Empirical roots of knot theory

Knot theory is a branch of (low-dimensional) topology, or the study of geometrical properties that are unaffected by continuous transformations. It particularly studies any closed lines that are embedded in three-dimensional spaces (spheres) while not intersecting themselves. The main questions in this domain are, first, whether, given a certain projection, it really represents a knot or can be untangled, second, whether, given two projections, these represent the same (un)knot or not, i.e. can be reduced to each other, and third, whether there exist any applicable algorithms to sort out the former two questions. Our main philosophical interest in this branch has to do with the presumed importance of the `empirical' contexts of inquiry and application throughout its history, through which might be challenged the absolute `purity' of mathematical method and content still defended by a vast majority of its philosophers. Indeed, in its concern for the foundational debate, twentieth century philosophy of mathematics has remained almost entirely a priori in nature. Only recently, openings have been made for more a posteriori, historical approaches. That is, after decades of unfruitful antagonism between the majority of foundationalist and a rather small group of squarely anti-foundationalist scholars, a phase of partial reconciliation has apparently set in. From one side, a group of post-analytically minded philosophers show a growing, serious interest in the history of mathematics, while trained historians aspiring to move beyond the mere narrative, start including questions about the nature of mathematics in their inquiries.