Reception of The Argument
At the start, Cantor's theory was controversial among mathematicians and (later) philosophers. As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there." Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.
Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence. "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'. In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets.
Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others. Hilbert predicted: "No one will drive us from the paradise which Cantor created for us". To which Wittgenstein replied "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.
Read more about this topic: Controversy Over Cantor's Theory
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