“I consider that a man’s brain originally is like a little empty attic, and you have to stock it with such furniture as you choose… It is a mistake to think that that little room has elastic walls and can distend to any extent. Depend upon it there comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.”
Sherlock Holmes famously told Watson that he didn’t care whether the Earth revolved around the Sun because astronomical knowledge served no purpose in detective work. Watson was scandalized, but Holmes understood something profound about intellectual efficiency: strategic ignorance has real value.
Mathematical research requires the same kind of strategic incompleteness. The most productive researchers aren’t those who master every detail of every tool they use, they’re those who develop excellent judgment about what to learn deeply, what to understand superficially, and what to ignore entirely.
Consider how Holmes approaches a case. He observes everything, but he doesn’t investigate everything equally. He knows which details matter for solving the crime and which are atmospheric noise. When he needs specialized knowledge—, about poisons, or handwriting, or tobacco ash, he knows where to find it without carrying encyclopedic knowledge about every possible topic.
Mathematical research works the same way. You need to distinguish between the “skeleton” and the “flesh” of mathematical knowledge. The skeleton is the essential logical structure, the minimal framework that makes everything work. The flesh is the detailed proofs, computational techniques, historical context, and elaborations that make the skeleton meaningful.
Holmes would have made an excellent mathematician. Here’s how to think like him:
Cultivate useful ignorance: Not all mathematical knowledge is equally valuable for your research. Holmes deliberately refused to learn facts that wouldn’t help him solve crimes. Similarly, you can often use powerful theorems without understanding their proofs, apply sophisticated techniques without mastering their derivations, and work with concepts without knowing their complete historical development.
Know where to find what you need: Holmes maintains a network of specialists, Wiggins and his Baker Street Irregulars, various experts in obscure fields. Build your own network of mathematical resources: which books explain technique X clearly, which papers bridge fields Y and Z, which colleagues understand topic W deeply enough to answer questions.
Distinguish observation from deduction: Holmes observes everything but deduces selectively. In mathematics, absorb broadly but focus your deep learning strategically. Read widely across fields to spot patterns and connections, but choose carefully where to invest the time needed for complete understanding.
Work with partial information: Holmes often begins investigating with incomplete data, developing theories that he refines as more evidence appears. Mathematical research requires the same comfort with uncertainty. You’ll often need to use tools you don’t fully understand, follow hunches based on incomplete analogies, and work with frameworks you haven’t mastered.
Recognize pattern across contexts: Holmes can spot when seemingly different crimes follow similar patterns. Develop the ability to recognize when mathematical structures appear in different guises across fields, even when the surface presentations are completely different.
The students who struggle most with research are often those who insist on complete understanding before proceeding, the mathematical equivalent of Watson’s horror at Holmes’s strategic ignorance. They want to master every lemma before using a theorem, understand every proof before applying a technique.
But mathematical research happens at the frontier of understanding. Complete knowledge is a luxury we rarely have. Instead, we work with provisional understanding, partial frameworks, and strategic gaps in our knowledge. The key is maintaining clear awareness of what you do and don’t understand, so you know when to dig deeper and when surface knowledge suffices.
Holmes succeeded because he understood that detective work isn’t about knowing everything, it’s about knowing the right things and knowing where to find everything else when you need it. Mathematical research follows the same principle. The goal isn’t omniscience; it’s strategic competence combined with excellent judgment about when to learn more.
This approach isn’t just about efficiency—it’s about creativity. Mathematical breakthroughs often come from seeing connections across fields you understand partially rather than understanding any single field completely. Like Holmes solving crimes by connecting seemingly unrelated observations, mathematical insight emerges from recognizing patterns across domains of incomplete knowledge.
But developing this comfort with uncertainty requires psychological adjustment that many students find difficult. That’s what we’ll explore in our final post.