If the first term taught me to compute, the second taught me to prove. It is a difference I did not know existed until I lived it.
Mathematical Thinking
Propositional logic, quantifiers, proof strategies, induction, equivalence relations, injective and bijective functions, integers, Euclid’s algorithm, congruences, Fermat’s little theorem and even an introduction to cryptography. The stated goal of the course is “to begin the path of formal thinking”, and that is exactly what it is: learning to write an argument that does not depend on the reader believing you.
My favourite assignment was proving by induction an identity I already “knew” was true. Knowing something is true and being able to prove it are different things, and the second is what counts in an exam, in a paper and in a code review.
Linear Algebra I
Gauss-Jordan, subspaces, linear independence, rank and nullity, linear transformations, determinants, and the heart of the course: eigenvalues and eigenvectors, diagonalization, with Markov chains as the application. Then Gram-Schmidt, orthogonal projections and least squares.
That last topic was a revelation. Linear regression, which in Economics appears as a technique, in Linear Algebra is a projection: you look for the point of the subspace spanned by your variables that lies closest to your data. Once something can be explained with geometry it stops being magic.
Calculus II
The Riemann integral, the fundamental theorem, integration techniques, improper integrals, sequences and series, convergence tests, power series. The series part is what later holds probability together: without knowing when an infinite sum converges you cannot define an expectation.
Economics II
Consumer and producer theory: utility, indifference curves, the budget constraint, demand, technology, short- and long-run costs, economies of scale, perfect competition and monopoly. Here the first term’s calculus became a tool: maximizing utility subject to a budget is constrained optimization.
Ideas and Institutions II and Problems of Contemporary Civilization I
Modernity: Descartes, Spinoza, Hobbes, Locke, Hume, Rousseau, Kant, Hegel. The thread is the social contract and the question of where legitimate authority comes from, “the passage from a world with God to a world without God”, as the founding text of General Studies puts it. Hobbes struck me as the most honest and Kant as the hardest.
In parallel, PCC I opened three fronts: education, civilization and culture, the individual and society. It is a course of problems, not answers, with a fixed structure per unit: statement, areas of conflict, interpretations, perspectives.
What I take with me
That proving is a skill, not a talent, and that it trains. And that Hobbes and least squares have something in common: both are attempts to impose order on something that, left alone, has none.