The Atlantic had an article "The Nobel Prize in Physics Is Really A Nobel Prize in Math".
There has been a debate as to how large a role mathematics plays in science. Gauss called mathematics the "Queen of the Sciences". On the other hand, E. O. Wilson says that great scientists do not need to be good at math. In fact, he goes as far as to say that mathematicians and statisticians should be subservient to scientists.
Great Scientist does not mean Good at Math
In physics, the interrelationship between mathematics and the science is clearly important. In experimental physics (especially, Big Science - lots of dollars), the clear need for integration of all aspects of STEM is clear. Feynman, Higgs and Einstein could not have done their work without mathematics.
The usual thing taught in classes in Special Relativity is that Einstein formulated Special Relativity to explain the results of the Michelson-Morley experiment. But, one professor who taught me relativity speculated that Einstein wasn't convinced that the data were accurate enough to better with that one experiment. He was driven by other concerns, like the problem of the inability for Maxwell's Equations to be consistent with Newton's Laws of Physics.
I am convinced that Einstein knew when very young that something like Special Relativity was required. As a young teenager, he imagined what it would be like to surf on a light wave. If you do that gedanken experiment, it is obvious that there is something very strange about the speed of light. At all other speeds, average electromagnetic field from a light source is zero. But, at the speed of light, the electric and magnetic field becomes static, energy is flowing in one way, not fluctuating back and forth. It is almost as if time just stands still. That is one fundamental characteristic of Relativity.
The work of Feynman and Higgs (among many others) made me convinced that the end of analysis as the major mathematics tool was near (analysis in the sense of solving partial differential equations, calculus) and that group theory and algebraic topology would dominate mathematics, especially what is used in science. My thinking on this dates back to the mid 1970's. We don't do geometry theorems the same way as in the time of Euclid. We aren't taught solid geometry and spherical trigonometry anymore because the mechanisms of analytic geometry and analysis solves these problems with much easier techniques.
But, linear algebra, group theory and algebraic topology has the ability to solve problems that can't begin to be solvable with standard analytic techniques. I was convinced that we would replace the teaching of the details of calculus at a deep level, replaced by modern algebra techniques. Why work on solutions of 6 order differential equations when the solution of most interesting problems to scientists are available much easier with the techniques of linear algebra? I am not saying that nobody should study the works of Edward Ince ("Ordinary Differential Equations"), but, it might not be necessary with more modern techniques from other fields of math.
Perhaps we have an analogy with classical music. In the history of music, the classical era is very short, from about 1750 through 1900. It surprises many people that classical music began about the same time as the American Revolution. There a lot of classical music listeners who dislike 20th century music and completely disdain the use of any new instruments (like saxophones and synthesizers). It is as though we froze music in the 19th century. I believe that this freeze of music genre is unique in history. Music in all other eras had much experimentation in new instruments, inclusive in who participated in music making (everybody made music). It is not that experimentation is completely dead, it is just absent in the circles of people who exclusively listen to classical music.
For me, it is extraordinarily weird to freeze music to a specific 150 year history along with the extremely stilted required behaviors of the audience. Make sure you clap only at the right points, nowhere else, stand during the Hallelujah Chorus, etc. No wonder classical music is having great trouble. It is eventually doomed, though will have an existence through a small set of enthusiasts, like those who love Plainsong chants and Renaissance music.
Back to the subject of Math, we are frozen in the same way to pre-1900 math. It is hard to induce people to embrace techniques developed in the 20th century. It is as though World War I completely froze pedagogy for most people. While we try to change how we teach and who we teach, what we teach is frozen, just like classical music.
