The esoteric fundamentals of mathematics originate in set theory and at its most fundamental level – symbolic logic. Theoretically, one could use symbolic logic to express all formulas and functions in mathematics. And much of what can be encoded into symbolic logic can be translated into mathematical notation as well.
Mathematics is a tool through which complex logical relationships can be encoded. It is not the only tool. Computer programming languages, for example, also use syntax that originates in logic but builds in a different direction.
A primary requirement for a field to move from a “soft science” to a “hard science” is the ability to develop a rich symbolic notation that can be used to encode, store and modify the information releveant to the field of study. As subjects such as psychology and biology move from the “soft science” category to the “hard science” category, they change from a reliance on jargon to an adaptation of symbolic notation from other disciplines (e.g., statistics, computer algorithms).
Ultimately, a discipline can mature only when it develops its own symbolism. The suitability of mathematics or computer science can only go so far, and a hodge podge of semantic tools will constrain advancement of a discipline.