History of Mathematics: From Euclid to Turing

A theorem proved in ancient Alexandria is as true today as the day it was written down. That permanence is what sets mathematics apart from every other science: physics revises its laws, biology redraws its trees, but a proved result stays proved. What changes is everything around the result: what counts as a proof, which objects are worth studying, what the notation looks like, and which questions seem answerable. This article follows that change from Babylonian clay tablets and Euclid's Elements to Gödel's incompleteness theorems and Turing's machines, and flags where the historical record is thinner than the legends suggest.

Key takeaways

Euclid's Elements (around 300 BCE) organized geometry as deduction from stated starting points, a method that dominated mathematics teaching for more than 2,000 years
Al-Khwarizmi's algebra book, written in Baghdad, gave us the word "algebra," and his name gave us "algorithm"
Newton and Leibniz developed calculus independently, and the modern consensus among historians is that neither copied the other
Euler's notation, including f(x), e, i, and Σ, is still in daily use
Gödel (1931) and Turing (1936) showed that proof and computation have built-in limits

Before the Greeks: calculation without proof

The earliest advanced mathematics we can read comes from Mesopotamia. The Babylonians used a base-60 place-value system, which survives today in our 60-minute hours and 360-degree circles. MacTutor, the University of St Andrews' history archive, notes that they solved quadratic equations of the forms x² + bx = c and x² − bx = c, taking the positive root, and treats their system as in some ways more advanced than ours.

The best-known tablet is Plimpton 322, a Babylonian clay tablet from about 1800 BCE now held at Columbia University. Its columns are usually read as the short side and hypotenuse of right triangles, which form Pythagorean triples. Scholars disagree about what it was for. Some see number theory, others see a teacher's aid for setting problems, and the trigonometric reading is now widely considered anachronistic. The honest summary is that we know the Babylonians could handle such numbers, and we do not know why this tablet exists.

What these cultures lacked, as far as the surviving sources show, was a tradition of proving that a method works for all cases. Their texts are procedures: do this, then this, and the answer appears. That is a different thing from showing why.

Greek proof: from "what works" to "why it must be true"

Greek mathematicians made a shift in what they asked for. A recipe says what to do. A proof says why it cannot fail. The tradition names Thales of Miletus (c. 624 to 547 BCE) as the first figure in this story, credited with theorems such as "a circle is bisected by any diameter" and "the base angles of an isosceles triangle are equal." MacTutor cautions that none of his writings survive, that the attributions come from authors writing centuries later, and that he may have used empirical rules rather than deductive proofs. The beginning of Greek proof is therefore a matter of tradition as much as of record.

Pythagoras (c. 570 to 490 BCE) is similarly hard to pin down. The relation between the sides of a right triangle was known to the Babylonians about a thousand years earlier, and MacTutor says only that Pythagoras "may have been the first to prove it." He founded a religious and philosophical community at Croton in southern Italy, whose members shared work and kept their findings secret, so separating his contributions from those of his followers is difficult. The discovery that the square root of 2 cannot be written as a ratio of whole numbers is attributed to the Pythagoreans, though MacTutor considers it unlikely that Pythagoras himself made it. Our profile of Pythagoras separates the surviving evidence from the legends.

Euclid and the architecture of a proof

Euclid worked in Alexandria, probably around 300 BCE, though MacTutor gives only approximate dates for him and notes that historians debate whether a single person or a team produced the works attributed to him. His Elements has 13 books: books 1 to 6 cover plane geometry, 7 to 9 number theory, 10 irrational magnitudes, and 11 to 13 solid geometry. Much of the material came from earlier mathematicians. Euclid's achievement was the organization.

The structure is the point. He starts from five postulates (geometric starting assumptions, such as the ability to draw a straight line between any two points) and a set of common notions (general principles, such as "things equal to the same thing are equal to each other"). Every later proposition is derived from those and from earlier propositions. Think of a building made of load-bearing bricks: if one brick is bad, everything above it is suspect, and if every brick is sound, the building stands. That discipline, in which each step must rest on an earlier step, is what mathematicians still mean by a proof.

The book was a central teaching text for more than 2,000 years, and MacTutor counts more than 1,000 editions since printing began in 1482.

The postulate that would not behave

Euclid's fifth postulate, the parallel postulate, says that through a point not on a given line, exactly one line can be drawn parallel to it. It looks less obvious than the other four, and mathematicians spent centuries trying to derive it from them. Omar Khayyam tried in the 11th century, and MacTutor notes that his attempt accidentally proved properties of figures in non-Euclidean geometries.

In the 19th century, mathematicians including Bolyai, Lobachevsky, and Riemann showed that consistent geometries exist in which the postulate fails. Euclid's results were not overturned. They were shown to describe one geometry among several. Einstein's general relativity later used a curved, non-Euclidean geometry to describe spacetime.

Archimedes: nearly calculus, two millennia early

Archimedes (c. 287 to 212 BCE) of Syracuse used the method of exhaustion, trapping a curved shape between inscribed and circumscribed polygons, to compute areas and volumes. He bounded pi between 3 10/71 and 3 1/7 using 96-sided polygons, and showed that a sphere's volume is two-thirds that of its circumscribing cylinder, a result he reportedly wanted on his tomb. His treatise The Method was recovered from a 10th-century manuscript in 1906. His techniques anticipate integration, though without the general machinery that came later. His life and the legends around it are covered in our profile of Archimedes.

Numbers themselves: zero and place value

It is easy to forget that writing numbers is a technology. The history of zero, as MacTutor lays it out, is slow and uneven rather than a single invention. The Babylonians had a place-value system for over a thousand years before they used a symbol for an empty place, and when they did, around 400 BCE, it was a placeholder rather than a number. Greek astronomers used the symbol O as a placeholder, but treated it as punctuation.

In India, zero was in use as a number by about 650 CE, and Brahmagupta in the seventh century gave rules for arithmetic with it, though he made an error on dividing zero by zero. The Maya developed a base-20 system with a zero symbol by 665, independently of the Old World. Al-Khwarizmi carried Indian numerals into the Islamic world, and Fibonacci introduced zero to Europe around 1200. MacTutor adds that zero became widely accepted in Europe only in the 1600s. Every calculation in the rest of this article depends on that slow acceptance.

The Islamic Golden Age: algebra and the algorithm

Muhammad ibn Musa al-Khwarizmi (c. 780 to 850) worked in Baghdad under the caliph al-Mamun, who founded the House of Wisdom to translate Greek scientific and philosophical works. His book Hisab al-jabr w'al-muqabala is often called the first book on algebra. MacTutor says it reduces linear and quadratic equations to six standard forms and solves them by two operations: al-jabr (moving subtracted terms to the other side) and al-muqabala (balancing like terms). He used words rather than symbols, and handled practical problems such as inheritance and land measurement.

Two English words trace to him. "Algebra" comes from al-jabr in the book's title, and "algorithm" comes from his name. His treatise on Indian numerals survives only in a Latin translation, Algoritmi de numero Indorum, which described the place-value system to European readers.

Omar Khayyam (1048 to 1131) is better known as a poet in the West, but MacTutor credits him with a complete classification of cubic equations, solved geometrically by intersecting conic sections. He also argued that such cubics cannot be solved by ruler and compass alone, a result proved about 750 years later. Of the roughly 600 four-line poems in Edward FitzGerald's 1859 translation, MacTutor says only about 120 verses can be attributed to him with certainty.

The Renaissance: solving cubics and meeting imaginary numbers

In 16th-century Italy, algebraists competed in public contests, and winning formulas were guarded as trade secrets. Girolamo Cardano published the solution of the cubic in Ars Magna in 1545, after learning that Scipione del Ferro had found it earlier. Niccolò Tartaglia had given Cardano his method in 1539 under an oath not to publish it, a promise Cardano considered void once he learned of del Ferro's work. Cardano's student Ludovico Ferrari solved the quartic.

Their formulas forced a problem. Cubic solutions sometimes required taking square roots of negative numbers as intermediate steps, even when the final answer was an ordinary real number. Cardano multiplied 5 + √−15 by 5 − √−15 to get 40 and called it "arithmetical subtlety." Rafael Bombelli's Algebra (1572) went further, working out rules for adding, subtracting, and multiplying these quantities, and showing that the cubic formula could yield correct real answers even when the steps passed through square roots of negatives. The numbers were not yet respectable, but the mathematics worked.

Coordinates, and a margin note

René Descartes (1596 to 1650) published La Géométrie in 1637, as one of three appendices to his Discours de la méthode. It applied algebra to geometry, which is the origin of Cartesian coordinates. The method turns a curve into an equation and an equation into a curve, and it later became a standard tool for working in physics. Pierre de Fermat, a lawyer and government official in Toulouse, worked on an algebraic approach to geometry at the same time, and Descartes was uneasy about Fermat's work on tangents and maxima.

Fermat (1601 to 1665) is remembered for a note he wrote in the margin of a translation of Diophantus's Arithmetica: that no whole-number solutions exist to xⁿ + yⁿ = zⁿ for n greater than 2, and that he had a remarkable proof the margin was too small to hold. MacTutor says that this proof is now believed to have been wrong. The statement, known as Fermat's Last Theorem, resisted proof for more than 350 years. Andrew Wiles announced a proof in June 1993, withdrew the claim when a problem appeared, and in November 1994 presented a corrected version, which has since been accepted. In 1654, Fermat and Blaise Pascal exchanged letters about gambling problems that MacTutor describes as the foundation of probability theory.

Newton and Leibniz: the calculus

Calculus gave mathematics two tools that complement each other. The derivative measures how fast a quantity is changing at an instant, like a car's speedometer. The integral adds up accumulated change, like the odometer. The fundamental theorem of calculus says these are inverse operations: the odometer's reading can be recovered from the speedometer's record and vice versa.

According to Wikipedia's summary of the dispute, Isaac Newton developed his "method of fluxions" from 1666 but did not publish his notation until 1693 in part and 1704 more fully. Gottfried Wilhelm Leibniz began his own version in 1674 and published first, in 1684. The two systems were largely translatable into each other. Leibniz's notation, with dy/dx for derivatives and ∫ for integrals, proved easier to use and is the one taught today. Newton's most famous application was in Principia (1687), where he used calculus-like reasoning to explain Kepler's laws of planetary motion by gravity, and our profile of Isaac Newton covers his work in more detail.

The priority fight

The dispute began around 1699, when Nicolas Fatio de Duillier accused Leibniz of plagiarism. It peaked in 1712, when the Royal Society, of which Newton was president, ruled in Newton's favor in the Commercium Epistolicum. Leibniz was never asked for his side and did not see the report until 1714. He died in 1716 with the matter unresolved. Historians now generally agree that the two invented calculus independently. The episode is mainly a caution about how institutions can be bent around a reputation.

Euler: the notation maker

Leonhard Euler (1707 to 1783) wrote 866 publications, according to Wikipedia's account of the Opera Omnia, and some researchers credit him with a quarter to a third of the century's output in mathematics and closely related sciences. He worked in Saint Petersburg and Berlin. After an illness in 1766 he became almost entirely blind, and by 1771, following a fire and a failed cataract operation, he was totally blind. He kept working with the help of his memory and assistants, including his sons and colleagues named Krafft, Lexell, and Fuss.

He shaped the way mathematics looks on the page. MacTutor credits him with introducing f(x) for functions in 1734, e for the base of natural logarithms in 1727, Σ for summation in 1755, and i for the square root of −1 in 1777. In 1735 he showed that the sum of the reciprocals of the squares, 1 + 1/4 + 1/9 + 1/16 and so on, equals π²/6, solving a problem that had defeated the Bernoulli family. Also in 1735, he presented his solution to the Seven Bridges of Königsberg problem, which Wikipedia calls the first theorem of graph theory. He showed that no walk can cross each of the city's seven bridges exactly once.

His formula e^(ix) = cos x + i sin x, published in 1748, contains the special case e^(iπ) + 1 = 0, often called the most beautiful equation in mathematics for joining five fundamental constants. Our profile of Leonhard Euler goes deeper into his work.

The 19th century: rigor, symmetry, and the infinite

The 19th century turned mathematics from a toolbox for physics into a self-contained discipline that examined its own foundations. Carl Friedrich Gauss (1777 to 1855) contributed to number theory, geometry, statistics, and analysis, and is covered in our profile. Mathematicians such as Cauchy and Weierstrass worked to give calculus precise definitions of limits and continuity, because the infinitesimals of the 17th century had never been put on a fully logical footing.

Galois: symmetry as the key to equations

Évariste Galois was born on 25 October 1811, fought a duel on 30 May 1832, and died the next day, aged 20. The reason for the duel is unclear, though MacTutor says it was probably linked to a woman. The legend says he spent the night before writing out everything he knew about group theory. MacTutor says that account appears exaggerated and instead points to a note in the margin of a manuscript: "There is something to complete in this demonstration. I do not have the time."

What he left was a method for deciding when an equation can be solved by radicals (formulas using roots and ordinary arithmetic). His approach looks at the symmetries of an equation's solutions, in the way that rotating or flipping a square shuffles its corners without changing the square. Niels Henrik Abel had earlier shown that no general radical formula exists for the fifth-degree equation, and Galois provided the framework to say exactly which equations are solvable and why. Joseph Liouville announced his papers to the Paris Academy in 1843 and published them in 1846. Read more in our profiles of Galois and Abel.

Cantor: infinities of different sizes

Georg Cantor (1845 to 1918) proved that the real numbers cannot be counted, in work he had completed by December 1873 and published in 1874. This implies that infinity comes in more than one size: there are more real numbers than whole numbers. He spent years trying to settle the continuum hypothesis, the question of whether any size of infinity lies between the whole numbers and the real numbers. MacTutor says that he twice thought he had a proof, once of the hypothesis and once of its negation, and each time found his mistake the next day.

His work met fierce opposition, notably from Leopold Kronecker, and MacTutor describes their antagonism as bitter. Cantor's first recorded attack of depression came in May 1884, and he spent time in sanatoria from 1899 onward. MacTutor says his mathematical worries and difficult relationships were magnified by depression but were not its cause. He entered a sanatorium for the last time in June 1917 and died of a heart attack in Halle on 6 January 1918. Hilbert later called his work "the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity." Today set theory is the standard foundation for most of mathematics.

Gödel and Turing: the limits of proof and computation

In 1900, David Hilbert presented 23 problems for the new century. The second asked for a direct proof that the axioms of arithmetic could never lead to a contradiction. Hilbert's wider program aimed to put all of mathematics on such a secure, finite foundation.

Kurt Gödel (1906 to 1978) published his incompleteness results in 1931. MacTutor summarizes them this way: in any axiomatic system of this kind, some propositions can be neither proved nor disproved from the axioms, and the system's consistency cannot be proved within it. His second theorem shows that Peano arithmetic cannot prove its own consistency. Whether this settles Hilbert's second problem is still debated, since Gerhard Gentzen proved arithmetic consistent in 1936 using methods outside the system. Gödel also showed in 1940 that the continuum hypothesis cannot be disproved from the standard axioms of set theory. His story is in our profile of Kurt Gödel.

Turing's machine

Alan Turing (1912 to 1954) published "On Computable Numbers, with an Application to the Entscheidungsproblem" in 1936, with the paper appearing in 1937 after delays. It described an abstract machine, now called a Turing machine: a tape divided into cells, a head that reads and writes symbols, and a finite table of rules. Think of a clerk with an endless roll of paper and a short rulebook, who can only look at one square at a time. The paper also described a universal machine that can carry out any computation given the right instructions, which is the conceptual ancestor of the stored-program computer.

The paper showed that some problems are not computable. The best-known example is the halting problem: no general procedure can decide, for every program and input, whether the program will eventually stop. Alonzo Church reached a related result in 1936 by a different route. The Church-Turing thesis states that the machines and Church's lambda calculus capture everything that can be computed. It is a thesis rather than a theorem, because it ties a precise mathematical definition to the informal idea of "mechanical procedure," so it cannot be proved in the ordinary sense.

During the Second World War, Turing worked at Bletchley Park on German Enigma messages. MacTutor says that with W. G. Welchman he developed the Bombe, which from late 1940 decoded Luftwaffe Enigma traffic, and that by mid-1941 his statistical methods had helped break German naval signals. Wikipedia adds that he led Hut 8, which handled naval Enigma. Many people contributed to that work. In 1950, in the journal Mind, he proposed what became known as the Turing test.

In 1952, Turing was tried and found guilty under British laws that criminalized homosexual acts, and he was given probation conditional on hormone treatment, which Wikipedia describes as "chemical castration." His security clearance was withdrawn. He died on 7 June 1954 of cyanide poisoning. An inquest ruled it a suicide, a verdict that some biographers and his mother questioned. Prime Minister Gordon Brown apologized on behalf of the government in 2009, Queen Elizabeth II granted a pardon in 2013, and a 2017 law retroactively pardoned others convicted under the same statutes. Our profile of Alan Turing and the broader history of computing pick up the story from here.

A timeline at a glance

Approximate date	Development	Key figure or source	Certainty
c. 1800 BCE	Plimpton 322 tablet; Babylonian quadratic methods	Anonymous scribes	Date secure; purpose debated
c. 600 BCE	Early deductive geometry	Thales (tradition)	Attributions come from later authors
c. 300 BCE	Elements, 13 books	Euclid	Dates approximate
c. 250 BCE	Method of exhaustion, bounds on pi	Archimedes	Well documented
7th century CE	Rules for arithmetic with zero	Brahmagupta	Included a known error on 0/0
Early 9th century CE	Book on al-jabr and al-muqabala	Al-Khwarizmi	Exact year of composition not recorded
1545	Cubic and quartic solutions published	Cardano, Tartaglia, Ferrari, del Ferro	Priority disputed
1572	Rules for imaginary numbers	Bombelli	Well documented
1637	La Géométrie	Descartes	Well documented
1674 to 1684	Calculus develops; Leibniz publishes first	Newton and Leibniz	Independent invention
1735	Königsberg bridges; Basel problem	Euler	Well documented
1873 to 1874	Reals are uncountable	Cantor	Well documented
1931	Incompleteness theorems	Gödel	Well documented
1936	Computability and the halting problem	Turing, Church	Well documented
1994	Proof of Fermat's Last Theorem accepted	Wiles	Well documented

What this history gets misread as

"Math is a straight line of progress." The record is full of detours. Zero took centuries to be accepted in Europe, imaginary numbers were used before anyone understood them, and the parallel postulate occupied people for two millennia before it turned out to be independent of the others.

"Euclid was proved wrong." Non-Euclidean geometry did not refute Euclid. It showed that his geometry is one consistent system among several, and his theorems remain valid within it.

"One genius, one discovery." The Newton and Leibniz case is only the most famous example. Cubic equations had at least three claimants, and the Church and Turing results arrived almost together. Mathematics tends to ripen collectively.

"The founders were lone eccentrics." Many of the stories, such as Galois writing all night, are compressed or embellished. The primary evidence is more modest and, in some cases, more interesting.

Open problems that continue the story

Mathematics is still unfinished. In 2000, the Clay Mathematics Institute announced seven Millennium Prize Problems, with $1 million offered for each solution. One, the Poincaré conjecture, has been solved by Grigori Perelman. The other six remain open, including the Riemann hypothesis, which concerns how prime numbers are distributed and claims that the nontrivial zeros of the zeta function all have real part 1/2, and P versus NP, which asks whether every problem whose solution is easy to check is also easy to solve.

Fermat's margin note, Hilbert's list, and the Millennium Problems share a pattern: a clearly stated question can drive work for generations, and the tools built along the way are often more valuable than the answer. Wiles's proof relied on ideas that did not exist in Fermat's century.

The takeaway

The permanence of a mathematical truth and the instability of mathematics itself are the same fact seen from two sides. Theorems do not expire, so each generation builds on the last, but what gets built changes the field's shape. Greek proof turned recipes into arguments. Algebra and then calculus turned geometry into machinery. Cantor, Gödel, and Turing turned the machinery on itself and found limits: some truths cannot be proved within a given system, and some questions no algorithm can answer.

The limits turned out to be productive. Turing's analysis of what cannot be computed also defined what computation is, and that definition underlies every digital device. A history that began with scribes pressing wedges into clay ends with the discovery that mathematics can describe its own boundaries.

FAQ

Q: Was mathematics discovered or invented? A: Philosophers have disagreed for centuries. Platonists hold that mathematical objects exist independently of people and are discovered. Formalists treat mathematics as a game with chosen rules. Intuitionists hold that mathematical objects exist only as mental constructions. The historical record supports both sides in part: results like the infinitude of primes feel discovered, while notation, definitions, and even the acceptance of zero were clearly human choices made over centuries.

Q: Why did Newton and Leibniz invent calculus independently? A: By the late 17th century, the problems that call for it, such as tangents to curves, areas under curves, and motion under changing speed, had been worked on for decades, and the needed pieces were available. Historians generally agree that the two developed it independently, and that their methods were largely translatable into each other. Similar independent discoveries recur across science.

Q: What is the most important unsolved problem in mathematics? A: There is no official ranking. The Clay Mathematics Institute's Millennium Prize list includes the Riemann hypothesis, P versus NP, the Hodge conjecture, the Birch and Swinnerton-Dyer conjecture, the Navier-Stokes problem, and the Yang-Mills mass gap, each with a $1 million prize. Of the original seven, only the Poincaré conjecture has been solved. P versus NP has the most direct connection to everyday computing.

Q: What did Turing's machine look like? A: It was a mathematical abstraction rather than a built device. A Turing machine has an unbounded tape divided into cells, a head that reads and writes symbols and moves one cell at a time, and a finite table of rules that says what to do given the current state and symbol. Real computers have finite memory, but they can compute the same class of functions as Turing machines, assuming the Church-Turing thesis.

Q: Who was the greatest mathematician of all time? A: Historians do not agree, and the question depends on the criteria. Archimedes, Euler, and Gauss appear on most lists. Euler is notable for sheer output, with 866 publications. Gauss worked across number theory, geometry, and statistics. Archimedes anticipated integration by about 1,900 years. Later figures such as Emmy Noether and Srinivasa Ramanujan are often added; see our profiles of Emmy Noether and Srinivasa Ramanujan.

Q: Did Pythagoras prove the Pythagorean theorem? A: Nobody can say for certain. The relationship was known to Babylonian scribes about a thousand years earlier, and Pythagoras left no writings. MacTutor says only that he "may have been the first to prove it." Attribution rests on authors who wrote centuries after his death.

Sources

MacTutor History of Mathematics Archive, University of St Andrews: biographies of Al-Khwarizmi, Euclid, Archimedes, Thales, Pythagoras, Omar Khayyam, Cardano, Bombelli, Descartes, Fermat, Euler, Galois, Cantor, Gödel, and Turing
MacTutor, A history of Zero and Babylonian mathematics
Clay Mathematics Institute, Millennium Prize Problems
Wikipedia: Plimpton 322, Leibniz-Newton calculus controversy, Leonhard Euler, Hilbert's second problem, Alan Turing

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Before the Greeks: calculation without proof