori constraints on the aims and practices of science. The metaphysical essentialism of this theory is shown to be compatible with the requirements of naturalistic epistemology. The theory is nevertheless unable to forestall the problem of incommensurability for scientific terms, because it misrepresents the conditions under which their reference is fixed. The resources of the Causal Theory of Reference and of the traditional cluster or “network” theory of meaning for handling problems of commensurability are compared, and an alternative approach is recommended.

This paper presents a critical examination of claims advanced by several philosophers to the effect that ‘time travel’ represents a physical possibility and that the interpretation of certain actually observed phenomena in terms of ‘time travel’ is both legitimate and advantageous. It is argued that (a) no convincing motivation for the introduction of the time travel hypothesis has been presented; (b) no coherent and interesting sense of ‘going backward in time’ has been supplied which makes ‘time travel’ compatible with Special Relativity; (c) even the conceptual possibility of ‘time travel’ is an unsettled and somewhat nebulous question.

The general concept of a causal system has been basic in scientific thought. It may be formulated as follows. The system in question possesses certain measurable attributes such as those of dimensions, temperature, and so forth. In the case of a causal system, it is affirmed that the subsequent development of the system from a known initial condition—that is, a condition in which the measurable variables have known values—is uniquely determined by these values. More definitely, the value of these same variables after any interval of time depends only on the initial values of the variables and the time interval which has elapsed.

Comparativism—the view that mass ratios are not grounded in absolute masses—faces a challenge by Baker which suggests that absolute masses are empirically meaningful. Regularity comparativism uses a liberalized version of the Mill-Ramsey-Lewis Best Systems Account to have both the laws of Newtonian gravity and the absolute mass scale supervene on a comparativist Humean mosaic as a package deal. I discuss three objections to this view and conclude that it is untenable. The most severe problem is that once we have reduced away the absolute masses, there is nothing that stops us from also reducing the mass ratios.

In the human sciences, experimental research is used to establish causal relationships. However, the extrapolation of these results to the target population can be problematic. To facilitate extrapolation, we propose to use the statistical technique Latent Class Regression Analysis in combination with the analogical reasoning theory for extrapolation. This statistical technique can identify latent classes that differ in the effect of X on Y. In order to extrapolate by means of analogical reasoning, one can characterize the latent classes by a combination of features and then compare these features to features of the target.

This article examines the issue of whether consideration of so-called minimal models can prompt learning about real-world targets. Using a widely cited example as a test case, it argues against the increasingly popular view that consideration of minimal models can prompt learning about such targets. The article criticizes influential defenses of this view for failing to explicate by virtue of what properties or features minimal models supposedly prompt learning. It then argues that consideration of minimal models cannot prompt learning about real-world targets unless one supplements these models with additional information or presuppositions concerning such targets.

The cosmological constant problem stems from treating quantum field theory and general relativity as an effective field theory (EFT). We argue that the problem is a reduction ad absurdum and that one should reject the assumption that general relativity can generically be treated as an EFT. This marks a failure of naturalness and provides an internal signal that EFT methods do not apply in all spacetime domains. We then take an external view, showing that the assumptions for using EFTs are violated in general relativistic domains where Λ is relevant. We highlight some ways forward that do not depend on naturalness.

Aldo Leopold’s land ethic, an extremely influential view in environmental ethics and conservation biology, is committed to the claim that interdependence between humans, other species, and abiotic entities plays a central role in our ethical responsibilities. Thus, a robust understanding of “interdependence” is necessary for evaluating the viability of the land ethic and related views, including ecological ones. I characterize and defend a Leopoldian concept of “interdependence,” arguing that it ought to include both negative and positive causal relations. I also show that strength and type of interdependence can vary with time, space, and context.