Informal mathematics
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Informal mathematics, also called naive mathematics, has historically the been the predominant form of mathematics at most times and in most cultures, and is the subject of many ethno-cultural studies of mathematics.
Another term used for informal mathematics is folk mathematics, which is ambiguous; the mathematical folklore article is dedicated to the usage of that term among professional mathematicians.
Informal mathematics means any informal mathematical practices, as used in everyday life, or by aboriginal or ancient peoples, without historical or geographical limitation. Modern mathematics, exceptionally from that point of view, emphasizes formal and strict proofs of all statements from given axioms. This can usefully be called therefore formal mathematics. Informal practices are usually understood intuitively and justified with examples — there are no axioms. This is of direct interest in anthropology and psychology: it casts light on the perceptions and agreements of other cultures. It is also of interest in developmental psychology as it reflects a naïve understanding of the relationships between numbers and things. The field of naïve physics is concerned with similar understandings of physics. People do use mathematics and physics in everyday life, without really understanding (or caring) how mathematical and physical ideas were historically derived and justified.
Some defend the modern sense of the term mathematics, as meaning only those systems justified with reference to axioms. This sense is very much a modern one: most cultures historically have used methods and principles of mathematics with no great concern for axiomatic proof. Several ancient societies have built rather impressive mathematical systems and carried out complex and fragile calculations based on proofless heuristics and practical approaches. Mathematical facts were accepted on a pragmatic basis. Empirical methods, as in science, provided the justification for a given technique. Sophisticated commerce, engineering, calendar creation and the prediction of eclipses and stellar progression were quite accurately practiced by several ancient cultures, on at least three continents.
Informality may not discern between statements given by inductive reasoning (as in approximations which are deemed "correct" merely because they are useful), and statements derived by deductive reasoning. There has long been a standard account of the development of geometry in ancient Egypt, followed by Greek mathematics and the emergence of deductive logic.
