First and foremost, a strong background in Mathematics pertinent to his or her field of study, and more importantly, a thorough understanding of the concepts related to the fields of science of interest to the individual.

Why specifically mathematics? And which Mathematics branches one should focus on?

Mathematics provides the essential quantitative tools needed for any Quantitative science, either to express physical laws or properties of the system under study.

I believe, a person involved in science should have essentially a strong background in single and multi-variable calculus and Linear and Abstract algebra. Modern day Quantitative sciences usually involve a lot of data analysis and interpretation, to understand the statistical data analysis tools and apply them one need the expertise in above mentioned subjects.

other that above mentioned branches, other branches of mathematics too need to be focused on depending on the specific subject of study, for example, any one studying Crystallography need a decent background in Group theory and Concept of Vector spaces, and people working on Quantum mechanics need a strong background in the theory of Hilbert spaces and Operator theory.

Development of writing and speaking skills.

A scientist has to know the rules of the logic.

phantasy

@ Alexander Krebs

Would you agree to go to a physician and instead of consulting you, he would give you a diagnosis according to a phantasy?

Thank you all!