Informal mathematics, also callednaïve mathematics, has historically been the predominant form ofmathematics at most times and in most cultures, and is the subject of modernethno-cultural studies of mathematics. The philosopherImre Lakatos in hisProofs and Refutations aimed to sharpen the formulation of informal mathematics, by reconstructing its role in nineteenth century mathematical debates and concept formation, opposing the predominant assumptions ofmathematical formalism.[1] Informality may not discern between statements given byinductive reasoning (as inapproximations which are deemed "correct" merely because they are useful), and statements derived bydeductive reasoning.
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 strictproofs of all statements from givenaxioms. This can usefully be called thereforeformal mathematics. Informal practices are usually understood intuitively and justified with examples—there are no axioms. This is of direct interest inanthropology andpsychology: it casts light on the perceptions and agreements of other cultures. It is also of interest indevelopmental psychology as it reflects a naïve understanding of the relationships between numbers and things. Another term used for informal mathematics isfolk mathematics, which is ambiguous; themathematical folklore article is dedicated to the usage of that term among professional mathematicians.
The field ofnaïve physics is concerned with similar understandings of physics. People use mathematics and physics in everyday life, without really understanding (or caring) how mathematical and physical ideas were historically derived and justified.
There has long been a standard account of the development ofgeometry in ancient Egypt, followed byGreek mathematics and the emergence of deductive logic. The modern sense of the termmathematics, as meaning only those systems justified with reference to axioms, is however ananachronism if read back into history. Several ancient societies built impressive mathematical systems and carried out complex calculations based on prooflessheuristics and practical approaches. Mathematical facts were accepted on apragmatic basis.Empirical methods, as in science, provided the justification for a given technique. Commerce,engineering,calendar creation and the prediction ofeclipses andstellar progression were practiced by ancient cultures on at least three continents.
