10 April 2021

On Generalization (1980-1989)

"An aphorism is a generalization of sorts, and our present-day writers seem more at home with the particular." (Anatole Broyard, "Wisdom of Aphorisms", 1983)

"There is no agreed upon definition of mathematics, but there is widespread agreement that the essence of mathematics is extension, generalization, and abstraction [… which] often bring increased confidence in the results of a specific application, as well as new viewpoints."  (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"When a theory is sufficiently general to cover many fields of application, it acquires some 'truth' from each of them. Thus [...] a positive value for generalization in mathematics." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"We generalize from one situation to another not because we cannot tell the difference between the two situations but because we judge that they are likely to belong to a set of situations having the same consequence." (Roger N Shepard, "Toward a Universal Law of Generalization for Psychological Science", Science 237 (4820), 1987)

"Physicists are all too apt to look for the wrong sorts of generalizations, to concoct theoretical models that are too neat, too powerful, and too clean. Not surprisingly, these seldom fit well with data. To produce a really good biological theory, one must try to see through the clutter produced by evolution to the basic mechanisms. What seems to physicists to be a hopelessly complicated process may have been what nature found simplest, because nature could build on what was already there." (Francis H C Crick, "What Mad Pursuit?: A Personal View of Scientific Discovery", 1988)

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