And we’ve killed it.

Picture by Eyasu Etsub on Unsplash

In an airplane interview released in 2009, Bobby Fischer, arguably the most talented chess player of all time, exclaimed

He argued that the game is irredeemably broken due to the influence of engines. The reason for this is that now, less and less depends on originality, creativity and individual skill of a player, whilst more and more depends on learning engine variations by heart till the 20th move. Memory substitutes tactical intuition and strategic thinking.

Though the grandmasters that make it to the top are surely masters of calculation and indeed have a deepened understanding of the…


How to interpret the same data in a completely different way.

Werner Heisenberg and Niels Bohr (Boston Arts Diary)

Realism in regards to mathematical objects claims that they are abstract (i.e. non-concrete and atemporal) and objective (i.e. independent on the subjective impression of a person). There were dozens of gallons of ink spilled over the dispute about the justification for committing oneself to such abstruse objects, and a certain kind of arguments bears on the indisputable effectiveness of mathematics in natural sciences. The claim boils down to:

Mathematical objects must be real, since they are what essentially is used to describe the physical world.

This style of arguing is popular among philosophers and some mathematicians. It starts with an…


Looking for a good reason to help others.

Photo by Toa Heftiba from Unsplash

There are at least two notions of altruism: a superficial and a profound one. Superficial altruism could be defined simply as a general willingness to assist others. It views helping people as a form of self-fulfillment.

But this understanding of the term is often criticized as too short-sighted, as we sometimes do good to feel better about ourselves, not necessarily to help others. Hence the more profound notion of altruism: a truly altruistic deed requires genuine selflessness and secrecy. Profound altruism views helping as a form of self-sacrifice. …


i.e. a historical case against Platonism in philosophy of mathematics

Almost everybody, I assume, would agree that mathematics, generally understood as a set of definitions, rules and theorems, is an a priori field. Following the Kantian nomenclature, for the last two centuries there was however no consensus if it was analytic or synthetic. Frege, Russell or the Vienna Circle believed that it was analytic and that it could be reduced to logic (side note: of course the scope of “logic” changed throughout time; that is why Frege was a logicist and a Platonist). On the other hand, Kant, Hilbert and Gödel believed that it was synthetic, each for different reasons.

Gödel in Vienna (Getty Images)


A brief survey of recent insights from cognitive science

R. Opałka’s OPALKA 1965 /1 — ∞”

In the recent “ask me anything” streaming, Noam Chomsky pointed to an interesting problem.

Darwin and Wallace were very puzzled and debated the fact that all humans have arithmetical capacity. [That] it’s just a part of our nature to understand that there are infinitely many natural numbers; that when you add them it works this way and not some other way, and so on. This seems to be a part of universal human nature. They were very puzzled by that because it couldn’t possibly have been selected — since it was never used! …


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…


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…


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.

One need not to be a practicing mathematician in order…


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…

Jan Gronwald

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