Informal mathematics, also called naïve mathematics, has historically been the predominant form of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics. The philosopher

Imre Lakatos Imre Lakatos (, ; hu, Lakatos Imre ; 9 November 1922 – 2 February 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pr ...

in his ''

Proofs and Refutations ''Proofs and Refutations: The Logic of Mathematical Discovery'' is a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students wh ...

'' aimed to sharpen the formulation of informal mathematics, by reconstructing its role in nineteenth century mathematical debates and concept formation, opposing the predominant assumptions of mathematical formalism.Imre Lakatos, ''Proofs and Refutations'' (1976), especially the Introduction. Informality may not discern between statements given by ''

inductive reasoning Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' re ...

'' (as in

approximation An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very ...

s which are deemed "correct" merely because they are useful), and statements derived by ''

deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...

''.

Terminology

''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 Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

of all statements from given

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s. 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 Anthropology is the scientific study of humanity, concerned with human behavior, human biology, cultures, societies, and linguistics, in both the present and past, including past human species. Social anthropology studies patterns of behavi ...

and

psychology Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries betwe ...

: it casts light on the perceptions and agreements of other cultures. It is also of interest in

developmental psychology Developmental psychology is the science, scientific study of how and why humans grow, change, and adapt across the course of their lives. Originally concerned with infants and children, the field has expanded to include adolescence, adult deve ...

as it reflects a naïve understanding of the relationships between numbers and things. Another term used for informal mathematics is folk mathematics, which is ambiguous; the

mathematical folklore In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, folk mathematics, or ...

article is dedicated to the usage of that term among professional mathematicians. The field of

naïve physics Naïve physics or folk physics is the untrained human perception of basic physical phenomena. In the field of artificial intelligence the study of naïve physics is a part of the effort to formalize the common knowledge of human beings. Many ideas ...

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.

History

There has long been a standard account of the development of geometry in ancient Egypt, followed by

Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...

and the emergence of deductive logic. The modern sense of the term ''mathematics'', as meaning only those systems justified with reference to axioms, is however an

anachronism An anachronism (from the Ancient Greek, Greek , 'against' and , 'time') is a chronology, chronological inconsistency in some arrangement, especially a juxtaposition of people, events, objects, language terms and customs from different time per ...

if read back into history. Several ancient societies built impressive mathematical systems and carried out complex calculations based on proofless

heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...

s and practical approaches. Mathematical facts were accepted on a

pragmatic Pragmatism is a philosophical movement. Pragmatism or pragmatic may also refer to: *Pragmaticism, Charles Sanders Peirce's post-1905 branch of philosophy *Pragmatics, a subfield of linguistics and semiotics *''Pragmatics'', an academic journal in ...

basis.

Empirical method Empirical research is research using empirical evidence. It is also a way of gaining knowledge by means of direct and indirect observation or experience. Empiricism values some research more than other kinds. Empirical evidence (the record of one ...

s, as in science, provided the justification for a given technique. Commerce,

engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A date is the designation of a single and specific day within such a system. A calendar is also a physi ...

creation and the prediction of

eclipse An eclipse is an astronomical event that occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three ce ...

s and stellar progression were practiced by ancient cultures on at least three continents. N.C. Ghosh included informal mathematics in the list of Folk Mathematics.

References

{{Areas of mathematics Philosophy of mathematics Critical pedagogy Sociology of scientific knowledge Mathematics and culture Scientific folklore

Terminology

History

See also

References