Editing Actual infinity (section)

==Scholastic, Renaissance and Enlightenment thinkers==
The overwhelming majority of [[Scholasticism|scholastic philosophers]] adhered to the motto ''Infinitum actu non datur''. This means there is only a (developing, improper, "syncategorematic") ''potential infinity'' but not a (fixed, proper, "categorematic") ''actual infinity''. There were exceptions, however, for example in England.

<blockquote>It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle. ([[Georg Cantor|G. Cantor]])<ref>{{Cite book|title=Gesammelte abhandlungen: Mathematischen und philosophischen inhalts.|last=Cantor|first=Georg|publisher=Georg Olms Verlag|year=1966|editor-last=Zermelo|editor-first=Ernst|pages=174}}</ref></blockquote>

<blockquote>Actual infinity exists in number, time and quantity. (J. Baconthorpe [9, p.&nbsp;96])</blockquote>

During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.

<blockquote>The continuum actually consists of infinitely many indivisibles ([[Galileo Galilei|G. Galilei]] [9, p.&nbsp;97])</blockquote>

<blockquote>I am so in favour of actual infinity. ([[Gottfried Wilhelm Leibniz|G.W. Leibniz]] [9, p.&nbsp;97])</blockquote>

However, the majority of pre-modern thinkers{{Citation needed|date=August 2009}} agreed with the well-known quote of Gauss:

<blockquote>I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.<ref>Stephen Kleene 1952 (1971 edition):48 attributes the first sentence of this quote to (Werke VIII p. 216).</ref> ([[Carl Friedrich Gauss|C.F. Gauss]] [in a letter to Schumacher, 12 July 1831])</blockquote>