## Final Report Summary - TMIRP (Topological Methods for Intertheoretic Reduction in Physics)

Distinct theories in physics often concern overlapping domains of phenomena. For example, both Newton’s theory of gravitational attraction and Einstein’s theory of general relativity concern how massive bodies attract one another. But these two describe the nature of that attraction in very different ways: in Newton’s theory, massive bodies exert a mutually attractive force, while in Einstein’s theory, matter warps space and time themselves so that massive bodies come together without force, just as two marbles rolling down a wide bowl will get closer together. Despite the radically different worldviews they suggest, and the experimentally verified superiority of Einstein’s, Newton’s theory is still used to great effect all the time, even in sending satellites or probes (or automobiles!) into orbit or on journeys across the solar system. How can a theory that is so wrong still be so right?

The standard sort of answer to this question is that, from the perspective of Einstein’s theory, we can explain why and under what circumstances Newton’s theory was and continues to be so successful. Among philosophers of science, such a relationship, known as reduction, is important because it informs us about how scientific theories are (or are not) unified, and the nature of scientific theory change. Among physicists, it is often known as a correspondence limit, and is equally important because it informs us about when we may simplify our treatment of a physical phenomenon and as a consistency check on the development of new theories, such as attempts to combine together Einstein’s theory with another great pillar of modern physics, the quantum theory of matter.

The main goal of the project Topological Methods for Intertheoretic Reduction in Physics (TMIRP) is to develop new methodological tools from the branch of mathematics called topology to better close the gap between philosophers’ and physicists’ understanding of reduction. Although much of the mathematics of these tools has been available for some time, they have received little attention from physicists and philosophers for this particular use, perhaps because, in their high degree of abstraction, they had not received a satisfactory interpretation until previous work in which I had provided such an interpretation and begun to apply it to the case of gravitational physics mentioned before: Newton’s and Einstein’s theories.

Indeed, one of the accomplishments of the project was increased clarity on how those two theories should be related and compared, in particular the need for precision and clarity in the description of observers, as described within those theories, and what exactly they can observe. These insights were then extended in two direct ways. The first was to the relationship between the general theory of relativity, already mentioned, and the special theory of relativity: Einstein’s earlier theory that did not incorporate gravitation, but still implemented his ideas about the relativity of motion and the special status of light. This has revealed two distinct ways in which the special theory is related to the general: one involving observing only small regions of space and another with relatively light matter. This in turn has revealed connections to certain versions of Einstein’s famous but vague principle of equivalence and the so-called requirement of “minimal coupling” between matter and the geometry of space and time that have hitherto been unexplored, despite them being venerable topics in the foundations of relativity theory. The second extension is to a broad class of theories in physics called gauge theories, which provide a general framework for theories of fundamental fields, particles, and interactions.

Another goal accomplished was to understand in general the relationship between philosophers’ and physicists’ notion of reduction better. On the one hand, physicists’ notion of reduction often invokes some sort of limiting relationship between two theories—e.g. the limit of vanishing gravitational interaction undergirds the reduction between special and general relativity. On the other hand, philosophers’ notion of reduction, following the influential account due to Ernest Nagel in the 1960s, takes reduction to involve a logical relationship between theories, in particular the possibility of a deduction of the laws of one theory by another (perhaps under special conditions). One finds in the literature both affirmative and negative answers to the question of whether the former can be fit into a (perhaps only slightly modified) version of the latter. A detailed investigation has revealed that it can be fit more closely than a detractor might have expected, but not completely. The reason for the lack of complete fit draws significant attention to the role of topology (and related mathematical structures) in describing how theories are related. Previous work done by the philosophical school of scientific structuralists shares with the present project a common concern with using topological-like structures to understand approximation of one theory by another, this was never a focus and was not really explored in any detail as is done in the current project.

One of the exciting developments of the TMIRP project has been further clarity on applications of the methods developed within it to other topics in philosophy of science, connections that were only dimly understood when the project was originally conceived. One such application involves so-call counterfactual reasoning in science, reasoning not about what has or will happen, but what would have happened or would happen if circumstances were different. Philosophers are keenly interested in the logic of this sort of reasoning, in part because of its intimate connection with causal reasoning, used throughout the sciences. Just as with the case of reduction, the scientific and philosophical treatments of counterfactual reasoning are not as clearly connected or mutually insightful as they could be, but the topological tools developed in the TMIRP project suggest ways of connecting them together. In particular, it shows how particular scientific theories, along with careful consideration of which observation are relevant for certain sorts of counterfactual claims, can be used rigorously to provide the truth-conditions (semantics) for those claims. With further work this in turn may have a salutary effect on causal and counterfactual reasoning in the sciences generally, including applications to cases of biomedicine and policy.

Another application involves the concept of emergence. Emergence is often understood as some kind of comparative novelty or apparent inexplicability. For example, the collective organization of a flock of birds, such as the dramatic undulating murmurations of starlings, is entirely surprising from the behavior of a single bird alone. Put in terms closer to scientific description, a good theory of flocking will predict properties that are novel and apparently inexplicable with respect to a good theory of a single bird. Traditionally philosophers have seen emergent properties like this as obstructions to reductive relationships between theories, but recently some have reconsidered this dogma. If one does, one can use the same mathematical tools for understanding reduction to also understand emergence without simplifying emergence as the negation of reduction. The upshot is that we now have better tools for understanding both how to explain the relationships between different scientific theories and how these theories describe novel phenomena and properties.

A third application is to the epistemology of modeling—that is, how we can gain knowledge from scientific models of phenomena. In particular, how can inferences from models to the phenomena they represent be justified when those models represent only imperfectly? I have shown how in the early twentieth century the physicist, historian and philosopher Pierre Duhem considered just this problem, arguing that inferences from mathematical models of phenomena to real physical applications must also be demonstrated to be approximately correct when the assumptions of the model are only approximately true. The same tools used to better understand reduction can be applied here, yielding a novel and rich mathematical theory of stability with epistemological consequences.

The standard sort of answer to this question is that, from the perspective of Einstein’s theory, we can explain why and under what circumstances Newton’s theory was and continues to be so successful. Among philosophers of science, such a relationship, known as reduction, is important because it informs us about how scientific theories are (or are not) unified, and the nature of scientific theory change. Among physicists, it is often known as a correspondence limit, and is equally important because it informs us about when we may simplify our treatment of a physical phenomenon and as a consistency check on the development of new theories, such as attempts to combine together Einstein’s theory with another great pillar of modern physics, the quantum theory of matter.

The main goal of the project Topological Methods for Intertheoretic Reduction in Physics (TMIRP) is to develop new methodological tools from the branch of mathematics called topology to better close the gap between philosophers’ and physicists’ understanding of reduction. Although much of the mathematics of these tools has been available for some time, they have received little attention from physicists and philosophers for this particular use, perhaps because, in their high degree of abstraction, they had not received a satisfactory interpretation until previous work in which I had provided such an interpretation and begun to apply it to the case of gravitational physics mentioned before: Newton’s and Einstein’s theories.

Indeed, one of the accomplishments of the project was increased clarity on how those two theories should be related and compared, in particular the need for precision and clarity in the description of observers, as described within those theories, and what exactly they can observe. These insights were then extended in two direct ways. The first was to the relationship between the general theory of relativity, already mentioned, and the special theory of relativity: Einstein’s earlier theory that did not incorporate gravitation, but still implemented his ideas about the relativity of motion and the special status of light. This has revealed two distinct ways in which the special theory is related to the general: one involving observing only small regions of space and another with relatively light matter. This in turn has revealed connections to certain versions of Einstein’s famous but vague principle of equivalence and the so-called requirement of “minimal coupling” between matter and the geometry of space and time that have hitherto been unexplored, despite them being venerable topics in the foundations of relativity theory. The second extension is to a broad class of theories in physics called gauge theories, which provide a general framework for theories of fundamental fields, particles, and interactions.

Another goal accomplished was to understand in general the relationship between philosophers’ and physicists’ notion of reduction better. On the one hand, physicists’ notion of reduction often invokes some sort of limiting relationship between two theories—e.g. the limit of vanishing gravitational interaction undergirds the reduction between special and general relativity. On the other hand, philosophers’ notion of reduction, following the influential account due to Ernest Nagel in the 1960s, takes reduction to involve a logical relationship between theories, in particular the possibility of a deduction of the laws of one theory by another (perhaps under special conditions). One finds in the literature both affirmative and negative answers to the question of whether the former can be fit into a (perhaps only slightly modified) version of the latter. A detailed investigation has revealed that it can be fit more closely than a detractor might have expected, but not completely. The reason for the lack of complete fit draws significant attention to the role of topology (and related mathematical structures) in describing how theories are related. Previous work done by the philosophical school of scientific structuralists shares with the present project a common concern with using topological-like structures to understand approximation of one theory by another, this was never a focus and was not really explored in any detail as is done in the current project.

One of the exciting developments of the TMIRP project has been further clarity on applications of the methods developed within it to other topics in philosophy of science, connections that were only dimly understood when the project was originally conceived. One such application involves so-call counterfactual reasoning in science, reasoning not about what has or will happen, but what would have happened or would happen if circumstances were different. Philosophers are keenly interested in the logic of this sort of reasoning, in part because of its intimate connection with causal reasoning, used throughout the sciences. Just as with the case of reduction, the scientific and philosophical treatments of counterfactual reasoning are not as clearly connected or mutually insightful as they could be, but the topological tools developed in the TMIRP project suggest ways of connecting them together. In particular, it shows how particular scientific theories, along with careful consideration of which observation are relevant for certain sorts of counterfactual claims, can be used rigorously to provide the truth-conditions (semantics) for those claims. With further work this in turn may have a salutary effect on causal and counterfactual reasoning in the sciences generally, including applications to cases of biomedicine and policy.

Another application involves the concept of emergence. Emergence is often understood as some kind of comparative novelty or apparent inexplicability. For example, the collective organization of a flock of birds, such as the dramatic undulating murmurations of starlings, is entirely surprising from the behavior of a single bird alone. Put in terms closer to scientific description, a good theory of flocking will predict properties that are novel and apparently inexplicable with respect to a good theory of a single bird. Traditionally philosophers have seen emergent properties like this as obstructions to reductive relationships between theories, but recently some have reconsidered this dogma. If one does, one can use the same mathematical tools for understanding reduction to also understand emergence without simplifying emergence as the negation of reduction. The upshot is that we now have better tools for understanding both how to explain the relationships between different scientific theories and how these theories describe novel phenomena and properties.

A third application is to the epistemology of modeling—that is, how we can gain knowledge from scientific models of phenomena. In particular, how can inferences from models to the phenomena they represent be justified when those models represent only imperfectly? I have shown how in the early twentieth century the physicist, historian and philosopher Pierre Duhem considered just this problem, arguing that inferences from mathematical models of phenomena to real physical applications must also be demonstrated to be approximately correct when the assumptions of the model are only approximately true. The same tools used to better understand reduction can be applied here, yielding a novel and rich mathematical theory of stability with epistemological consequences.