Editing Finitism (section)

== Views regarding infinite mathematical objects ==
[[Leopold Kronecker]] remained a strident opponent to Cantor's set theory:{{sfn|Eriksson|Estep|Johnson|2004|pages=230–232}}
{{blockquote|text=
{{lang|de|Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.}}
God created the integers; all else is the work of man.
|source=1886 lecture at the {{lang|de|Berliner Naturforscher-Versammlung}}{{refn|according to {{harvnb|Weber|1893|page=19}}}}}}

[[Reuben Goodstein]] was another proponent of finitism.  Some of his work involved building up to [[Mathematical analysis|analysis]] from finitist foundations.

Although he denied it, much of [[Ludwig Wittgenstein]]'s writing on mathematics has a strong affinity with finitism.{{sfn|Rodych|2018}}

If finitists are contrasted with [[Transfinite number|transfinitists]] (proponents of e.g. [[Georg Cantor]]'s hierarchy of infinities), then also [[Aristotle]] may be characterized as a finitist. Aristotle especially promoted the [[potential infinity]] as a middle option between strict finitism and [[actual infinity]] (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite [[cardinal number|cardinal]] and [[ordinal number|ordinal]] numbers, which have nothing to do with the things in nature):

{{blockquote
|text=But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.
|source=Aristotle, Physics, Book 3, Chapter 6}}