An advanced abstract algebra course that requires prior proof experience. 18.901 (Introduction to Topology):
Before 18.090, students harbor several dangerous intuitions. The course is designed to systematically demolish them.
While the syllabus evolves slightly depending on the instructor (notable past instructors include Dr. Paul Bamberg and Prof. Haynes Miller), the core of 18.090 revolves around four fundamental pillars. Let’s explore each in detail.
Very few students work on these problems individually; most utilize TAs, professors, and peer study groups to navigate the material. Final Verdict
The honest answer: You will feel lost. You will erase entire proofs. You will question if you belong in a math major.
Learning to distinguish between "inclusive or" (standard in math) and "exclusive or" (common in everyday English). Academic Role Within the MIT Mathematics Department
18.090 (Introduction to Mathematical Reasoning) is a foundational undergraduate course that teaches students how to think, write, and argue like mathematicians. Unlike computational or technique-focused classes, its core goal is to develop the habits and language required for rigorous mathematical thought: precise definitions, clear logical structure, correct proof techniques, and effective mathematical communication. Mastery of these skills is essential for success in higher-level mathematics, theoretical computer science, and any discipline that demands formal reasoning.
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For many students entering the hallowed halls of the Massachusetts Institute of Technology, there is a silent, often terrifying, academic barrier. It is not calculus—most MIT freshmen have already mastered differentiation and integration in high school. It is not linear algebra or differential equations. The true hurdle is .
If you feel confident in your computational skills but "shaky" when asked to write a proof from scratch, 18.090 is an excellent investment. It provides a safer environment to fail and learn the "language of math" before the pace and abstraction accelerate in the 18.10x or 18.70x sequences.
Students often ask: "Will I ever prove that the square root of 2 is irrational again in real life?" Probably not. But here is what you will use:
