My parents introduced me to the wonders of the public library at a young age. It held an immense number of books on every subject you could imagine.
But wait: with so many books on so many subjects by so many authors, how can you possibly find what you want or even know what’s available? The answer is an index. In those days, libraries typically had three different indexes found in the card catalog (hundreds of drawers full of index cards). These cards allowed you to find books (a) by author, (b) by title, or (c) by subject. Beyond that, the books were shelved according to their subjects, using either the Dewey decimal system or the Library of Congress system. If you were familiar with these systems, you could easily find what you were looking for: a particular book that had been mentioned in the newspaper, books written by your favorite author, or books on a particular subject you were researching for school. Nowadays, of course, the indexes are computerized, allowing you to find books more flexibly and easily than we ever imagined in the days of the card catalog.
Could you have a library without an index? Yes, but it would be much less useful. It would be harder to find what you want, and every search would take significantly longer. How to best catalog information so it’s easily findable is so important that an entire branch of academia, library science, is dedicated to it.