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Learning how to read, write, and critique mathematical statements.
Unlike calculus recitations where a TA works through problems, 18.090 recitations are often student-driven . A student is called to the blackboard to present their proof. The TA and peers then act as hostile (but constructive) reviewers. They will ask:
The hardest part of 18.090 to replicate is the blackboard defense. Find a study partner. You write a proof. They try to break it. Do not accept your own proof until your partner has failed to find a loophole.
That bridge is officially called .
The course title is deliberate. is broader than proof. In research mathematics, you spend 90% of your time reasoning—exploring examples, finding counterexamples, guessing a pattern—and only 10% writing the final polished proof. 18.090 introduction to mathematical reasoning mit
Learning to write proofs is a skill that takes practice.
It can be taken early in an undergraduate career. Crucially, it lists 18.02 (Multivariable Calculus) as a corequisite rather than a prerequisite, meaning students can enroll in it concurrently with their foundational calculus sequence.
It is ideal for math majors, minors, or students in related fields (like computer science or physics) who want a rigorous introduction to abstract mathematical reasoning. How to Prepare and Succeed
| Misconception | Reality (Taught in 18.090) | | :--- | :--- | | "A proof is just a sequence of equations." | A proof is a narrative. It requires words like "therefore," "assume," "note that," and "suppose." | | "One example proves a universal statement." | No. One example disproves a universal statement. To prove it, you need a general argument. | | "If you can't find a counterexample, the statement is true." | Absence of evidence is not evidence of absence. You must prove impossibility. | | "Proof by contradiction is the most powerful method." | Often, it's a crutch that obscures a constructive direct proof. Use it sparingly. | Learning how to read, write, and critique mathematical
: Concepts taught in this course, such as logic, induction, and graph theory basics, directly apply to algorithm design and theoretical computer science.
Other texts occasionally referenced include:
For official materials, you can check the MIT Mathematics Department or browse related lecture notes on MIT OpenCourseWare . 18.0x - MIT Mathematics
covering basic logic or induction to test your current level? 18.0x - MIT Mathematics The TA and peers then act as hostile
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In calculus, if you spent 30 minutes on a problem, you were doing it wrong. In pure math, spending three days on a single proof is completely normal. Give your brain time to simmer on difficult concepts. Be Specific with Quantifiers: "For every there is a " is completely different from "There is a ." Treat your logical symbols with absolute precision.
Proving base cases and inductive steps to show a property holds for all infinite elements of a set (e.g., all natural numbers). 3. Set Theory and Relations
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