Reasoning in Science and Mathematics
Background: Claims, Inferences, Arguments, Explanations
Models and Theories
Models and theories arise by a process of abstraction. They do not codify truths of the world, but rather set out ways in which to reason about the world through ignoring what we consider to be extraneous. By better understanding the process of abstraction we can proceed more clearly in creating theories and evaluating them.
Experiments in science are meant to produce observations that confirm or disconfirm a theory or to lead to new conjectures. By looking at examples of experiments we can get a better idea of what scientists expect of a good experiment.
Mathematics as the Art of Abstraction
Mathematics proceeds by a process of abstraction, so that mathematical claims like scientific claims are neither true nor false, but only true or false in an application of the theory to which they belong. A proof in mathematics is meant to show that a claim follows from the assumptions of a particular mathematical theory.