RICHARD DEDEKIND (1831-1916) published a small booklet in 1888 in which he gave a sophisticated and novel answer to the fundamental question of what the (natural) numbers really are.

A formalist, in the colloquial sense of the word, is someone who follows the prescribed formalities exactly and does not question them. Formalists are therefore not well-liked in all areas of life.

PLATO (Πλάτων) was born around 428 B.C.E. in Athens (or Aigina?). He descended from an old, highly respected Athenian family.

We want to discuss the question as to what we can expect from a proof of existence in mathematics. Are we allowed to expect that an object that fulfills the claim of existence is explicitly given by name and also clearly described, - or do we have to rest satisfied with the hint that the assumption of non-existence of such an object would lead to contradictions? So, the question is which information the existential quantifier ∃x (read: there exists an x) can and should provide.

After ALEXANDER’s death in 323, ARISTOTLE was accused of godlessness in Athens. He fled to Chalkis on Euboea, where he died shortly afterwards (approx. in the year 322 B.C.E.).

From the early 18th century onwards, it was often held that psychology is a ‘general science of the mind’ (“eine allgemeine Wissenschaft des Geistes”, as it was called by WILHELM WUNDT, 'Logik', I, p. 1, op. cit.), and therefore the basis of all philosophy, logic and mathematics. In the 19th century, this view was even adopted by many mathematicians. It became customary in the introductory chapters of textbooks to introduce the basic mathematical terms in the terminology of psychology. It was believed that the soul (or mind) was capable of producing mathematical objects and that the laws of logic, arithmetic, geometry, etc., could be traced back to psychic laws.

The Age of Enlightenment produced two great philosophical schools of thought, ‘rationalism’ (RENÉ DESCARTES, GOTTFRIED WILHELM LEIBNIZ and others) and ‘empiricism’ (GEORGE BERKELEY, DAVID HUME and others). The fundamental difference between these two schools of thought ‘concerned the measurement of the extent of our knowledge a priori, that is, the knowledge we can have independently of our senseperception.

Philosophy of Mathematics addresses questions raised within mathematics but answered by philosophy, as ontology and epistemology of mathematical objects. .

The term ‘Platonism’ in the present Philosophy of Mathematics does not refer to the entire philosophy represented by PLATO, but only to a general tendency of Platonic philosophy.

The concept of infinity has been considered a difficult and problematic concept from antiquity to modern times. The view that DESCARTES occasionally expressed in his letters to MERSENNE was widespread.

As mathematics developed rapidly in the modern age, it became necessary to consolidate its foundations. In particular, it became necessary to define the concepts of positive, negative, real and complex numbers, free from objections, and to consider their ontological and epistemological status. The philosophers' traditional views on the foundations of mathematics were studied diligently, but were not well received.

The question of the status of Euclidean geometry among the sciences was the subject of an intensive debate in the 16th century, the age of the Renaissance. There was widespread agreement that Euclidean geometry did not meet the Aristotelian requirements for a science, but nevertheless had the highest degree of clarity and certainty of all disciplines.

Empiricism is the position in philosophy in which all knowledge is ultimately derived from sensory experience. Accordingly, all that we humans can know is ultimately based on sensory perception. All questions relating to the origin and justification of our knowledge can ultimately only be decided by referring to our sensory perceptions.

In ancient times, epic poets repeatedly invoked the Muses to inspire them and guide them in the writing of their songs. The Muses are the daughters of Zeus and Mnêmosynê; they are the goddesses of song, knowledge and memory. The invocation of the Muses is based on the conviction that they, as goddesses, attend all events, and therefore have proper knowledge of everything.

In today’s mathematics, the infinite is present almost everywhere, and it could hardly be expressed better than the way HERRMANN WEYL described it:However, mathematics seems to have only gradually found this ‘living centre’ in modern times. In earlier times, the concept of infinity was considered a difficult and problematic concept.

In this introductory chapter, we want to discuss one of the earliest mathematical discoveries, namely, the discovery of the existence of incommensurable quantities by the Pythagoreans about two and a half thousand years ago.

DAVID HILBERT (1862-1943) was an excellent mathematician who did outstanding work in many areas of mathematics and mathematical physics. He was also one of the few who thought intensively about the foundations of mathematics. Thereby, he became the pioneer of many new developments.

JOHN LOCKE was born on August 29, 1632, in Wrington (Somerset/England). He studied medicine, natural sciences and philosophy at Oxford from 1652 onwards. In 1658, he became magister artium and, from then on, taught as a tutor at Christchurch College in Oxford.