Men of Mathematics

1. Mathematics: A Human Endeavor Rooted in Ancient Thought

The great mathematicians have played a part in the evolution of scientific and philosophic thought comparable to that of the philosophers and scientists themselves.

Humanity's intellectual journey. Mathematics, far from being a cold, abstract discipline, is a vibrant and evolving creation of the human spirit. Its history is intertwined with the lives and personalities of the individuals who shaped it, reflecting their struggles, triumphs, and the broader intellectual currents of their times.

Ancient foundations. Modern mathematics builds upon the insights of ancient civilizations, particularly the Greeks. Figures like Pythagoras, Zeno, Eudoxus, and Archimedes laid the groundwork for concepts such as proof, irrational numbers, and the calculus, which continue to resonate in contemporary mathematics.

• Pythagoras: Proof and the discovery of irrational numbers
• Zeno: Paradoxes of motion and the infinite
• Eudoxus: Method of exhaustion and the theory of proportion
• Archimedes: Integral calculus and mechanical genius

Modern mathematics. The focus is on modern mathematics, beginning with the advances of the 17th century and continuing to the present day. This era saw the development of analytic geometry, calculus, and other powerful tools that transformed the landscape of scientific and philosophical thought.

2. Analytic Geometry and Calculus: Cornerstones of Modern Mathematics

Modern mathematics began with two great advances, analytic geometry and the calculus.

Revolutionary tools. Analytic geometry and calculus, emerging in the 17th century, marked a turning point in mathematical history. These methods provided unprecedented power for solving problems in geometry, physics, and other fields.

Analytic geometry. Descartes' analytic geometry, by bridging algebra and geometry, allowed mathematicians to express geometric relationships algebraically and vice versa. This opened up new avenues for discovery and proof.

• Ordinary boys could prove results that baffled the greatest Greek geometers.
• Descartes and Fermat are credited with its creation.

Calculus. The calculus, developed by Newton and Leibniz, provided a framework for understanding change and motion. It enabled mathematicians to calculate areas, volumes, tangents, and other quantities that were previously inaccessible.

• Archimedes had the fundamental notion of limiting sums.
• Fermat invented the method of Cartesian geometry independently of Descartes.

These two advances, analytic geometry and calculus, set the stage for the explosive growth of mathematics in the centuries that followed.

3. Descartes: From Doubt to Geometry

In saying that Descartes was responsible for the creation of analytic geometry we do not mean to imply that the new method sprang fullarmed from his mind alone.

The power of doubt. Descartes, a philosopher, soldier, and mathematician, sought tranquility and repose amidst the turmoil of 17th-century Europe. His method of systematic doubt led him to the fundamental truth "Cogito ergo sum" (I think, therefore I am), which became the cornerstone of his philosophy.

Analytic geometry. Descartes' greatest contribution to mathematics was the development of analytic geometry, a method that combined algebra and geometry. This allowed mathematicians to express geometric shapes using algebraic equations and to solve geometric problems using algebraic techniques.

• Descartes' method was so powerful that ordinary boys could use it to prove results that would have baffled the greatest of the Greek geometers.
• Descartes shared the honor with Fermat.

A life of the mind. Despite his active life as a soldier and courtier, Descartes remained committed to intellectual pursuits. He spent years in solitary retreat, meditating and developing his philosophical and mathematical ideas.

4. Fermat: The Amateur's Profound Flick to the Calculus

Greatest mathematician of the seventeenth century.

A hobbyist's genius. Fermat, a lawyer by profession, pursued mathematics as a hobby. Despite his amateur status, he made profound contributions to number theory, calculus, and analytic geometry.

Calculus and optics. Fermat's "flick to the calculus" involved developing methods for finding maxima and minima of functions, as well as tangents to curves. He also formulated a physical principle related to optics.

• Fermat conceived and applied the leading idea of the differential calculus thirteen years before Newton was born and seventeen before Leibniz was born.
• Fermat invented the method of Cartesian geometry independently of Descartes.

Number theory. Fermat's greatest achievement was in number theory, where he made groundbreaking discoveries about the properties of integers. His "Last Theorem," a deceptively simple statement about the impossibility of certain equations, has challenged mathematicians for centuries.

5. Pascal: Bridging Geometry and the Divine

An infant prodigy buries his talent.

Prodigy and piety. Pascal, a brilliant mathematician and physicist, experienced a profound religious conversion that led him to abandon scientific pursuits. He devoted his life to religious contemplation and writing.

Geometry and probability. Despite his religious fervor, Pascal made significant contributions to geometry, including Pascal's theorem on conic sections. He also co-created the mathematical theory of probability with Fermat.

• Pascal's wonderful theorem: Pascal's theorem.
• Pascal creates the theory of probability with Fermat.

The wager. Pascal's famous "wager" argued that it is rational to believe in God, even if there is no proof, because the potential reward (eternal life) outweighs the potential loss (finite earthly pleasures).

6. Newton: The Seashore and the Ocean of Truth

On the shoulders of giants.

Humility and genius. Newton, one of the greatest scientists of all time, saw himself as a boy playing on the seashore, while the vast ocean of truth lay undiscovered before him. This humility belied his unparalleled achievements in mathematics, physics, and astronomy.

Calculus and Principia. Newton's invention of the calculus provided a powerful tool for understanding the physical world. His Principia Mathematica, a masterpiece of scientific thought, laid down the laws of motion and universal gravitation.

• The calculus.
• Newton unsurpassed in pure mathematics, supreme in natural philosophy.

A life of science. Newton's life was marked by intense intellectual activity, controversy, and public service. He served as a professor at Cambridge, a member of Parliament, and Master of the Mint.

7. Leibniz: The Universalist's Unfinished Dream

Two superb contributions.

A polymath's ambition. Leibniz, a philosopher, diplomat, and mathematician, sought to create a "universal characteristic," a symbolic language that would encompass all knowledge. He made significant contributions to the calculus, logic, and metaphysics.

Calculus and diplomacy. Leibniz independently developed the calculus, creating a notation that is still used today. He also served as a diplomat for the Duke of Brunswick, navigating the complex political landscape of 17th-century Europe.

• The “universal characteristic.”
• Symbolic reasoning.

An unfinished dream. Leibniz's vision of a universal symbolic reasoning system remained largely unrealized during his lifetime. However, his ideas inspired later generations of mathematicians and logicians, who have made significant progress in this field.

8. The Bernoullis: Heredity's Hand in Mathematical Talent

Eight mathematicians in three generations.

A family of mathematicians. The Bernoulli family, spanning three generations, produced eight mathematicians, providing clinical evidence for the role of heredity in mathematical talent.

Calculus of variations. The Bernoullis made significant contributions to the calculus of variations, a field that deals with finding the maximum or minimum values of integrals. This work had important applications in physics and engineering.

• The calculus of variations.

A family feud. Despite their shared talent, the Bernoulli brothers were often embroiled in bitter rivalries and disputes over priority. This highlights the competitive nature of scientific discovery.

9. Euler: Analysis Incarnate and the Power of Abstraction

The most prolific mathematician in history.

Prolific genius. Euler, the most prolific mathematician in history, made groundbreaking contributions to analysis, number theory, geometry, and physics. His work laid the foundation for much of modern mathematics.

Abstraction and application. Euler demonstrated the power of abstraction in mathematics, developing general methods that could be applied to a wide range of problems. He also showed the practicality of seemingly abstract mathematical concepts.

• Celestial mechanics and naval warfare.
• Master and inspirer of masters for a century.

A life of dedication. Despite facing numerous challenges, including blindness in his later years, Euler remained committed to his mathematical pursuits. He continued to produce a vast body of work, dictating his findings to his sons.

10. Lagrange: Modesty, Mechanics, and the Revolution

Greatest and most modest mathematician of the eighteenth century.

Modesty and mastery. Lagrange, considered one of the greatest mathematicians of the 18th century, was known for his modesty and his profound contributions to mechanics, analysis, and number theory.

Analytical mechanics. Lagrange's Mécanique Analytique, a masterpiece of mathematical thought, presented a unified and elegant framework for understanding mechanics. It relied solely on analytical methods, dispensing with geometric arguments.

• The Mécanique analytique a living masterpiece.
• A landmark in the theory of equations.

Revolution and reflection. Lagrange lived through the French Revolution, witnessing both its horrors and its transformative potential. He contributed to the development of the metric system and served on various scientific committees.

11. Monge and Fourier: Geometry, Heat, and the Emperor's Friends

A knife grinder’s son and a tailor’s boy help Napoleon to upset the aristocrats’ applecart.

Revolutionary contributions. Monge and Fourier, both of humble origins, rose to prominence during the French Revolution and made significant contributions to mathematics and physics.

Descriptive geometry and the Machine Age. Monge's invention of descriptive geometry provided a powerful tool for representing three-dimensional objects in two dimensions. This had a profound impact on engineering and manufacturing.

• Monge’s descriptive geometry and the Machine Age.

Fourier analysis and modern physics. Fourier's work on heat conduction led to the development of Fourier analysis, a technique for representing functions as sums of trigonometric series. This has become an indispensable tool in physics and engineering.

• Fourier’s analysis and modern physics.

12. Poncelet: Geometry Resurrected from Napoleonic Shambles

Resurrected from a Napoleonic shambles.

From war to geometry. Poncelet, a soldier in Napoleon's army, was captured during the disastrous Russian campaign of 1812. While imprisoned, he developed projective geometry, a new approach to geometry that emphasized invariant properties.

Projective geometry. Poncelet's projective geometry provided a more general and flexible framework for studying geometric shapes. It emphasized properties that remain unchanged under projection, such as incidence and cross-ratio.

• Poncelet’s projective geometry.
• Principles of continuity and duality.

Continuity and duality. Poncelet's principles of continuity and duality provided powerful tools for discovering new geometric theorems and unifying seemingly disparate results.

13. Gauss: The Prince of Mathematicians and the Arithmetical Queen

Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics.

Unequalled intellect. Gauss, often called the "Prince of Mathematicians," made fundamental contributions to number theory, analysis, geometry, and physics. His work has had a profound and lasting impact on science and mathematics.

Number theory and Ceres. Gauss's Disquisitiones Arithmeticae, a masterpiece of number theory, laid the foundation for modern arithmetic. His calculations of the orbit of Ceres, a newly discovered asteroid, demonstrated the power of his mathematical methods.

• The Disquisitiones Arithmeticae.
• The Ceres disaster.

A sage of sages. Gauss was known for his intellectual rigor, his reluctance to publish incomplete work, and his profound insights into the nature of mathematics. He was revered by his contemporaries and continues to inspire mathematicians today.

14. Cauchy: Rigor, Religion, and Revolution in Analysis

Change in nature of mathematics with nineteenth century.

Rigor in analysis. Cauchy, a leading mathematician of the 19th century, brought a new level of rigor to mathematical analysis. He insisted on precise definitions and rigorous proofs, transforming the field from a collection of intuitive techniques into a solid logical structure.

The pious hippopotamus. Cauchy's deep religious beliefs influenced his mathematical work and his personal life. He was a devout Catholic and a staunch defender of traditional values.

• The pious hippopotamus.

Drawbacks of an irreproachable character. Cauchy's strong convictions and uncompromising nature sometimes led to conflicts with his colleagues and the French government. He was exiled for refusing to take an oath of allegiance to the July Monarchy.

15. Lobachevsky: Challenging Euclid, Expanding the Mind

The widow’s mite.

Beyond Euclid. Lobachevsky, a Russian mathematician, challenged the foundations of geometry by developing non-Euclidean geometry. This revolutionary work shattered the long-held belief that Euclid's geometry was the only possible description of space.

Non-Euclidean geometry. Lobachevsky's geometry, based on the hypothesis of the acute angle, led to a consistent and logically sound system that differed from Euclid's in its treatment of parallel lines. This opened up new avenues for mathematical exploration.

• Non-Euclidean geometry.
• A Copernicus of the intellect.

A Copernicus of the intellect. Lobachevsky's work, initially met with ridicule and incomprehension, eventually gained recognition as a profound contribution to mathematics and a testament to the power of human reason.

16. Abel: Genius and Poverty

Norway in 1802.

Smothered by clerical fecundity. Abel, a Norwegian mathematician, overcame extreme poverty and lack of recognition to make groundbreaking contributions to algebra and analysis. His work on elliptic functions and the unsolvability of the quintic equation revolutionized mathematics.

Abel and the quintic. Abel proved that there is no general algebraic solution for equations of degree five or higher. This settled a long-standing problem and opened up new avenues for algebraic research.

• Abel and the quintic.
• Something to keep mathematicians busy 500 years.

Crowning a corpse. Abel's genius was recognized only after his death at the age of twenty six. His work continues to inspire mathematicians today.

17. Jacobi: Elliptic Functions and the Honor of the Human Mind

Galvanoplastics versus mathematics.

The great algorist. Jacobi, a German mathematician, was renowned for his exceptional skill in manipulating formulas and his contributions to elliptic functions, number theory, and other areas of mathematics.

Elliptic functions. Jacobi's work on elliptic functions, building on the work of Abel, led to a deeper understanding of these important functions and their applications in geometry and physics.

• Elliptic functions.
• Their place in the general development.

Jacobi’s retort. Jacobi believed that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world.

18. Hamilton: An Irish Tragedy of Quaternions and Alcohol

Ireland’s greatest.

Ireland’s greatest. Hamilton, an Irish mathematician and physicist, made significant contributions to optics, dynamics, and algebra. His most famous invention was quaternions, a system of numbers that extends complex numbers to three dimensions.

Quaternions. Hamilton's discovery of quaternions, a non-commutative algebra, challenged the long-held belief that multiplication must be commutative. This opened up new avenues for algebraic research.

• Quaternions.
• Mountains of paper.

Marriage and alcohol. Hamilton's personal life was marred by disappointment in love and a struggle with alcoholism. Despite these challenges, he continued to produce groundbreaking work in mathematics and physics.

19. Galois: Genius and Stupidity

An all-time world record in stupidity.

A life cut short. Galois, a French mathematician, made revolutionary contributions to the theory of equations before his untimely death at the age of twenty. His work laid the foundation for modern group theory.

The riddle of equations solved. Galois' theory provided a way to determine whether a given algebraic equation can be solved by radicals. This settled a long-standing problem and opened up new avenues for algebraic research.

• The riddle of equations solved.

Buried like a dog. Galois' life was marked by political turmoil, academic ineptitude, and personal tragedy. His genius was not recognized until after his death.

20. Cayley and Sylvester: Invariant Twins and the Fabric of Modern Algebra

Cayley’s contributions.

Collaborative genius. Cayley and Sylvester, two English mathematicians, formed a close friendship and collaborated on groundbreaking work in the theory of invariants. Their work laid the foundation for modern algebra and geometry.

The theory of invariants. Cayley and Sylvester developed the theory of invariants, which deals with expressions that remain unchanged under certain transformations. This has had a profound impact on mathematics and physics.

• The theory of invariants.

Called to Johns Hopkins University. Sylvester’s mission to the Virginians. Further false steps.

21. Weierstrass and Kowalewski: Master, Pupil, and the Arithmetization of Analysis

The father of modern analysis.

The father of modern analysis. Weierstrass, a German mathematician, is considered the father of modern analysis. He brought a new level of rigor to the foundations of calculus and developed the theory of analytic functions.

Arithmetization of analysis. Weierstrass sought to base all of analysis on the common whole numbers, eliminating reliance on geometric intuition. This led to a more precise and rigorous foundation for calculus.

• Arithmetization of analysis.
• Doubts.

A woman’s gratitude. Sonja Kowalewski, a Russian mathematician, was Weierstrass's student and friend. She made significant contributions to analysis and mechanics, overcoming significant obstacles as a woman in science.

22. Boole: Logic, Laws of Thought, and the Dawn of Symbolic Reasoning

British mathematics.

Damned at birth by snobbery. Boole, an English mathematician and logician, developed symbolic logic, a system of algebra for representing logical relationships. His work laid the foundation for modern computer science.

Symbolic logic. Boole's "Laws of Thought" presented a new way of representing logical statements using mathematical symbols. This allowed for the manipulation of logical expressions using algebraic techniques.

• Symbolic logic.
• Its mathematical significance.

Dead in his prime. Boole's work was initially met with skepticism, but it eventually gained recognition as a profound contribution to logic and mathematics.

23. Hermite: Mysticism, Transcendence, and the Power of Internationalism

Old problems and new methods.

Old problems and new methods. Hermite, a French mathematician, made significant contributions to number theory, algebra, and analysis. He is best known for his proof that the number e is transcendental.

Transcendental numbers. Hermite's proof that e is transcendental was a major breakthrough in number theory. It paved the way for Lindemann's proof that π is also transcendental.

• Transcendental numbers.
• A hint to circle-squarers.

Hermite’s internationalism. Hermite's internationalism.

24. Kronecker: The Doubter

Legend of an American saint.

Lucky Kronecker. Kronecker, a German mathematician, was known for his skepticism and his insistence on basing mathematics on the common whole numbers. His work challenged the foundations of analysis and inspired new approaches to number theory.

His skepticism his most original contribution. Kronecker's skepticism his most original contribution.

25. Anima Candida

Poor but happy.

Riemann’s chronic shyness. Riemann, a German mathematician, made groundbreaking contributions to geometry, analysis, and number theory. His work on the foundations of geometry paved the way for Einstein's theory of general relativity.

Riemann’s landmark in geometry. Riemann's landmark in geometry.

26. Arithmetic the Second

Aged in the wood.

Dedekind, last pupil of Gauss. Dedekind, last pupil of Gauss.

27. The Last Universalist

Poincaré’s universality and methods.

Poincaré’s account. Poincaré’s account.

28. Paradise Lost?

Old foes with new faces.

Cantor’s artistic inheritance and father-fixation. Cantor’s artistic inheritance and father-fixation.

Last updated: March 1, 2025