Developing Mathematics and Science Anew

ion comes from the Latin abstractio (detachment, division, retention), introduced by the Roman philosopher and theologian Boethius when translating Aristotle’s term aphairesis. Since the relevant to us meaning of the term abstraction has to do with the process of forming a general mental image of an object—note the root Form—we are entirely justified in interpreting this meaning as directly related to the process of induction. In that sense, abstraction is the process of forming the ‘idea’ of the class—via the class representation—on the basis of a small number of its members. So, although there are other uses of “abstraction”, in one sense, especially in science or philosophy, “induction” captures more accurately its meaning. Moving on to abduction, it is the term introduced by Charles Peirce (1839–1914), possibly the greatest American philosopher, for the logical process that works in the direction opposite to deduction: inferring A from B, where A is a possible cause of B (there could be other causes for B). For example, if coming back home you found the grass to be wet, you conclude that it had rained. Over the years, Peirce also called the same process retroduction, hypothesis, and presumption. Moreover, Peirce suggests that abductive reasoning from B to A should involve not just the inference that B follows from A, but also that A is one of the most “economical” explanations of B. The interesting question is this: Is abduction based on induction? It appears, without fully realizing it, Peirce himself gave a positive answer to the question when he admitted that the “first emergence of this new element [A] into consciousness must be regarded as a perceptive judgment”. It appears that what he calls abduction is, in fact, a logical elaboration of the results of “perceptive judgment”, i.e. of induction. I will address the central role of induction in the “perceptive judgment” in Chapter 9. Some researcher agree with this conclusion, see for example the view of Francis Reilly 53 or of John Holland and colleagues, who state that abduction is “induction in the service of explanation, in which a new empirical rule is created to render predictable what would otherwise be mysterious”. 54 Lastly, we deal with universals and particulars, a topic which dates back to at least Plato, but became increasingly prominent starting with the medieval philosophers—Porphyry (c. 232–305), Boethius (c. 475/480–524), Abelard (1079–1142), Aquinas (c. 1225–74), Duns Scotus (c. 1266– 1308), Ockham (c. 1285–1349)—all the way to the present philosophers. Basically, a particular refers to a concrete object, while a universal refers to the characteristics—or properties, or features, or qualities, or attributes—that are shared by the particulars. So we are dealing with concrete objects and their features. Philosophers who believe in the reality of universals are “realists”, while those denying it are “nominalists”: nominalists believe that universals are just names that do not stand for anything