Historically, there are two kinds of induction, called here the postulational and the hypothetical.
Postulational induction (cf. material induction) is the induction practiced in ancient and early modern times in which empirical induction leads to essential definitions and universal postulates for subsequent deduction. This is the Socratic view of induction: “in modern philosopher’s technical terms—the Socratic view of induction holds that in human cognition, ampliation takes place fundamentally at the conceptual, not the propositional, level.” This induction is described by John McCaskey; see his website here.
See for example, The History of Induction. In his dissertaion McCaskey gives “A account of how philosophical induction was conceived in the ancient world and how that conception was later rediscovered by, especially, Francis Bacon.” (see here)
A history of ancient postulational induction is given by Lucio Russo in The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn, although he mistakenly calls it the “hypothetico-deductive method”. As McCaskey shows, that term is from J. S. Mill, and it refers to what is called here hypothetical induction. Russo says what he means by it when he describes examples (p.17):
- Their statements are not about concrete objects, but about specific theoretical entities.
- The theory has a rigorously deductive structure.
- Applications to the real world are based on correspondence rules between the entities of the theory and concrete objects.
This is postulational induction. It is still practiced in physics and chemistry, the gold standard of science.
Hypothetical induction (cf. formal induction) is the so-called “hypothetico-deductive method” practiced since the 19th century in which empirical induction produces particular hypotheses, which are tested as propositions. A collection of confirmed hypotheses leads to ‘deductions’ about other hypotheses, which are further tested. As one author noted, “a better name might be the hypothetico-inferential method” (see here) since they need not be logical deductions. Consequences may be based on statistical models or intuitive expectations.
McCaskey has pointed out that this form of induction requires a principle of uniformity, quoting J. S. Mill: “the uniformity of the course of nature, will appear as the ultimate major premise of all inductions.” (see here) That is the principle of scientism because it makes uniformity universal, excluding particularities that do not conform.
Although the hypothetico-deductive method has precursors, it particularly arose in the 19th century as science expanded into statistical methods, historical inquiry, and the social sciences. For example, Ernst Mayr describes it as a method in which “scientists are satisfied to consider as true either that which appears most probable on the basis of the available evidence, or that which is consistent with more, or more compelling, facts than competing hypotheses.” (see here, p.26)
What is the best induction? Mayr says whatever method is used, “Scientists in their actual research often go back and forth between a phase in which they collect material or conduct purely descriptive or classificatory research and another phase of concept formation or testing of theories.” (p.28) But the hypothetico-deductive method leads to a philosophical “problem of induction” in how to justify induction (see here and here), which has defied solution. Postulational induction does not have a problem of induction.
Hypothetical induction grounds science in “what scientists believe” rather than in universal concepts and logic. Postulational induction grounds science in causal, essential concepts and universal postulates. Postulational induction is mathematics backed by empirical research. It is the best induction.