For much of human history, since the days of Plato, classical mathematics has been held in a sacrosanct regard unparalleled by the other sciences. The sterile paradigm of objective knowledge and easily manipulated symbols was a model of intellectual infallibility. During the eighteenth century, however, the world of classical mathematics was turned upside down, and subject to critical reevaluation.
A catalyst for this drastic shift in thinking was the emergence of non-Euclidean geometry. Much of mathematics was heretofore based upon geometrical observations, giving an otherwise theoretical science ties to the physical realm. As mathematicians began to work in non-Euclidean situations, however, tackling problems and attempting solutions which did not adhere to the formally laid out geometrical axioms, a degree of physical grounding was lost. This divergence from strictly Euclidean modes of thinking facilitated a shift in overall mathematical work, which was to rely more heavily on the development of rigorous analysis as opposed to ideas based solely on empirical knowledge.
During the nineteenth century, such mathematicians as Dedekind and Weierstrass began to formulate new theories concerning the foundations of mathematics foundations based upon arithmetic instead of geometrical fact. This search, which was to continue throughout the century with the establishment and refinement of set theory by the likes of Frege and Russell, gave rise to two distinct schools of mathematic-philosophical thought, both with very different subscribers. Foundationalism was a branch generally comprised of philosophers dabbling in mathematics, whilst constructivism (which would prove to be the more impacting of the two) was/is supported by the practicing mathematician.
As its name suggests, foundationalism is an overarching term given to the search for the immutable foundations of mathematics. Even though such luminaries as the logician Gottlob Frege engaged in this field of study, it has been increasingly discarded by the mathematical community. As Reuben Hersh explains, foundationalism can unnecessarily alienate mathematicians from further philosophical investigation, as an unstated consensus that the philosophy of mathematics is research on the foundation of mathematics. If I find research in foundations uninteresting or irrelevant, I conclude that I’m simply not interested in the philosophy (thereby depriving myself of any chance of confronting my own uncertainties about the meaning, nature, purpose, or significance of mathematical research).
Alternatively, constructivism takes the stance that the mathematician constructs his or her own pragmatic sphere of truth. Thomas Tymoczko summarizes well that constructivism insists that any mathematical reality is conditioned by the actual and potential constructions of mathematicians who invent mathematics. Almost by definition, this is a philosophy tailored to the needs of the practicing mathematician, who must confront problems of both substance and technique on a regular basis.
Thus do foundationalism and constructivism offer two diametrically opposed starting points for further investigations into the philosophical underpinnings of mathematics.