Centipede mathematics is a term used, sometimes derogatorily, to describe the

generalisation A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set theory, set of elements, as well as one or more commo ...

and study of mathematical objects satisfying progressively fewer and fewer restrictions. This type of study is likened to studying how a

centipede Centipedes (from New Latin , "hundred", and Latin , " foot") are predatory arthropods belonging to the class Chilopoda (Ancient Greek , ''kheilos'', lip, and New Latin suffix , "foot", describing the forcipules) of the subphylum Myriapoda, an ...

behaves when its legs are removed one by one. Magma to group4

The term is attributed to Polish mathematician

Antoni Zygmund Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...

. Zygmund is said to have described the metaphor of the centipede thus: "You take a centipede and pull off ninety-nine of its legs and see what it can do."Thus, Zygmund has been known by many mathematicians as the "Centipede Surgeon". The study of

semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...

s is cited as an example of doing centipede mathematics. One starts with the notion of an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

. First delete the commutativity restriction to obtain the concept of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

. The restriction of existence of inverses is then removed. This produces a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

. If one now removes the restriction regarding the existence of identity, the resulting object turns out to be a semigroup. Still more legs can be removed. If the

associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

restriction is also discarded one gets a

magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...

or a groupoid. The restrictions that define an abelian group may be removed in different orders also. The study of

ternary ring In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Mars ...

has been cited as an example of centipede mathematics. The progressive removal of axioms of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

and studying the resulting geometrical objects also illustrate the methodology of centipede mathematics. The following quote summarises the value and usefulness of the concept: "The term ‘centipede mathematics’ is new to me, but its practice is surely of great antiquity. The binomial theorem (tear off the leg that says that the exponent has to be a natural number) is a good example. A related notion is the importance of good notation and the importance of overloading, aka abuse of language, to establish useful analogies." — Gavin Wraith.

References

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External links

Procrustean Mathematics
Philosophy of mathematics