Philosophy of Logic and Mathematics

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    How mathematics figures differently in exact solutions, simulations, and physical models
    (Springer Nature, 2023-03-29) Sterrett, Susan G.
    The role of mathematics in scientific practice is too readily relegated to that of formulating equations that model or describe what is being investigated, and then finding solutions to those equations. I survey the role of mathematics in: 1. Exact solutions of differential equations, especially conformal mapping; and 2. Simulations of solutions to differential equations via numerical methods and via agent-based models; and 3. The use of experimental models to solve equations (a) via physical analogies based on similarity of the form of the equations, such as Prandtl's soap-film method, and (b) the method of physically similar systems. Two major themes emerge: First, the role of mathematics in science is not well described by deduction from axioms, although it generally involves deductive reasoning. Creative leaps, the integration of experimental or observational evidence, synthesis of ideas from different areas of mathematics, and insight regarding analogous forms are required to find solutions to equations. Second, methods that involve mappings or transformations are in use in disparate contexts, from the purely mathematical context of conformal mapping where it is mathematical objects that are mapped, to the use of concrete physical experimental models, where one concrete thing is shown to correspond to another.
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    How many thoughts can fit in the form of a proposition?
    (2004--06-24) Sterrett, Susan G.
    I argue here that Frege’s eventual view on the relation between sentences and the thoughts they express is that, ideally, a sentence expresses exactly one thought, and a thought is expressed by exactly one (canonical) sentence. This may clash with some mainstream views of Frege, for it has the consequence of de-emphasizing the philosophical significance of the question of how it is possible for someone to regard one sentence as true yet regard another sentence that expresses the same thought as false. This account of Frege was developed by taking a long-range look at his writings over the course of his life.
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    Frege and Hilbert on the Foundations of Geometry
    (1994-10-14) Sterrett, Susan G.
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    Three views of logic: Mathematics, Philosophy, Computer Science
    (Princeton University Press, 2014) Loveland, Donald W.; Hodel, Richard E.; Sterrett, Susan G.
    Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity. The proof theory portion presents classical propositional logic and first-order logic using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time. Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings. Gives an exceptionally broad view of logic. Treats traditional logic in a modern format. Presents relevance logic with applications. Provides an ideal text for a variety of one-semester upper-level undergraduate courses.