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Wittgenstein was a notorious critic of set theory, calling it "laughable nonsense". However, he also wholeheartedly rejected intuitionist logic of Brouwer and Weyl, saying "it is nonsense, all of it".

After reading his 1939 lectures at Cambridge, I got the impression that he was a finitist. True mathematics is finite mathematics. "1, 2, 3" is ok, "1, 2, 3 and so on" is ok, but "1, 2, 3... endlessly" is not ok.

Was Wittgenstein's mathematical philosophy finitist?

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    Can you give the definition of "finitist" you have in mind? Do you means that there are only a finite number of things, including numbers, or do you mean that one cannot refer successfully to a completed infinity?
    – David Gudeman
    Commented Mar 22, 2023 at 22:48
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    How much mathematics did Wittengtein actually know?
    – Frank
    Commented Mar 23, 2023 at 0:02
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    Sort of, he did not neatly fit into isms and changed his tenets a couple of times. "Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986: 300–301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite (RFM V, §21; Zettel §274, 1947) and infinite mathematical extensions," SEP.
    – Conifold
    Commented Mar 23, 2023 at 3:44
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    See e.g. Alice Ambrose, Finitism in Mathematics I (Mind, 1935) as well as Mathieu Marion, Wittgenstein and finitism (Synt. 1995).
    – Mauro ALLEGRANZA
    Commented Mar 23, 2023 at 7:12
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    @Frank: He studied engineering, from which questions about the mathematics diverted him into philosophy, where he was taught by and later had his PhD submitted by Russell, who co-wrote Principia Mathematica. Wittgenstein was extremely well versed on disputes about the foundations of mathematics inc writing Remarks on the Foundations of Mathematics. His writings from 1929-44 were devoted to the subject, a fact that he himself emphasized in 1944 by writing that his “chief contribution has been in the philosophy of mathematics”
    – CriglCragl
    Commented Mar 23, 2023 at 15:12

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The SEP article Wittgenstein’s Philosophy of Mathematics puts it:

"Wittgenstein’s Philosophy of Mathematics begins in a rudimentary way in the Tractatus, develops into a finitistic constructivism in the middle period (Philosophical Remarks (1929–30) and Philosophical Grammar (1931–33), respectively), and is further developed in new and old directions in the manuscripts used for Remarks on the Foundations of Mathematics"

In the section 3.2 Wittgenstein’s Later Finitistic Constructivism the article specifically goes into evidence for his views on infinity, which it identifies as finitist.

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