Srinivasa Ramanujan. 1887 – 1920
Absolutely! There are a variety of excellent textbooks in mathematics up to the graduate level that will enable a highly intelligent person to master mathematics to that level. Beyond that, you will need to delve into research journals to find the latest developments in particular branches of mathematics.
A remarkable example of a person who learned mathematics at an advanced level from textbooks is captured in the story of Srinivasa Ramanujan, a relatively unschooled person of high intelligence from southern India who, a century ago, with a modicum of formal instruction, rose to the top of the mathematical world. Ramanujan’s deep intuitive understanding of numbers and their representation as infinite series spawned identities that were previously unknown to the world of mathematics. One of the remarkable infinite series he discovered for the reciprocal of pi is:
Especially interesting is that this infinite series converges very quickly, with the first term yielding the correct value of pi to 6 decimal digits. In 1991, American biographer Robert Kanigel celebrated the genius of Ramanujan in his award-winning book, The Man Who Knew Infinity: A Life of the Genius Ramanujan (made into a movie in 2015.) It is particularly noteworthy that Ramanujan learned most of his mathematics from the few textbooks that he was able to garner as a poor clerk in Madras.
Today, universities provide their students with access to virtually all the important research journals, so if you enrol in a university course, you can usually gain free access to these journals. Alternatively, you can pay a fee to read any particular journal article.
However, if you wish to earn a Ph.D. in mathematics, then your best path is to enrol at a university and work under the guidance of a mentor who can familiarize you with the problems that are of current interest.
My personal belief is that attending lectures and making notes are an inefficient way to learn mathematics. In my opinion it’s far more efficient to learn mathematical concepts from a textbook and complete a lot of exercises to consolidate the concepts. Then summarize the ideas in some brief notes for future reference. After that, extend what you have learned by reflecting on the implications of various theorems and conjecturing other relationships. Then attempt to prove your conjectures. If you wish to reach the upper echelons of mathematical mastery, you have to migrate from textbooks to research journals. By following this process, most people of high intelligence should be able to learn mathematics at an advanced level.
