11/4/2023 0 Comments Working Toward Solutions in Fluid Dynamics and Astrophysics: What the Equations Don't SayLink to the book Introduction By Lydia Patton Systems of differential equations describe, model, explain, and predict states of physical systems. Experimental and theoretical branches of physics including general relativity, climate science, and particle physics have differential equations at their center. Newton’s law of gravitation, Hooke’s law, the Einstein field equations for the gravitational field in general relativity, Maxwell’s equations for electromagnetism, and the Navier-Stokes equations for fluid motion are all systems of differential equations. Philosophical questions arise from the use of differential equations in physical science and in mathematics, and a systematic, ground-level treatment would provide a crucial framework for this area of research. In particular, there is a need for sustained attention to the questions in the philosophy of science that arise from the use of differential equations in physical science. Wendy Parker (2017), Margaret Morrison (1999), Nicolas Fillion (2012), Erik Curiel (2010), Robert Batterman (2013), and James Mattingly and Walter Warwick (2009) are among those who have contributed philosophical studies of the use of differential equations in the analysis of physical systems. The papers in this volume analyze the use of differential equations in fluid dynamics and astronomy. The central problem at stake is the fact that direct solutions to differential equations are not available in many domains for which the systems of equations are constructed. Lack of a direct or immediate solution means that an equation or system of equations does not have a solution in a domain without employing a method that either restricts the domain, or extends the equations, or both. Mathematicians may refer to the lack of an 'analytic' solution, which often means that there is no unique function that is a solution to the equations. Or they may refer to the lack of an 'exact' solution. An equation's being without an exact solution can mean a number of things: that no closed form solution can be written, for instance, or that the available solution does not provide unique exact values for variables of interest, but provide approximate or perturbative solutions instead. (These are not the only definitions in common use. See Fillion and Bangu 2015 and McLarty 2023 for discussion of types of solutions.) The papers that follow focus on the Navier-Stokes equations in fluid dynamics and Einstein's field equations of general relativity as they are employed in astrophysics. The Navier-Stokes equations do not have immediate exact solutions for many physical fluid systems. Similarly, the Einstein field equations of general relativity do not have direct solutions in many domains of interest, including the merger of astronomical bodies, which emanates gravitational waves. And yet, scientists and engineers work with these equations every day. Ingenious methods have been devised, which may involve finding simulated solutions for artificial or idealized situations and then extending those solutions to actual cases; or finding 'weak' solutions which may not have derivatives (solutions) everywhere, but which nonetheless satisfy the equations in some restricted sense; or finding another strategy to extend the equations to the domain of interest. In many cases, solutions that initially apply only in restricted or idealized contexts are extended to a wider set of physically realistic situations. Equations may be used in scientific reasoning to yield predictions and explanations, to predict states of a system and uniquely specify the values of variables. Equations can play these roles directly when immediate solutions are available. This collection goes beyond that familiar case to explore what happens when direct solutions are absent, or when the task at hand is precisely to find a way to connect the equations to a specific physical situation. One of the most striking results learned early on in real analysis is the fact that there are many more irrationals than rational numbers. Early education often presents irrational numbers as exceptions, so it comes as a surprise to find that they are more common than the rational numbers. The application of differential equations to physical contexts reveals a similar situation. Physical reasoning using differential equations usually is presented as follows: scientists select an appropriate equation or system of equations, find a solution, and thereby determine the evolution and properties of a physical system. But cases in which a direct solution is not available and scientists must work out a solution, or work in the absence of a solution, are far from the exception. Working with equations can involve reasoning toward, from, and around equations, as well, in the absence of a solution in the domain of interest. The papers focus on how scientists reason with, around, and toward differential equations in fluid dynamics and astrophysics in the absence of immediate solutions. The process of reasoning may involve extending available solutions to differential equations to initially inaccessible domains. (A conversation with Hasok Chang put a clear focus on 'stretching' or 'extending' equations, which is a helpful way of describing one aspect of these situations.) Or, it may involve finding novel ways of simulating or modeling the domain so that it can make contact with the equations, or new methods for calculating, or new ways of measuring. McLarty (this volume) focuses serious attention on how methods of calculation can make a difference in methods for solving, but also for reasoning more broadly with, differential equations. McLarty draws an analogy between the Navier-Stokes equations and Emmy Noether's theorems on this score. A target physical phenomenon, say turbulent fluids or merging black holes, must be characterized in a certain way to be tractable using differential equations in the first place. One must choose a way to represent and measure physical variables. Beyond that, scientists must determine ways to represent a physical domain as a system, including how to determine initial conditions, boundary conditions, and constraints on system evolution. Curiel has argued that "it is satisfaction of the kinematical constraints — fixed, unchanging relations of constraint among the possible values of a system’s physical quantities — that ground the idea of the individual state of a system as represented by a given theory. If the individual quantities a theory attributes to a system do not stand in the minimal relations to each other required by the theory, then the idea of a state as representing that kind of system cannot be cogently formulated, and without the idea of an individual state of a system one can do nothing in the theory to try to represent the system" (Curiel 2016). In many cases, establishing initial and boundary conditions and kinematical constraints to characterize a given problem allows for differential equations to be applied to solve that problem. For this collection, Susan Sterrett builds on her earlier work on symbolic reasoning, physical analogy, dimensional analysis, and modeling. She argues here that there is a stronger and broader role for mathematics in characterizing physical systems than simply solving equations that are already known. Sterrett, McLarty, and Patton (this volume) urge that we evaluate the mathematics used in reasoning about physical systems in a more flexible, creative way - including the reasoning used to characterize and to measure those systems. Setting up a solution to differential equations in a specific physical domain requires finding precise ways to determine the conditions and constraints under which a solution is possible in that domain. Even in highly theoretical fields of physics, setting up a problem involves deep understanding of the physical situation at hand, as Elder's and Sterrett's papers for this collection show brilliantly. Abstract theoretical research may require increasingly precise understanding of a physical system, because counterfactual reasoning is based on knowing the exact parameters and constraints to alter when departing from the concrete properties of the system. Patton (this volume) argues that weak and simulated solutions to equations can allow for heuristic extension of structural, physical reasoning into situations where the equations lack direct solutions. On the other side, reasoning about the physical properties of a system may require significant theoretical or modelling resources. For instance, as Elder (this volume) explains, the data gathered by the new generation of gravitational wave detectors is not independently informative. Data is filtered through a bank of waveform templates generated using both empirical and theoretical reasoning, after which post-data analysis allows for estimates of the physical parameters of the target system. (Elder (this volume) and Patton (2020) provide details, including references to the scientific literature on this topic.) The numerical simulations and waveform templates used by the LIGO-Virgo-Kagra Scientific Collaboration are constructed and validated using vast theoretical and empirical resources including probabilistic methods, methods of approximation, dynamical equations and reasoning, and empirical input. Elder shows that the parameters, dynamical reasoning, and empirical information crystallized in the waveform templates supports their extraordinary flexibility in application. An essential tenet of this volume is that a great deal of creative mathematical reasoning can go on in physics without direct solutions to the equations in the field of interest. Our aim is to integrate this fact, known to the scientists themselves and to historians of mathematics for some time, into the philosophical analysis of physical reasoning. (A recent, much-anticipated history of differential equations is Gray (2021); this and much of Gray's earlier work, including his work on Poincaré, explores the development of equations in satisfying detail.) Lydia Patton Virginia Tech Link to the book References Batterman, Robert. 2013. "The Tyranny of Scales", pp. 255-86 in The Oxford Handbook of Philosophy of Physics. Oxford: Oxford University Press. Curiel, Erik. 2010 (preprint). "On the Formal Consistency of Theory and Experiment, with Applications to Problems in the Initial-Value Formulation of the Partial-Differential Equations of Mathematical Physics." Preprint, PhilSciArchive. Url = {http://philsci-archive.pitt.edu/8660/}. Gray, Jeremy. 2021. Change and Variations: A History of Differential Equations to 1900. Dordrecht: Springer. Fillion, Nicolas. 2012. The Reasonable Effectiveness of Mathematics in the Natural Sciences. Dissertation, The University of Western Ontario. Mattingly, James and Walter Warwick. 2009. "Projectible Predicates in Analogue and Simulated Systems." Synthese 8 (3): 465-482. Morrison, Margaret. 1999. "Models as Autonomous Agents." In Mary Morgan and Margaret Morrison. Models as Mediators. Cambridge University Press. Parker, Wendy. 2017. "Computer Simulation, Measurement, and Data Assimilation." British Journal for the Philosophy of Science 68 (1): 273-304.
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