The course departs from lecture-only formats. Common practices include:

A distinctive MIT feature is the use of LaTeX for final projects. Students write a short paper (3–5 pages) proving a non-trivial theorem of their choice, from Cantor’s diagonal argument to the infinitude of primes in arithmetic progressions (special case).

The final major unit tackles the natural numbers. Induction is a proof technique for infinite sequences of statements. 18.090 deconstructs the induction machine:

Students practice "strong induction" (where you assume P(1) through P(k) to prove P(k+1)) and explore its connection to recursion. Classic problems include: proving the sum of the first n integers is n(n+1)/2, proving the Fundamental Theorem of Arithmetic, and analyzing the Tower of Hanoi.

This course is the bridge from computational calculus to rigorous proof-based mathematics. It covers logic, sets, functions, proof techniques (induction, contradiction), and basic number theory/analysis.

The primary goal is not to memorize facts, but to master the methodology of mathematics. By the end of the course, you should be able to:

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:

This ritual is terrifying but transformative. It destroys the illusion that mathematics is about getting the right answer. It reveals that mathematics is about justification.

MIT’s course 18.090, Introduction to Mathematical Reasoning, serves as a foundational bridge between computational calculus and abstract, proof-based mathematics. This paper explores the course’s objectives, typical syllabus, pedagogical methods, and its role in preparing undergraduates for higher-level courses in analysis, algebra, and topology. Special emphasis is placed on how the course demystifies mathematical logic, set theory, and proof techniques, thereby transforming students from passive formula-users into active mathematical thinkers.