In order to know what we are talking about, it's important to define our terms. Here is my attempt to define the terms foundations and foundations of mathematics. I am trying to make the meaning of these terms crystal clear to anybody who wants to understand them.
Reference: Leonard Peikoff, Objectivism: The Philosophy of Ayn Rand, pp. 121-141.
2. Within the integrated whole of human knowledge, we may focus on what are usually called "fields of study" or "specialties". A field of study is distinguished by a certain conceptual unity: the concepts that make up the field are closely related to each other and are sufficiently self-contained so that the field lends itself to study in isolation for some purposes. Usually if not always, this kind of conceptual unity follows from the existence of a specific subject matter, the real-world object of study.
Example: The theory of electrical circuits is a field of study characterized by a subject matter (electrical circuits) and a set of basic concepts (resistance, impedance, voltage, etc) that are closely related and in a sense complete, so that electrical circuit theory lends itself to study as a subject which may be regarded as more-or-less self-contained for some purposes. But this does not mean that the subject exists in isolation, because (1) the principles of electrical circuit theory must be formulated in such a way as to be consistent with the rest of human knowledge, and (2) electrical circuit theory has many applications and connections to other fields of study (electromagnetic theory, quantum physics, acoustics, etc).
Reference: H. G. Apostle, Aristotle's Philosophy of Mathematics.
This definition of mathematics may seem too old-fashioned, but I believe it can be stretched to cover not only ancient but also modern mathematics, and it has the additional advantage of being well-linked to applications and the rest of human knowledge.
4. Like any large field of study, mathematics has a number of subfields, for instance functional analysis and algebraic geometry. Each of these subfields has its own conceptual framework, but they are all part of mathematics and there are many links among them, just as there are many links between mathematics and the rest of human knowledge. In accordance with point 1 above, all of this has to be consistent.
In the history of particular fields of study, the foundations often take time to develop. At first the concepts and their relationships may not be very clear, and the foundations are not very systematic. As time goes on, certain concepts may emerge as more fundamental, and certain principles may become apparent, so that a more systematic approach becomes appropriate. An example is the gradual clarification of the concept of real number through the centuries, culminating in axioms for the real number system.
The foundations of X are not necessarily the most interesting part of field X. But foundations help us to focus on the conceptual unity of the field, and provide the links which are essential for applications and for integration into the context of the rest of human knowledge.
One of the important early developments was Cartesian geometry, which showed that much of geometry could be reduced to algebra, so algebra took on a more foundational role. Another important development was the "arithmetization of analysis" (Weierstrass, Dedekind). Thus it was no longer necessary to regard real numbers and continuous functions as basic, unanalyzed concepts; instead they could be reduced to the natural numbers. This made possible the axiomatization of analysis in terms of second order arithmetic (carried out systematically by Hilbert and Bernays).
Reference: Hilbert and Bernays, Grundlagen der Mathematik, Vols. I and II.
The Frege-Russell-Hilbert-Gödel-... line is what I would regard as the systematic phase of foundations of mathematics. Most of the systematic phase took place in the late 19th and 20th centuries.
7. What is properly regarded as foundational in one context should not necessarily be regarded as foundational in another. For instance, if Y is a subfield of X, then "foundations of Y" is not necessarily part of "foundations of X", because the most basic concepts of Y may be reducible to even more basic concepts of X, since X provides a wider context.
Example: Zariski's work on foundations of algebraic geometry, although interesting and important, is not part of foundations of mathematics.
-- Stephen G. Simpson
email@example.com / 7 April 2006