My answer to a question on Quora: How many years could it take me to study and understand all the mathematics fields that exist so far?
If you mean understanding at the level of ability to do research work in every field of mathematics, then, I am afraid, there is no hope to achieve this goal. Mathematics expands, and the cutting edge of mathematical research moves further and further away from any fixed reference point, say, undergraduate mathematics. From the point of view of an aspiring PhD student, mathematics looks like New York in the Capek Brothers’ book A Long Cat Tale:
And New York – well, houses there are so tall that they can’t even finish building them. Before the bricklayers and tilers climb up them on their ladders, it is noon, so they eat their lunches and start climbing down again to be in their beds by bedtime. And so it goes on day after day.
It was written in the first half of the 20th century, but Joseph and Karel Capek understood thing or two about futurology (although the term “futurology”, most likely, did not exist in their time: they were the people who coined the word “robot”. We live in the world where, in almost every field of human endeavor, no-one can understand everything. The human civilization that we transform and build is immensely complex, and mathematics is perhaps its most complex part.
[For this post, I cannibalized some bits of my paper Mathematics for makers and mathematics for users; it discusses some relevant themes.]