How is spending time studying pure sciences such as mathematics and physics considered unbeneficial or a waste of time, even though they contribute to development, sharpen the mind, and foster thinking?

Imam Ibn Rajab's words pertain to beneficial knowledge, which is what Muslims need and what does not harm their religion, such as the science of arithmetic to the extent that it benefits in matters of inheritance (fara'id) and bequests (wasaya). Anything beyond that is unnecessary.

Accordingly, modern mathematics and its advancements were not present in Ibn Rajab's era. As for now, there is no harm in studying sciences that have worldly benefits as a communal obligation (fard kifaya), such as medicine and arithmetic, for they are essential for the proper conduct of worldly affairs.

Whoever learns something from these permissible sciences can purify their intention therein by seeking good for themselves and for people, or by fulfilling the needs of the needy, or by assisting the weak, or by enriching their nation, thereby earning reward, by Allah's permission.

Nevertheless, one should not overstate the importance of a university specialization, for the advancement of nations is not tied to the specialization of a single individual but to a combination of intertwined factors. Therefore, one should consider what benefit they can provide, what suits their abilities, and what benefits them in their religion.