In much theorizing about inference from evidence during the last 50 years and more there has been much talk about the role of "generalizations." I have long thought that the ordinary connotation of "generalization," though not necessarily its definition, fails to do justice to the complexity and depth of the complexes of ancillary theoretical propositions and arguments that bear on inferences from evidence. A nifty illustration of this might be the discovery or rediscovery that medieval Iranian craftsmen may have had an understanding of "an advanced math of quasi crystals, which was not understood by modern scientists until three decades ago." John Noble Wilford, In Medieval Architecture, Signs of Advanced Math, NY Times (Feb. 27, 2007). This understanding is inferred by Peter J. Lu and Paul J. Steinhardt from the patterns they observed in medieval Islamic architecture. See Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture, Science 23 February 2007: Vol. 315. no. 5815, pp. 1106 - 1110. One practically imagines Umberto Eco noting that one tile does not fit the usual pattern and attempting to infer some message that the designer of the ill-fitting tile might have been trying to send to some observer eight centuries later.The trouble (if any) with "generalization" is that it connotes "relative frequency statement." But is there an adequate substitute for "generalization"? "Law-like statement"? Are all generalizations "like" natural laws? "Nomological construct"? Such a phrase is redolent of all the ugly academic language that causes nightmares and public scorn. "Principles"? This is too broad and fuzzy.
N.B. Scientists are not the only people who use complex theoretical constructs to draw inferences about the world. The difference between scientists and ordinary people with ordinary generalizations in their heads (and in their hearts?) may be mainly or only that (i) ordinary theoretical constructs are not usually (if ever) fully spelled out and (ii) ordinary theoretical constructs -- "generalizations" -- are not systematic in the way that principles in a scientific theory such as Newtonian mechanics are systematic (but this does not mean that "ordinary," or everyday, theoretical constructs are necessarily invalid or somehow "bad" -- because if that were the case most of us would have been dead long ago).
2 comments:
NYT: "In Medieval Architecture, Signs of Advanced Math" (title of the article)
-- Could one produce the tile patterns with empirically found rules (trial and error, step-by-step refinement)? Is really advanced math (in sense of XXth century) absolutely necessary to produce these patterns?
NYT: "Mr. Lu found that the interlocking tiles were arranged in predictable ways to create a pattern that never repeats — that is, quasi crystals."
-- Can one say that the tile patterns necessarily are an embodiment of quasi crystals (abstract concepts of Penrose's theoretical development)? Perhaps not. Seven stones, under certain circumstances, can represent a prime number, but most often - are just stones.
NYT: "Mr. Lu said that it would be “incredible if it were all coincidence.”"
-- This statement needs a critical proof! Coincidences may have different reasons!
Yes, Priit is quite right -- it is easy to make too much out of coincidences -- but it remains true that complex theoretical constructs -- such as a mathematical theory, beliefs about the springs of human action -- sometimes, perhaps frequently, perhaps almost always -- play an important role in factual inference.
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