**Ivan Śleszyński**'s first name is sometimes written as 'Jan', which is the Polish version, while his last name is either given by the Polish 'Śleszyński' or the Russian versions 'Sleshinskii' or 'Sleshinsky'. Although Ivan was born in the Ukraine, he was ethnically Polish, being born into a Polish family living in Lysianka, a town about 160 km due south of Kiev. He studied mathematics at Odessa University and graduated from there in 1875. He then travelled to Germany where he studied under Karl Weierstrass at the University of Berlin, receiving his doctorate in 1882. Returning to Odessa, he became professor of mathematics at the University, holding the position from 1883 to 1909. The year 1909 was significant in another way, for it was the one in which he published his translation of Louis Couturat's famous book

*The algebra of logic*. This work by Śleszyński was more than a translation since it contained Śleszyński's own very useful commentary. This text had a major influence on the development of mathematical logic in Russia since it became the main textbook used by students of the subject over many years.

Śleszyński left Odessa and went to Poland in 1911 where he was appointed as an extraordinary professor at the Jagellonian University of Krakow. We should note that in fact Krakow was at this time in the Austro-Hungarian Empire but, remembering Śleszyński's Polish background, it is fair to say that he was moving to Poland. In 1919 he was promoted from extraordinary professor to become the full Professor of Logic and Mathematics the Jagellonian University. He continued to teach at Krakow until, having reached the age of seventy, he retired in 1924. In fact we note that the university decided not to fill his chair after he retired.

Śleszyński's main work was on continued fractions, least squares and axiomatic proof theory based on mathematical logic. In a paper of 1892, based on his doctoral dissertation, he examined Cauchy's version of the Central Limit Theorem using characteristic function methods, and made several significant improvements and corrections. Because of the work, he is recognised as giving the first rigorous proof of a restricted form of the Central Limit Theorem.

In 1898 Alfred Pringsheim proved that the condition

*b*

_{n}| ≥ |

*a*

_{n}| + 1,

*a*

_{n}≠ 0,

*n*≥ 1,

*K*(

^{an}/

_{bn}), where

*a*

_{n}and

*b*

_{n}are complex numbers; a result now known as the Pringsheim criterion. W J Thron states in [6] that this result was established ten years earlier by Śleszyński. Thron demonstrates that Pringsheim was aware of Śleszyński's work, though Pringsheim himself claims that he only became aware of Śleszyński after his article was completed. Six papers by Śleszyński on continued fractions are discussed in [6] where a complete bibliography of Śleszyński's mathematical papers is given. His work on continued fractions is also discussed in [4].

In [2] Bednarowski discusses Śleszyński's book *O Logice Tradycyjnej* published in Krakow in 1921:-

Śleszyński then represents the five different situations by using Venn diagrams. InŚleszyński assumes that the part of traditional logic created by Aristotle is a theory of relations which may hold between two classes. He then askes the following question. Having two non-empty classes A and B, what are the possible relations between them so far as having elements in common is concerned? His answer is that between A and B there holds one and only one of five relations which he symbolises by a, b, g, d, e.

*a*the two classes

*A*and

*B*coincide, in

*b*the class

*A*is properly contained in

*B*, in

*g*the class

*B*is properly contained in

*A*, in

*d*the classes

*A*-

*B*,

*B*-

*A*, and

*A*intersect

*B*are all non-empty, and in the final case

*e*,

*A*and

*B*are disjoint. Śleszyński then argues as follows. First he says that either

*A*and

*B*have common elements or they do not. If they do not then we have the situation

*e*. Next Śleszyński looks at the situation where common elements exist. Either one of

*A*or

*B*contains an element not in the other, or they do not. If they do not, then we have the situation

*a*. There remains the case where either one of

*A*or

*B*contains an element not in the other. If

*A*fails to contain an element not in

*B*we have

*b*. Otherwise

*A*contains an element not in

*B*. If also

*A*contains an element not in

*B*then

*d*otherwise

*g*. Śleszyński also goes on to consider what happens when empty classes are allowed and shows that three further relations occur.

We should mention another interesting work by Śleszyński, namely *On the significance of logic for mathematics* (Polish) published in 1923. However, despite the interesting publications we have mentioned, Śleszyński did not publish much of his work. This was rectified by a major two-volume publication in the years following his retirement. One of Śleszyński's most famous students at the Jagellonian University of Krakow was Stanisław Zaremba. In 1925 Zaremba, acting as editor, published the first of two volumes of *The theory of proof* based on Śleszyński's lectures at Krakow. A second volume appeared in 1929. McCall writes in [1]:-

We end this brief biography by giving the following quote by Śleszyński:-Much indeed can be learned from the rich collection of[Śleszyński's]papers on various subjects in the realm of formal logic, and of mathematical logic and its history ... Introduction to mathematical logic, complete proof, mathematical proof, exposition of the theory of propositions, the Boolean calculus, Grassmann's logic, Schröder's algebra, Poretsky's seven laws, Peano's doctrine, Burali-Forti's doctrine - these are some of the themes pursued in this work, from which I personally have learned a great deal and thanks to which I have got a clear idea of many an unclear thing.

The point of civilization is the exchange of ideas. And where is this exchange, if everybody writes and nobody reads?

**Article by:** *J J O'Connor* and *E F Robertson*