For centuries, mathematics seemed to exist in service to the natural

It is always easier to summarize the argument in one line and then approach the question.

Here is what the argument is trying to say -

Primary object of mathematics = abstract creations --> Mathematics is not a natural science.

Immediately, we can spot that there is a jump in the argument from the LHS to RHS. "abstract creations" are somehow linked to "natural sciences".
Assumptions must link the LHS of the argument(premise) to the RHS of the argument (conclusion).

Any answer option that does this must be the correct answer.

A - we are concerned with whether mathematics is one of the natural sciences or not. "Social sciences" are not relevant to the argument.

B - correct answer. abstract = not tangible/not perceptible to the senses. this option negates a potential flaw in the argument. If natural sciences can also be something abstract/intangible, the argument falls apart.

C - this has nothing to do with my conclusion - mathematics is not a natural science.

D - whether mathematics rises to same rank or not is irrelevant to the conclusion - mathematics is not a natural science. Since mathematics is a independent field, it can probably rise to the same level as other fields - does not mean that it is not a natural sciences field.

E - what times are we referring to? Not clear. Currently, the focus of mathematics is on abstract concepts -> hence not a natural science - this is the argument. Also, if there are breakthroughs in physics or not through mathematical concepts does not have any impact on the claim that the primary object of mathematics is abstract concepts.
This option also slightly weakens the argument by stating that mathematics is indeed used in natural sciences such as physics.

B is the correct answer

Conclusion: mathematics is not properly one of the natural sciences.
Premise: the primary object of mathematical study were the abstract creations of mathematics.....

Based on above info, it can be seen that, for anything to be one of the natural science, the 'anything' must deal with non - abstract work.

'Abstract' means non-tangible or something, that exists only in mind. It means, natural science deals only with tangible subjects.

Official Explanation

The jump from the third sentence to the fourth is interesting in this argument. The third states: math studies "numbers and shapes and functions and their more abstruse extensions" and that the implications for science aren't decisively important.

The fourth sentence states: math is not a natural science. To make this jump, we need some assumption that tells us what the criterion would be for discipline to be a "natural science," and why a field that studies "numbers and shapes and functions and their more abstruse extensions" would not qualify.

The whole issue of whether one discipline "serves" another is a big distraction to the main argument.

(B) is the credited answer. One can count physical things, but a pure number itself is not a sense object. There are physical objects in the shape of, say, a triangle, but the triangle itself is not a sense object. Certainly functions and "their more abstruse extensions" are not sense objects. If focus on tangible sense objects is the criterion that qualifies a discipline as a natural science, it's easy to see why mathematics does not meet this criterion. This completely explains the argument.

(BTW, the vast majority of mathematicians and scientists would wholeheartedly agree both with the conclusion as well as with the credited answer here.)

(A) & (D) & (E) all focus on the distractor issue. Whether or not math "serves" another discipline does not, in and of itself, speak to whether it qualifies as a natural science. In addition, (D) introduces the idea of "rank," implying that the natural sciences occupy a certain "rank" and that mathematics, in serving them, would be below this "rank"—none of this finds any support in the prompt.

(C) is anecdotal evidence: what is true for just one individual or for a handful of individuals does not reflect on the nature of the discipline of mathematics itself. (BTW, Einstein really did struggle with mathematics!)

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