It kinds of depends on your personality type, too. Although I'd argue that mathematics is ideally suited to both ends of the spectrum. Namely, whether you enjoy exploring all details of a very specific niche and like specific detail, or if you prefer coming to understand the very general pictures.
I think I'm at the extreme end of the latter personality type; once I understand the general concepts behind something I will probably become bored of the specifics. I can't stand noodling around with details, to the extent that it can sometimes even makes me physically uncomfortable. And I seem inherently compelled to see things in terms of general structures; this is why I'm engaged with science (the big pictures which explain reality, not little details like the moons of our solar system or specific chemical structures or things), that's why I'm studying mathematics (which is essentially a way of codifying any aspect of physical reality and is really the most general activity possible), and that's why my favourite activity is philosophy.
I imagine you are this kind of person too, but I would guess that you have so far only been exposed to one kind of mathematics, namely the specific kind, for applications (are you well versed in calculus?), which is also the kind of thing you basically do all of the way through school. Now, knowing this mathematics can be pretty cool, in the sense that it vastly increases your competence at manipulating reality (via applied physics), so it's satisfying in the same sense that, say, having the skills to build a suspension bridge is satisfying. And personally I was definitely good at it, although like I said previously, only because I realised it's not intellectually different, and is actually just a tower of very simple elements. But I wouldn't say that hammers or welding are interesting in themselves.
But there is a second half to mathematics, which is that which concerns itself with generalisation, insight, and underlying, overarching structures, which you probably haven't encountered before, which I suppose is roughly that which is called 'pure'.
One example of something studied in pure is group theory. This is the mathematical encoding of symmetry. For instance, a beer mat has a symmetry called D8, which means you can rotate it by right angles, or flip it over. One of the aims of group theory is to examine the structure of these symmetries, and classify them according to a general taxonomy. Group theory can also be applied to physical reality. For instance, Noether's theorem is an extremely powerful and general theorem, which says that whenever a bunch of laws have a symmetry, there will be a corresponding conservation law. One symmetry of the laws of physics is time translation invariance. This turns out to be what's behind conservation of energy.
Another example of something studied in pure is cardinality. The amazing fact is that infinity (the 'number') comes in different flavours; there's actually a hierarchy of infinities. The first is that of the natural numbers, {1, 2, 3, ...}. This turns out to be exactly the same infinity as the number of fractions. However, it is not the same as the infinity of the continuum (the decimals); this infinity turns out to be strictly larger. You can make an even bigger infinity out of that. A natural question is whether there are any infinities inbetween the first and the second. Turns out that this is impossible to prove in standard mathematics, and so must either be assumed to be true or false (which generates two separate branches of mathematical truths). This in turn is directly related to Godel's incompleteness theorem, which states that there exist propositions in any reasonably powerful formal system which cannot be proven by that formal system, unless the formal system is selfcontradictory, in which case it is useless.
Does this stuff sound rather more interesting to your sensibilities?


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