Editing Locus (mathematics) (section)

==History and philosophy==
Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a [[circle (mathematics)|circle]] in the [[Euclidean plane]] was defined as the ''locus'' of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is the set of points that are at a given distance from the center.<ref>{{citation |title=The History of Mathematics: A Brief Course |first=Roger L. |last=Cooke |edition=3rd |publisher=John Wiley & Sons |year=2012 |isbn=9781118460290 |url=https://books.google.com/books?id=CFDaj0WUvM8C&pg=PT534 |contribution=38.3 Topology |quote=The word locus is one that we still use today to denote the path followed by a point moving subject to stated constraints, although, since the introduction of set theory, a locus is more often thought of statically as the set of points satisfying a given collection.}}</ref>

In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as avoiding the [[actual infinity|actual infinite]] was an important philosophical position of earlier mathematicians.<ref>{{citation |title=Elements of the History of Mathematics |first=N. |last=Bourbaki |author-link=Nicolas Bourbaki |translator=J. Meldrum|translator-link=John D. P. Meldrum |publisher=Springer |year=2013 |isbn=9783642616938 |page=26 |url=https://books.google.com/books?id=4JprCQAAQBAJ&pg=PA26 |quote=the classical mathematicians carefully avoided introducing into their reasoning the 'actual infinity'}}.</ref><ref name="microscope"/>

Once [[set theory]] became the universal basis over which the whole mathematics is built,<ref>{{citation |title=The Foundations of Mathematics in the Theory of Sets |volume=82 |series=Encyclopedia of Mathematics and its Applications |first=John P. |last=Mayberry |publisher=Cambridge University Press |year=2000 |isbn=9780521770347 |url=https://books.google.com/books?id=mP1ofko7p6IC&pg=PA7 |page=7 |quote=set theory provides the foundations for all mathematics}}.</ref> the term of locus became rather old-fashioned.<ref>{{citation |title=Combinatorics and Geometry, Part 1 |volume=5 |series=Handbook of Applicable Mathematics |first1=Walter |last1=Ledermann |first2=S. |last2=Vajda |publisher=Wiley |year=1985 |isbn=9780471900238 |page=32 |quote=We begin by explaining a slightly old-fashioned term}}.</ref> Nevertheless, the word is still widely used, mainly for a concise formulation, for example:
* ''[[Critical locus]]'', the set of the [[critical point (mathematics)|critical points]] of a [[differentiable function]].
* ''Zero locus'' or ''vanishing locus'', the set of points where a function vanishes, in that it takes the [[Value (mathematics)|value]] zero.
* ''Singular locus'', the set of the [[singular points of an algebraic variety]].
* ''[[Connectedness locus]]'', the subset of the parameter set of a family of [[rational function]]s for which the [[Julia set]] of the function is connected.

More recently, techniques such as the theory of [[Scheme (mathematics)|schemes]], and the use of [[category theory]] instead of [[set theory]] to give a foundation to mathematics, have returned to notions more like the original definition of a locus as an object in itself rather than as a set of points.<ref name="microscope">{{citation |title=Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice |first=Alexandre |last=Borovik |publisher=American Mathematical Society |year=2010 |isbn=9780821847619 |contribution=6.2.4 Can one live without actual infinity? |url=https://books.google.com/books?id=hEPSAwAAQBAJ&pg=PA124 |page=124}}.</ref>