Developed by Peirce, James, and Dewey, pragmatism holds that the meaning of a concept is constituted by its practical effects. Truth is not a static correspondence to reality but an evolving, functional property of beliefs that guide effective action. James extended the theory: a belief is true if it works satisfactorily in the context of our experiences.

Gödel's two incompleteness theorems (1931) shattered Hilbert's program of finding a complete and consistent set of axioms for all of mathematics. The first theorem: in any consistent formal system F capable of expressing arithmetic, there exist statements that can neither be proved nor disproved within F. The second theorem: such a system cannot prove its own consistency. The proofs use self-referential Gödel numbering to encode "This statement is unprovable."

Bourdieu distinguished social capital from economic and cultural capital: it consists of actual or potential resources linked to durable networks of mutual acquaintance and recognition. Putnam later broadened the concept into "bridging" (connecting across groups) and "bonding" (reinforcing within groups) social capital. Strong social capital correlates with better health, economic outcomes, and civic participation.

Developed through contributions from Heisenberg, Schrödinger, Bohr, Dirac, and Born, quantum mechanics describes the behavior of matter and energy at atomic and subatomic scales. The Schrödinger equation governs the time evolution of quantum states. The measurement problem — how superpositions "collapse" to definite values upon observation — remains philosophically unresolved. Interpretations include Copenhagen, Many-Worlds, and Pilot Wave.

Category theory abstracts mathematical structure to its bare essence: objects and the structure-preserving maps (morphisms) between them. Functors map between categories, natural transformations map between functors. Eilenberg and Mac Lane developed it to formalize relationships between algebraic topology constructions. It has since become the lingua franca of mathematics, and its influence extends to theoretical computer science (type theory) and physics (TQFT).

Rational choice theory assumes that individuals make decisions by comparing costs and benefits to maximize utility. Von Neumann and Morgenstern formalized this into expected utility theory. Arrow's impossibility theorem and behavioral economics (Kahneman, Tversky) have both challenged its descriptive accuracy, but it remains the dominant normative framework in economics, political science, and sociology.

Popper's criterion of falsifiability distinguishes science from non-science (pseudoscience, metaphysics). A theory is scientific only if it makes predictions that could in principle be shown false by experiment. Science proceeds through conjectures and refutations: we never verify theories, only fail to falsify them. This resolves the problem of induction — we don't need induction; we need only severe tests. Critics (Kuhn, Lakatos) noted that scientists rarely immediately abandon theories under anomalies.

Kahneman and Tversky's prospect theory (1979) showed that people's choices systematically violate expected utility theory. Key features: (1) outcomes are evaluated relative to a reference point, not in absolute terms; (2) the value function is concave for gains and convex for losses, with losses weighted roughly twice as heavily as gains; (3) probability weighting overweights small probabilities and underweights large ones. This earned Kahneman the 2002 Nobel Prize in Economics.

The four laws of thermodynamics: the zeroth establishes thermal equilibrium; the first, conservation of energy; the second, that entropy in a closed system never decreases; the fourth (Nernst), that absolute zero is unattainable. The second law gives time a direction — the increase of entropy. Boltzmann's statistical mechanics grounded thermodynamics in the behavior of atoms, connecting macroscopic observables to microscopic probabilities.