Editing Gottfried Wilhelm Leibniz (section)

===Topology===
Leibniz was the first to use the term ''analysis situs'',<ref>Loemker §27</ref> later used in the 19th century to refer to what is now known as [[topology]]. There are two takes on this situation. On the one hand, Mates, citing a 1954 paper in German by [[Jacob Freudenthal]], argues:

{{blockquote|Although for Leibniz the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer [[Euler]], in the famous 1736 paper solving the [[Seven Bridges of Königsberg|Königsberg Bridge Problem]] and its generalizations, used the term ''geometria situs'' in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ... [It] is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics.<ref>Mates (1986), 240</ref>}}

But Hideaki Hirano argues differently, quoting [[Benoit Mandelbrot|Mandelbrot]]:<ref>{{cite web|url=http://www.t.hosei.ac.jp/~hhirano/academia/leibniz.htm |title=Leibniz's Cultural Pluralism And Natural Law |last=Hirano |first=Hideaki |access-date=10 March 2010 |url-status=dead |archive-url=https://web.archive.org/web/20090522130455/http://www.t.hosei.ac.jp/~hhirano/academia/leibniz.htm |archive-date=22 May 2009 }}</ref>

{{blockquote|To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in "packing",&nbsp;... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In ''Euclidis Prota''&nbsp;..., which is an attempt to tighten Euclid's axioms, he states&nbsp;...: "I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets." This claim can be proved today.<ref>Mandelbrot (1977), 419. Quoted in Hirano (1997).</ref>}}

Thus the [[fractal|fractal geometry]] promoted by Mandelbrot drew on Leibniz's notions of [[self-similarity]] and the principle of continuity: ''[[Natura non facit saltus]]''.<ref name="Saltus"/> We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole", he was anticipating topology by more than two centuries. As for "packing", Leibniz told his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.

He envisioned the field of [[combinatorial topology]] as early as 1679, in his work titled ''Characteristica Geometrica,'' as he "tried to formulate basic geometric properties of figures, to use special symbols to represent them, and to combine these properties under operations so as to produce new ones."<ref name=":2" />