The Little Book of Mathematical Principles, Theories & Things

1. Numbers Evolved: From Symbols to Place Value and Zero

The value of each digit is shown by its place in the number.

Early systems. Ancient civilizations developed various ways to write numbers, often using unique symbols for different powers of ten, like the Egyptians. The Chinese system mirrored spoken language, indicating how many hundreds, tens, and units were present. These systems required an ever-increasing number of symbols for larger values.

Place value revolution. The Indo-Arabic system, which came to the West via Arab countries, introduced the revolutionary concept of place value, using only ten symbols (0-9). The position of a digit determines its value (units, tens, hundreds, etc.), allowing any number to be written efficiently. The Babylonians also used a place value system, but based on powers of 60, which is still seen today in time measurement.

The power of zero. Crucially, the Indo-Arabic system included zero, a concept that took centuries to be accepted as a number in its own right. While early counting boards used spaces for missing digits, a symbol for zero became essential for written place value systems. Indian mathematicians, like Brahmagupta in 628 AD, were pioneers in defining rules for arithmetic involving zero, paving the way for modern calculations and algebra.

2. Ancient Geometry's Triumphs and Three Impossible Problems

It destroyed the notion that everything could be described in terms of whole numbers.

Pythagorean insights. The Pythagoreans, a mystical Greek sect, believed "All Things Are Numbers," meaning everything could be explained by whole numbers and their ratios (fractions). They are credited with major discoveries, including the Earth's sphericity and the famous Pythagorean Theorem relating the sides of a right-angled triangle. This theorem, known numerically to Babylonians earlier, became a cornerstone of geometry.

The shock of irrationality. A profound crisis hit the Pythagoreans when they discovered numbers that could not be expressed as a ratio of two whole numbers – irrational numbers. The square root of 2, the length of the hypotenuse of a right triangle with sides of length 1, was proven to be irrational. This discovery shattered their core belief and demonstrated that geometry contained concepts beyond simple numerical ratios.

The three great problems. Greek geometers, valuing exactness with only a straight edge and compasses, posed three famous construction problems: trisecting an angle, doubling the cube (constructing a cube with twice the volume), and squaring the circle (constructing a square with the same area as a given circle). Despite centuries of ingenious attempts, these problems resisted solution, hinting at limitations that would only be understood much later with advanced algebra.

3. Calculus: Unlocking Change and Area with Infinitesimals

Calculus was, and still is, the most powerful tool in science.

Archimedes' early methods. Long before formal calculus, Archimedes in ancient Greece devised methods to tackle problems involving curves, such as finding the area under a parabola or the volume of a sphere. He used a technique called the "method of exhaustion," essentially summing infinitely many smaller areas or volumes, anticipating later integral calculus. His work showed that infinite sums could yield finite, rigorous results.

Newton and Leibnitz's invention. In the 17th century, independently, Isaac Newton and Gottfried Leibnitz developed infinitesimal calculus, providing a systematic way to handle changing quantities and areas. Differentiation allowed finding the instantaneous rate of change (gradient) of a curve, crucial for understanding speed and acceleration. Integration provided a method for finding the area under a curve by summing infinitely thin strips.

The fundamental connection. The profound link between differentiation and integration was revealed by the Fundamental Theorem of Calculus. It showed these two seemingly different operations are inverses of each other, much like addition and subtraction. Despite initial philosophical doubts about using "infinitely small" quantities, calculus proved immensely powerful, becoming indispensable for physics and engineering, describing everything from planetary motion to fluid dynamics.

4. Probability: Quantifying Chance and Predicting the Future

The theory of probability began with Fermat and Pascal’s work in this field.

Gambling sparks theory. The formal study of probability arose in the 17th century from questions about gambling posed by a French aristocrat, the Chevalier de Méré, to mathematician Blaise Pascal. Pascal corresponded with Pierre de Fermat about problems like how to divide stakes fairly if a game is interrupted, leading to the foundational ideas of calculating expected outcomes and probabilities. This marked a shift from superstition to mathematical analysis of chance.

Pascal's Triangle and distributions. Pascal's Triangle, known for centuries for its algebraic properties, was utilized by Pascal and Fermat to calculate probabilities, particularly for repeated events like coin spins. This led to the binomial distribution, a formula for finding the probability of a specific number of successes in a series of independent trials. This distribution is fundamental in statistics for analyzing outcomes with two possibilities.

Laws of chance. Later mathematicians formalized probability theory. Jacob Bernoulli's Law of Large Numbers showed that as an experiment is repeated many times, the observed proportion of successes will approach the theoretical probability. Abraham de Moivre and others discovered the normal distribution (the bell curve), which approximates the binomial distribution for large numbers of trials and models data clustering around an average, becoming central to statistics and error analysis.

5. Expanding Numbers: Negative, Complex, and Beyond

With complex numbers, mathematics is richer, more regular, and more satisfying.

The need for negatives. While intuitive for concepts like debt or temperature, negative numbers were not universally accepted in early mathematics. Greek mathematicians often rejected negative solutions to equations. Chinese mathematicians were among the first to use them systematically, employing different colored rods for positive and negative values. Their practical utility in simplifying algebraic rules eventually led to their widespread acceptance.

Inventing the imaginary. Equations like x² = -1 had no solution within the realm of ordinary (real) numbers. The need to solve such equations, particularly those arising from cubic formulas, led to the invention of the imaginary unit, i, defined as the square root of -1. Complex numbers, of the form a + bi, combine real and imaginary parts, initially viewed with suspicion but proving incredibly useful.

Richer algebra. Complex numbers brought a new completeness to algebra. While a quadratic equation might have zero, one, or two real solutions, it always has exactly two complex solutions (counting multiplicity). This elegance extends to all polynomial equations, as stated by the Fundamental Theorem of Algebra: an equation of degree n has exactly n complex solutions. Complex numbers are now essential in fields like electrical engineering and quantum mechanics.

6. Geometry's New Frontiers: Beyond Euclid's Parallel Lines

...if God created the world, he created it in accordance with Euclidean geometry, and yet there are mathematicians, men of extraordinary genius, who doubt whether the whole universe was created only in Euclid’s geometry …

Euclid's enduring system. Euclid's Elements provided a rigorous, axiomatic foundation for geometry that lasted over 2000 years. Starting from definitions and postulates, theorems were logically deduced. However, one postulate, the fifth (parallel postulate), was far less obvious than the others, leading to centuries of attempts to prove it from the rest. These attempts ultimately failed.

Birth of non-Euclidean geometry. In the 19th century, mathematicians like Lobachevsky and Bolyai explored what happens if the fifth postulate is denied. Assuming either that no parallel lines exist through a point (elliptic geometry, like on a sphere) or that multiple parallel lines exist (hyperbolic geometry), they created consistent, alternative geometries. This revolutionary discovery showed that geometry was not a single, divinely ordained system but could take different forms, impacting philosophy and even theology.

Topology's qualitative view. Moving beyond measurement, topology emerged as the study of shapes' properties that remain unchanged under continuous deformation (stretching, bending, but not tearing or gluing). This led to counter-intuitive ideas like a doughnut being topologically equivalent to a coffee cup (both have one hole). Topology focuses on connectivity and structure, providing a different lens through which to view geometric objects, including strange forms like the Möbius strip or the theoretical Klein bottle.

7. The Foundations Shaken: Paradoxes and Limits of Proof

There is a theorem of arithmetic which cannot be proved from the axioms of arithmetic.

Logic and sets. George Boole showed that logic could be formalized algebraically, creating Boolean algebra, now fundamental to computer design. Building on this, mathematicians like Frege, Russell, and Whitehead attempted to reduce all of mathematics to logic, aiming for a complete, consistent axiomatic system from which all mathematical truths could be derived. This ambitious project sought to eliminate ambiguity and paradoxes.

Russell's paradox. This foundational program was severely challenged by paradoxes, most famously Bertrand Russell's paradox concerning sets that contain themselves. Defining R as the set of all sets that do not contain themselves leads to a contradiction: if R contains itself, it shouldn't; if it doesn't, it should. This revealed flaws in the naive definition of a set and necessitated a more rigorous axiomatic approach, like Zermelo-Fraenkel set theory, which restricts what can be considered a set.

Gödel's incompleteness. The most profound blow to the idea of a complete, formal system came from Kurt Gödel in 1931. His incompleteness theorem proved that for any consistent set of axioms strong enough to include basic arithmetic, there will always be true statements within that system that cannot be proven from those axioms. This demonstrated that mathematics is inherently richer than any formal logical system can capture, showing the limits of purely mechanical deduction and ensuring the continued need for human insight.

8. Mathematics Describes the Cosmos: From Earth-Centered to Relativity

The most famous story in science is that of Sir Isaac Newton and the apple: it fell from a tree in his garden and inspired the theory of gravity.

Earth vs. Sun centered. For over a thousand years, Ptolemy's Earth-centered model, with planets moving in complex circles (epicycles), accurately predicted celestial movements. Nicolaus Copernicus proposed a simpler Sun-centered model, but it faced objections, including the lack of observable stellar parallax. Johannes Kepler refined the Sun-centered view with his laws of planetary motion, showing planets move in ellipses, not perfect circles, a crucial step towards greater accuracy.

Newton's universal laws. Isaac Newton synthesized centuries of physics and mathematics with his laws of motion and universal gravitation. The famous apple story illustrates the core idea: the same force pulling the apple to Earth governs the moon's orbit and the planets' paths around the Sun. His law of gravity, stating that attraction is proportional to mass and inversely proportional to the square of distance, provided a single mathematical framework explaining diverse phenomena.

Einstein's relativity. Newton's laws held for centuries but were challenged by phenomena at extreme scales. Albert Einstein's Special Relativity (1905) showed that motion is relative and the speed of light is constant for all observers, leading to counter-intuitive consequences for time, length, and mass. General Relativity (1915) extended this to acceleration and gravity, proposing that mass warps spacetime itself, a revolutionary geometric view of gravity confirmed by observations like the bending of starlight.

9. The Dawn of Computing: Machines, Logic, and Algorithms

A Turing machine is a computer, stripped down to its bare essentials.

Mechanical calculators. The desire to automate calculation led to mechanical devices. Wilhelm Schickard designed one in the 17th century, and Blaise Pascal built the Pascaline for addition and subtraction. Gottfried Leibnitz created a machine for all four operations. These early efforts faced technical hurdles, particularly with carrying digits, but paved the way for later, more reliable mechanical calculators used widely until the electronic age.

Babbage's vision. Charles Babbage, in the 19th century, envisioned more sophisticated machines. His Difference Engine could automate the calculation of polynomial functions for tables. More remarkably, his Analytical Engine was conceived as the first programmable, general-purpose computer, using punched cards for instructions and data, and incorporating concepts like branching and looping. Though never fully built in his lifetime, it anticipated modern computing architecture.

Electronic and theoretical foundations. The 20th century brought electronic computing. Machines like Colossus and ENIAC, initially built for wartime calculations, demonstrated the power of electronic valves for speed and programmability. Alan Turing formalized the concept of computation with the Turing machine, an abstract model defining what is computable. His work, alongside others like Claude Shannon on binary logic circuits, laid the theoretical groundwork for the digital age, where information is processed using simple 0s and 1s.

10. Modern Mysteries: Primes, Complexity, and Unsolved Problems

The Riemann hypothesis is one of the most famous, and perhaps the most important, unsolved problem of mathematics.

The enigma of primes. Prime numbers, the building blocks of integers, have fascinated mathematicians since antiquity. Euclid proved there are infinitely many, but finding a pattern or formula to generate them has been elusive. Conjectures like Goldbach's (every even number > 2 is sum of two primes) remain unproven despite extensive testing. The distribution of primes is irregular but follows statistical patterns described by the Prime Number Theorem.

Hilbert's challenges. At the turn of the 20th century, David Hilbert posed 23 influential problems that guided much of the century's mathematical research. These included questions about the foundations of mathematics, number theory (like the Riemann Hypothesis and Goldbach's Conjecture), and the solvability of equations. Solving a Hilbert problem became a major achievement, though some, like the Riemann Hypothesis and the P=NP? question, remain open.

Complexity and computation limits. The rise of computing brought new questions about the limits of computation. The P=NP? conjecture asks if problems whose solutions can be quickly verified can also be quickly solved. This question is central to theoretical computer science and has major implications for cryptography; if P=NP, many modern codes based on the difficulty of factoring large numbers could be broken easily. Unsolved problems like these continue to drive mathematical innovation.

Last updated: May 19, 2025