In Pursuit of the Unknown

How can a few symbols on a page explain the world? In In Pursuit of the Unknown, mathematician Ian Stewart argues that equations are not abstract ornaments but compact machines of insight. They reveal relationships, predict outcomes, and connect hidden structures across science, technology, and daily life. To understand how mathematics built modern civilization, you need to see equations not as obstacles but as stories — each one turning uncertainty into understanding.

Equations as dual engines: identity and discovery

Stewart begins with the distinction between two kinds of equations. Some are pure identities — statements that two expressions always yield the same result (like Euclid’s presentation of the Pythagorean theorem). Others are discovery engines — rules linking quantities to uncover the unknown, as Newton’s law of gravitation does. Recognizing this difference changes how you read equations: one tells you what is true, the other tells you what to do.

When Robert Recorde invented the “=” sign in 1557, he made equality communicable. That symbol allowed science to express balance and symmetry in a portable shorthand. From Maxwell’s field equations predicting radio waves to Einstein’s energy-mass relation, each formula becomes a compressed poem of physical order.

The expanding map of mathematical ideas

Stewart traces how equations evolve into conceptual compasses. The Pythagorean theorem moves from measuring land into defining curvature and spacetime; logarithms turn laborious multiplication into simple addition and later describe sound perception and radioactive decay. Calculus makes change measurable; Maxwell’s laws unify electricity and magnetism; Riemannian geometry bends space to describe gravity. Each new formula extends the reach of human thought, turning observation into prediction.

Far from being dry abstractions, these equations changed how you navigate, communicate, heal, and understand time itself. They drive GPS satellites, power climate models, and explain chaos and risk. The book unfolds as a grand adventure—from triangles to quantum waves, from ordered formulas to the unpredictable beauty of chaos—showing how the language of mathematics continually deepens reality’s meaning.

Certainty, approximation, and the limits of modeling

Stewart’s broader claim is that mathematics succeeds because the universe itself is lawful enough to be captured symbolically yet complex enough to challenge those laws. Newton’s gravity collapses under Einstein’s curvature for extremes; the Navier–Stokes equations mirror oceans and arteries but remain unsolved in theory; quantum mechanics predicts outcomes exactly yet hides them behind probability. Equations are neither divine truths nor mere guesses — they are continually refined approximations to nature’s behavior.

In doing so, Stewart reminds you that mathematics is historical and human. From Napier’s logarithms easing navigators’ work to Shannon encoding information on Voyager’s journey to the stars, each advance answers a specific pain point — too slow calculation, too chaotic prediction, too noisy communication. The book’s connective tissue is utility born of beauty.

Why equations matter to you

Understanding equations changes how you see the modern world. You live inside Maxwell’s fields every time you use Wi‑Fi; you experience Einstein’s equations with each GPS correction; you depend on Navier–Stokes when weather forecasts appear; and you rely on Shannon’s bits when your phone transmits messages flawlessly through noise. Mathematical laws create invisible infrastructures around you.

Mathematics makes the invisible visible

Equations reveal hidden symmetries, patterns, and forecasts beneath everyday events. Once you read them fluently, you see continuity between the falling apple and the orbiting moon, between the violin’s wave and the JPEG on your screen.

Stewart’s theme is both practical and philosophical: mathematics gives you coherence in a complex world. It connects local truths to global patterns, shows where precision meets approximation, and demonstrates that the history of ideas is written not merely in words but in equations. Each chapter builds a new bridge between symbol and experience, proving that when you learn to read equations, you learn to read the universe itself.

The Pythagorean theorem, a² + b² = c², begins as a simple geometric identity. Yet its reach stretches from early Babylonian tablets to Einstein’s curved universe. Ian Stewart paints this equation as a cornerstone that bridges geometry, measurement, and cosmology. When you learn what the theorem truly says, you start to see how distance becomes an organizing idea for all of space.

Ancient origins and geometric insight

Babylonians recorded Pythagorean triples like 3–4–5 centuries before Pythagoras. Euclid’s proof formalized the relationship between areas of squares on triangle sides. That simplicity enabled practical triangulation—surveying fields, mapping cities, measuring Earth’s circumference (Eratosthenes). Descartes later converted geometry to algebraic coordinates, embedding Pythagoras into every distance formula.

Curvature and new geometries

Carl Friedrich Gauss noticed that Pythagoras holds perfectly only on flat surfaces. Deviations mark curvature; Riemann extended this insight to higher dimensions using metric tensors (ds² formulas). By redefining local distance rules, he made curvature a property you can compute without leaving the surface. Einstein later used Riemannian geometry to show gravity is curvature of spacetime itself—the warped continuation of the triangle’s logic into the cosmos.

Modern uses and intuitive payoff

From GPS positioning to aircraft navigation, your life depends on refinements of the Pythagorean distance law adjusted for relativity and curvature. Stewart’s historical path—from rope-measured fields to warped space—exemplifies how abstract geometry governs practical systems. (Note: It echoes Martin Gardner’s philosophy that simple mathematical games yield profound insight into structure.)

Curvature as the generalized Pythagoras

If space were flat, distance follows ordinary Pythagoras; if it isn’t, the theorem becomes local—showing how geometry encodes the universe’s shape.

Stewart uses the theorem to argue that mathematics grows by generalizing familiar statements. Once you allow triangles to live on spheres or distort space, you open the gateway to Einstein’s field equations and modern cosmology. The farmer’s rope and the physicist’s tensor measure the same concept—distance—but one seeds the other’s universe.

Calculus emerged from the need to measure continual change. Stewart shows how Galileo’s rolling balls led Newton and Leibniz to formalize differentiation and integration. Their insight: change itself can be quantified by comparing infinitesimal differences and considering limits. This single method became the foundation of almost every physical theory after 1700.

From motion to method

Newton treated quantities as fluids, describing their rates (“fluxions”) in time, while Leibniz developed the algebraic form dy/dx. Despite philosophical disputes (Bishop Berkeley mocked infinitesimals as “ghosts of departed quantities”), later mathematicians such as Bolzano and Weierstrass made the method rigorous using limits.

Why limits matter

The limit concept expresses what “approach” means precisely: no division by zero, just controlled convergence. That foundation ensures all calculus operations stand on logic rather than intuition. Stewart likens calculus to a reliable machine—one you can trust once you understand how limits resolve apparent paradoxes.

Applications and universality

Calculus builds differential equations: motion (Newton), heat (Fourier), fluids (Navier–Stokes), and quantum dynamics (Schrödinger). In everyday terms, it powers engineering design, economics, epidemiology, and AI optimization. Stewart’s narrative connects those diverse forms into one theme—the logic of infinitesimal change organizes your world, whether mapping a planet’s orbit or a cell’s growth.

Calculus is structured curiosity

The method doesn’t just compute—it asks how any quantity behaves when tweaked, enlarged, or shrunk. Mastery means learning to ask those questions with precision.

Stewart turns calculus into an allegory for science itself: start with observation, build symbolic tools, refine logic, and you achieve insight that reshapes your worldview. From Galileo’s slope to computer simulations of flight, the calculus of change never stops changing you.

Equations for waves unify physics, communication, and art. Stewart follows the thread from a vibrating violin string to the electromagnetic spectrum, to Fourier analysis and modern digital technology. The hidden connection: waves carry energy, and their mathematical analysis carries information.

Maxwell’s electromagnetic revolution

James Clerk Maxwell condensed electricity and magnetism into four equations based on Faraday’s experiments. The curl and divergence operators tie electric and magnetic fields together, implying self-propagating waves. Hertz later observed radio waves, confirming theory and giving rise to the era of communication—from Marconi’s signal to Wi‑Fi.

Fourier and the language of signals

Fourier’s insight—that any shape or signal can be represented as a sum of sinusoids—led to transforms used in acoustics, seismology, and digital imaging. Discrete cosine transforms compress photos; wavelets capture localized patterns in speech and medicine. Stewart uses this link to illustrate mathematics’ reach: the same principle that explains heat diffusion lets your camera store thousands of pictures efficiently.

Information and communication

Claude Shannon translated signal variability into measurable information. His entropy formula H = –Σ p log2 p parallels thermodynamic entropy and defines the limits of compression and transmission. Error‑correcting codes (Hamming, Reed–Solomon) protect against noise, keeping probes like Voyager connected across billions of miles.

Mathematics turns vibration into meaning

From violin strings to satellite data, the same wave equations enable sound, vision, and digital preservation.

Stewart’s synthesis shows that waves and information are two faces of one concept: structured oscillation. Studying them reveals how nature encodes messages—whether energy or bits—and how humans learned to read and replicate that code through mathematics.

The 20th century reshaped reality with relativity and quantum mechanics. Stewart explains how equations for spacetime and wave functions turned philosophical puzzles into precise frameworks that now govern technology and cosmology.

Einstein’s curved spacetime

Starting from Michelson–Morley’s ether failure, Einstein redefined motion and simultaneity. Special relativity set light’s speed as invariant; general relativity made gravity geometry itself. Schwarzschild’s black-hole solution and Hubble’s expanding universe extended those ideas to cosmic scale. Stewart traces ongoing puzzles—dark matter, dark energy, inflation—as reminders that even giant equations invite fresh interpretation.

Quantum mechanics and the measurement challenge

Planck’s quanta, Einstein’s photons, de Broglie’s waves, and Schrödinger’s equation produce a world ruled by probability. Superposition and measurement (“collapse”) divide physicists between Copenhagen pragmatists and many‑worlds realists. Decoherence explains why you never see cats both alive and dead; environment interaction erases quantum phases. Despite philosophical uncertainty, quantum prediction drives semiconductors, lasers, and emerging quantum computers.

Predictive success beyond intuition

GPS needs relativistic time correction; transistor miniaturization depends on quantum tunnelling models. The practical irony: equations once accused of metaphysical excess now define hardware precision. Stewart presents relativity and quantum theory not as rival visions but as evidence that mathematics continually expands your conceptual reach.

Physics demands humility

When equations succeed experimentally but defy intuition, you glimpse that reality may be more mathematical than mental.

Stewart demonstrates that modern physics grew from earlier mathematical styles—Pythagoras’ geometry and calculus’ continuous change—into probabilistic and geometric frameworks that let you comprehend extremes: near light speed and near zero scale. Understanding those equations teaches you to accept precision without certainty.

Not all laws yield predictable outcomes. Stewart’s chapters on chaos, fluid equations, and financial risk reveal that mathematics also maps failure, turbulence, and unpredictability—turning ignorance into structured awareness.

Fluid motion and computation

Navier–Stokes equations describe fluid dynamics from blood to climate. Because analytical solutions are rare, scientists rely on computational fluid dynamics (CFD). Medical engineers at EPFL and Suncica Canic use it to test stents; climate scientists use it to simulate atmospheric circulation. Higher resolution consumes exponentially more computing power, showing mathematics’ balance between theory and numerical ingenuity.

Chaos and deterministic unpredictability

Robert May’s logistic map and Lorenz’s weather study proved that deterministic equations can behave randomly. As initial conditions shift minutely, outcomes diverge exponentially—the butterfly effect. Chaos appears in population cycles, weather, and plankton flows, limiting long‑range forecasts. Stewart highlights this as mathematics’ moral: understanding limits of predictability is as important as predicting itself.

Equations, risk, and systemic fragility

Financial modeling illustrates another form of chaos. The Black–Scholes formula priced options elegantly under ideal assumptions. When markets ignored those limits, massive derivative exposure magnified crises (LTCM, 2008). Haldane and May’s ecological analogy suggests resilience comes from diversity and modularity—concepts borrowed from nonlinear dynamics and network theory.

Mathematics teaches humility

Equations empower prediction only within their assumptions. True expertise means knowing when models break, not just when they work.

Stewart closes these themes by turning mathematical precision into prudence. When you study chaos and turbulence, you learn that uncertainty is natural, not a flaw. The pursuit of understanding continues—but now with respect for complexity as part of reality’s design.

Beyond numbers and curves lie equations that describe qualitative structure—the shape and direction of processes. Stewart’s later chapters on topology and entropy explore how mathematics captures invariance and irreversibility, the skeletons of both form and time.

Topology and invariant structure

Euler’s simple F – E + V = 2 links geometry to topology, revealing traits that survive bending and stretching. It generalizes to F – E + V = 2 – 2g for surfaces of genus g and leads to classification of all orientable shapes. Through invariants like knot polynomials (Alexander, Jones), topology became a tool for distinguishing DNA configurations and analyzing molecular structures.

Entropy and time’s asymmetry

Carnot’s and Clausius’s equations define entropy as measurable disorder; Boltzmann’s S = k log W transforms it into a counting of microstates. Stewart uses everyday analogies—a shuffled deck—to show why irreversible processes dominate nature. Entropy gives time an arrow: while microscopic laws can run backward, macroscopic probabilities overwhelmingly prefer forward evolution.

Linking order, life, and information

Schrödinger’s notion of “negative entropy” bridges thermodynamics and biology: organisms maintain structure by exporting entropy. Shannon’s information measure parallels this by quantifying uncertainty in communication. Mathematics thus connects physical disorder, informational uncertainty, and biological self‑organization under one conceptual umbrella.

Order hides in invariants

Whether in a knotted DNA loop or a thermodynamic system, conserved quantities reveal internal structure that persists amid change.

Stewart ends by connecting geometry, probability, and information as manifestations of one idea: mathematics maps stability within flux. Topology explains what survives deformation; entropy explains why change has a direction. The result is a deep symmetry of insight—the study of invariants becomes the study of life’s persistence and time’s flow.