Bedürftig T., Murawski R. Philosophy of mathematics

Berlin: De Gruyter, 2018. — 474 p.The present book is an introduction to the philosophy of mathematics. It represents a special approach to it – one that does not look philosophically at mathematics from the outside but rather attempts to ask philosophical questions from the standpoint of a mathematical (research and teaching) practice concerning fundamental concepts, constructions and methods. It looks for answers both in mathematics and in the philosophy (of mathematics) from their beginnings till today. We think that without the historical component the philosophy of mathematics would be empty. Moreover, the historical approach is important for we should consider (even current) mathematics as being in the process of permanent development.
Philosophy of mathematics is a field between philosophy and mathematics. This is in fact a difficult position. Those being outside mathematics will ask: what at all has mathematics – the firmly stated and infallible system of numbers, formulas and methods – in common with philosophy. On the other hand philosophy – to which the philosophy of mathematics belongs – and philosophers themselves have and had always problems with mathematics, the discipline forming in a sense a paradigm of the scientific thinking and developed to such a degree that it sometimes appears to be completely unmanageable and unclear. Finally, the subject of the philosophy of mathematics is just mathematics – and mathematicians are generally not inclined to consider it in a philosophical way. This is in some sense a rational attitude because all that mathematicians are doing seems to be far away from any philosophy