Mathematics - curse or blessing?!

With the begining of this semester I also started my bachelor in the topic of mathematics.
By studying  a lot of very dry definitions of algebraic sets and topological features of closed set the question emerges:

Do we need all of this as theoretical physicists?!

I.e. does "nature" really care about e.g. closed or open sets?
This was and is an absolute non-trival question. Most physicists would argue that these mathematical concepts only emerge from our logical understanding as a human. But on the other side: Humanity has build up logical concepts from watching at nature itself. Consider for example the integer numbers \mathbb{N}. These arise naturally from counting stuff like apples or so.
But the question is then, what is couting? It is the ability to see similarities between two things and to say, that these two or more objects are equal. This then give us naturally the concept of \mathbb{N} in order to specialize how many "equal" things there are.

This was just ment to give a little example of how mathematical concepts arise from nature and this is also the reason why I think that mathematics is much more a blessing as a cure for any theoretical physicists. It gives the ability to "see" structures, similarities and relations between objects and theories, that somehow might look completly different.
Due to this face I would encourage everyone, who would love to work in theoretical physics, to put his effort a little bit more towards mathematics. Of course one should not lose his mind in the big sea of defintions, lemma and propositions but being a little bit more rigorious about math will benifit a lot. At least that is was I think and I hope to get from my studies in these fields.

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