There are some types of math that are applicable to the real world, and some that aren’t. That’s why a person can spend their entire mathematical career on theoretical maths, and a different person can spend all that time on applied maths. Therefore it’s hard to figure out whether ‘math is applicable in the real world’.
On one hand math is intrinsic in what we do, even before we knew what math was we started counting, math existed before we created it. Math arises from the world naturally, and as such there are naturally a lot of applications of it. When we go to more abstract concepts it becomes less clear as to how they can be applied in the real world but maybe we’ll find a way they can be applied in the future. We get so caught up in notation we forget that math is very intrinsic to human nature and our world. In natural sciences we can use math to model everything. Because math arises from nature and nature has those mathematical patterns we can make use of those and apply them to different aspects of nature that may not seem mathematical. Part of the reason it may seem that some math doesn’t have practical applications is that the advancement of human mathematical development is very far ahead of technology and sciences. For example computer scientists use a lot of the factorization theorums first developed by Euclid who is long dead. The fact is that when Euclid formulated these theorums he didn’t know they would have such far reaching consequences, he had no idea what a computer is. That shows how many application math can have not just now, but far into the future. Maybe our ideas of research in modular theory, like solving for Fermat’s last theorum could potentially have far reaching consequences in modeling our world and certain things in the future of this world. This shows how math has many applications not only in the present, but also in things we haven’t even thought of. Even things that don’t seem to have applications in the real world may have applications in the future. In addition, it doesn’t necessarily need to have practical applications in that sense as math can help you develop your mental faculties, which can be applicable in may different things.
However there’s always the argument that math is a social construct and isn’t inherent on the world. In that sense, math isn’t real and can’t truly be applied to the real world. Math is perfect and the real world is not. Sometimes math can have models for things but they tend not to be perfect. That’s why uncertainties are so important and common in math. A lot of science is math, and in things like theoretical physics, it may be all math. In that sense, math is used to solve questions in physics, but it can’t be tested in the real world as of yet, so there’s no way of knowing whether certain maths are applicable. That’s why we can’t prove string theory. In a lot of math, data is required, but it’s never enough to have a full truth.
Overall, I think that if certain areas of math don’t seem to have applications in the real world, they may in the future, and that’s why I can’t say that math does or doesn’t have real-world applications with absolute certainty.