Herman Wyle (1885-1955) was a renowned and creative mathematician and theoretical physicist. His mathematical work was instrumental in forming the foundations for modern quantum physics. This 1917 book is an essay against the set theory foundations for mathematics that was starting to become widely accepted. His view was that set theory leads to contradictions and requires just as much human intuition as does working directly with geometry and natural numbers:
“A set-theoretic treatment of the natural numbers such as that offered in Dedekind (1888) may indeed contribute to the systematization of mathematics; but it must not be allowed to obscure the fact that our grasp of the basic concepts of set theory depends on a prior intuition of iteration and of the sequence of natural numbers.” (p 24)
His alternate approach was based upon objects and iteration but in the end he was not satisfied with the result. An “object” is a form of super-variable in which spaces for various sorts of data are reserved for use by operations inherent to the object itself. His conceptual use of objects in this book may be one of the first uses of an idea that has come to dominate computer science.
“The foundation consists of, first, one or more individual categories of object (the ‘basic categories’) and, second, certain individual immediately exhibited properties of and relations between objects of the basic categories (the ‘primitive relations’).” ( p 41)
His preface to a 1932 reprint sums up his view of the state of mathematical foundations:
“The point of view adopted in this monograph continues to strike me as a natural transitional stage in the development of foundational research. However, in the period since its appearance, my work has been superseded by two trends identified by the catchwords Intuitionism and Formalism. Still, this deeper grounding of the foundation has not lead to an even moderately satisfying or defensible conclusion; things remain in a state of flux.” (p 3)