If you want, I can:
This course serves as the bridge between computational calculus (like 18.01/18.02) and abstract mathematics (like 18.100 Real Analysis or 18.701 Algebra). It is designed to teach students how to write rigorous proofs and think abstractly.
When starting out, try to separate your "scratch work" from your "proof."
While MIT often cycles through different variations of this course (sometimes combined with Discrete Math), the best resource on MIT OCW is:
Proof techniques
Sets, functions, and relations
Number theory basics
Combinatorics & counting
Elementary structures and examples
Proof-writing practice
Problem: Prove that if $n$ is an integer and $n^2$ is even, then $n$ is even.
Hint: Try a proof by contradiction.
18.090: Introduction to Mathematical Reasoning at MIT is a foundational bridging course designed to transition students from computational "plug-and-chug" math to the rigorous, proof-oriented thinking required for upper-level mathematics. Course Overview 18.090 introduction to mathematical reasoning mit
The primary goal of 18.090 is to teach you how to understand and construct formal mathematical arguments. While many introductory calculus or linear algebra courses focus on solving for a numerical value, this class shifts the focus to why a statement is true and how to prove it definitively. Key Content & Curriculum
The course covers a mix of foundational logic and specific mathematical structures to give you a "test flight" in various areas of pure math:
Foundational Logic: Methods of proof (direct, contradiction, induction), quantifiers ( ), and infinite sets.
Algebraic Concepts: Exploration of permutations, fields, and vector spaces.
Analysis: Introduction to sequences of real numbers, which serves as a gateway to 18.100 (Real Analysis). Who Should Take It?
Proof Novices: It is particularly suitable for students who want more experience with proofs before tackling "heavyweight" subjects like 18.100 (Real Analysis), 18.701 (Algebra I), or 18.901 (Introduction to Topology). If you want, I can:
Non-Math Majors: Students in related fields with significant mathematical content (like Course 6/Computer Science) often find it a helpful intermediate step.
Future Pure Math Majors: If you are planning on the "Pure Option" for Course 18, this is a frequently recommended starting point to build the necessary "mathematical maturity". The Student Experience
Taking a class at the MIT Department of Mathematics means facing a significant jump in difficulty from high school. Students often report:
Conceptual Shift: Unlike 18.01 or 18.02, where you might learn an algorithm and repeat it, 18.090 requires reading additional sources and collaborating with peers on complex problem sets (Psets).
Humbling Rigor: It is common for students used to straight-As to find their first Psets or exams significantly more challenging than expected.
Collaboration is Key: Very few students work on these problems individually; most utilize TAs, professors, and peer study groups to navigate the material. Final Verdict This course serves as the bridge between computational
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.
Are you planning to take this as a prerequisite for a specific advanced course, or as an elective to strengthen your general reasoning skills? Course 18: Mathematics Fall 2025 (Archive)