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Advantageous Approximation: The Importance of Feasible Reasoning in Pure and Applied MathematicsUnderstanding the nature of scientific inference is a central problem in philosophy of science, with a long history of attempts to represent scientific inference using logical and computational models. A traditional approach that is very influential in philosophy represents scientific theories as deductive, formal systems, where from a proof-theoretic perspective reasoning is entirely mechanical. This deductive-computational view of theories has had greater traction for philosophical accounts of pure mathematics than applied mathematics, as increasingly many philosophers point to the non-deductive character of mathematical methods in the sciences. There is, however, reason to be critical of this formal representation of reasoning in pure mathematics also, as very little work in pure mathematics involves direct reasoning from axioms, and theorems of interest often involve reference to subclasses of models or relations to structures of other kinds.
The purpose of this poster is to outline an argument that a major limitation of the deductive-computational view is its attempt to represent scientific inference as deductive and mechanical in principle, without regard to what determines epistemic needs and drives methodological choices in scientific practice. I argue that a key determinant of method choice, and hence of how and what we can know, is the need for feasible solutions to scientific problems, where feasible means possible under the contextual constraints of scientific practice. Feasibility of inference is an objective epistemic property, not merely a pragmatic one, since it marks the difference between possibility and impossibility of knowing in the real world.
To make the argument, I show that a particular form of inference characterized by feasibility needs is replicated widely across both pure and applied mathematics, from mathematical modeling and analysis, numerical computation, and data handling in applied mathematics, to computer algebra and algebraic topology in pure mathematics. The pattern holding these cases together involves problem-solving situations where solutions are not feasibly accessible in a contextually relevant sense, and feasible solutions are obtained by transforming the problem into an approximately equivalent one, where computed solutions are used to solve approximately the original problem. This type of model can not only represent but also explain the use of non-deductive inference methods in the sciences. I show how this inference pattern can be regarded as a higher-order strategy of approximate computation in contrast to the standard deductive-computational view of reasoning within a formal system. As a result, I argue that feasible inference models can allow us to better comprehend the epistemology and methodology of science than can an epistemology based primarily on formal systems.
Robert H C Moir