What inductive explanations could not be

Marc Lange argues that proofs by mathematical induction are generally not explanatory because inductive explanation is irreparably circular. He supports this circularity claim by presenting two putative inductive explanantia that are one another’s explananda. On pain of circularity, at most one of this pair may be a true explanation. But because there are no relevant differences between the two explanantia on offer, neither has the explanatory high ground. Thus, neither is an explanation. I argue that there is no important asymmetry between the two cases because they are two presentations of the same explanation. The circularity argument requires a problematic notion of identity of proofs. I argue for a criterion of proof individuation that identifies the two proofs Lange offers. This criterion can be expressed in two equivalent ways: one uses the language of homotopy type theory, and the second assigns algebraic representatives to proofs. Though I will concentrate on one example, a criterion of proof identity has much broader consequences: any investigation into mathematical practice must make use of some proof-individuation principle

Separability in gauge theories

Much of the philosophical literature on classical Yang--Mills theories is concerned with the differences between interpretations that represent the state of the world in terms of fields and those that represent the state of the world in terms of properties assigned to curves in spacetime. In the philosophical literature on Yang-Mills theories, field formulations are taken to have more structure and to be local in a particular sense, while the alternative curve-based formulations are taken to have less structure at the cost of nonlocality. I formalize the notion of locality at issue and show that theories with less structure are indeed nonlocal. However, the amount of structure had by some formulation is independent of whether it uses fields or curves. This leads to a general lesson about structure in mathematized theories. The difference in structure corresponding to the difference in locality is not a difference in set-theoretic structure. Rather, it is a difference in the structure of the collection of models of the theory considered as a category

The non-ideal theory of the Aharonov–Bohm effect

Elay Shech and John Earman have recently argued that the common topological interpretation of the Aharonov–Bohm (AB) effect is unsatisfactory because it fails to justify idealizations that it presupposes. In particular, they argue that an adequate account of the AB effect must address the role of boundary conditions in certain ideal cases of the effect. In this paper I defend the topological interpretation against their criticisms. I consider three types of idealization that might arise in treatments of the effect. First, Shech takes the AB effect to involve an idealization in the form of a singular limit, analogous to the thermodynamic limit in statistical mechanics. But, I argue, the AB effect itself features no singular limits, so it doesn’t involve idealizations in this sense. Second, I argue that Shech and Earman’s emphasis on the role of boundary conditions in the AB effect is misplaced. The idealizations that are useful in connecting the theoretical description of the AB effect to experiment do interact with facts about boundary conditions, but none of these idealizations are presupposed by the topological interpretation of the effect. Indeed, the boundary conditions for which Shech and demands justification are incompatible with some instances of the AB effect, so the topological interpretation ought not justify them. Finally, I address the role of the non-relativistic approximation usually presumed in discussions of the AB effect. This approximation is essential if—as the topological interpretation supposes—the AB effect constrains and justifies a relativistic theory of the electromagnetic interaction. In this case the ends justify the means. So the topological view presupposes no unjustified idealizations

The non-ideal theory of the Aharonov–Bohm effect

Elay Shech and John Earman have recently argued that the common topological interpretation of the Aharonov–Bohm (AB) effect is unsatisfactory because it fails to justify idealizations that it presupposes. In particular, they argue that an adequate account of the AB effect must address the role of boundary conditions in certain ideal cases of the effect. In this paper I defend the topological interpretation against their criticisms. I consider three types of idealization that might arise in treatments of the effect. First, Shech takes the AB effect to involve an idealization in the form of a singular limit, analogous to the thermodynamic limit in statistical mechanics. But, I argue, the AB effect itself features no singular limits, so it doesn’t involve idealizations in this sense. Second, I argue that Shech and Earman’s emphasis on the role of boundary conditions in the AB effect is misplaced. The idealizations that are useful in connecting the theoretical description of the AB effect to experiment do interact with facts about boundary conditions, but none of these idealizations are presupposed by the topological interpretation of the effect. Indeed, the boundary conditions for which Shech and demands justification are incompatible with some instances of the AB effect, so the topological interpretation ought not justify them. Finally, I address the role of the non-relativistic approximation usually presumed in discussions of the AB effect. This approximation is essential if—as the topological interpretation supposes—the AB effect constrains and justifies a relativistic theory of the electromagnetic interaction. In this case the ends justify the means. So the topological view presupposes no unjustified idealizations

Accessory cell activity of tumour-associated macrophages

On the basis of Fc receptor expression and phagocytic activity, approximately 20 -25% of the cells present within the highly immunogenic, methylcholanthrene- induced, murine fibrosarcoma FSA -R could be classified as macrophages. These cells did not express the Mac -1 antigen and were approximately 50% I -Ak- positive. The expression of I -Ak required the presence of mature T cells, and macrophages obtained from tumours grown in nu /nu hosts were I -Ak negative. Both the percentage of macrophages present within the tumour and I -Ak expression by these cells remained constant during the observed period of tumour growth and during serial passage of the tumour in vivo.Macrophages were enriched from enzymatically disaggregated tumour cell suspensions by virtue of their capacity to adhere tightly to glass or plastic surfaces and the accessory cell activity of the adherent cell population was investigated in various well defined in vitro assay systems. Thus tumour - associated macrophages were shown to be fully capable of reconstituting the primary anti -CRBC PFC response of Sephadex G -10 passed normal spleen cells. This function required the presence of I -Ak- positive cells, and tumour- associated macro- phages treated with anti -Ia serum plus complement or obtained from tumours grown in nu /nu hosts, were inactive. Tumour - associated macrophages were also able to supply the essential cell activity required for effective cooperation between antigen- primed TH cells and normal B cells in the generation of PFC responses in vitro. Finally, tumour -associated macrophages were found to secrete a soluble factor(s) which considerably enhanced the primary anti -CRBC PFC response of whole normal spleen cells. Cell separation studies indicated that this activity was primarily a function of large macrophages. Furthermore, since macrophages treated with anti -Ia serum plus complement', or obtained from tumours grown in nu /nu hosts were as efficient as normal tumour -associated macrophages at enhancing the response, it would seem that I -Ak- positive cells were not involved in this function.The possible significance of tumour -associated macrophage accessory cell activity is discussed