Xjyuk

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 are two prominent examples I know of:

• 
  The representation theory of finite groups: I believe the representation theory of finite groups (which at that time might have been called "character theory") was introduced by Frobenius with the purpose of understanding the structure of finite group in general. While there are certainly important structural results about finite groups that were first proved using representation theory (e.g., Burnside's theorem), the tremendous advances in our understanding of finite groups in the mid-to-late 20th century (e.g., the classification of finite simple groups) did not heavily use representation theory. Nevertheless, the representation theory of the symmetric group, of finite groups of Lie type, positive characteristic phenomenon, etc. continue to be important topics of research.


• 
  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?

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.

"The modern study of knots grew out an attempt by three 19th-century Scottish
physicists to apply knot theory to fundamental questions about the universe".

This is a copy of a copy of some history of the origins of free probability by Dan Voiculescu extracted from a response by Roland Speicher, a developer of the field, to an MO-Q:

This is from his article "Background and Outlook" in the Lectures Notes
"Free Probability and Operator Algebras", see
http://www.ems-ph.org/books/book.php?proj_nr=208

Just before starting in this new direction, I had worked with Mihai Pimsner,
computing the K-theory of the reduced $C^*$-algebras of free groups. From the
K-theory work I had acquired a taste for operator algebras associated with free
groups and I became interested in a famous problem about the von Neumann
algebras $L(mathbbF_n)$ generated by the left regular representations of free groups,
which appears in Kadison's Baton-Rouge problem list. The problem, which
may have already been known to Murray and von Neumann, is:are $L(mathbbF_m)$ and $L(mathbbF_n)$ non-isomorphic if $m not= n$?

This is still an open problem. Fortunately, after trying in vain to solve it,
I realized it was time to be more humble and to ask: is there anything I can
do, which may be useful in connection with this problem? Since I had come
across computations of norms and spectra of certain convolution operators on
free groups (i.e., elements of $L(mathbbF_n)$), I thought of finding ways to streamline
some of these computations and perhaps be able to compute more complicated
examples. This, of course, meant computing expectations of powers of such
operators with respect to the von Neumann trace-state $tau(T) = langle T e_e,e_erangle$, $e_g$
being the canonical basis of the $l^2$-space.

The key remark I made was that if $T_1$, $T_2$ are convolution operators on $mathbbF_m$
and $mathbbF_n$ then the operator on $mathbbF_m+n = mathbbF_m ast mathbbF_n$ which is $T_1 + T_2$, has moments $tau((T_1 + T_2)^p)$ which depend only on the moments $tau(T_j^k)$, $j = 1, 2$ , but not
on the actual $T_1$ and $T_2$. This was like the addition of independent random
variables, only classical independence had to be replaced by a notion of free
independence, which led to a free central limit theorem, a free analogue of
the Gaussian functor, free convolution, an abstract existence theorem for one
variable free cumulants, etc.

Logic and set theory were developed by Frege, Russell and Whitehead, Hilbert and others in the late 19th, early 20th centuries with the goal of providing a firm foundation for all of Mathematics. In this they failed miserably, but nevertheless they have continued to develop and to be studied for other reasons.

Multiplication of quaternions was introduced for use in physics for purposes for which cross-products of vectors came to be used and have been used ever since.

But today quaternions are used in computer graphics. I suspect they also have other applications.

Ronald Fisher's theory of fiducial inference was introduced around 1930 or so (I think?), for the purpose of solving the Behrens–Fisher problem. It turned out that fiducial intervals for that problem did not have constant coverage rates, or in what then came to be standard terminology, they are not confidence intervals. That's not necessarily fatal in some contexts, since Bayesian credible intervals don't have constant coverage rates, but everyone understands that there are good reasons for that. Fisher wrote a paper saying that that criticism is unconvincing, and I wonder if anyone understands what Fisher was trying to say. Fisher was brilliant but irascible. (He was a very prolific author of research papers in statistical theory and in population genetics, a science of which he was one of the three major founders. I think he may have single-handedly founded the theory of design of experiments, but I'm not sure about that.)

However, fiducial methods seem to be undergoing some sort of revival:

https://statistics.fas.harvard.edu/event/4th-bayesian-fiducial-and-frequentist-conference-bff4

The typical oracle methods of Computability theory AKA Recursion theory were shown to be insufficient to settle the P vs. NP problem by Baker, Gill and Solovay 1975.

Thus recursion theory became divorced from the problems of efficient computability and experienced a bit of a setback (not as many papers in Ann.Math. anymore etc.).

Nevertheless it continued as the study of in principle computability.