[Math] Examples of “unsuccessful” theories with afterlives

Question

I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and continued to be studied for other reasons.

Here is a prominent example I know of:

• Lie theory: It is my understanding that Lie introduced Lie groups with the idea that they would help in solving differential equations (I guess, by consideration of the symmetries of these equations). While symmetry techniques for differential equations to some extent continue to be studied (see differential Galois theory), they remain far from the mainstream of DE research. But of course Lie theory is nonetheless now seen as a central topic in mathematics.

Are there some other examples along these lines?

Answer 1

I quote at length from the Wikipedia essay on the history of knot theory:

In 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther. Chemical elements would thus correspond to knots and links. Tait's experiments were inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids. Thomson and Tait believed that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do. For example, Thomson thought that sodium could be the Hopf link due to its two lines of spectra.

Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures on alternating knots. (The conjectures were proved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and Thomas Kirkman.

James Clerk Maxwell, a colleague and friend of Thomson's and Tait's, also developed a strong interest in knots. Maxwell studied Listing's work on knots. He re-interpreted Gauss' linking integral in terms of electromagnetic theory. In his formulation, the integral represented the work done by a charged particle moving along one component of the link under the influence of the magnetic field generated by an electric current along the other component. Maxwell also continued the study of smoke rings by considering three interacting rings.

When the luminiferous æther was not detected in the Michelson–Morley experiment, vortex theory became completely obsolete, and knot theory ceased to be of great scientific interest. Modern physics demonstrates that the discrete wavelengths depend on quantum energy levels.