In this course, words have extremely precise meanings. You cannot prove a function is "continuous" if you cannot write down the exact epsilon-delta definition.
MIT’s 18.090 Introduction to Mathematical Reasoning is more than a prerequisite — it is a cognitive rite of passage. By systematically teaching the grammar of mathematical arguments, the course empowers students to engage with advanced mathematics not as a collection of procedures, but as a living discipline of discovery and justification. For any undergraduate considering a major in mathematics, physics, computer science, or engineering, 18.090 provides the logical compass needed to navigate rigorous theoretical work.
As one MIT course evaluation comment read: “Before 18.090, I could solve for x. After 18.090, I could prove why x must exist.”
Not everyone at MIT takes 18.090. Some arrive with AP credit in BC Calculus and a strong background in math competitions (IMO, USAMO). For those students, 18.090 might be redundant. However, for the following archetypes, 18.090 is non-negotiable: 18.090 introduction to mathematical reasoning mit
The "A in Calculus, F in Proofs" Student: You can compute derivatives in your sleep, but when asked, "Prove that if n is odd, then n² is odd," you freeze. Take 18.090.
The Physics Refugee: Physics uses math as a tool. You are comfortable with hand-waving and infinitesimals. Mathematics demands absolute precision. 18.090 will rewire your brain.
The Future Computer Scientist: Algorithms, complexity theory (P vs. NP), and program correctness all rely on induction and logic. 18.090 is a secret weapon for technical interviews at quant funds or FAANG. In this course, words have extremely precise meanings
The Math Minor: You don't need to become a pure mathematician, but you want to understand math from the inside. This is the most efficient way to gain that intuition.
While the exact syllabus evolves, a representative semester includes:
| Week | Topic | |------|-------| | 1 | Logical connectives, truth tables, tautologies | | 2 | Quantifiers, negations, converse/inverse | | 3 | Proof techniques: direct, contrapositive, contradiction | | 4 | Mathematical induction (ordinary and strong) | | 5 | Sets: union, intersection, power sets, Cartesian products | | 6 | Functions: injective, surjective, bijective, inverses | | 7 | Relations: equivalence relations, partitions | | 8 | Midterm review & exam | | 9 | Number theory: divisibility, primes, GCD, Euclidean algorithm | | 10 | Modular arithmetic and proofs | | 11 | Real numbers: least upper bound property, sequences | | 12 | Countability: finite, countably infinite, uncountable sets | | 13 | Introduction to combinatorial proofs | | 14 | Final review and project presentations | "How to Read and Do Proofs" by Daniel Solow (6th Edition)
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.
MIT is famous for intensity, but 18.090 is often described as "difficult but fair."
One student quipped: "In 18.01, I could check my answer by plugging it back in. In 18.090, I have to check my soul for logical consistency."