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Reviews of the book ``The World of Mathematics and its Material Shadows'' by Józef Życiński

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Are mathematical concepts just creations of the human mind, or are they a reflection of the real world? Do mathematicians create mathematical concepts and theories or do they discover them? We do not know the full answer to these questions, but many mathematicians are sympathetic to the view that mathematics describes a certain reality, independent of humankind. In the academic year 2006/2007, the recently deceased Archbishop of Lublin, the Rev. Prof. Józef Życiński\footnote{Józef Życiński (1948-2011) was a philosopher, theologian, an Archbishop of Lublin and the grand chancellor of the Catholic University of Lublin. He was the author of more than 50 books and hundreds of papers on topics related to philosophy of science, philosophy of nature, cosmology and evolution. He was a member of the Polish Academy of Science, the Pontifical Council for Culture, the European Academy of Science and Art in Salzburg and the Russian Academy of the Natural Sciences.} gave a series of lectures entitled "Elements of Platonic in the fundamentals of mathematics" and the book reviewed here is based on these lecture notes, edited by the Rev. Prof. Michał  Heller. The book begins by presenting two contrasting answers to the questions posed above: "{\it Empiricism states that mathematics is an expression of the world constructed by human minds, $\dots$; Platonic states that mathematics describes a reality which predates any human activity $\dots$\ \/}."  The book gives various arguments in favour of both of these views, although the author does not hide the fact that he supports Plato's approach. In one of the chapters the author considers the question regarding the existence of mathematical objects and describes the views of Brouwer, Hilbert, Popper and Ellis. Ellis proposed that "{\it a given object should be viewed as being onitically real, if its presence leads to observable consequences in the real world \/}". In turn, according to Brouwer, only those mathematical objects which can be effectively constructed should be viewed as real. Another chapter is devoted to controversies regarding the fundamentals of mathematics. In this chapter the author talks about Hilbert's work on the proof of the consistency of arithmetic as proposed by Gentzen, which was later overthrown by the discoveries of G\"{o}del, as well as Bourbaki's approach to the fundamentals of mathematics. The next chapter considers the concept of truth in mathematics, including comments on Tarski's theorem regarding the undefinability of truth. Although the book is of a philosophical nature, we find many examples from the discipline of mathematics. As such, the reader may learn about the intuitionism of Brouwer, as well as the  G\"odel and L\"owenheim-Skolem theorems and their philosophical aspects. The author also writes in an accessible style on many problems in pure mathematics, including Fermat's last theorem and the Riemann hypothesis.While reading this book, I only found a few minor errors. For example, on page 63, instead of "the roots of the equation $\zeta(z)=0$", the author should have written "the roots of the equation $\zeta(1/2+iz)=0$", since this is the equation that Riemann had in mind in the context of this reference.
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