Ethno-cultural studies of mathematics
Encyclopedia
Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics
at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics
. The philosopher Imre Lakatos
in his Proofs 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 of mathematical formalism. Informality may not discern between statements given by inductive reasoning
(as in approximation
s which are deemed "correct" merely because they are useful), and statements derived by deductive reasoning
.
of all statements from given axiom
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
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. 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.
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.
in ancient Egypt
, followed by Greek mathematics
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
if read back into history. Several ancient societies built impressive mathematical systems and carried out complex calculations based on proofless heuristic
s and practical approaches. Mathematical facts were accepted on a pragmatic basis. Empirical method
s, as in science, provided the justification for a given technique. Commerce, engineering
, calendar
creation and the prediction of eclipse
s and stellar progression were practiced by ancient cultures on at least three continents.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics
Ethnomathematics
In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture . Often associated with "cultures without written expression" , it may also be defined as "'the mathematics which is practised among identifiable cultural groups'" In mathematics education,...
. The philosopher Imre Lakatos
Imre Lakatos
Imre Lakatos 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 pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his...
in his Proofs and Refutations
Proofs and Refutations
Proofs and Refutations is a book by the philosopher Imre Lakatos expounding his view ofthe progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron...
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. Informality may not discern between statements given by inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
(as in approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
s which are deemed "correct" merely because they are useful), and statements derived by deductive reasoning
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...
.
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 proofsMathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
of all statements from given axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
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
Anthropology is the study of humanity. It has origins in the humanities, the natural sciences, and the social sciences. The term "anthropology" is from the Greek anthrōpos , "man", understood to mean mankind or humanity, and -logia , "discourse" or "study", and was first used in 1501 by German...
and psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
: it casts light on the perceptions and agreements of other cultures. It is also of interest in developmental psychology
Developmental psychology
Developmental psychology, also known as human development, is the scientific study of systematic psychological changes, emotional changes, and perception changes that occur in human beings over the course of their life span. Originally concerned with infants and children, the field has expanded to...
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 article is dedicated to the usage of that term among professional mathematicians.
The field of naïve physics
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....
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.
History
There has long been a standard account of the development of geometryGeometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
in ancient Egypt
Ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
, followed by Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
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
Anachronism
An anachronism—from the Greek ανά and χρόνος — is an inconsistency in some chronological arrangement, especially a chronological misplacing of persons, events, objects, or customs in regard to each other...
if read back into history. Several ancient societies built impressive mathematical systems and carried out complex calculations based on proofless heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
s and practical approaches. Mathematical facts were accepted on a pragmatic basis. Empirical method
Empirical method
The empirical method is generally taken to mean the approach of using a collection of data to base a theory or derive a conclusion in science...
s, as in science, provided the justification for a given technique. Commerce, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, calendar
Calendar
A calendar is a system of organizing days for social, religious, commercial, or administrative purposes. This is done by giving names to periods of time, typically days, weeks, months, and years. The name given to each day is known as a date. Periods in a calendar are usually, though not...
creation and the prediction of eclipse
Eclipse
An eclipse is an astronomical event that occurs when an astronomical object is temporarily obscured, either by passing into the shadow of another body or by having another body pass between it and the viewer...
s and stellar progression were practiced by ancient cultures on at least three continents.
See also
- Folk psychologyFolk psychologyFolk psychology is the set of assumptions, constructs, and convictions that makes up the everyday language in which people discuss human psychology...
- Mathematical Platonism
- PseudomathematicsPseudomathematicsPseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models...
- EthnomathematicsEthnomathematicsIn mathematics education, ethnomathematics is the study of the relationship between mathematics and culture . Often associated with "cultures without written expression" , it may also be defined as "'the mathematics which is practised among identifiable cultural groups'" In mathematics education,...
- NumeracyNumeracyNumeracy is the ability to reason with numbers and other mathematical concepts. A numerically literate person can manage and respond to the mathematical demands of life...