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The fathers of constructivist thought in philosophy of mathematics (from left to right): Leopold Kronecker, Henri Poincaré, L.E.J. Brouwer and David Hilbert

The foundational crisis in mathematics along with roughly four decades following it, was likely the most fertile period in the history of logic and studies in the foundations. After discovering the set-theoretic paradoxes, such as the paradox of the set of all sets, together with the logical ones, like Russell’s paradox, mathematicians, realizing that the naïve set teory, or Frege’s Grundlagen der Arithmetik cannot do for a consistent basis for mathematics, began to seek for another, more solid foundation. This came also with the justified skepticism in regard to the means employed by Georg Cantor and Richard Dedekind in their work in set theory and analysis respectively. In this article I set out to tell a story of various paths taken by mathematicians in the first half of the 20th century in an attempt to set boundaries on the hitherto unrestricted Cantor’s paradise. My aim is to distinguish as clearly as I can between the tags like “constructivism”, “finitism”, “intuitionism”, since they have often been conflated and I struggled for some time with understanding their boundaries myself. …


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Mathematics is often believed to bring people to madness. We hear many stories like those about Gödel, Cantor, Nash, and Grothendieck, describing geniuses haunted by insanity that is developing along with their mathematics. And there is something to it. A certain psychologist said that

A paranoid person is irrationally rational. . . . Paranoid thinking is characterized not by illogic, but by a misguided logic, by logic run wild

Mathematics is the paradigm of rationality and maybe if the rationality takes over all of the aspects of life, we can talk of a mental issue. But this time I want to bring to light an opposite example. This time I want to share a story about a mathematician who was the voice of reason and sanity in the world that has run wild. And one whose mathematics was the model of his approach in social life. …


Canadian mathematician Simon Kochen recalled in his tribute to Kurt Gödel how during his PhD exam, he was asked to name five of Gödel’s theorems. The essence of the question was that each of the theorems either gave birth to a new branch of, or revolutionized, modern mathematical logic. “Proof theory, model theory, recursion theory, set theory, intuitionistic logic - all had been transformed by, or, in certain cases, had gotten their inception from, Gödel’s work” (Goldstein, 2005). But among the brilliant achievements of Kurt Gödel one stands out exceptionally.

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One need not to be a practicing mathematician in order to grasp the basic idea and message of the Incompleteness Theorem. And maybe that is why this result gained so much audacity in the popular scientific debate. But, of course, this ingenious simplicity is only one of many aspects of the 1931 work that distinguish it from other outstanding works of the Austrian intellectual giant. …


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Source: “Understanding Cantor’s Mathematical Infinity

On August 2. 2020 Cantor’s Paradise published Bruno Campello’s brief critique of Cantor’s approach in transfinite mathematics. The author raises some doubts about Cantor’s debunking Euclid’s 5th principle stating that the whole is greater than the part. Campello gives an interesting argument against Cantor’s reasoning, but it itself raises a number of question marks to say the least. Below I comment on his discussion of the alleged Cantor’s fallacy.

To begin with, Campello says that “Cantor and his epigones believed that, along with a principle of ancient geometry, he was also breaking down an established belief of common sense and one of the pillars of classical logic”, but it seems that not only Cantor and his mob believed it — the article comes to a close with these…

Jan Gronwald

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