I just came back from attending the 1052nd AMS (sectional) meeting at Penn State, last weekend, and realized that the Kingdom of Mathematics is dead. Instead we have a disjoint union of narrow specialties, and people who know everything about nothing, and nothing about anything (except their very narrow acre). Not only do they know nothing besides their narrow expertise, they don't care!–Doron Zeilberger, Oct. 28, 2009
Here's a hot take: all the ideas we describe in math are relatively easy to grasp, but mathematicians make things complicated because of our poor short term memory. To give an example, let's take Calculus— the ideas behind derivatives/integration are really simple (limit of sums, limit of slopes) but we make them difficult by introducing levels of rigor and notation that leave a first year Calculus student behind in the dust.
This applies to higher level concepts as well: I believe that groups, fields, metric spaces, topological spaces, etc are fundamentally simple, but they suddenly aren't because of the level of rigor needed to formally approach these topics.
Rigor arises as a process to justify logical reason, but if our long term memory capacity was higher and we remembered much more of concepts learnt earlier, then there would be no need to obfuscate math with weird symbols and definitions. We add the rigor to remind ourselves that our reasoning related to past subjects is correct, which leads to rigor being required to have further areas of study utilize the current topic.
Of course, without rigor there would be no certainty, which is pretty much the point of math itself. So I guess we wouldn't be studying math, it would just be another branch of science, or maybe even just pseudoscience. On the flip side, solving this issue could possibly lead to the Pre-20th century level of unification in mathematics, one that hasn't been seen since David Hilbert.
Ever since the 20th century, mathematics has become so vast and fractured that specialization is the only way to survive in the field. No one can claim to truly understand more than a couple of hyper-specialized fields anymore due to the immense time and effort it takes to read enough papers to become knowledgable in a field. To quote Doron Zeilberger once again, we no longer have "mathematicians", but instead we have "topological algebraic Lie theorists, algebraic analytic number theorists, pseudo-spectral graph theorists etc".
This is, of course, not the fault of our mathematicians, but a natural consequence of the direction the field has been going, (and I argue) as well as our natural short term memory capacity. In an ideal world where our brains were constructed differently, we would have mathematicians consulting and working with a plethora of other mathematicians of topics at the highest level, research and developments realidly accessible to the layman that puts in 2 weeks as opposed to 20 years to understand the work, and a rapid flow of major breakthroughs (after all, great proofs like the proof of Fermat's Last Theorem utilized the hyper-specialized work of many many fields).
But of course, this is just wishful thinking and entertaining a fantasy "what-if" situation. Or maybe it's just BS, and if we truly could remember more, mathematics would still end up the way it is to preserve the aspect of exclusivisity and prestige. However, it's something interesting to think about: if we were given a choice between psuedoscience and progress as opposed to rigor and fragmentation, which would mathematicians pick?