My professor says: If you be a mathematician then you can easily learn any discipline like physics, chemistry, engineering and even medical! Is he right? I am an enthusiast of all scientific disciplines. Should I study mathematics? I know it is impossible to be an expert in any field, but if I want to know about everything (at least a bit about everything), should I choose mathematics?
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In my personal experience, my degree in mathematics has allowed me to move very freely through applied fields such as engineering and healthcare. This comes with a caveat: I also have a substantial prior background in aeronautical and mechanical engineering, and although I did not degree in these fields, I studied them concurrently and in principle could obtain my degree there with little extra work (but not little extra money!).
This experience has given me an analytical edge over my peers in engineering fields, but I am behind my peers in mathematical fields. Nonetheless, I can continue to study and learn, and in fact I am doing so.
My decisions were motivated by my desire to leave academia. So if you want to stay in academia, your experience will differ. I also had a significant edge on the business/career maturity side of things compared to my peers. So that affects things, too.
In the end, I consider myself a mathematician, and one of my great joys in my career is being able to understand ideas and theorems in engineering math that even the best engineering professors do not get. It makes me feel as if I can contribute.
At the same time, I feel inadequate in my mathematical abilities. I do not get to exercise them regularly. They atrophy. I forget theorems and definitions. This is part of my life. I can remember either theorems from abstract algebra or regulations governing the safety of medical devices, but not both. There is a trade-off, or at least there is for most people.
In the end, I made my decision because it best fit my vision of my future.
You need to figure out what that vision is for you, and when you have it, figure out how to get there. The decision will make itself.
Math gives you a different, logic driven, analytic approach to all kinds of things. And that helps a lot concerning all kinds of tasks. So what your professor says is kind of right.
However (and that is the part of the statement that bothers me) other scientific disciplines are also challenging in their very own way. A huge part of medicine or biology is knowledge of that discipline (rather than a specific way of thinking). And physics students of early semesters can easily solve integrals that look like torture for most math graduates. Although math is a good foundation for almost anything, this doesn't mean you can simply do everything else without a (probably) huge additional effort.
I myself am no real mathematician, but a computer scientist and although I love all that weird, theoretical stuff from computability over complexity theory to cryptography (which also would be perfect fields of study for a mathematician), I got the feeling that I'm better off with my choice, as my knowledge feels a bit closer to the real-world application.
And, what I should add: If you are not staying in the "academic context" it is quite unlikely that you will actually work as a mathematician. While that does not need to be a bad thing, you should be aware that your field of study probably won't be equal to your field of work.
Yes. More specifically, take classes in PDEs, numerical linear algebra, computer science, differential geometry, and optimization, and you can easily make a tremendous impact in all of the fields you named.