"if one person can see it as a paradise of mathematicians, why should not another see it as a joke?" The rejection of Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism and intuitionism. Wittgenstein did not object to mathematical formalism wholesale, but had a finitist view on what Cantor's proof meant. The philosopher maintained that belief in infinities arises from confusing the intensional nature of mathematical laws with the extensional nature of sets, sequences, symbols etc. A series of symbols is finite in his view: In Wittgenstein's words: "...A curve is not composed of points, it is a law that points obey, or again, a law according to which points can be constructed." He also described the diagonal argument as "hocus pocus" and not proving what it purports to do. Objection to the axiom of infinity A common objection to Cantor's theory of infinite number involves the axiom of infinity (which is, indeed, an axiom and not a logical truth). Mayberry has noted that "... the set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them—indeed, the most important of them, namely Cantor's Axiom, the so-called Axiom of Infinity—has scarcely any claim to self-evidence at all …" Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl wrote: The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics (for example, that includes real analysis). See also Preintuitionism Notes References "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." Translated in (address to the Fourth International Congress of Mathematicians) External links Doron Zeilberger's 68th Opinion Philosopher Hartley Slater's argument against the idea of "number" that underpins Cantor's set theory Wolfgang Mueckenheim: Transfinity - A Source Book Hodges "An editor recalls some hopeless papers" Discovery and invention controversies History of mathematics Philosophy of mathematics Set theory Georg Cantor