Analogies are the unseen threads connecting disparate ideas — the bridge between the known and the unknown. Critical thinking supplies the rigor; creative analogy supplies the leap. Together they drive discovery across statistics, science, and technology.
The AI age is defined by the value of questions over answers, adaptability over stability, and connections over isolated knowledge.
This opening chapter introduces the book's twin engines — critical thinking and creative analogy — and the similarity principle that unites imitation, analogy, and innovation. It ranges across engineering, science, mathematics, AI, design, and philosophy, showing how analogies simplify complexity and spark breakthroughs. These simulators let you run its central ideas: move through the imitation→creativity→illogicality continuum, watch a geometric proof assemble itself, estimate π by throwing darts, gauge a hidden population by mark–recapture, let a colony of ants find the shortest path, map one domain onto another, and feel decision paralysis in Fredkin's Paradox.
Imitation, analogy, and innovation are all degrees of similarity. Extreme likeness is imitation; partial, principled likeness is analogy and creativity; too little likeness and an idea tips into "illogical" nonsense. Slide from high similarity toward pure novelty and watch which mode dominates (Figure 1.1).
The more equally attractive two options appear, the harder it is to choose between them — even though, paradoxically, the choice then matters less. Minsky noted that rational agents waste disproportionate time on trivial decisions. Tune the gap between two options and watch deliberation explode as they converge.
The model. Two options differ in value by Δ. Deliberation time grows as the difference shrinks — model it as T = c/(|Δ| + ε) — while the regret of choosing wrong is only |Δ|. As Δ→0 the agonizing time diverges but the stakes vanish. The resolution (Klein, 2001): treat deliberation time itself as a cost, and once its marginal cost exceeds the possible regret, just pick — for near-ties, flip a coin.
Mathematical induction, seen as a picture: each odd number is an L-shaped layer (a "gnomon") that wraps around the previous square to make the next one. The inductive step is literally visible — add the (n+1)-th layer of 2n+1 cells to an n×n square and you get an (n+1)×(n+1) square (Figure 1.3).
Since the area of a circle ties to π, random sampling can approximate it: scatter points in a square with an inscribed circle, and the fraction landing inside, times 4, estimates π. The estimate homes in as ~1/√N — analogy connecting integration, area, and constants like π.
How many fish are in a pond you can't drain? Mark M of them, let them mix, then recapture a sample of n and count how many m carry a mark. Since the marked fraction in the sample should match the marked fraction overall, N ≈ M·n/m — the Lincoln–Petersen estimator.
No ant knows the map. Each wanders, and on finding food lays down pheromone; shorter routes get reinforced faster, so the colony converges on a good path — then re-routes when the world changes. Click the grid to drop or clear a wall and watch the trail adapt (Figure 1.4).
The model. Ants leave the nest (green) and head for food (amber), choosing each step by pheromone strength and a pull toward the goal. On reaching food they deposit pheromone back along their path — more for shorter paths. Pheromone evaporates over time, so stale routes fade and the colony keeps only what works. This decentralized "stigmergy" solves shortest-path and routing problems without any central planner.
Chang, M. (2025). Critical Thinking and Creative Analogies in Statistics, Science, and Technology: Essential Skills for the AI Era. Chapman and Hall/CRC — Chapter 1.
Holyoak, K.J. & Thagard, P. (1995), Mental Leaps: Analogy in Creative Thought. · Minsky, M. (1986), The Society of Mind (Fredkin's Paradox). · Klein, G. (2001), on decision cost. · Lincoln, F.C. (1930) & Petersen, C.G.J. — mark–recapture. · Dorigo, M. (1992), Ant Colony Optimization. · Metropolis & Ulam (1949), Monte Carlo.