The History of Mathematics and Its Applications

The Map of Mathematics

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BCE the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by <a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Ancient Egypt</a> and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.The earliest mathematical texts available are from <a href='https://history-maps.com/story/History-of-Iraq' rel='noopener noreferrer' target='_blank'>Mesopotamia</a> and Egypt – Plimpton 322 (Babylonian c. 2000 – 1900 BCE),[1] the Rhind Mathematical Papyrus (Egyptian c. 1800 BCE)[2] and the Moscow Mathematical Papyrus (Egyptian c. 1890 BCE). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.The study of mathematics as a 'demonstrative discipline' began in the 6th century BCE with the Pythagoreans, who coined the term 'mathematics' from the ancient <a href='https://history-maps.com/story/History-of-Greece' rel='noopener noreferrer' target='_blank'>Greek</a> μάθημα (mathema), meaning 'subject of instruction'.[3] Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.[4] Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. <a href='https://history-maps.com/story/History-of-China' rel='noopener noreferrer' target='_blank'>Chinese</a> mathematics made early contributions, including a place value system and the first use of negative numbers.[5] The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium CE in <a href='https://history-maps.com/story/History-of-India' rel='noopener noreferrer' target='_blank'>India</a> and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī.[6] <a href='https://history-maps.com/story/History-of-Iran' rel='noopener noreferrer' target='_blank'>Islamic</a> mathematics, in turn, developed and expanded the mathematics known to these civilizations.[7] Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of <a href='https://history-maps.com/story/History-of-Mexico' rel='noopener noreferrer' target='_blank'>Mexico</a> and Central America, where the concept of zero was given a standard symbol in Maya numerals.Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation.[8] Beginning in Renaissance <a href='https://history-maps.com/story/History-of-Italy' rel='noopener noreferrer' target='_blank'>Italy</a> in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century.

Ancient <a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Egyptian</a> mathematics was developed and used in Ancient Egypt c. 3000 to c. 300 BCE, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical problems, often involving multiplication and fractions. Evidence for Egyptian mathematics is limited to a scarce amount of surviving sources written on papyrus. From these texts it is known that ancient Egyptians understood concepts of geometry, such as determining the surface area and volume of three-dimensional shapes useful for architectural engineering, and algebra, such as the false position method and quadratic equations.Written evidence of the use of mathematics dates back to at least 3200 BCE with the ivory labels found in Tomb U-j at Abydos. These labels appear to have been used as tags for grave goods and some are inscribed with numbers.[18] Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners.[19] Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.[20] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.[20]The earliest true mathematical documents date to the 12th Dynasty (c. 1990–1800 BCE). The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus 6619 all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (c. 1650 BCE) is said to be based on an older mathematical text from the 12th dynasty.[22]

Ancient Egyptian Mathematics

The ancient <a href='https://history-maps.com/story/History-of-Iraq' rel='noopener noreferrer' target='_blank'>Sumerians</a> of Mesopotamia developed a complex system of metrology from 3000 BCE. From 2600 BCE onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[9]

Sumerian Mathematics

The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, <a href='https://history-maps.com/story/History-of-China' rel='noopener noreferrer' target='_blank'>China</a>, and Russia, millennia before the adoption of the Hindu-Arabic numeral system.[127] The exact origin of the abacus has not yet emerged. It consists of rows of movable beads, or similar objects, strung on a wire. They represent digits. One of the two numbers is set up, and the beads are manipulated to perform an operation such as addition, or even a square or cubic root. The <a href='https://history-maps.com/story/History-of-Iraq' rel='noopener noreferrer' target='_blank'>Sumerian</a> abacus appeared between 2700 and 2300 BCE. It held a table of successive columns which delimited the successive orders of magnitude of their sexagesimal (base 60) number system.[128]

Abacus

<a href='https://history-maps.com/story/History-of-Iraq' rel='noopener noreferrer' target='_blank'>Babylonian</a> mathematics were written using a sexagesimal (base-60) numeral system.[12] From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.[12] Also, unlike the <a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Egyptians</a>, <a href='https://history-maps.com/story/History-of-Greece' rel='noopener noreferrer' target='_blank'>Greeks</a>, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the decimal system.[13] The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation.[13] The notational system of the Babylonians was the best of any civilization until the Renaissance,[14] and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.[14] The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.[13] By the <a href='https://history-maps.com/story/Seleucid-Empire' rel='noopener noreferrer' target='_blank'>Seleucid</a> period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.[13] This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.[13]Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs.[15] The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.[16] Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem.[17] However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.[13]They also used a form of Fourier analysis to compute an ephemeris (table of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.[11] To make calculations of the movements of celestial bodies, the Babylonians used basic arithmetic and a coordinate system based on the ecliptic, the part of the heavens that the sun and planets travel through.

Old Babylonian Mathematics

Greek mathematics allegedly began with Thales of Miletus (c. 624-548 BCE). Very little is known about his life, although it is generally agreed that he was one of the Seven Wise Men of Greece. According to Proclus, he traveled to Babylon from where he learned mathematics and other subjects, coming up with the proof of what is now called Thales' Theorem.[23] Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.[30]

Thales's Theorem

An equally enigmatic figure is Pythagoras of Samos (c. 580–500 BCE), who supposedly visited <a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Egypt</a> and <a href='https://history-maps.com/story/History-of-Iraq' rel='noopener noreferrer' target='_blank'>Babylon</a>,[24] and ultimately settled in Croton, Magna Graecia, where he started a kind of brotherhood. Pythagoreans supposedly believed that 'all is number' and were keen in looking for mathematical relations between numbers and things.[25] Pythagoras himself was given credit for many later discoveries, including the construction of the five regular solids.Almost half of the material in Euclid's Elements is customarily attributed to the Pythagoreans, including the discovery of irrationals, attributed to Hippasus (c. 530–450 BCE) and Theodorus (fl. 450 BCE).[26] It was the Pythagoreans who coined the term 'mathematics', and with whom the study of mathematics for its own sake begins. The greatest mathematician associated with the group, however, may have been Archytas (c. 435-360 BCE), who solved the problem of doubling the cube, identified the harmonic mean, and possibly contributed to optics and mechanics.[26] Other mathematicians active in this period, not fully affiliated with any school, include Hippocrates of Chios (c. 470–410 BCE), Theaetetus (c. 417–369 BCE), and Eudoxus (c. 408–355 BCE).

Pythagoras

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),[39] who probably discovered them while identifying sides of the pentagram.[40] The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus, in the 5th century BCE, however, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.'[41] Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.

Discovery of Irrational Numbers

Plato is important in the history of mathematics for inspiring and guiding others.[31] His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BCE, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came.[32] Plato also discussed the foundations of mathematics,[33] clarified some of the definitions (e.g. that of a line as 'breadthless length'), and reorganized the assumptions.[34] The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.[32]

Plato

The oldest existent work on geometry in <a href='https://history-maps.com/story/History-of-China' rel='noopener noreferrer' target='_blank'>China</a> comes from the philosophical Mohist canon c. 330 BCE, compiled by the followers of Mozi (470–390 BCE). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.[77] It also defined the concepts of circumference, diameter, radius, and volume.[78]

Chinese Geometry

The Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BCE and is perhaps the oldest surviving mathematical text of <a href='https://history-maps.com/story/History-of-China' rel='noopener noreferrer' target='_blank'>China</a>.[68] Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called 'rod numerals' in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.[69] Thus, the number 123 would be written using the symbol for '1', followed by the symbol for '100', then the symbol for '2' followed by the symbol for '10', followed by the symbol for '3'. This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the <a href='https://history-maps.com/story/History-of-India' rel='noopener noreferrer' target='_blank'>Indian</a> numeral system.[76] Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the suan pan, or Chinese abacus. It is presumed that officials used the multiplication table to calculate land surface area, yields of crops and the amounts of taxes owed.[68]

Chinese Decimal System

The Hellenistic era began in the late 4th century BCE, following <a href='https://history-maps.com/story/Conquests-of-Alexander-the-Great' rel='noopener noreferrer' target='_blank'>Alexander the Great's</a> conquest of the Eastern Mediterranean, <a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Egypt</a>, <a href='https://history-maps.com/story/History-of-Iraq' rel='noopener noreferrer' target='_blank'>Mesopotamia</a>, the <a href='https://history-maps.com/story/History-of-Iran' rel='noopener noreferrer' target='_blank'>Iranian</a> plateau, Central Asia, and parts of <a href='https://history-maps.com/story/History-of-India' rel='noopener noreferrer' target='_blank'>India</a>, leading to the spread of the Greek language and culture across these regions. Greek became the lingua franca of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics.[27]Greek mathematics and astronomy reached its acme during the Hellenistic and early Roman periods, and much of the work represented by authors such as Euclid (fl. 300 BCE), Archimedes (c. 287–212 BCE), Apollonius (c. 240–190 BCE), Hipparchus (c. 190–120 BCE), and Ptolemy (c. 100–170 CE) was of a very advanced level and rarely mastered outside a small circle.Several centers of learning appeared during the Hellenistic period, of which the most important one was the Mouseion in Alexandria, Egypt, which attracted scholars from across the Hellenistic world (mostly Greek, but also Egyptian, Jewish, Persian, among others).[28] Although few in number, Hellenistic mathematicians actively communicated with each other; publication consisted of passing and copying someone's work among colleagues.[29]

Hellenistic Greek Mathematics

In the 3rd century BCE, the premier center of mathematical education and research was the Musaeum of Alexandria.[36] It was there that Euclid (c. 300 BCE) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time.[35]Considered the 'father of geometry', Euclid is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[37] In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry,[37] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.[38]The Euclidean algorithm is one of the oldest algorithms in common use.[93] It appears in Euclid's Elements (c. 300 BCE), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm is formulated for integers, whereas in Book 10, it is formulated for lengths of line segments. Centuries later, Euclid's algorithm was discovered independently both in India and in China,[94] primarily to solve Diophantine equations that arose in astronomy and making accurate calendars.

Euclid

Archimedes of Syracuse is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,[42] Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.[43] These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.[44]Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever,[45] the widespread use of the concept of center of gravity,[46] and the enunciation of the law of buoyancy or Archimedes' principle.Archimedes died during the <a href='https://history-maps.com/story/Second-Punic-War' rel='noopener noreferrer' target='_blank'>siege of Syracuse</a>, when he was killed by a Roman soldier despite orders that he should not be harmed.

Archimedes

Apollonius of Perga (c. 262-190 BCE) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.[47] He also coined the terminology in use today for conic sections, namely parabola ('place beside' or 'comparison'), 'ellipse' ('deficiency'), and 'hyperbola' ('a throw beyond').[48] His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.[49] While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.[50]

Apollonius's Parabola

In 212 BCE, the Emperor Qin Shi Huang commanded all books in the <a href='https://history-maps.com/story/Qin-dynasty' rel='noopener noreferrer' target='_blank'>Qin Empire</a> other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient <a href='https://history-maps.com/story/History-of-China' rel='noopener noreferrer' target='_blank'>Chinese</a> mathematics before this date. After the book burning of 212 BCE, the <a href='https://history-maps.com/story/Han-Dynasty' rel='noopener noreferrer' target='_blank'>Han dynasty</a> (202 BCE–220 CE) produced works of mathematics which presumably expanded on works that are now lost. After the book burning of 212 BCE, the Han dynasty (202 BCE–220 CE) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by CE 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles.[79] It created mathematical proof for the Pythagorean theorem,[81] and a mathematical formula for Gaussian elimination.[80] The treatise also provides values of π,[79] which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 CE) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724,[82] as well as 3.162 by taking the square root of 10.[83]Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art but may well contain much older material.[84] The mathematician Liu Hui (c. 3rd century) established rules for the addition and subtraction of negative numbers.

Nine Chapters on the Mathematical Art

The 3rd century BCE is generally regarded as the 'Golden Age' of Greek mathematics, with advances in pure mathematics henceforth in relative decline.[51] Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers.[51] Hipparchus of Nicaea (c. 190–120 BCE) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.[52]

Hipparchus & Trigonometry

In the 2nd century CE, the Greco-<a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Egyptian</a> astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.[63]

Almagest of Ptolemy

In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The earliest known statement of the theorem is by the Chinese mathematician Sun-tzu in the Sun-tzu Suan-ching in the 3rd century CE.

Chinese Remainder Theorem

Following a period of stagnation after Ptolemy, the period between 250 and 350 CE is sometimes referred to as the 'Silver Age' of Greek mathematics.[53] During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as 'Diophantine analysis'.[54] The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations.[55] The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).[56] Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.[55]

Diophantine Analysis

Ancient <a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Egyptian</a> numerals were of base 10. They used hieroglyphs for the digits and were not positional. By the middle of the 2nd millennium BCE, the Babylonian mathematics had a sophisticated base 60 positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system.The concept of zero as a written digit in the decimal place value notation was developed in India.[65] A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.[66] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from CE 224–383, CE 680–779, and CE 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together.[67] Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and incorrectly division by zero as:<blockquote>A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.</blockquote>

Story of Zero

The first woman mathematician recorded by history was Hypatia of Alexandria (CE 350–415). She wrote many works on applied mathematics. Because of a political dispute, the <a href='https://history-maps.com/story/History-of-Christianity' rel='noopener noreferrer' target='_blank'>Christian</a> community in Alexandria had her stripped publicly and executed. Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus, Simplicius and Eutocius.[57] Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by the emperor <a href='https://history-maps.com/story/Byzantine-Empire-Justinian-dynasty' rel='noopener noreferrer' target='_blank'>Justinian</a> in 529 CE is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia.[58] Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.[59]

Hypatia

The modern sine convention is first attested in the Surya Siddhanta (showing strong Hellenistic influence)[64], and its properties were further documented by the 5th century (CE) Indian mathematician and astronomer Aryabhata.[60] The Surya Siddhanta describes rules to calculate the motions of various planets and the moon relative to various constellations, diameters of various planets, and calculates the orbits of various astronomical bodies. The text is known for some of earliest known discussion of sexagesimal fractions and trigonometric functions.[61]

Indian Trigonometry

Around 500 CE, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration.[62] Though about half of the entries are wrong, it is in the Aryabhatiya that the decimal place-value system first appears.

Indian Decimal System

In the 9th century, the mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote an important book on the Hindu–Arabic numerals and one on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[87] and he was the first to teach algebra in an elementary form and for its own sake.[88] He also discussed the fundamental method of 'reduction' and 'balancing', referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.[89] His algebra was also no longer concerned 'with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study.' He also studied an equation for its own sake and 'in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.'[90]

Muhammad ibn Musa al-Khwarizmi

Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ was a prominent <a href='https://history-maps.com/story/History-of-Egypt' rel='noopener noreferrer' target='_blank'>Egyptian</a> mathematician during the Islamic Golden Age. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations.[91] His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.[92]

Abu Kamil

In the Pre-Columbian Americas, the Maya civilization that flourished in <a href="https://history-maps.com/story/History-of-Mexico" target="_blank" rel="noopener noreferrer">Mexico</a> and Central America during the 1st millennium CE developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, <a href="https://history-maps.com/story/History-of-Egypt" target="_blank" rel="noopener noreferrer">Egyptian</a>, and Asian mathematics.[92] Maya numerals used a base of twenty, the vigesimal system, instead of a base of ten that forms the basis of the decimal system used by most modern cultures.[92] The Maya used mathematics to create the Maya calendar as well as to predict astronomical phenomena in their native Maya astronomy.[92] While the concept of zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.[92]

Mayan Mathematics

Abū Bakr Muḥammad ibn al Ḥasan al-Karajī was a 10th-century <a href='https://history-maps.com/story/History-of-Iran' rel='noopener noreferrer' target='_blank'>Persian</a> mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: Al-Badi' fi'l-hisab (Wonderful on calculation), Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi fi'l-hisab (Sufficient on calculation).Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus) but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala, which survives from the medieval era in at least four copies. His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.

Al-Karaji

The high-water mark of <a href='https://history-maps.com/story/History-of-China' rel='noopener noreferrer' target='_blank'>Chinese</a> mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.[70] The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.[71] The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (CE 1238–1298).[71]<a href='https://history-maps.com/story/History-of-Japan' rel='noopener noreferrer' target='_blank'>Japanese</a> mathematics, <a href='https://history-maps.com/story/History-of-Korea' rel='noopener noreferrer' target='_blank'>Korean</a> mathematics, and <a href='https://history-maps.com/story/History-of-Vietnam' rel='noopener noreferrer' target='_blank'>Vietnamese</a> mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian-based East Asian cultural sphere.[72] Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's <a href='https://history-maps.com/story/Ming-dynasty' rel='noopener noreferrer' target='_blank'>Ming dynasty</a> (1368–1644).[73] For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed the Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers.[74] Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers, whereas in Japan it was more prevalent in the realm of private schools.[75]

Chinese Algebra

Europeans learned of Arabic numerals about the 10th century, though their spread was a gradual process. Two centuries later, in the Algerian city of Béjaïa, the Italian scholar Fibonacci first encountered the numerals; his work was crucial in making them known throughout Europe. European trade, books, and colonialism helped popularize the adoption of Arabic numerals around the world. The numerals have found worldwide use significantly beyond the contemporary spread of the Latin alphabet, and have become common in the writing systems where other numeral systems existed previously, such as Chinese and Japanese numerals. The first mentions of the numerals from 1 to 9 in the West are found in the Codex Vigilanus of 976, an illuminated collection of various historical documents covering a period from antiquity to the 10th century in Hispania.[68]

Hindu-Arabic Numerals

In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.[95] These and other new sources sparked a renewal of mathematics.Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using Roman numerals.) There, he observed a system of arithmetic (specifically algorism) which due to the positional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. He soon realised the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system. Leonardo wrote Liber Abaci in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the Fibonacci sequence (known to Indian mathematicians for hundreds of years before that)[96] which Fibonacci used as an unremarkable example.

Leonardo Fibonacci

Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.[86] The Kerala school has made a number of contributions to the fields of infinite series and calculus.

Infinite Series

The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the 'problem of points').[105] Christiaan Huygens published a book on the subject in 1657.[106] In the 19th century, what is considered the classical definition of probability was completed by Pierre Laplace.[107]Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory.This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.[108]

Probability Theory

The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe. Galileo observed the moons of Jupiter in orbit about that planet, using a telescope based Hans Lipperhey's. Tycho Brahe had gathered a large quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinates.

Logarithms

The Cartesian refers to the French mathematician and philosopher René Descartes, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery.[109] The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.[110]Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.[111]The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[112] The two-coordinate description of the plane was later generalized into the concept of vector spaces.[113]Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.

Cartesian Coordinate System

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.[97]Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.[98] Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in science, engineering, and social science.Isaac Newton developed the use of calculus in his laws of motion and universal gravitation.These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.[99]Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today.[100] The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.

Calculus

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.[114] This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy[115] and L'Huilier,[116] and represents the beginning of the branch of mathematics known as topology.

Graph Theory

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[124] Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.[125] Some authors[126] attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his 'The Doctrine of Chances' the study of the coefficients in the binomial expansion of (a + b)n.

Normal Distribution

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation 'our jewel' and 'the most remarkable formula in mathematics'. When x = &pi;, Euler's formula may be rewritten as ei&pi; + 1 = 0 or ei&pi; = -1, which is known as Euler's identity.

Euler's Formula

In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.[122] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.[122] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.

Bayes' Theorem

In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.The law was first[101] formulated by Joseph-Louis Lagrange in 1773,[102] followed by Carl Friedrich Gauss in 1835,[103] both in the context of the attraction of ellipsoids. It is one of Maxwell's equations, which forms the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law,[104] and vice versa.

Gauss's Law

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups.

Group Theory

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis.

Fourier Analysis

Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.The equations have two major variants. The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The macroscopic equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic-scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials. The term 'Maxwell's equations' is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved spacetime, commonly used in high-energy and gravitational physics, are compatible with general relativity. In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences.The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.

Maxwell's Equations

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

Set Theory

Game theory is the study of mathematical models of strategic interactions among rational agents.[117] It has applications in all fields of social science, as well as in logic, systems science and computer science. The concepts of game theory are used extensively in economics as well.[118] The traditional methods of game theory addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by the losses and gains of other participants. In the 21st century, the advanced game theories apply to a wider range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers.Game theory did not exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928.[119] Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern.[120] The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.[121]

Game Theory

Story of Mathematics

Thales of Miletus (c. 626/623 – c. 548/545 BCE) was an Ancient Greek pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He is recognized as one of the Seven Sages of Greece and is often regarded as the first philosopher in the Greek tradition, moving away from mythological explanations of the world to natural philosophy. Thales proposed that all of nature is based on a single ultimate substance, which he theorized to be water, and he believed the Earth floated on water.Thales is credited with significant contributions to mathematics and science. He is associated with Thales's theorem, which pertains to the properties of triangles and circles. He reportedly calculated the heights of the pyramids and the distance of ships from the shore. As an astronomer, Thales is said to have predicted a solar eclipse in 585 BCE and is attributed with identifying the constellation Ursa Major and understanding the timings of solstices and equinoxes. He also demonstrated engineering skills by enabling the Lydian army to cross the Halys River.The primary source of information about Thales's life is Diogenes Laërtius, who wrote in the third century CE, long after Thales's death. According to Herodotus, Thales predicted the solar eclipse during a battle between the Lydians and the Medes. The historian Apollodorus of Athens placed Thales's birth around 625 BCE, based on the assumption that he reached his peak at age 40.Thales's ancestry is debated, with some sources suggesting he was of Phoenician descent, while others assert he was a native Milesian of noble birth. Diogenes Laërtius mentions conflicting accounts regarding Thales's marital status, with some stating he married and had a son, while others claim he remained single and later adopted a nephew.Thales is believed to have traveled to Egypt, where he may have learned geometry, although evidence of such visits is debated among scholars. He is also associated with Babylonian mathematics, with some historians arguing that Babylonian influence on Greek mathematics is evident, while others dispute this claim.As one of the Seven Sages, Thales was known for his wisdom and practical advice. He is credited with the maxim 'know thyself' and was recognized for his contributions to philosophy and mathematics. Thales's theories included the idea that all things are full of gods, suggesting a belief in a divine presence in nature.Thales's death is attributed to heat stroke and thirst while watching the Olympic games, as noted by Diogenes Laërtius. His influence on subsequent Greek philosophers, particularly Anaximander and Anaximenes, is significant, marking the beginning of the Milesian school of thought. Thales's legacy as a foundational figure in Western philosophy and science continues to be acknowledged.

Thales of Miletus

Pythagoras of Samos (c. 570 – c. 495 BCE) was an ancient Greek philosopher and the founder of Pythagoreanism. He is known for his influence on mathematics, philosophy, and religious thought, particularly in Magna Graecia. Pythagoras traveled to Croton in southern Italy around 530 BCE, where he established a school that emphasized secrecy and communal living. His teachings included the concept of metempsychosis, the belief in the immortality of the soul and its reincarnation into new bodies. He is also associated with the idea of musica universalis, which posits that celestial bodies move according to mathematical ratios, creating a cosmic harmony.Pythagoras is credited with various mathematical and scientific discoveries, including the Pythagorean theorem, the theory of proportions, and the identification of the morning and evening stars as the planet Venus. However, modern scholars debate the extent of his contributions, suggesting that many achievements attributed to him may have originated from earlier sources or his followers, such as Hippasus and Philolaus. Pythagoras was reputed to be the first to call himself a philosopher, meaning 'lover of wisdom.'Born on Samos, Pythagoras was the son of Mnesarchus, a gem-engraver or merchant. His early life coincided with a period of cultural flourishing on the island, which was a hub for trade and intellectual exchange. Influences from Near Eastern cultures likely shaped his philosophical outlook. Pythagoras is said to have traveled extensively, learning from various cultures, including the Egyptians, Persians, and possibly even Indian sages. He also studied under notable Greek thinkers, including Pherecydes of Syros and possibly Thales and Anaximander.Upon arriving in Croton, Pythagoras quickly gained political influence and served as an advisor to local elites. His followers, known for their ascetic lifestyle, faced opposition from democratic factions in the city. Following a military victory by Croton over Sybaris around 510 BCE, tensions escalated, leading to violent confrontations. Pythagoras's fate during this period is uncertain; accounts suggest he may have been killed or escaped to Metapontum, where he died.Pythagoreanism was characterized by a communal lifestyle, with members sharing possessions and adhering to strict ethical guidelines. The community was divided into two groups: the mathematikoi, who focused on intellectual pursuits, and the akousmatikoi, who emphasized religious teachings. Pythagorean teachings included dietary restrictions, such as abstaining from certain foods, possibly linked to beliefs in metempsychosis.Legends surrounding Pythagoras depict him as a wonder-worker with supernatural abilities. He was said to have had a golden thigh and to have performed miraculous feats, such as calming animals and predicting events. Despite the lack of authentic writings, Pythagoras's teachings and the Pythagorean way of life significantly influenced later philosophers, including Plato and Aristotle, and continued to resonate throughout antiquity and the Middle Ages.In later periods, Pythagorean ideas were revived and integrated into various philosophical and scientific frameworks. His influence extended to early Christian thought, where he was respected for his moral teachings and belief in the immortality of the soul. Pythagoreanism also impacted the development of mathematics and music, with later scholars attributing significant discoveries to him, although many of these contributions are now debated. Pythagoras's legacy endures in various fields, including philosophy, mathematics, and esotericism, reflecting the lasting impact of his teachings on Western thought.

Euclid, an ancient Greek mathematician active around 300 BCE, is recognized as a prominent figure in geometry and logic. He is best known for his work, the Elements, which laid the foundational principles of geometry and influenced the field for centuries. Euclid's approach combined earlier theories from mathematicians such as Eudoxus, Hippocrates, Thales, and Theaetetus, creating a systematic framework now known as Euclidean geometry. His contributions place him alongside Archimedes and Apollonius of Perga as one of the greatest mathematicians of antiquity.Little is known about Euclid's life, with most information derived from later scholars like Proclus and Pappus of Alexandria. It is generally accepted that he spent his career in Alexandria, living after the time of Plato's students and before Archimedes. Speculation suggests he may have studied at the Platonic Academy and later taught at the Musaeum in Alexandria, a center of learning established under Ptolemy I. The exact dates of his birth and death remain uncertain, though he is believed to have died around 270 BCE.Euclid's Elements consists of thirteen books that systematically present geometric principles, beginning with definitions and postulates that serve as the basis for subsequent theorems. The work is divided into three main topics: plane geometry, number theory, and solid geometry. The first book introduces fundamental concepts and includes the famous parallel postulate, while later books explore number theory and solid geometry. The Elements is characterized by its logical structure, with propositions supported by rigorous proofs and diagrams.In addition to the Elements, Euclid authored several other works, including Optics, which addresses perspective, and Data, which discusses the implications of given information in geometry. Other texts attributed to him, such as On Divisions of Figures and Catoptrics, have uncertain authorship. Several works are lost, including Conics, which dealt with conic sections, and Pseudaria, which focused on common fallacies in geometry.Euclid's influence on mathematics is profound, with the Elements being one of the most published and studied texts in history. It served as a standard textbook in both the Medieval Arab and Latin worlds, shaping mathematical education for centuries. The legacy of Euclid endures, as his name is often synonymous with geometry itself, and his systematic approach to mathematics laid the groundwork for future developments in the field.

Archimedes of Syracuse (c. 287 – c. 212 BCE) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from Syracuse, Sicily. His life details are sparse, but his surviving works establish him as a prominent figure in classical antiquity and one of the greatest mathematicians. Archimedes made significant contributions to geometry, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola, employing the method of exhaustion and concepts of infinitesimals. He approximated pi (π), defined the Archimedean spiral, and developed a system for expressing large numbers through exponentiation.In the realm of physics, Archimedes applied mathematics to physical phenomena, formulating the law of the lever and the principle of buoyancy, known as Archimedes' principle. He conducted astronomical measurements, including the apparent diameter of the Sun, and is credited with constructing a planetarium device to model celestial movements. Archimedes also designed innovative machines, such as the screw pump and various war machines to defend Syracuse from Roman invasion.Archimedes died during the siege of Syracuse, reportedly killed by a Roman soldier despite orders for his protection. Cicero later described visiting his tomb, which featured a sphere and cylinder, symbols of his mathematical achievements. Although his inventions gained recognition, his mathematical writings were less known in antiquity. They were first compiled comprehensively around 530 CE by Isidore of Miletus, with Eutocius' commentaries later broadening their readership. Archimedes' works were translated into Arabic in the 9th century and into Latin in the 12th century, influencing scientists during the Renaissance and the Scientific Revolution. The discovery of the Archimedes Palimpsest in 1906 revealed lost works and insights into his methodologies.Archimedes was born around 287 BCE, likely in Syracuse, a self-governing colony in Magna Graecia. His father, Phidias, was an astronomer, but little else is known about his family. Plutarch noted a possible connection to King Hiero II of Syracuse, though other sources suggest he may have come from humble origins. His educational background remains uncertain, but he maintained correspondence with scholars in Alexandria, indicating a network of intellectual exchange.One notable story involves the 'wreath problem,' where Hiero II suspected a goldsmith of fraud. Archimedes discovered a method to determine the purity of a crown by measuring water displacement, famously exclaiming 'Eureka!' upon his realization. Another account describes his involvement in the design of the Syracusia, a large ship, where he demonstrated the principle of leverage to move heavy objects.During the Siege of Syracuse in 214 BCE, Archimedes reportedly oversaw the defense of the city using advanced war machines, including improved catapults and cranes. Accounts of his use of 'burning mirrors' to set Roman ships ablaze are less substantiated and debated among historians. His death occurred amidst the chaos of the siege, with various accounts detailing his final moments, often emphasizing his focus on mathematical problems even in peril.Archimedes' mathematical contributions include the method of exhaustion, which he used to derive the area of a circle and the volumes of various geometric shapes. He also developed a system for representing large numbers in his work, The Sand Reckoner, and posed complex problems, such as the Cattle Problem, which involved solving simultaneous equations. His treatises, including On the Sphere and Cylinder and Quadrature of the Parabola, showcase his innovative approaches to geometry and mechanics.Many of Archimedes' works have been lost or exist only in fragments. The Archimedes Palimpsest, discovered in 1906, contains several of his treatises, including the only surviving copy of On Floating Bodies. This palimpsest has provided valuable insights into his mathematical techniques and theories. Archimedes' legacy endures as a foundational figure in mathematics and science, influencing generations of scholars and researchers.

Claudius Ptolemy, known as Ptolemy, was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who lived from approximately 100 CE to the 160s or 170s CE. He authored several scientific treatises, three of which significantly influenced later Byzantine, Islamic, and Western European science. His most notable works include the Almagest, originally titled Mathēmatikḗ Syntaxis, which is a comprehensive astronomical treatise; the Geography, which discusses maps and geographic knowledge of the Greco-Roman world; and the Tetrábiblos, an astrological treatise that adapted horoscopic astrology to the Aristotelian natural philosophy of his time.Ptolemy's work was promoted by the Catholic Church, particularly his geocentric model of the Solar System, which remained influential throughout late antiquity and the Middle Ages. His writings, especially the Almagest, were widely copied and commented upon, although mastery of the mathematics required to fully understand them was limited. Ptolemy's models, particularly his use of epicycles, were complex theoretical constructs based on faith rather than empirical evidence.The exact date and place of Ptolemy's birth are unknown, though some later sources suggest he was born in Ptolemais Hermiou, a Greek city in Egypt. He lived in Alexandria, a major center of learning in the Roman province of Egypt. Ptolemy's Latin name, Claudius, indicates he was likely a Roman citizen, and he was familiar with Greek philosophy and Babylonian astronomical observations. He addressed a figure named Syrus in some of his works, suggesting a shared interest in astronomy.Ptolemy's death occurred in Alexandria, with estimates of the year ranging from around 165 CE to 175 CE. His name, Ptolemaeus, is a Latinized version of his Greek name, Πτολεμαῖος, and was common among the Macedonian elite during the time of Alexander the Great. The name Claudius indicates his Roman citizenship, possibly granted to an ancestor by the emperors Claudius or Nero.Ptolemy's Almagest is the only surviving comprehensive ancient treatise on astronomy, merging Babylonian arithmetical techniques with Greek geometrical models. He derived his models from over 800 years of astronomical observations, presenting them alongside tables for calculating celestial positions. The Almagest also includes a star catalogue, which influenced the modern system of constellations. For over a millennium, it served as the authoritative text on astronomy in Europe, the Middle East, and North Africa, preserved in Arabic manuscripts and translated into Latin during the 12th century.His Handy Tables provided astronomical data for calculating celestial positions and eclipses, while the Planetary Hypotheses offered a physical model of the universe with nested spheres. Other works include the Analemma, which specified the Sun's location, and the Geography, which detailed how to draw maps using geographical coordinates. The Geography cataloged 8,000 localities, significantly advancing the field of cartography.Ptolemy's Tetrabiblos systematically organized astrological knowledge, influencing astrological writers for centuries. His Harmonics explored the mathematics of musical scales, while the Optics examined properties of sight and vision. Ptolemy also engaged in philosophical discussions about knowledge and the human psyche, arguing for the superiority of mathematical knowledge over other forms.Despite the historical confusion surrounding his lineage, Ptolemy's contributions to science and philosophy have left a lasting legacy, influencing various fields and scholars throughout history.

Claudius Ptolemy

Aryabhata (476–550 CE) was a prominent mathematician and astronomer from ancient India. He is known for his works, including the Āryabhaṭīya and the Arya-siddhanta. The Āryabhaṭīya indicates that he was born in 476 CE, as he states he was 23 years old in 499 CE, which corresponds to 3600 years into the Kali Yuga. Aryabhata identified himself as a native of Kusumapura, which is associated with Pataliputra, present-day Patna in Bihar.There are alternative theories regarding his origins, with some sources suggesting he belonged to the Aśmaka region, located between the Narmada and Godavari rivers. Aryabhata's education likely took place in Kusumapura, where he may have led an institution. Although some speculate he was connected to the university of Nalanda, there is no definitive evidence to support this claim. Aryabhata is also credited with establishing an observatory at the Sun temple in Taregana, Bihar.The Āryabhaṭīya is the only surviving work attributed to Aryabhata, consisting of 108 verses and 13 introductory verses divided into four chapters. It covers various topics in mathematics and astronomy, including arithmetic, algebra, and trigonometry. Aryabhata's work introduced significant innovations, such as the concept of the relativity of motion, which he illustrated through the analogy of a moving boat. He also made advancements in the place-value system and approximated the value of π.Aryabhata's contributions to trigonometry included the introduction of the sine function, referred to as 'jya'. His methods for solving indeterminate equations and his results for summation of series were influential in the development of algebra. In astronomy, Aryabhata proposed a geocentric model of the solar system, where the Earth rotates on its axis, and he provided scientific explanations for solar and lunar eclipses.His calculations of the sidereal rotation and the length of the sidereal year were remarkably accurate for his time. Aryabhata's work had a lasting impact on Indian astronomy and mathematics, influencing subsequent scholars and cultures through translations, particularly during the Islamic Golden Age. His definitions of trigonometric functions and methods of astronomical calculations were widely adopted and contributed to the development of calendars in both India and the Islamic world.Aryabhata's legacy is commemorated in various ways, including the naming of India's first satellite and a lunar crater after him. His contributions continue to be recognized in modern educational institutions and competitions.

Aryabhata

Brahmagupta (c. 598 – c. 668 CE) was an Indian mathematician and astronomer born in Bhillamāla, in the Gurjaradesa region, during the reign of the Chavda dynasty. He was the son of Jishnugupta and identified as a Shaivite Hindu. Brahmagupta became a prominent figure in the Brahmapaksha school of astronomy, which was one of the four major schools in India at the time. He studied various traditional Siddhantas and the works of earlier astronomers, including Aryabhata I and Varahamihira.In 628 CE, at the age of 30, Brahmagupta composed the Brāhmasphuṭasiddhānta, a significant treatise that revised existing astronomical knowledge and introduced original concepts. This work consists of 24 chapters and 1008 verses, covering topics in astronomy, algebra, geometry, and trigonometry. He is credited with the first clear description of the quadratic formula and the concept of gravity as an attractive force, using the term 'gurutvākarṣaṇam' in Sanskrit.Later in his life, Brahmagupta moved to Ujjaini, a major center for astronomy, where he wrote the Khandakhadyaka in 665 CE, a practical manual for students. He died in 668 CE, likely in Ujjain.Brahmagupta's works include the Brāhmasphuṭasiddhānta and the Khandakhadyaka. His mathematical contributions included solutions to linear and quadratic equations, rules for arithmetic operations involving zero and negative numbers, and methods for generating Pythagorean triples. He also provided a formula for the area of cyclic quadrilaterals and made significant advancements in geometry and trigonometry, including the development of a sine table.His ideas influenced later mathematicians, including Bhāskara II, and his texts were translated into Arabic, contributing to the spread of Indian mathematics and astronomy in the Islamic world. Brahmagupta's work laid foundational principles that would be built upon by subsequent scholars in both India and the broader mathematical community.

Brahmagupta

Muhammad ibn Musa al-Khwarizmi, active from approximately 780 CE to 850 CE, was a mathematician during the Islamic Golden Age, known for his contributions to mathematics, astronomy, and geography. He worked at the House of Wisdom in Baghdad around 820 CE, a center for scientific study established by the Abbasid Caliph al-Ma'mun. Al-Khwarizmi's influential treatise on algebra, Al-Jabr, was compiled between 813 CE and 833 CE. This work presented systematic solutions for linear and quadratic equations and introduced methods such as 'reduction' and 'balancing.' His approach to algebra established it as an independent discipline, earning him the title of the father of algebra. The term 'algebra' itself is derived from the title of his treatise.Al-Khwarizmi's work also included a textbook on Indian arithmetic, which introduced the decimal-based positional number system to the Western world through Latin translations in the 12th century. His contributions to geography involved revising Ptolemy's 2nd-century work, providing longitudes and latitudes for various cities and localities. He produced astronomical tables and wrote about calendrical calculations, the astrolabe, and sundials. His accurate sine and cosine tables advanced trigonometry.Details about al-Khwarizmi's life are limited. He is believed to have originated from Khwarazm, a region in present-day Turkmenistan and Uzbekistan. His name indicates his geographical roots, and he was likely of Persian descent. Al-Khwarizmi's works were pivotal in the transmission of knowledge from the Islamic world to Europe, influencing mathematical education for centuries. His systematic methods and clear exposition laid the groundwork for future developments in mathematics.

Al-Khwarizmi

Leonardo Bonacci, commonly known as Fibonacci, was an Italian mathematician born around 1170 in the Republic of Pisa. He was the son of Guglielmo, a merchant and customs official who managed a trading post in Bugia, present-day Béjaïa, Algeria. Fibonacci traveled with his father during his youth and received his education in Bugia, where he was introduced to the Hindu-Arabic numeral system.Fibonacci's travels around the Mediterranean allowed him to interact with various merchants and learn different arithmetic systems. He recognized the advantages of the Hindu-Arabic numeral system, which facilitated calculations through a place-value system, unlike the Roman numerals prevalent at the time. In 1202, he completed his influential work, Liber Abaci, which introduced the Hindu-Arabic numeral system to Europe and demonstrated its practical applications in commerce, including bookkeeping, currency conversion, and interest calculations.Fibonacci was invited to the court of Emperor Frederick II, where he engaged in mathematical discussions and solved problems posed by John of Palermo, a member of the court. In 1240, the Republic of Pisa honored him with a salary for his contributions as an advisor on accounting and education. Fibonacci is believed to have died between 1240 and 1250 in Pisa.In Liber Abaci, Fibonacci presented the modus Indorum, detailing the Hindu-Arabic numeral system, which included ten digits and positional notation. The book was well-received and significantly impacted European mathematics, facilitating the transition from Roman numerals to a more efficient numerical system that supported the growth of banking and accounting.Additionally, Liber Abaci contained a problem about rabbit population growth, leading to the sequence of numbers now known as Fibonacci numbers. This sequence, where each number is the sum of the two preceding ones, was described by Fibonacci starting from 1 and 2, extending to the thirteenth term, 233. He did not reference the golden ratio associated with the sequence.Fibonacci's legacy includes numerous mathematical concepts named after him, such as the Brahmagupta–Fibonacci identity and the Fibonacci search technique. His works, including Liber Abaci and Practica Geometriae, contributed to the development of mathematics in Europe during the Middle Ages.

Fibonacci

Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī, known as Naṣīr al-Dīn al-Ṭūsī, was born in 1201 in Tus, Khorasan. He began his studies early, focusing on the Quran, Hadith, jurisprudence, logic, philosophy, mathematics, medicine, and astronomy. After losing his father, he pursued knowledge diligently, traveling to Nishapur to study under notable scholars such as Farid al-Din Damad and Muhammad Hasib. He also interacted with the Sufi master Attar of Nishapur and attended lectures by Qutb al-Din al-Misri.Al-Tusi authored numerous works across various fields, including mathematics, astronomy, and philosophy. He is credited with significant advancements in astronomy, creating accurate tables of planetary motion and critiques of Ptolemaic models. His contributions to trigonometry established it as an independent mathematical discipline. Al-Tusi's writings include the 'Nasirean Ethics' and 'Zij-i Ilkhani,' which contained astronomical tables and models that influenced later scholars.During the Mongol invasion, al-Tusi was associated with the Nizari Ismaili state. After the fall of Alamut in 1256, he became an advisor to Hulegu Khan, who respected his scientific expertise. Al-Tusi's role during the Mongol conquest of Baghdad in 1258 is controversial, with some sources suggesting he aided the Mongols. Following the siege, he was appointed to manage religious finances and promoted Twelver Shi'a Islam in Persia and Iraq.Al-Tusi's works, numbering around 150, include treatises on ethics, logic, and astronomy. His 'Sayr wa-Suluk' serves as an autobiography, while 'Akhlaq-i Nasiri' compares Islamic ethics with those of Aristotle and Plato. His influence extended to various fields, including biology and chemistry, where he proposed early concepts of conservation of mass and biological classification.Al-Tusi's legacy is marked by his contributions to science and philosophy, with his methods and ideas potentially influencing later European scholars, including Nicolaus Copernicus. His observatory in Maragheh became a center for astronomical research, and his techniques, such as the Tusi couple, were significant in the development of later astronomical models.

Nasir al-Din al-Tusi

René Descartes was born on 31 March 1596 in La Haye en Touraine, France. His mother died shortly after his birth, and he was raised by his father, Joachim, a member of the Parlement of Rennes, along with his grandmother and great-uncle. Descartes attended the Jesuit Collège Royal Henry-Le-Grand in La Flèche, where he was introduced to mathematics and physics. He graduated in 1614 and studied law at the University of Poitiers, earning degrees in canon and civil law in 1616, as per his father's wishes.In 1618, Descartes joined the Protestant Dutch States Army as a mercenary, where he studied military engineering and mathematics. He met Isaac Beeckman, who influenced his mathematical pursuits. During his military service, he experienced a series of dreams that he interpreted as a divine revelation, leading him to develop his ideas on analytic geometry and the application of mathematics to philosophy. He left the army in 1620 and traveled through various countries before settling in France, where he began writing his first essay on method.Descartes returned to the Dutch Republic in 1628, where he continued his studies at the University of Franeker and Leiden University. He published his major works during this period, including the 'Discourse on the Method' in 1637, which outlined his philosophical approach and introduced the famous phrase 'Cogito, ergo sum' ('I think, therefore I am'). He also published 'La Géométrie,' which laid the groundwork for analytic geometry. His works gained attention, but he faced challenges, including the condemnation of Cartesian philosophy at the University of Utrecht in 1643.In 1649, Descartes accepted an invitation from Queen Christina of Sweden to tutor her and establish a scientific academy. He moved to Sweden in winter, where he faced harsh conditions. Descartes contracted pneumonia and died on 11 February 1650. His remains were initially buried in Stockholm but were later transferred to France in 1666. His philosophical and mathematical contributions, particularly in epistemology and the development of analytic geometry, have had a lasting impact on modern philosophy and science.Descartes's philosophical work emphasized the importance of doubt and reason as foundations for knowledge. He proposed a dualistic view of the mind and body, arguing that they are distinct substances that interact. His theories on emotions and the nature of the soul were articulated in 'The Passions of the Soul,' published in 1649. He also contributed to various fields, including physics, optics, and meteorology, proposing theories that challenged existing beliefs and laid the groundwork for future scientific inquiry.In mathematics, Descartes introduced the convention of using letters to represent unknowns and developed the Cartesian coordinate system, which revolutionized the relationship between algebra and geometry. His work influenced later mathematicians, including Isaac Newton, and provided a foundation for calculus. Descartes's mechanical philosophy sought to explain natural phenomena through mathematical principles, marking a significant shift in scientific thought.

René Descartes

Pierre de Fermat was born on 17 August 1601 in Beaumont-de-Lomagne, France, to Dominique Fermat, a wealthy leather merchant, and Claire de Long. He had one brother and two sisters and was raised in his hometown. Fermat attended the University of Orléans, where he earned a bachelor in civil law in 1626. He later moved to Bordeaux, where he began serious mathematical research and interacted with local mathematicians, including Beaugrand. In 1629, he presented a restoration of Apollonius's work to a mathematician in Bordeaux and was influenced by François Viète's work.In 1630, Fermat purchased a councilor position at the Parlement de Toulouse, which he held for life. He married Louise de Long on 1 June 1631, and they had eight children, five of whom survived to adulthood. Fluent in six languages, Fermat was also a poet and a skilled Latinist. He communicated his mathematical findings primarily through letters, often without formal proofs. His work laid the groundwork for calculus, and he engaged in priority disputes with contemporaries like Descartes and Wallis.Fermat's significant contributions to analytic geometry included his manuscript 'Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum,' circulated in 1636. This work introduced a method for determining maxima, minima, and tangents to curves, which was akin to differential calculus. He was the first known individual to evaluate integrals of general power functions, aiding later developments in calculus by Newton and Leibniz.In number theory, Fermat studied various concepts, including Pell's equation and perfect numbers. He discovered Fermat's little theorem and developed a factorization method known as Fermat's factorization method. He also popularized the proof by infinite descent and proved Fermat's right triangle theorem, which included a corollary for Fermat's Last Theorem for the case n=4. His Last Theorem was noted in the margin of a copy of Diophantus' Arithmetica, with Fermat stating that the margin was too small for a proof.Fermat corresponded with Blaise Pascal in 1654, contributing to the foundation of probability theory. He conducted the first rigorous probability calculation, addressing a gambler's question about the odds of rolling dice. In optics, Fermat refined the principle of least time, stating that light travels between two points along the shortest time path, which contributed to the development of the principle of least action in physics.Pierre de Fermat died on 12 January 1665 in Castres, France. His legacy includes significant advancements in mathematics, particularly in number theory and calculus, and he is recognized as one of the leading mathematicians of the 17th century.

Pierre de Fermat

Sir Isaac Newton was born on 25 December 1642 (Julian calendar) at Woolsthorpe Manor in Lincolnshire, England. His father died three months before his birth, and he was raised by his maternal grandmother after his mother remarried. Newton attended The King's School in Grantham, where he developed a foundation in mathematics. He later enrolled at Trinity College, Cambridge, in 1661, initially as a subsizar. He earned a scholarship in 1664 and graduated with a BA in 1665. The university closed temporarily due to the Great Plague, prompting Newton to return home, where he made significant discoveries in mathematics and physics.During the years 1665 to 1667, Newton developed the foundations of calculus, formulated the laws of motion, and conducted experiments in optics. He returned to Cambridge in 1667 and was elected a fellow of Trinity College. In 1669, he succeeded Isaac Barrow as the Lucasian Professor of Mathematics. Newton's work in mathematics included the generalization of the binomial theorem and contributions to the study of cubic curves and finite differences. He also engaged in a dispute with Gottfried Wilhelm Leibniz over the invention of calculus, with both men independently developing the subject.Newton's major work, the 'Philosophiæ Naturalis Principia Mathematica,' was published in 1687. In this book, he articulated the laws of motion and universal gravitation, providing a comprehensive framework for classical mechanics. His work unified terrestrial and celestial mechanics, demonstrating that the same principles govern both. Newton's theories accounted for planetary motion, tides, and the precession of the equinoxes, establishing a heliocentric view of the Solar System.In addition to his contributions to physics, Newton made significant advancements in optics. He discovered that white light is composed of a spectrum of colors, which he demonstrated using prisms. His work on light culminated in the publication of 'Opticks' in 1704, where he explored the nature of color and light. He also constructed the first practical reflecting telescope, which addressed issues of chromatic aberration found in refracting telescopes.Newton's career included roles beyond academia. He served as a Member of Parliament for the University of Cambridge and was appointed Warden and later Master of the Royal Mint, where he implemented reforms to combat counterfeiting and improve coinage. He was knighted by Queen Anne in 1705 and became president of the Royal Society in 1703.Newton's religious views were unorthodox; he rejected the doctrine of the Trinity and engaged in extensive theological studies, much of which remained unpublished during his lifetime. He also pursued alchemical research, which he considered integral to his scientific inquiries.Newton died on 20 March 1727 (Julian calendar) in London. He was buried in Westminster Abbey, becoming the first scientist to receive such an honor. His legacy includes foundational contributions to mathematics, physics, and optics, and he is often regarded as one of the most influential figures in the history of science.

Isaac Newton

Gottfried Wilhelm Leibniz was born on 1 July 1646 in Leipzig, in the Electorate of Saxony, to Friedrich Leibniz and Catharina Schmuck. After his father's death when he was six, he was raised by his mother and had access to his father's extensive library, which fostered his early intellectual development. He enrolled at the University of Leipzig at age 14, earning a bachelor's degree in Philosophy by December 1662 and a master's degree in February 1664. He published a dissertation on philosophical problems related to law in December 1664 and completed his legal studies, obtaining a bachelor's degree in Law in September 1665. In 1666, he wrote his first book, De Arte Combinatoria, which included a geometrical proof of the existence of God.Leibniz's career began as a secretary to an alchemical society in Nuremberg, where he met Johann Christian von Boyneburg, who later introduced him to the Elector of Mainz. Leibniz was appointed assessor in the Court of Appeal in 1669 and began to gain recognition for his writings. He proposed a diplomatic plan to distract France from German territories, which gained the Elector's cautious support. In 1672, he moved to Paris, where he studied mathematics and physics under Christiaan Huygens and made significant contributions to calculus, independently of Isaac Newton. He also developed a calculating machine capable of performing basic arithmetic operations.In 1676, Leibniz accepted a position as a counselor to the House of Brunswick, where he served as a historian, political adviser, and librarian. He promoted various projects, including the use of windmills in mining operations, although these were not successful. His correspondence with influential figures, including the Electress Sophia of Hanover, helped him maintain a network of support. Leibniz's work on calculus continued, and he published important mathematical papers in the Acta Eruditorum, a journal he co-founded.Leibniz's philosophical contributions included his optimism, which posited that our world is the best possible world created by an all-knowing God. He was a key figure in 17th-century rationalism and idealism, and his ideas on monads and pre-established harmony influenced later philosophical thought. He engaged in a priority dispute with Newton over the invention of calculus, which overshadowed his later years. In 1712, he became an Imperial Court Councillor in Vienna, but his relationship with the new British King George I was strained due to the calculus controversy.Leibniz died on 14 November 1716 in Hanover, where he was buried without much recognition. His contributions to mathematics, philosophy, and various sciences were largely unappreciated during his lifetime. However, his reputation began to recover in the 18th century, and he became recognized as a significant figure in the history of philosophy and mathematics. His writings, which included works on law, ethics, and logic, were published posthumously, and his ideas continue to influence contemporary thought.

Gottfried Wilhelm Leibniz

Thomas Bayes (c. 1701 – 7 April 1761) was an English statistician, philosopher, and Presbyterian minister known for formulating Bayes' theorem. He was the son of Joshua Bayes, a Presbyterian minister, and likely born in Hertfordshire. Bayes came from a prominent nonconformist family in Sheffield. In 1719, he enrolled at the University of Edinburgh to study logic and theology. After returning around 1722, he assisted his father at his chapel in London before moving to Tunbridge Wells, Kent, around 1734, where he served as minister of the Mount Sion Chapel until 1752.During his lifetime, Bayes published two works: 'Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures' (1731) and 'An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst' (published anonymously in 1736). The latter defended the logical foundation of Isaac Newton's calculus against criticism from George Berkeley. Bayes was elected a Fellow of the Royal Society in 1742, likely due to his work on calculus, as he did not publish other mathematical works during his lifetime.In his later years, Bayes developed an interest in probability. Historian Stephen Stigler suggests that this interest arose while reviewing Thomas Simpson's work from 1755, while George Alfred Barnard posits that he learned from Abraham de Moivre's writings. Others speculate that he aimed to counter David Hume's arguments against miracles based on testimony. His findings on probability theory were shared in manuscript form with his friend Richard Price after his death.By 1755, Bayes was ill, and he died in Tunbridge Wells in 1761. He was buried in Bunhill Fields burial ground in Moorgate, London, where many nonconformists are interred. Bayes' solution to a problem of inverse probability was presented posthumously in 'An Essay Towards Solving a Problem in the Doctrine of Chances,' which was read to the Royal Society in 1763 and published in the Philosophical Transactions of the Royal Society of London the following year. This essay introduced a theorem concerning the conditional probability distribution of a binomial parameter.Bayes' theorem addresses problems of inverse probability, exemplified by questions regarding the likelihood of drawing specific colored balls from an urn. His essay contained solutions to problems posed by Abraham de Moivre. A paper on asymptotic series was also published posthumously.Bayesian probability, a term that emerged around 1950, refers to interpretations of probability as a measure of epistemic confidence rather than frequency. While Bayes himself may not have embraced this broad interpretation, his work laid the groundwork for modern Bayesian statistics, which has applications across various fields. The rigorous axiomatic framework for probability theory was developed much later, in the early to mid-20th century.

Thomas Bayes

Leonhard Euler was born on 15 April 1707 in Basel, Switzerland, to Paul III Euler and Marguerite Brucker. He was the eldest of four children. The family moved to Riehen shortly after his birth, where his father served as a pastor. Euler received early education in mathematics from his father and later attended the University of Basel at the age of 13, where he studied under Johann Bernoulli. He earned a Master of Philosophy in 1723 and later completed a dissertation on sound propagation in 1726.In 1727, Euler moved to Saint Petersburg to work at the Imperial Russian Academy of Sciences, where he quickly advanced to a position in the mathematics department. He became proficient in Russian and took on additional roles, including serving as a medic in the Russian Navy. Euler's career in Saint Petersburg was marked by significant contributions to various fields of mathematics and science, including mechanics and optics.In 1741, Euler accepted an invitation to join the Berlin Academy, where he remained for 15 years. During this period, he published numerous works, including 'Introductio in analysin infinitorum' and 'Institutiones calculi differentialis.' Euler also tutored Princess Friederike Charlotte and maintained connections with the Saint Petersburg Academy, publishing many papers during his time in Berlin.In 1766, Euler returned to Saint Petersburg after a period of political stability under Catherine the Great. He continued to produce significant work, despite deteriorating eyesight that led to near-total blindness. Euler's productivity remained high, and he produced an average of one mathematical paper per week.Euler's contributions to mathematics include the introduction of modern notation for functions, the use of the Greek letter π for the ratio of a circle's circumference to its diameter, and the development of graph theory through his solution to the Seven Bridges of Königsberg problem. He also made significant advances in number theory, including the formulation of the Euler product formula for the Riemann zeta function and the introduction of the totient function.Euler's work in physics encompassed mechanics, fluid dynamics, and optics. He reformulated Newton's laws of motion and contributed to the understanding of celestial mechanics. Euler's theories in optics supported the wave theory of light, and he predicted phenomena such as cavitation in fluid dynamics.Euler married Katharina Gsell in 1734, and they had 13 children, of whom five survived. After Katharina's death in 1773, he married her half-sister, Salome Abigail Gsell. Euler died on 18 September 1783 in Saint Petersburg after suffering a brain hemorrhage. He was buried next to his first wife. Euler's extensive body of work, comprising over 800 publications, has had a lasting impact on mathematics and science, establishing him as one of the most prolific mathematicians in history.

Leonhard Euler

Joseph-Louis Lagrange, born Giuseppe Luigi Lagrangia on 25 January 1736, was an Italian and naturalized French mathematician, physicist, and astronomer. He made notable contributions to analysis, number theory, and mechanics. In 1766, he succeeded Leonhard Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he remained for over twenty years, producing extensive work and receiving several prizes from the French Academy of Sciences. His treatise, Mécanique analytique, published in 1788, provided a comprehensive treatment of classical mechanics and laid the groundwork for mathematical physics in the nineteenth century.At the age of 51, Lagrange moved to Paris in 1787, becoming a member of the French Academy of Sciences. He played a significant role in the decimalization process during the French Revolution, became the first professor of analysis at the École Polytechnique in 1794, and was a founding member of the Bureau des Longitudes. He was appointed senator in 1799.Lagrange was a pioneer in the calculus of variations, deriving the Euler–Lagrange equations and extending the method to include constraints through Lagrange multipliers. He developed methods for solving differential equations, applied calculus to probability theory, and proved that every natural number can be expressed as a sum of four squares. His work in analytical functions laid foundations for group theory, and he made significant contributions to interpolation and Taylor's theorem. He studied the three-body problem and discovered the solutions known as Lagrangian points.Born in Turin, Lagrange was the first of eleven children. His family had French roots, and his father was a law doctor at the University of Turin. Initially, Lagrange showed little interest in mathematics until he encountered a paper by Edmond Halley at the age of seventeen, which sparked his passion for the subject. He became an assistant professor of mathematics at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught calculus and mechanics.In 1757, Lagrange co-founded the Turin Academy of Sciences, where many of his early writings were published. His work on the tautochrone problem and variational calculus gained the attention of Euler and Maupertuis, who encouraged him to move to Berlin. After initially declining, he accepted an invitation from Frederick the Great in 1766, following Euler's departure from Berlin. During his time in Berlin, Lagrange produced numerous papers and completed his Mécanique analytique.In 1786, Lagrange moved to Paris, where he was honored and became a member of the French Academy of Sciences. He experienced a period of melancholy but later married Renée-Françoise-Adélaïde Le Monnier. Despite the turmoil of the French Revolution, he was exempted from expulsion and continued to receive honors from various governments. He contributed to the development of the metric system and was involved in the reform of weights and measures.Lagrange was appointed to a mathematical chair at the École Normale in 1795 and later became a professor at the École Polytechnique. His lectures were well-regarded, although he faced challenges in teaching. He continued to work on his Mécanique analytique until his death on 10 April 1813 in Paris. He was buried in the Panthéon, and his contributions to mathematics and science are commemorated in various ways, including the naming of streets and celestial bodies after him.

Joseph-Louis Lagrange

Pierre-Simon, Marquis de Laplace, was born on 23 March 1749 in Beaumont-en-Auge, Normandy. His father, Pierre de Laplace, was a farmer and cider merchant. Laplace attended a school run by a Benedictine priory, where he was initially prepared for the priesthood. However, he shifted his focus to mathematics, studying at the University of Caen, where he was mentored by Christophe Gadbled and Pierre Le Canu. By the age of 20, he had already published a paper in the Royal Society of Turin's Mélanges.In 1771, Laplace moved to Paris with a letter of introduction to Jean le Rond d'Alembert. He quickly gained recognition for his mathematical abilities and was elected to the French Academy of Sciences in 1773. His early work included significant contributions to differential equations and the foundations of probability and statistics. He collaborated with Antoine Lavoisier on experiments related to heat and published their findings in 1783.Laplace's major work, Mécanique céleste, published between 1799 and 1825, translated classical mechanics into calculus-based formulations. He extended Newton's work and introduced concepts such as Laplace's equation and the Laplace transform, which became fundamental in mathematical physics. He also developed the nebular hypothesis regarding the formation of the Solar System and proposed ideas akin to black holes.In addition to his contributions to astronomy and mathematics, Laplace made advancements in the understanding of tides through his dynamic theory, which accounted for various physical factors affecting tidal movements. He formulated Laplace's tidal equations in 1776, which are still used in modern tidal analysis.Laplace's work in probability culminated in his 1812 publication, Théorie analytique des probabilités, where he laid the groundwork for Bayesian probability. He articulated principles of inductive reasoning and introduced the method of least squares, which became essential in statistical analysis.Throughout his career, Laplace held various positions, including a brief term as Minister of the Interior under Napoleon Bonaparte. He was later elevated to the Senate and received the title of marquis in 1817. Laplace died on 5 March 1827 in Paris, and his contributions to science have left a lasting legacy in multiple fields.

Pierre-Simon Laplace

Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel. His father, Gebhard Dietrich Gauss, held various low-status jobs, while his mother, Dorothea, was nearly illiterate. Gauss displayed prodigious mathematical talent from a young age, which led to his education at the Collegium Carolinum and later at the University of Göttingen, where he studied under notable figures such as Abraham Gotthelf Kästner and Georg Christoph Lichtenberg. He graduated with a Doctor of Philosophy in 1799 from the University of Helmstedt, after which he received financial support from the Duke of Brunswick to pursue his studies as a private scholar.During his early career, Gauss made significant contributions to mathematics, including the proof of the fundamental theorem of algebra and the publication of his influential work, Disquisitiones Arithmeticae, in 1801. He also developed the law of quadratic reciprocity and made advancements in number theory, including the construction of the heptadecagon and the theory of binary and ternary quadratic forms. His work laid the foundation for many mathematical concepts that bear his name.In 1807, Gauss became a professor and director of the Göttingen Observatory, where he focused on astronomy and geodesy. He played a crucial role in the identification of the dwarf planet Ceres and developed methods for calculating the orbits of celestial bodies, including the Gaussian gravitational constant and the method of least squares. Gauss's contributions to geodesy included leading a major survey of the Kingdom of Hanover and inventing the heliotrope for precise measurements.Gauss's research extended to various fields, including differential geometry, where he published groundbreaking work on the curvature of surfaces. He also explored non-Euclidean geometry, coining the term and conducting studies that would later influence the field. His work in physics included significant contributions to magnetism and electromagnetism, collaborating with Wilhelm Weber to establish foundational principles in these areas.Throughout his life, Gauss was known for his perfectionism and reluctance to publish incomplete work, leaving many of his findings to be published posthumously. He was not an enthusiastic teacher, preferring to focus on his research, although he mentored several students who became prominent mathematicians. Gauss married twice, first to Johanna Osthoff, who died shortly after childbirth, and later to Wilhelmine Waldeck, with whom he had three more children. He faced personal challenges, including the illnesses of family members and financial difficulties after the death of the Duke of Brunswick.Gauss died on 23 February 1855 in Göttingen, where he was buried. His legacy includes numerous mathematical and scientific concepts named after him, reflecting his profound impact on mathematics, astronomy, and physics.

Carl Friedrich Gauss

Carl Gustav Jacob Jacobi was born on 10 December 1804 in Potsdam to Ashkenazi Jewish parents. He was the second of four children of banker Simon Jacobi. His elder brother, Moritz, later became an engineer and physicist. Jacobi received his early education at home from his uncle Lehman, who taught him classical languages and mathematics. In 1816, he attended the Potsdam Gymnasium, where he excelled in various subjects and advanced to the senior year within six months. However, he had to wait until 1821 to enroll at the University of Berlin due to age restrictions. During this time, he began researching the quintic equation.At Berlin University, Jacobi initially pursued both philology and mathematics but found the mathematics curriculum too basic. He focused on advanced works by Euler, Lagrange, and Laplace. By 1823, he decided to concentrate solely on mathematics. He qualified to teach secondary school and was offered a position at the Joachimsthal Gymnasium but chose to aim for a university position instead. In 1825, he earned a Doctor of Philosophy degree with a dissertation on partial fraction decomposition and subsequently completed his habilitation. He converted to Christianity and began lecturing on the theory of curves and surfaces at the University of Berlin.Jacobi became a private lecturer in 1826, an extraordinary professor in 1827, and a tenured professor at Königsberg University in 1829, where he remained until 1842. He experienced a breakdown from overwork in 1843 and spent time in Italy to recover. Upon returning to Berlin, he lived as a royal pensioner. During the Revolution of 1848, he attempted to enter politics but faced setbacks after the revolution was suppressed. His royal grant was temporarily revoked but was later restored through the intervention of Alexander von Humboldt.Jacobi died on 18 February 1851 from smallpox. He was buried in a cemetery in the Kreuzberg section of Berlin, near the grave of astronomer Johann Encke. His birth name was Jacques Simon, which was later Germanized to Carl Gustav Jacob Jacobi and published in Latin as Carolus Gustavus Jacobus Jacobi. He is sometimes referred to as C. G. J. Jacobi.Jacobi made significant contributions to mathematics, particularly in elliptic functions, dynamics, differential equations, determinants, and number theory. His work on elliptic functions and their relation to the elliptic theta function was detailed in his treatise 'Fundamenta nova theoriae functionum ellipticarum' published in 1829. He established important connections between elliptic functions and various physical problems, including the motion of pendulums and planetary motion.Jacobi's research extended to differential equations and classical mechanics, notably the Hamilton–Jacobi theory. He emphasized the importance of inverting known results to explore new research avenues. In 1835, he classified periodic functions, proving that a univariate single-valued function cannot have more than two periods. He discovered fundamental properties of theta functions and contributed to number theory by proving Fermat's two-square theorem and Lagrange's four-square theorem, among other results.Jacobi was a pioneer in the theory of determinants, introducing the Jacobian determinant, which is essential in multiple integrals. He also studied symmetric polynomials, leading to the development of Schur polynomials and the Jacobi–Trudi identities. His work in celestial mechanics included the introduction of the Jacobi integral for sidereal coordinate systems. He left behind numerous manuscripts, with many published in Crelle's Journal. His collected works were published by the Berlin Academy between 1881 and 1891.

Carl Gustav Jacobi

Évariste Galois was born on 25 October 1811 in Bourg-la-Reine, France, to Nicolas-Gabriel Galois and Adélaïde-Marie Demante. His father, a Republican, served as the mayor of the village after the restoration of the monarchy in 1814. Galois's mother, educated in classical literature, was responsible for his early education. In October 1823, he entered the Lycée Louis-le-Grand, where his mathematical talent was recognized by his teacher, Louis Paul Émile Richard. By age 14, Galois had developed a keen interest in mathematics, studying works by Adrien-Marie Legendre and Joseph-Louis Lagrange.In 1828, Galois attempted the entrance examination for the École Polytechnique but failed due to inadequate explanations during the oral exam. He then enrolled in the École Normale, where he found some supportive professors. His first paper on simple continued fractions was published in 1829, and he began making significant contributions to polynomial equation theory. He submitted two papers to the Academy of Sciences, which were not published during his lifetime, despite being recognized for their importance by Augustin-Louis Cauchy.Galois's father died by suicide in July 1829, which deeply affected him. He made a second attempt to enter the Polytechnique but failed again, possibly due to his frustration with the examination process. After passing the Baccalaureate examinations, he received his degree in December 1829. Galois continued to submit his work on equation theory, but it remained unpublished. In 1830, he published three papers, one of which laid the groundwork for Galois theory.During this period, France experienced political upheaval. The July Revolution of 1830 led to the abdication of Charles X and the rise of Louis Philippe I. Galois, a staunch Republican, became involved in political activism, leading to his expulsion from the École Normale. He joined the National Guard and participated in protests. Galois was arrested multiple times for his political activities, including a notable incident in which he was detained after a toast perceived as a threat against the king.In April 1831, Galois was acquitted of charges related to his toast but was arrested again on Bastille Day for wearing the uniform of a disbanded artillery unit. He was sentenced to six months in prison for this offense. During his imprisonment, he continued to develop his mathematical ideas. After his release in April 1832, he attempted to establish a private class in advanced algebra but struggled to maintain interest due to his political commitments.On 30 May 1832, Galois was involved in a duel, the reasons for which remain unclear. He had written letters alluding to a broken love affair shortly before his death. Galois was shot in the abdomen and died the following morning, on 31 May 1832, at the age of 20. His funeral was marked by riots, and he was buried in a common grave in Montparnasse Cemetery. Galois's mathematical contributions, particularly in group theory and Galois theory, were recognized posthumously, with his work published in 1846 by Joseph Liouville, leading to significant developments in abstract algebra.

Évariste Galois

Charles Hermite was born on 24 December 1822 in Dieuze, Moselle, France. He was the sixth of seven children in his family. His father, Ferdinand Hermite, worked in the drapery business and pursued a career as an artist. The family relocated to Nancy in 1828. Hermite's education began at Collège de Nancy, followed by studies at Collège Henri IV and Lycée Louis-le-Grand in Paris. He was influenced by the works of Joseph-Louis Lagrange and Carl Friedrich Gauss.Hermite aimed to attend École Polytechnique, a prestigious military academy known for its focus on mathematics and science. After a year of preparation under the guidance of Eugène Charles Catalan, he gained admission in 1842. However, due to a deformity in his right foot, he was expelled after one year. Despite attempts to regain his place, he left the institution without graduating.In 1842, Hermite published his first mathematical work in Nouvelles Annales de Mathématiques, providing a proof of Niels Abel's proposition regarding the impossibility of solving fifth-degree equations algebraically. His correspondence with Carl Jacobi led to the inclusion of his articles in Jacobi's complete works, focusing on elliptic functions.After five years of private study, Hermite passed the baccalauréat examination in 1847. He married Louise Bertrand, sister of mathematician Joseph Bertrand, in 1848. That same year, he returned to École Polytechnique as a répétiteur and examinateur d'admission. He was elected to the French Academy of Sciences in July 1848. In 1856, he contracted smallpox, which prompted a return to his Catholic faith, influenced by Augustin-Louis Cauchy and a nun who cared for him.From 1862 to 1873, Hermite served as a lecturer at École Normale Supérieure. In 1869, he became a professor of mathematics at both École Polytechnique and the University of Paris, positions he held until his death. He was promoted to grand officer in the French Legion of Honour on his 70th birthday and was elected an honorary member of the Manchester Literary and Philosophical Society in 1892.Hermite made significant contributions to mathematics, particularly in number theory, elliptic functions, and the theory of numbers. In 1858, he demonstrated that fifth-degree equations could be solved using elliptic functions. In 1873, he proved that e, the base of natural logarithms, is a transcendental number. His methods were later utilized by Ferdinand von Lindemann to prove the transcendence of π in 1882.Hermite passed away in Paris on 14 January 1901 at the age of 78. His legacy includes various mathematical concepts named after him, such as Hermite polynomials and Hermitian operators.

Charles Hermite

Georg Friedrich Bernhard Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor who had served in the Napoleonic Wars. Riemann was the second of six children, and his mother, Charlotte Ebell, died in 1846. He displayed exceptional mathematical talent from an early age but struggled with timidity and frail health.Riemann's education began in 1840 when he moved to Hanover to live with his grandmother and attend lyceum. After his grandmother's death in 1842, he transferred to the Johanneum Lüneburg, where he excelled in mathematics. Initially, he studied philology and theology to support his family, but in 1846, he was sent to the University of Göttingen to study theology. There, he shifted his focus to mathematics under Carl Friedrich Gauss, who encouraged him to pursue a mathematical career. Riemann transferred to the University of Berlin in 1847, where he studied under prominent mathematicians and returned to Göttingen in 1849.Riemann began lecturing in 1854, establishing the field of Riemannian geometry, which later influenced Albert Einstein's theory of relativity. In 1857, he was granted a regular salary at the University of Göttingen, and in 1859, he became head of the mathematics department following Dirichlet's death. He married Elise Koch in 1862, and they had a daughter.In 1866, Riemann fled Göttingen due to the conflict between Hanover and Prussia. He died of tuberculosis on 20 July 1866 during his third trip to Italy, in Selasca, where he was buried. Riemann was a devout Christian, and at the time of his death, he was reciting the Lord's Prayer with his wife.Riemann's contributions to mathematics include the Riemann integral, which provided a rigorous formulation of integration, and his work on Fourier series. He introduced Riemann surfaces in complex analysis, allowing multi-valued functions to be treated as single-valued. His 1859 paper on the prime-counting function introduced the Riemann hypothesis, a significant conjecture in analytic number theory. Riemann's work in differential geometry laid the groundwork for the mathematics of general relativity.His habilitation thesis established the Riemann integral and showed that every piecewise continuous function is integrable. He also proved the Riemann–Lebesgue lemma regarding Fourier series. Riemann's exploration of the zeta function led to the formulation of the Riemann hypothesis, which concerns the distribution of prime numbers. His work on abelian functions and theta functions advanced the understanding of complex analysis and algebraic curves.Riemann's published works include his inaugural dissertation in 1851, his 1857 paper on abelian functions, and his 1859 paper on the prime-counting function. His 1868 lecture on the foundations of geometry, delivered in Göttingen, is now recognized as a pivotal work in geometry, although it was published posthumously. Riemann's contributions have had a lasting impact on various fields of mathematics, including analysis, number theory, and geometry.

Bernhard Riemann

Georg Ferdinand Ludwig Philipp Cantor was born on 3 March 1845 in Saint Petersburg, Russian Empire. He was the eldest of six children in a family with a musical background; his father was a Danish trader and his mother an Austro-Hungarian artist. The family moved to Germany in 1856 due to his father's illness, first settling in Wiesbaden and later in Frankfurt. Cantor graduated with distinction from the Realschule in Darmstadt in 1860 and from the Höhere Gewerbeschule Darmstadt in 1862. He then attended the Swiss Federal Polytechnic in Zurich before transferring to the University of Berlin, where he studied under notable mathematicians such as Leopold Kronecker and Karl Weierstrass. He earned his doctoral degree in 1867.After submitting his dissertation on number theory, Cantor briefly taught at a girls' school in Berlin before joining the University of Halle, where he spent his entire academic career. He married Vally Guttmann in 1874, and they had six children. Cantor was promoted to extraordinary professor in 1872 and became a full professor in 1879. Despite his achievements, he faced significant opposition from contemporaries like Kronecker, who disapproved of Cantor's theories on transfinite numbers and set theory.Cantor's groundbreaking work established the foundations of set theory, introducing concepts such as one-to-one correspondence, cardinal and ordinal numbers, and the distinction between countable and uncountable sets. His 1874 paper demonstrated that the set of real numbers is more numerous than the set of natural numbers, leading to the conclusion that there are different sizes of infinity. This work was met with resistance, particularly from Kronecker, who labeled Cantor a 'scientific charlatan' and opposed his ideas on the existence of actual infinity.Throughout his career, Cantor experienced bouts of depression, which some attribute to the hostility he faced from the mathematical community. Despite this, he continued to make significant contributions, including the diagonal argument and Cantor's theorem, which states that the power set of any set has a strictly greater cardinality than the set itself. His work laid the groundwork for modern mathematics, influencing various fields and leading to the establishment of set theory as a foundational theory.In 1904, Cantor received the Sylvester Medal from the Royal Society, recognizing his contributions to mathematics. He continued to engage with the mathematical community, founding the German Mathematical Society and participating in the first International Congress of Mathematicians in 1897. However, his mental health deteriorated, and he spent time in sanatoria due to chronic depression.Cantor's later years were marked by personal tragedy, including the death of his youngest son in 1916. He retired in 1913 and lived in poverty during World War I. Cantor died of a heart attack on 6 January 1918 in a sanatorium, leaving behind a legacy that transformed the understanding of infinity and set theory in mathematics.

Georg Cantor

Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin. His father, Christian Ferdinand Frobenius, was a Protestant parson, and his mother was Christine Elizabeth Friedrich. Frobenius began his education at the Joachimsthal Gymnasium in 1860 and graduated in 1867. He then attended the University of Göttingen for one semester before returning to Berlin, where he studied under notable mathematicians such as Leopold Kronecker, Ernst Kummer, and Karl Weierstrass. He received his doctorate in 1870, with a thesis focused on the solution of differential equations. After teaching at secondary schools, he was appointed as an extraordinary professor of mathematics at the University of Berlin in 1874.Frobenius moved to Zürich in 1875 to become an ordinary professor at the Eidgenössische Polytechnikum, where he worked for seventeen years. During this time, he married and raised a family while making significant contributions to various fields of mathematics. Following the death of Kronecker in December 1891, Frobenius returned to Berlin in 1893, where he was elected to the Prussian Academy of Sciences.Frobenius made substantial contributions to group theory, particularly in the second half of his career. He proved the Sylow theorems for abstract groups and established a fundamental theorem regarding the number of solutions to the equation xn = 1 in finite groups. He introduced the theory of group characters and representations, leading to the concept of Frobenius reciprocity and the definition of Frobenius groups. His work on characters included constructing the character table for the group PSL(2,p) for all odd primes p.In number theory, Frobenius developed a method for associating primes with conjugacy classes in Galois groups over the rational numbers. He defined the Frobenius conjugacy class of a prime p in a finite Galois extension K/Q, establishing a connection between the distribution of these classes and classical results about primes in arithmetic progressions. His construction of Frobenius elements has been crucial for the study of Galois groups in various contexts.

Georg Frobenius

David Hilbert was born on 23 January 1862 in the Province of Prussia, either in Königsberg or Wehlau. He was the first of two children and only son of Otto Hilbert, a county judge, and Maria Therese Hilbert. His family moved to Königsberg after his father became a city judge. Hilbert began his schooling at the age of eight and graduated from the Wilhelm Gymnasium in early 1880. He enrolled at the University of Königsberg later that year, where he developed a friendship with Hermann Minkowski.Hilbert obtained his doctorate in 1885 with a dissertation on invariant properties of binary forms. He served as a Privatdozent at the University of Königsberg from 1886 to 1895. In 1895, he became a Professor of Mathematics at the University of Göttingen, where he remained for the rest of his life. Göttingen became a leading institution in mathematics during his tenure.Among Hilbert's notable students were Hermann Weyl, Ernst Zermelo, and John von Neumann. He supervised 69 Ph.D. students, many of whom became prominent mathematicians. Hilbert was also the editor of the Mathematische Annalen from 1902 to 1939 and was elected an International Member of the United States National Academy of Sciences in 1907.Hilbert married Käthe Jerosch in 1892, and they had one son, Franz, who suffered from mental illness. Hilbert was raised a Calvinist but later became agnostic, arguing that mathematical truth is independent of religious beliefs. He developed a close friendship with Minkowski, who influenced his work in physics.In his later years, Hilbert faced health issues, including pernicious anemia, which affected his scientific output. He witnessed the purge of Jewish faculty members at Göttingen during the Nazi regime and expressed his discontent with the situation. Hilbert passed away on 14 February 1943, with few attendees at his funeral, reflecting the significant changes at the university.Hilbert made substantial contributions to various fields of mathematics, including invariant theory, algebraic number theory, and the foundations of geometry. He is known for his work on the axiomatization of geometry, particularly in his book Grundlagen der Geometrie, published in 1899. This work introduced a formal set of axioms that replaced Euclid's traditional axioms.In 1900, Hilbert presented a list of 23 unsolved problems at the International Congress of Mathematicians in Paris, which shaped mathematical research in the 20th century. His approach to mathematics emphasized formalism, advocating for a solid logical foundation for the discipline. He proposed Hilbert's program in 1920, aiming to establish a complete and consistent axiomatic system for all of mathematics.Hilbert's work in functional analysis led to the concept of Hilbert space, which became fundamental in quantum mechanics. He also contributed to the mathematical formulation of general relativity, collaborating with Einstein on the field equations. Hilbert's influence extended to various areas of mathematics and physics, and his legacy continues to impact the fields today.

David Hilbert

Bertrand Arthur William Russell, 3rd Earl Russell, was born on 18 May 1872 at Ravenscroft in Trellech, Monmouthshire, into a prominent British aristocratic family. His parents were Viscount and Viscountess Amberley, who were early advocates of birth control. Russell's paternal grandfather, Lord John Russell, served as Prime Minister of the United Kingdom twice. Russell's early life was marked by tragedy; his mother died of diphtheria when he was two, followed by the death of his sister Rachel. His father died in 1876, leaving Russell and his brother Frank in the care of their paternal grandparents. Russell's grandmother, the Countess Russell, played a significant role in his upbringing, instilling progressive views alongside her religious conservatism.Russell's childhood was characterized by loneliness and a deep interest in mathematics and literature, particularly the works of Euclid and Percy Bysshe Shelley. He was educated at home and later attended Trinity College, Cambridge, where he excelled in mathematics and philosophy, graduating in 1893. He became a Fellow of Trinity College in 1895 and began publishing works that would establish him as a leading figure in logic and philosophy.In 1903, Russell published The Principles of Mathematics, advocating for logicism, the idea that mathematics is reducible to logic. He collaborated with Alfred North Whitehead on the three-volume Principia Mathematica, published between 1910 and 1913, which further solidified his reputation. Russell's essay 'On Denoting' was published in 1905 and is considered a significant contribution to analytic philosophy.Russell was a vocal pacifist, particularly during World War I, which led to his dismissal from Trinity College in 1916 after he was convicted under the Defence of the Realm Act for his anti-war activities. He was imprisoned for six months in 1918 for publicly opposing the war. After the war, he continued to engage in political activism, advocating for nuclear disarmament and opposing the Vietnam War.In 1950, Russell received the Nobel Prize in Literature for his significant writings promoting humanitarian ideals and freedom of thought. He was also awarded several other honors throughout his life, including the De Morgan Medal and the Jerusalem Prize. Russell's personal life included four marriages, with his third wife Patricia Spence bearing him a son, Conrad. He became the 3rd Earl Russell in 1931 after the death of his brother.Russell's later years were marked by continued political activism and philosophical writing. He was involved in various causes, including the anti-nuclear movement and the campaign for civil rights. He died on 2 February 1970 at the age of 97, leaving behind a legacy as one of the most influential philosophers of the 20th century.

Bertrand Russell

Godfrey Harold Hardy was born on 7 February 1877 in Cranleigh, Surrey, England, into a family with a strong educational background. His father served as Bursar and Art Master at Cranleigh School, while his mother was a senior mistress at Lincoln Training College. Hardy displayed an early aptitude for mathematics, writing numbers up to millions at the age of two and factorising hymn numbers during church services. He attended Cranleigh School and later received a scholarship to Winchester College for his mathematical abilities. In 1896, he entered Trinity College, Cambridge, where he initially struggled with his first tutor but found inspiration under Augustus Love, who introduced him to more rigorous mathematical concepts.Hardy excelled in the Mathematics Tripos examination, ranking fourth after two years of preparation. He was elected to a Prize Fellowship at Trinity College in 1900 and earned his M.A. in 1903. After his fellowship expired in 1906, he became a lecturer at Trinity, balancing teaching with research. In 1913, he received a letter from Indian mathematician Srinivasa Ramanujan, which he initially suspected to be a hoax but later recognized as genuine. This marked the beginning of a significant collaboration that would influence both their careers.In 1919, following the Bertrand Russell affair during World War I, Hardy left Cambridge for the Savilian Chair of Geometry at Oxford. He spent the academic year 1928–1929 at Princeton University, returning to Cambridge in 1931 to hold the Sadleirian Professorship until 1942. Hardy was involved in various academic societies and served on the governing body of Abingdon School from 1922 to 1935. His health declined after a coronary thrombosis in 1939, leading to a loss of creative mathematical ability. In 1947, he attempted suicide and subsequently died suddenly while listening to his sister read.Hardy is recognized for reforming British mathematics by emphasizing rigor, contrasting with the applied mathematics tradition that had dominated. He collaborated extensively with John Edensor Littlewood, contributing to mathematical analysis and number theory, including the Hardy–Littlewood circle method and conjectures in prime number theory. He also formulated the Hardy–Weinberg principle in population genetics independently of Wilhelm Weinberg. Hardy's work, while aimed at pure mathematics, found applications in various scientific fields, including physics and cryptography.Throughout his life, Hardy was known for his shyness and social awkwardness. He was associated with the Bloomsbury Group and maintained friendships with prominent intellectuals. A lifelong bachelor, he was cared for by his sister in his later years. Hardy was an avid cricket fan and expressed a disdain for the military applications of mathematics, preferring to focus on its aesthetic value. He published numerous works, including 'A Mathematician's Apology' and 'An Introduction to the Theory of Numbers,' and his collected papers were published in seven volumes by Oxford University Press.

G.H. Hardy

Amalie Emmy Noether was born on 23 March 1882 in Erlangen, Bavaria, to a Jewish family. Her father, Max Noether, was a mathematician. Initially, she aimed to teach languages but shifted her focus to mathematics at the University of Erlangen–Nuremberg, where she completed her doctorate in 1907 under Paul Gordan. For seven years, she worked unpaid at the Mathematical Institute of Erlangen due to the exclusion of women from academic positions. In 1915, she was invited to the University of Göttingen by David Hilbert and Felix Klein, although her appointment faced opposition from the philosophical faculty. She lectured under Hilbert's name until her habilitation was approved in 1919, allowing her to become a Privatdozent.Noether became a prominent member of the Göttingen mathematics department until 1933, mentoring students known as the 'Noether Boys.' Her work in abstract algebra began in the 1920s, leading to significant contributions such as the theory of ideals in commutative rings, which earned the term 'Noetherian' in her honor. She also proved Noether's first and second theorems, establishing a connection between symmetry and conservation laws in physics.In 1933, the Nazi regime dismissed Jewish professors from their positions, prompting Noether to emigrate to the United States. She accepted a position at Bryn Mawr College and also lectured at the Institute for Advanced Study in Princeton. During her time in the U.S., she continued to teach and mentor graduate students, contributing to the development of modern algebra.Noether's mathematical work is categorized into three epochs. The first epoch (1908–1919) focused on algebraic invariants and number fields. The second epoch (1920–1926) marked her groundbreaking work in abstract algebra, particularly in the theory of ideals. The third epoch (1927–1935) involved her research on noncommutative algebras and representation theory.Noether's contributions to mathematics and physics have had a lasting impact. She passed away on 14 April 1935 due to complications from surgery. Her legacy endures, with many mathematical concepts and objects named in her honor, and she is regarded as one of the most significant mathematicians of the twentieth century.

Emmy Noether

Srinivasa Ramanujan Iyengar was born on 22 December 1887 in Erode, Tamil Nadu, into a Tamil Brahmin family. His father worked as a clerk, and his mother was a housewife. Ramanujan's early life was marked by personal loss, including the death of his siblings and a bout of smallpox. He showed exceptional mathematical talent from a young age, mastering advanced trigonometry by age 13 and independently discovering sophisticated theorems.Ramanujan's formal education was limited, and he struggled to find recognition for his work in mathematics. In 1913, he began a correspondence with G. H. Hardy, a prominent mathematician at Cambridge University. Hardy recognized Ramanujan's extraordinary talent and invited him to England. Ramanujan arrived in Cambridge in 1914, where he collaborated with Hardy and J. E. Littlewood, producing groundbreaking results in number theory, infinite series, and continued fractions.Throughout his life, Ramanujan compiled nearly 3,900 results, many of which were novel and influential. His work included the Ramanujan prime, the Ramanujan theta function, and mock theta functions, which opened new areas of research. He became one of the youngest Fellows of the Royal Society and the first Indian to be elected a Fellow of Trinity College, Cambridge.Ramanujan's health deteriorated in England, and he returned to India in 1919 due to illness, believed to be hepatic amoebiasis. He continued to produce mathematical ideas until his death on 26 April 1920 at the age of 32. His notebooks, containing both published and unpublished results, have been extensively studied and continue to inspire mathematicians. The rediscovery of his 'lost notebook' in 1976 generated significant excitement in the mathematical community.Ramanujan's contributions to mathematics have been recognized posthumously, with his birthday celebrated as National Mathematics Day in India. His legacy endures through ongoing research inspired by his work, and he is regarded as one of the greatest mathematicians in history.

Srinivasa Ramanujan

Wilhelm Friedrich Ackermann was born on 29 March 1896 in Herscheid, Germany. He earned a Ph.D. from the University of Göttingen in 1925 for his thesis titled Begründung des 'tertium non datur' mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which provided a consistency proof of arithmetic without relying on Peano induction. This work was significant in the field of proof theory during the 1920s and aligned with the principles of Hilbert's school of thought.From 1929 to 1948, Ackermann taught at the Arnoldinum Gymnasium in Burgsteinfurt, later moving to Lüdenscheid, where he continued teaching until 1961. He was recognized as a corresponding member of the Akademie der Wissenschaften in Göttingen and held the position of honorary professor at the University of Münster.In 1928, Ackermann collaborated with David Hilbert to transform Hilbert's lectures on mathematical logic into the text Principles of Mathematical Logic. This work was notable for being the first comprehensive exposition of first-order logic and for introducing the completeness and decidability problem, known as the Entscheidungsproblem. Ackermann further contributed to the field by constructing consistency proofs for set theory in 1937, full arithmetic in 1940, type-free logic in 1952, and a new axiomatization of set theory in 1956.Throughout his later years, Ackermann remained active in research, publishing numerous contributions to the foundations of mathematics until his death on 24 December 1962 in Lüdenscheid, West Germany.

Wilhelm Ackermann

John von Neumann was born on December 28, 1903, in Budapest, Kingdom of Hungary, to a wealthy Jewish family. His father, Miksa Neumann, was a banker, and his mother, Margit Kann, came from a prominent family. Von Neumann displayed prodigious mathematical talent from a young age, mastering complex calculations and languages, including Ancient Greek, by the age of six. He attended the Lutheran Fasori Evangélikus Gimnázium, where he befriended fellow student Eugene Wigner. Despite his father's wishes for him to pursue a career in chemical engineering, von Neumann enrolled in the University of Budapest as a Ph.D. candidate in mathematics while also studying at ETH Zurich.He graduated in 1926 with a degree in chemical engineering and a Ph.D. in mathematics. His early work included significant contributions to set theory, where he provided an axiomatization that addressed issues such as Russell's paradox. He also made foundational contributions to functional analysis and ergodic theory. In 1929, he began his academic career at Princeton University, where he became a leading figure in mathematics and physics.During World War II, von Neumann contributed to the Manhattan Project, developing mathematical models for nuclear weapons. He played a crucial role in the design of explosive lenses for the implosion-type bomb used in Nagasaki. After the war, he continued to influence military and defense strategies, serving on various committees related to nuclear weapons and missile defense.Von Neumann's work extended beyond mathematics and physics into economics, where he co-authored the influential book 'Theory of Games and Economic Behavior' with Oskar Morgenstern, establishing game theory as a formal discipline. He also contributed to the development of the digital computer, proposing the architecture that would become known as the von Neumann architecture.In his later years, von Neumann's health declined due to cancer, which he believed may have been caused by radiation exposure during his work at Los Alamos. He died on February 8, 1957, at the age of 53. His legacy includes numerous contributions across various fields, and he is widely regarded as one of the most influential mathematicians and scientists of the 20th century.

John von Neumann

Andrey Nikolaevich Kolmogorov was born on 25 April 1903 in Tambov, Russia. His mother, Maria Yakovlevna Kolmogorova, died during childbirth, and he was raised by his aunts in Tunoshna, near Yaroslavl, at his grandfather's estate. His father, Nikolai Matveyevich Katayev, an agronomist, was exiled for revolutionary activities and disappeared in 1919 during the Russian Civil War.Kolmogorov's early education took place in a village school, where he developed an interest in mathematics. By age six, he recognized patterns in the sums of odd numbers. In 1910, he moved to Moscow with his adopted aunt and graduated from high school in 1920. He then enrolled at Moscow State University and the Mendeleev Moscow Institute of Chemistry and Technology, where he gained a reputation for his broad knowledge and published his first research paper on landholding practices in the Novgorod Republic.In 1922, Kolmogorov gained international recognition for constructing a divergent Fourier series and decided to pursue mathematics as a career. He graduated from Moscow State University in 1925 and studied under Nikolai Luzin, forming a close friendship with fellow student Pavel Alexandrov. Kolmogorov's interest in probability theory grew during this time, and he published work on intuitionistic logic in 1925. He earned his Doctor of Philosophy degree in 1929.Kolmogorov traveled to Göttingen, Munich, and Paris in 1930, establishing scientific contacts and publishing his work on probability theory in 1931. He became a professor at Moscow State University that same year. In 1933, he published 'Foundations of the Theory of Probability,' which established the axiomatic foundations of probability theory. He became the first chairman of the probability theory department at the university in 1935.During the Great Purge in 1936, Kolmogorov testified against his doctoral advisor, Nikolai Luzin, amid accusations of academic misconduct. The motivations behind his testimony remain debated among historians. In 1938, he published a paper on stationary stochastic processes, which had military applications during the Cold War. He was elected a full member of the USSR Academy of Sciences in 1939.During World War II, Kolmogorov applied statistical theory to artillery fire and developed a stochastic distribution scheme for barrage balloons to protect Moscow. He independently developed pivotal equations in stochastic processes, known as the Chapman–Kolmogorov equations, alongside British mathematician Sydney Chapman.After the war, Kolmogorov focused on turbulence and classical mechanics, presenting the Kolmogorov–Arnold–Moser theorem in 1954. He contributed to algorithmic complexity theory and married Anna Dmitrievna Egorova in 1942. Throughout his career, he held various positions at Moscow State University, including Dean of the Department of Mechanics and Mathematics.In 1971, Kolmogorov participated in an oceanographic expedition aboard the research vessel Dmitri Mendeleev. He wrote articles for the Great Soviet Encyclopedia and explored the relationship between abstract and applied probability theory in his later years. Kolmogorov died on 20 October 1987 in Moscow and was buried in Novodevichy cemetery.

Andrey Kolmogorov

Kurt Friedrich Gödel was born on April 28, 1906, in Brünn, Austria-Hungary, into a wealthy German-speaking family. His father, Rudolf Gödel, was a managing director of a textile firm, and his mother, Marianne, was of Protestant background. Gödel's early education took place at a Lutheran school and later at a realgymnasium, where he excelled in mathematics and languages. He developed a keen interest in mathematics during his teenage years, influenced by his brother's studies in Vienna.At 18, Gödel enrolled at the University of Vienna, initially intending to study theoretical physics. He attended courses in mathematics and philosophy, engaging with the Vienna Circle and adopting ideas of mathematical realism. His doctoral dissertation, completed in 1929 under Hans Hahn, established the completeness theorem for first-order logic. Gödel's incompleteness theorems, published in 1931, demonstrated the limitations of formal axiomatic systems, showing that such systems could not prove their own consistency or decide the truth of all statements about natural numbers.Gödel's work also included significant contributions to set theory, proving that the axiom of choice and the continuum hypothesis could not be disproved from Zermelo–Fraenkel set theory, assuming its consistency. He emigrated to the United States in 1939 to escape the rise of Nazi Germany, settling in Princeton, New Jersey, where he accepted a position at the Institute for Advanced Study.In Princeton, Gödel formed a close friendship with Albert Einstein. He continued to work on foundational issues in mathematics and logic, including the independence of the axiom of choice and the continuum hypothesis. Gödel's later work involved exploring the implications of general relativity, notably demonstrating solutions involving closed timelike curves, which suggested the possibility of time travel.Gödel married Adele Nimbursky in 1938, and they had no children. His later years were marked by mental health struggles, including paranoia and obsessive fears. He died on January 14, 1978, from malnutrition caused by his refusal to eat, believing his food was poisoned. He was buried in Princeton Cemetery.Gödel's contributions to logic and mathematics have had a lasting impact, influencing various fields and prompting further exploration of the foundations of mathematics and philosophy.

Kurt Gödel

André Weil was born on 6 May 1906 in Paris to agnostic Alsatian Jewish parents who had fled the annexation of Alsace-Lorraine by the German Empire. He had one sibling, a younger sister named Simone Weil, who later became a notable philosopher. Weil studied in Paris, Rome, and Göttingen, earning his doctorate in 1928. During his time in Germany, he formed a friendship with Carl Ludwig Siegel. From 1930, he spent two academic years at Aligarh Muslim University in India. Weil had a lifelong interest in classical Greek and Latin literature, as well as Hinduism and Sanskrit literature, having taught himself Sanskrit at the age of 14.After a year of teaching at Aix-Marseille University, he spent six years at the University of Strasbourg. He married Éveline de Possel in 1937. When World War II began, Weil was in Finland, having traveled in Scandinavia since April 1939. His wife returned to France without him. Weil was arrested in Finland on suspicion of spying at the outbreak of the Winter War, but his life was not in danger. He returned to France via Sweden and the United Kingdom, where he was detained in Le Havre in January 1940 for failing to report for military duty. He was imprisoned in Le Havre and then Rouen, where he completed significant work that established his reputation. He was tried on 3 May 1940 and sentenced to five years in prison, but he requested to join a military unit instead. After the fall of France in June 1940, he reunited with his family in Marseille and later joined his wife in Clermont-Ferrand.In January 1941, Weil and his family sailed from Marseille to New York, where he spent the remainder of the war. He received support from the Rockefeller Foundation and the Guggenheim Foundation. He taught undergraduate mathematics at Lehigh University for two years, where he faced challenges such as being unappreciated and poorly paid. He then moved to Brazil, teaching at the Universidade de São Paulo from 1945 to 1947. Weil and Éveline had two daughters, Sylvie and Nicolette. He returned to the United States and taught at the University of Chicago from 1947 to 1958 before moving to the Institute for Advanced Study, where he spent the rest of his career. He was a Plenary Speaker at the International Congress of Mathematicians in 1950, 1954, and 1978. In 1966, he was elected a Foreign Member of the Royal Society, and in 1979, he shared the second Wolf Prize in Mathematics with Jean Leray.Weil made significant contributions to various areas of mathematics, particularly in establishing connections between algebraic geometry and number theory. His doctoral work led to the Mordell-Weil theorem, which he applied in Siegel's theorem on integral points. He developed the separation of the infinite descent argument into two structural approaches using height functions and Galois cohomology. His proof of the Riemann hypothesis for zeta-functions of curves over finite fields in the 1940s laid the groundwork for algebraic geometry. The Weil conjectures, proposed around 1950, were influential and later proved by several mathematicians, including Pierre Deligne in 1973.Weil introduced the adele ring in the late 1930s and provided a proof of the Riemann-Roch theorem using this concept. His work anticipated later ideas in moduli spaces of bundles. He also contributed to Pontryagin duality, differential geometry, and general topology, introducing the concept of a uniform space. His collaboration with Nicolas Bourbaki, of which he was a founding member, influenced his work on sheaf theory. He chose the symbol ∅ to represent the empty set, derived from the Norwegian alphabet.Weil's first paper in 1926 established that the classical isoperimetric inequality holds on non-positively curved surfaces, contributing to what became known as the Cartan-Hadamard conjecture. He discovered the Weil representation, which connected representation theory and theta functions. Weil was a member of both the National Academy of Sciences and the American Philosophical Society.Weil's ideas significantly influenced the writings and seminars of Bourbaki, and he authored several books on the history of number theory. He was agnostic but respected various religions, particularly Hindu thought. He passed away on 6 August 1998.

André Weil

Alan Mathison Turing was born on 23 June 1912 in Maida Vale, London. He was raised in southern England and attended St Michael's primary school, where his talent in mathematics was recognized. Turing later studied at Hazelhurst Preparatory School and then Sherborne School, where he developed a keen interest in science and mathematics. He formed a significant friendship with Christopher Morcom, whose death in 1930 deeply affected Turing.Turing went on to King's College, Cambridge, where he graduated with first-class honours in mathematics in 1934. He earned a doctorate from Princeton University in 1938, where he worked on computability and developed the concept of the Turing machine, a foundational model for theoretical computer science. Turing's work established a formalization of algorithms and computation.During World War II, Turing worked at Bletchley Park, Britain's codebreaking center. He led Hut 8, focusing on German naval cryptanalysis. Turing improved existing methods for breaking the Enigma cipher and developed the bombe, an electromechanical machine that automated the decryption process. His contributions were crucial in intercepting German communications, significantly aiding the Allied war effort.After the war, Turing worked at the National Physical Laboratory, where he designed the Automatic Computing Engine, one of the first stored-program computers. He later joined the University of Manchester, contributing to early computer development and exploring mathematical biology. Turing published a paper on morphogenesis, proposing a reaction-diffusion model to explain pattern formation in biological organisms.In 1952, Turing was prosecuted for homosexual acts, which were illegal in the UK at the time. He accepted hormone treatment as an alternative to imprisonment. Turing died on 7 June 1954 from cyanide poisoning, with his death ruled a suicide, although some evidence suggested it could have been accidental. His contributions to mathematics and computing were not fully recognized during his lifetime due to the secrecy surrounding his wartime work.In subsequent years, Turing's legacy grew, leading to public recognition and memorials. In 2009, the British government issued an apology for his treatment, and in 2013, he received a posthumous pardon. The term 'Alan Turing law' refers to legislation that pardoned men convicted under historical laws against homosexual acts. Turing's work remains influential in computer science and artificial intelligence, and he is commemorated in various ways, including his portrait on the Bank of England £50 note.

Alan Turing

Paul Richard Halmos was born on 3 March 1916 in the Kingdom of Hungary into a Jewish family. He immigrated to the United States at the age of 13. Halmos earned his Bachelor of Arts degree from the University of Illinois, majoring in mathematics and fulfilling the requirements for a degree in philosophy. He graduated at the age of 19 after three years of study. Initially pursuing a Ph.D. in philosophy, he shifted to mathematics after failing his master's oral exams and completed his doctorate in 1938 under the supervision of Joseph L. Doob. His dissertation was titled 'Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems.'After graduation, Halmos joined the Institute for Advanced Study, where he worked under John von Neumann. This experience significantly influenced his career. During his time at the Institute, he authored his first book, 'Finite Dimensional Vector Spaces,' which established his reputation as a mathematical expositor. Halmos held various academic positions, including at Syracuse University, the University of Chicago from 1946 to 1960, the University of Michigan from approximately 1961 to 1967, the University of Hawaii in 1967-1968, Indiana University from 1969 to 1985, and the University of California at Santa Barbara from 1976 to 1978. After retiring from Indiana University in 1985, he was affiliated with the Mathematics department at Santa Clara University until his death in 2006.Halmos made significant contributions to several areas of mathematics, including mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis, particularly in Hilbert spaces. He developed polyadic algebras, an algebraic version of first-order logic, which was distinct from the cylindric algebras of Alfred Tarski and his students. Halmos was recognized for his clarity and engagement as a mathematical expositor, receiving the Lester R. Ford Award in 1971 and again in 1977. He chaired the American Mathematical Society committee that produced the AMS style guide for academic mathematics, published in 1973, and received the Leroy P. Steele Prize for exposition in 1983.In his 1968 article in the American Scientist, Halmos argued that mathematics is a creative art and that mathematicians should be viewed as artists. He discussed the division of the field into mathology and mathophysics, asserting that mathematicians and painters share similar thought processes. His 1985 book, 'I Want to Be a Mathematician,' is an account of his experiences as an academic mathematician in 20th-century America, focusing on his professional life rather than personal details. Halmos claimed to have invented the 'iff' notation for 'if and only if' and to have been the first to use the 'tombstone' symbol to indicate the end of a proof, which is sometimes referred to as a halmos.Halmos received the Deborah and Franklin Haimo Award for Distinguished College or University Teaching of Mathematics in 1994. He and his wife, Virginia Halmos, established the Euler Book Prize in 2005, awarded by the Mathematical Association of America for books that enhance the public's view of mathematics. Halmos authored numerous influential books, including 'Measure Theory' (1950), 'Introduction to Hilbert Space and the Theory of Spectral Multiplicity' (1951), 'Naive Set Theory' (1960), and 'Algebraic Logic' (1962), among others. His works have been widely reviewed and have contributed significantly to the field of mathematics.

Paul Halmos

Atle Selberg was born on 14 June 1917 in Langesund, Norway, to Anna Kristina Selberg and Ole Michael Ludvigsen Selberg. He had three brothers, two of whom, Sigmund and Henrik, were also mathematicians. Selberg was influenced by the work of Srinivasa Ramanujan during his school years and discovered an exact analytical formula for the partition function, although it was first published by Hans Rademacher. He studied at the University of Oslo and completed his doctorate in 1943.During World War II, Selberg worked in isolation due to the German occupation of Norway. He fought against the German invasion and was imprisoned multiple times. After the war, his mathematical achievements became recognized, including a proof that a positive proportion of the zeros of the Riemann zeta function lie on the critical line where the real part of s equals one-half.Following the war, Selberg focused on sieve theory, a previously overlooked area in mathematics. In 1947, he introduced the Selberg sieve, which provided auxiliary upper bounds and contributed to significant results such as Chen's theorem. In 1948, he published two papers in the Annals of Mathematics, proving theorems regarding primes in arithmetic progression and the density of primes using elementary methods. This challenged the prevailing belief that such results required complex analysis. His work led to a dispute with Paul Erdős, as both mathematicians independently proved the prime number theorem later that year.Selberg moved to the United States, where he worked as an associate professor at Syracuse University before joining the Institute for Advanced Study in Princeton, New Jersey, in the 1950s. He developed the Selberg trace formula, establishing a duality between the lengths of closed geodesics on compact Riemann surfaces and the eigenvalues of the Laplacian, paralleling the relationship between prime numbers and the zeros of the zeta function. He primarily worked alone, with Sarvadaman Chowla as his only coauthor.Selberg received numerous accolades for his contributions to mathematics, including the Fields Medal in 1950 and the Wolf Prize in Mathematics in 1986. He was awarded an honorary Abel Prize in 2002, the year it was established. He was elected to several prestigious academies, including the Norwegian Academy of Science and Letters and the American Academy of Arts and Sciences. In 1972, he received an honorary degree from the Norwegian Institute of Technology.Selberg was married to Hedvig, who passed away in 1995. They had two children, Ingrid and Lars. In 2003, he married Betty Frances Compton. Selberg died at home in Princeton, New Jersey, on 6 August 2007, due to heart failure, leaving behind his widow, children, and four grandchildren.His collected works were published in two volumes, containing numerous articles and lectures on sieve theory.

Atle Selberg

Imre Lakatos was born Imre Lipsitz on 9 November 1922 in Debrecen, Hungary, into a Jewish family. He earned a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944. Following the German invasion of Hungary in March 1944, Lakatos and his girlfriend, Éva Révész, formed a Marxist resistance group. In May 1944, they faced a moral dilemma regarding a young Jewish activist, Éva Izsák, whom Lakatos believed was at risk of betrayal. He decided that she should commit suicide to protect the group, leading to her being taken to Debrecen and given cyanide. During the Nazi occupation, Lakatos evaded persecution by adopting the surname Molnár. His mother and grandmother were murdered in Auschwitz, and he later changed his surname to Lakatos in honor of Géza Lakatos.After World War II, Lakatos worked as a senior official in the Hungarian Ministry of Education starting in 1947. He completed his PhD at Debrecen University in 1948 and attended private seminars led by György Lukács. In 1949, he studied at Moscow State University under Sofya Yanovskaya. Upon returning to Hungary, he became embroiled in internal conflicts within the communist party, resulting in his imprisonment from 1950 to 1953 on charges of revisionism. Despite his earlier hardline Stalinist stance, he became involved with dissident groups leading up to the 1956 Hungarian Revolution.Following the Soviet invasion of Hungary in November 1956, Lakatos fled to Vienna and subsequently to England, where he lived for the remainder of his life without obtaining British citizenship. He received a PhD in philosophy from the University of Cambridge in 1961, with a thesis titled 'Essays in the Logic of Mathematical Discovery.' His posthumously published book, 'Proofs and Refutations: The Logic of Mathematical Discovery,' is based on this work. In 1960, he joined the London School of Economics (LSE), where he focused on the philosophy of mathematics and science, collaborating with notable figures such as Karl Popper and Joseph Agassi.In 1965, Lakatos co-edited 'Criticism and the Growth of Knowledge,' which included discussions on Thomas Kuhn's 'The Structure of Scientific Revolutions.' He became the editor of the British Journal for the Philosophy of Science in January 1971, a position he held until his death in 1974. An international conference organized by Lakatos and Spiro Latsis in 1975, dedicated to his methodology of research programmes, took place posthumously. Lakatos passed away from a heart attack on 2 February 1974 at the age of 51, and the Lakatos Award was established in his memory.Lakatos's philosophical contributions include his views on the fallibility of mathematics and the methodology of proofs and refutations. His book 'Proofs and Refutations' illustrates the iterative nature of mathematical discovery through a fictional dialogue among students exploring the Euler characteristic. He introduced the concept of 'monsters' to describe counterexamples that challenge mathematical theorems, proposing strategies such as monster-barring and monster-adjustment to refine mathematical understanding.In the philosophy of science, Lakatos developed the model of 'research programmes' to reconcile Popper's falsificationism with Kuhn's views on scientific revolutions. A research programme consists of a hard core of theoretical assumptions, supported by auxiliary hypotheses that can be modified in response to empirical evidence. Lakatos distinguished between progressive and degenerative research programmes based on their ability to produce novel predictions and explanations. He argued that scientific theories should not be abandoned solely due to anomalies but should be evaluated based on their overall explanatory power.Lakatos also critiqued pseudoscience, defining it as theories that fail to make novel predictions or whose predictions are largely falsified. He identified several examples of pseudoscience, including Ptolemaic astronomy and Freudian psychoanalysis. His work emphasized the importance of historical context in evaluating scientific theories, asserting that philosophy of science must be informed by the history of science.Despite his contributions, Lakatos faced criticism, notably from Paul Feyerabend, who argued that Lakatos's methodology lacked the rigor of a true methodology. Lakatos's correspondence with Feyerabend revealed their differing perspectives on the nature of scientific practice and rationality.

Imre Lakatos

Olga Aleksandrovna Ladyzhenskaya was born on 7 March 1922 in Kologriv, Russia. She was the daughter of a mathematics teacher, which fostered her early interest in the subject. Her father, Aleksandr Ivanovich Ladýzhenski, was arrested by the NKVD in 1937 and executed, impacting her educational opportunities. Ladyzhenskaya completed high school in 1939, while her older sisters were not allowed to do so. Due to her father's political status, she was denied admission to Leningrad State University and instead attended a pedagogical institute. Following the German invasion in June 1941, she taught at a school in Kologriv. In 1943, she gained admission to Moscow State University, graduating in 1947.Ladyzhenskaya began her teaching career in the Physics department of Moscow State University in 1950. She defended her PhD in 1951 under the supervision of Sergei Sobolev and Vladimir Smirnov, and obtained a second doctorate in 1953. In 1954, she joined the mathematical physics laboratory at the Steklov Institute, becoming its head in 1961. Throughout her career, she published over two hundred scientific papers and authored six monographs, focusing on partial differential equations and fluid dynamics.Her notable contributions include rigorous proofs of the convergence of finite difference methods for the Navier–Stokes equations and analyses of the regularity of parabolic and quasilinear elliptic equations. Ladyzhenskaya was recognized for her work with several awards, including the P. L. Chebyshev Prize in 1966, the USSR State Prize in 1969, and the Lomonosov Gold Medal in 2002. She was also a member of the Russian Academy of Sciences and the Lincei National Academy in Rome.Ladyzhenskaya maintained friendships with prominent cultural figures, including writer Aleksandr Solzhenitsyn and poet Anna Akhmatova. She was involved in philanthropic activities and faced personal risks to support those opposing the Soviet regime. In her later years, she dealt with various eye problems that affected her work. Ladyzhenskaya passed away in her sleep on 12 January 2004, just before a planned trip to Florida.

Olga Ladyzhenskaya

Harish-Chandra, born on 11 October 1923 in Kanpur, was an American mathematician and physicist known for his contributions to representation theory and harmonic analysis on semisimple Lie groups. He received his education at B.N.S.D. College in Kanpur and the University of Allahabad, where he earned a master's degree in physics in 1940. He then pursued further studies at the Indian Institute of Science in Bangalore under Homi J. Bhabha.In 1945, Harish-Chandra moved to the University of Cambridge, where he worked as a research student under Paul Dirac. During his time at Cambridge, he attended lectures by Wolfgang Pauli and identified an error in Pauli's work, leading to a lifelong friendship. He completed his PhD in 1947, focusing on Infinite Irreducible Representations of the Lorentz Group.Harish-Chandra received several honors throughout his career, including membership in the National Academy of Sciences and fellowship in the Royal Society. He was awarded the Cole Prize by the American Mathematical Society in 1954 and the Srinivasa Ramanujan Medal by the Indian National Science Academy in 1974. In 1981, he received an honorary degree from Yale University and was awarded the Padma Bhushan in 1977. The Harish-Chandra Research Institute in India is named in his honor, and his birthday is celebrated annually at V.S.S.D. College in Kanpur.In 1952, he married Lalitha 'Lily' Kale, who was born in Warsaw and moved to Bangalore with her family during World War II. They had two daughters, Premala and Devika, with Premala also pursuing a career in physics.Harish-Chandra's health began to decline in 1969, when he experienced his first heart attack. He suffered additional heart attacks in 1970 and 1982, which affected his physical capabilities. He passed away on 16 October 1983, following a final heart attack after a conference held in his honor.

Harish-Chandra

Jean-Pierre Serre was born on 15 September 1926 in Bages, Pyrénées-Orientales, to pharmacist parents. He received his education at the Lycée de Nîmes and later attended the École Normale Supérieure in Paris from 1945 to 1948. He earned his doctorate from the Sorbonne in 1951. From 1948 to 1954, he worked at the Centre National de la Recherche Scientifique in Paris. In 1956, he became a professor at the Collège de France, a position he held until his retirement in 1994.Serre was an influential figure in the school of Henri Cartan, focusing on algebraic topology, several complex variables, commutative algebra, and algebraic geometry. His doctoral thesis addressed the Leray–Serre spectral sequence associated with a fibration. He collaborated with Cartan to apply Eilenberg–MacLane spaces for computing homotopy groups of spheres, a significant problem in topology at the time. In 1954, he received the Fields Medal, becoming the youngest recipient of this award.During the 1950s and 1960s, Serre collaborated with Alexander Grothendieck, contributing to foundational work motivated by the Weil conjectures. His notable papers include 'Faisceaux Algébriques Cohérents' (1955) and 'Géométrie Algébrique et Géométrie Analytique' (1956). He recognized the need for more refined cohomology theories to address the Weil conjectures, proposing theories based on Witt vector coefficients. In 1958, he highlighted the importance of isotrivial principal bundles on algebraic varieties, which inspired Grothendieck's development of étale topology and cohomology.From 1959, Serre's research interests shifted towards group theory and number theory, particularly Galois representations and modular forms. His contributions included 'Conjecture II' on Galois cohomology, the Borel–Serre compactification, and results on the number of points of curves over finite fields. He also introduced the concept of p-adic modular forms and established the Serre conjecture on mod-p representations, linking it to Fermat's Last Theorem. His inquiry into whether a finitely generated projective module over a polynomial ring is free led to significant developments in commutative algebra, culminating in the Quillen–Suslin theorem, proven independently by Daniel Quillen and Andrei Suslin in 1976.Serre received numerous honors throughout his career, including the Balzan Prize in 1985, the Steele Prize in 1995, and the Wolf Prize in Mathematics in 2000. He was the first recipient of the Abel Prize in 2003. He was recognized as a foreign member of several scientific academies and received multiple honorary degrees from prestigious institutions. In France, he was awarded the Grand Cross of the Legion of Honour and the Grand Cross of the Legion of Merit.

Jean-Pierre Serre

Stephen Smale was born on July 15, 1930, in Flint, Michigan. He began his higher education at the University of Michigan in 1948, where he initially excelled in an honors calculus sequence. However, his performance declined during his sophomore and junior years, resulting in mediocre grades. He earned a Bachelor of Science degree in 1952 and continued as a graduate student at the same university, where he struggled academically until he was motivated to improve his performance after a warning from the department chair. Smale completed his PhD in 1957 under the supervision of Raoul Bott and began his career as an instructor at the University of Chicago.Smale gained attention early in his career for his remarks about his work habits while proving the higher-dimensional Poincaré conjecture. He was politically active, participating in movements such as the Free Speech movement and the Fair Play for Cuba Committee. In 1966, while in Moscow to accept the Fields Medal, he held a press conference to criticize U.S. involvement in Vietnam and Soviet actions in Hungary. Following this, he faced difficulties in renewing his National Science Foundation grant and was subpoenaed by the House Un-American Activities Committee.In 1960, Smale received a Sloan Research Fellowship and joined the mathematics faculty at the University of California, Berkeley. He briefly moved to Columbia University in 1961 but returned to Berkeley in 1964, where he spent the majority of his career. He became a professor emeritus at Berkeley in 1995 and later took a position at the City University of Hong Kong. Smale also developed a notable private mineral collection, which is documented in the book 'The Smale Collection: Beauty in Natural Crystals.'Smale's research contributions include proving that the oriented diffeomorphism group of the two-dimensional sphere has the same homotopy type as the special orthogonal group of 3 × 3 matrices. He studied immersions of the two-dimensional sphere into Euclidean space, clarifying conditions under which two immersions can be deformed into one another. His work led to the concept of sphere eversion and extended to higher-dimensional spheres. He introduced the Morse–Smale system in dynamical systems, proving significant inequalities and constructing self-indexing Morse functions, which he used to resolve the generalized Poincaré conjecture in dimensions greater than four. He also established the h-cobordism theorem and classified simply-connected smooth five-dimensional manifolds.Smale introduced the horseshoe map, which inspired further research, and applied Morse theory to mathematical economics. In 1998, he compiled a list of 18 mathematical problems to be solved in the 21st century, echoing Hilbert's famous list from 1900. This list included several significant unsolved problems, such as the Riemann hypothesis and the Poincaré conjecture.Throughout his career, Smale received numerous accolades, including the Chauvenet Prize in 1988 and the Wolf Prize in mathematics in 2007. His work has had a lasting impact on various fields within mathematics.

Stephen Smale

Jacques Tits was born on 12 August 1930 in Uccle, Belgium, to Léon Tits, a professor, and Lousia André. He attended the Athénée of Uccle and the Free University of Brussels, where he completed his doctorate in 1950 under the supervision of Paul Libois. His dissertation focused on the generalization of projective groups based on the concept of transitivity.Tits held academic positions at the Free University of Brussels from 1962 to 1964, the University of Bonn from 1964 to 1974, and the Collège de France in Paris until he became emeritus in 2000. In 1974, he changed his citizenship to French to comply with the teaching requirements at the Collège de France, renouncing his Belgian citizenship due to the country's nationality laws at that time. He was an honorary member of the Nicolas Bourbaki group, contributing to the popularization of H.S.M. Coxeter's work and introducing terminology such as Coxeter number, Coxeter group, and Coxeter graph.Tits passed away on 5 December 2021 in the 13th arrondissement of Paris at the age of 91. Throughout his career, he received numerous awards, including the Wolf Prize in Mathematics in 1993, the Cantor Medal from the Deutsche Mathematiker-Vereinigung in 1996, and the Abel Prize in 2008, which he shared with John Griggs Thompson for their significant contributions to algebra and group theory. He became a member of the French Academy of Sciences in 1979 and was also a member of the Norwegian Academy of Science and Letters and a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1988.Tits made substantial contributions to mathematics, particularly in group theory and incidence geometry. He introduced the theory of buildings, known as Tits buildings, which are combinatorial structures on which groups act, especially in algebraic group theory. His classification of irreducible buildings of spherical type and rank at least three was significant, as was his work on the existence of groups of Lie type. He also classified polar spaces of rank at least three and developed the theory of affine buildings in collaboration with François Bruhat. His well-known theorem, the Tits alternative, states that if G is a finitely generated subgroup of a linear group, then G either has a solvable subgroup of finite index or a free subgroup of rank 2. The Tits group and the Kantor–Koecher–Tits construction are named in his honor, and he introduced the Kneser–Tits conjecture.Tits published several influential works, including 'Algebraic and abstract simple groups' in 1964, 'Buildings of spherical type and finite BN-pairs' in 1974, and 'Moufang polygons' in 2002. His collected works were published in four volumes by the European Mathematical Society in 2013.

Jacques Tits

John Griggs Thompson was born on October 13, 1932. He is an American mathematician recognized for his contributions to the field of finite groups. Thompson earned his Bachelor of Arts from Yale University in 1955 and completed his doctorate at the University of Chicago in 1959 under the guidance of Saunders Mac Lane. He initially joined the mathematics faculty at the University of Chicago before moving to the United Kingdom in 1970 to assume the Rouse Ball Professorship in Mathematics at the University of Cambridge. Later, he transitioned to the University of Florida, where he became a Graduate Research Professor. He holds the title of professor emeritus of pure mathematics at the University of Cambridge and continues to serve as a professor of mathematics at the University of Florida.Thompson's doctoral thesis introduced innovative techniques and resolved a long-standing problem in finite group theory regarding the nilpotency of Frobenius kernels. This achievement garnered attention, including coverage in The New York Times. He played a significant role in the classification of finite simple groups. In 1963, he and Walter Feit demonstrated that all nonabelian finite simple groups have even order, a result published in the Pacific Journal of Mathematics. This work earned them the 1965 Cole Prize in Algebra from the American Mathematical Society. Thompson's N-group papers classified finite simple groups where the normalizer of every non-identity solvable subgroup is solvable, leading to the classification of all minimal finite simple groups. His contributions were acknowledged in the citation for his Fields Medal awarded in 1970.Thompson is associated with the Thompson group Th, one of the 26 sporadic finite simple groups. He also made significant advancements in the inverse Galois problem, establishing a criterion for a finite group to be a Galois group, which implies that the monster simple group qualifies as a Galois group.Thompson was elected to the United States National Academy of Sciences in 1971. He received the Senior Berwick Prize from the London Mathematical Society in 1982 and was awarded an honorary Doctor of Science degree from the University of Oxford in 1988. In 1992, he received the Wolf Prize in Mathematics. He was awarded the United States National Medal of Science in 2000 and is a Fellow of the Royal Society, having received its Sylvester Medal in 1985. Additionally, he is a member of the Norwegian Academy of Science and Letters.

John G. Thompson

John Horton Conway was born on 26 December 1937 in Liverpool, England, to Cyril Horton Conway and Agnes Boyce. He developed an interest in mathematics at an early age, aspiring to become a mathematician by the age of 11. After completing his education at sixth form, he studied mathematics at Gonville and Caius College, Cambridge. During his time at Cambridge, he transformed from a reserved adolescent into an extroverted individual, earning the nickname 'the world's most charismatic mathematician.' He received a BA in 1959 and began research in number theory under the supervision of Harold Davenport. Conway solved an open problem regarding the representation of numbers as sums of fifth powers, which led him to explore infinite ordinals. His interest in games emerged during his studies, particularly through his engagement with backgammon.In 1964, Conway earned his doctorate and became a College Fellow and Lecturer in Mathematics at Sidney Sussex College, Cambridge. He left Cambridge in 1986 to accept the John von Neumann Chair of Mathematics at Princeton University, where he continued his academic career.Conway's work in mathematics spanned various fields, including finite groups, knot theory, number theory, combinatorial game theory, and coding theory. He is best known for inventing the Game of Life, a cellular automaton that gained popularity after being featured in Martin Gardner's Mathematical Games column in 1970. This exposure made Conway a prominent figure in recreational mathematics. He also collaborated with Gardner on various topics, including the game of Sprouts and the Penrose tilings.In combinatorial game theory, Conway co-authored the book 'Winning Ways for your Mathematical Plays' and developed the surreal numbers and Conway chained arrow notation. His contributions to geometry included establishing the existence of sixty-four convex uniform polychora and creating Conway polyhedron notation. In knot theory, he introduced the Conway polynomial and developed tangle theory, while in group theory, he co-authored the ATLAS of Finite Groups and contributed to the classification of finite simple groups, including the discovery of the Conway groups.Conway's work in number theory included proving a case of Edward Waring's conjecture and demonstrating the algorithmic undecidability of a generalization of the Collatz problem. He also invented the Doomsday algorithm for calculating the day of the week. His research extended into theoretical physics, where he and Simon B. Kochen proved the free will theorem, linking concepts in quantum mechanics to mathematical principles.Conway was married three times and had two sons and four daughters from his first two marriages, and a son with his third wife, Diana. He had three grandchildren and two great-grandchildren. On 8 April 2020, he developed symptoms of COVID-19 and died on 11 April 2020, in New Brunswick, New Jersey, at the age of 82.

John Horton Conway

Pierre René, Viscount Deligne, was born on 3 October 1944 in Etterbeek, Belgium. He attended Athénée Adolphe Max and later studied at the Université libre de Bruxelles, where he wrote a dissertation on the Lefschetz theorem and spectral sequences. He completed his doctorate at the University of Paris-Sud in 1972 under Alexander Grothendieck, with a thesis on Hodge theory.Deligne began his career in 1965 at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, collaborating with Grothendieck on the generalization of Zariski's main theorem within scheme theory. In 1968, he worked with Jean-Pierre Serre, contributing to significant results regarding l-adic representations and L-functions. His research also included Hodge theory, where he introduced the concept of weights and applied it to complex geometry. He collaborated with David Mumford on moduli spaces for curves, which influenced the theory of algebraic stacks and had applications in string theory.Deligne is renowned for proving the third and final Weil conjecture in 1973, completing a program initiated by Grothendieck. His proof included the Ramanujan–Petersson conjecture for modular forms of weight greater than one. His 1974 paper provided the first proof of the Weil conjectures, establishing estimates for the eigenvalues of the Frobenius endomorphism, which is analogous to the Riemann hypothesis. His 1980 paper presented a more general version of the Riemann hypothesis.From 1970 to 1984, Deligne was a permanent member of the IHÉS staff, during which he made significant contributions beyond algebraic geometry. He worked with George Lusztig on étale cohomology and representations of finite groups of Lie type, and with Michael Rapoport on moduli spaces from an arithmetic perspective. He received the Fields Medal in 1978 for his contributions.Deligne defined absolute Hodge cycles to address gaps in the theory of motives, which is still largely conjectural. He developed the theory of mixed Hodge structures, which generalizes classical Hodge theory and was instrumental in proving the Weil conjectures. His 1990 paper reworked Tannakian category theory, linking it to the theory of motives.Alongside Alexander Beilinson, Joseph Bernstein, and Ofer Gabber, Deligne contributed to the theory of perverse sheaves, which has implications for the Riemann–Hilbert correspondence. His joint paper with Phillip Griffiths, John Morgan, and Dennis Sullivan in 1974 addressed real homotopy theory of compact Kähler manifolds, resolving several significant questions in complex differential geometry. He also generalized Milnor maps in complex singularity theory and worked on abelian L-functions and their extensions.Deligne's research includes cohomological descent, motivic L-functions, mixed sheaves, and the geometry of braid groups. He provided a modern axiomatic definition of Shimura varieties and collaborated with George Mostow on examples of non-arithmetic lattices.Deligne received numerous awards, including the Fields Medal in 1978, the Crafoord Prize in 1988, the Wolf Prize in 2008, and the Abel Prize in 2013. He was elected a foreign member of the Académie des Sciences de Paris in 1978 and ennobled as viscount by the Belgian king in 2006. He became a member of the Royal Swedish Academy of Sciences and the American Philosophical Society in 2009.

Pierre Deligne

Shing-Tung Yau was born on April 4, 1949, in Shantou, Guangdong, Republic of China, to Hakka parents. His family relocated to British Hong Kong during the Communist takeover of mainland China when he was a few months old. Yau's father, Chen Ying Chiu, was a scholar, and his mother, Yeuk Lam Leung, was from Meixian District. Yau was the fifth of eight children. The family faced financial difficulties after losing their possessions, and Yau's father and second-oldest sister died when he was thirteen. He graduated from Pui Ching Middle School and studied mathematics at the Chinese University of Hong Kong from 1966 to 1969, leaving before receiving a degree.In 1969, Yau moved to the United States to pursue a Ph.D. in mathematics at the University of California, Berkeley. He was inspired by John Milnor's papers on geometric group theory and formulated a generalization of Preissman's theorem. He completed his Ph.D. in 1971 under Shiing-Shen Chern. Yau spent a year at the Institute for Advanced Study in Princeton before joining Stony Brook University as an assistant professor in 1972. He became an associate professor at Stanford University in 1974 and took a visiting faculty position at UCLA in 1976. He married physicist Yu-Yun Kuo in 1976 and returned to the Institute for Advanced Study in 1979, becoming a professor there in 1980. He held a chair professorship at the University of California, San Diego, in 1984 and moved to Harvard University in 1987.Yau was awarded the Fields Medal in 1982 for his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. His work significantly advanced modern differential geometry and geometric analysis. He resolved the Calabi conjecture in 1978, demonstrating the existence of Kähler-Einstein metrics on closed Kähler manifolds with nonpositive first Chern class. Yau's positive energy theorem, developed with Richard Schoen, established that the gravitational energy of an isolated physical system is nonnegative in general relativity.Yau's contributions extend to various fields, including convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory. He has authored over five hundred research articles and has been involved in establishing several mathematical institutes in China. Yau proposed the International Congress of Chinese Mathematicians, which began in 1998, and has served as an editor for several mathematical journals.In 1978, Yau became stateless after the British Consulate revoked his Hong Kong residency due to his permanent residency status in the United States. He remained stateless until 1990 when he obtained U.S. citizenship. Yau has received numerous honors, including the Oswald Veblen Prize in Geometry, the John J. Carty Award for the Advancement of Science, the Guggenheim Fellowship, and the Wolf Prize in Mathematics. He has also been recognized with honorary degrees from various universities and is a foreign member of several national academies.

Shing-Tung Yau

Grigori Yakovlevich Perelman was born on 13 June 1966 in Leningrad, Soviet Union, to Jewish parents Yakov and Lyubov. His mother, who had given up her graduate studies in mathematics to raise him, recognized his mathematical talent early, enrolling him in an after-school program at the age of 10. Perelman attended Leningrad Secondary School 239, excelling in mathematics and physics, and won a gold medal at the International Mathematical Olympiad in 1982. He continued his studies at Leningrad State University, where he completed his PhD in 1990 under the guidance of Aleksandr Aleksandrov and Yuri Burago.Perelman's early research included work in convex geometry and negatively curved surfaces. He published articles on combinatorial structures of convex polyhedra and established a measure-theoretic formulation of Helly's theorem. His graduate studies focused on negatively curved hypersurfaces, leading to significant results regarding the structure of such surfaces. In the 1990s, he collaborated with Yuri Burago, Mikhael Gromov, and Anton Petrunin on Alexandrov spaces, establishing foundational concepts and proving a stability theorem.In 1994, Perelman proved the soul conjecture in Riemannian geometry, resolving a problem that had remained open for 20 years. He later developed new techniques in Ricci flow analysis, culminating in his proof of the Poincaré conjecture and Thurston's geometrization conjecture between 2002 and 2003. These conjectures had been central problems in topology and geometry for decades. Perelman's work was published in a series of preprints on arXiv, which were initially difficult for the mathematical community to fully comprehend due to their terse nature.In 2006, Perelman was offered the Fields Medal for his contributions to geometry but declined the award, expressing disinterest in fame and recognition. He was later recognized by the journal Science as the Breakthrough of the Year for his proof of the Poincaré conjecture. In 2010, he was awarded the Millennium Prize for solving the conjecture but also rejected this prize, citing a belief that his contributions were not greater than those of Richard S. Hamilton, who had laid the groundwork for the Ricci flow.Perelman resigned from his position at the Steklov Institute in December 2005 and subsequently withdrew from professional mathematics, expressing disappointment with the ethical standards in the field. He has lived in seclusion in Saint Petersburg, avoiding media attention and declining interview requests since 2006. His later activities remain largely unknown, although he has been reported to have shown interest in other mathematical problems.

Grigori Perelman

Terence Chi-Shen Tao was born on 17 July 1975 to Chinese immigrant parents in Adelaide, South Australia. His father, Billy Tao, was a paediatrician from Shanghai, and his mother, Grace Leong, was a mathematics and physics teacher from Hong Kong. The family emigrated to Australia in 1972. Tao has two brothers, both of whom have represented Australia in mathematics and chess.From a young age, Tao exhibited exceptional mathematical talent, skipping five grades in school. He began attending university-level mathematics courses at the age of nine and achieved a score of 760 on the SAT math section at eight years old, making him one of only three children in the Johns Hopkins Study of Exceptional Talent program to do so. He participated in the International Mathematical Olympiad at the age of ten, winning a bronze medal in 1986, a silver in 1987, and a gold medal in 1988, becoming the youngest winner of each medal in the competition's history.At age 14, Tao attended the Research Science Institute, a summer program for gifted students. He completed both his bachelor's and master's degrees at Flinders University by the age of 16. In 1992, he won a Fulbright Scholarship to study at Princeton University, where he earned his PhD under Elias Stein at the age of 21. Tao joined the faculty at the University of California, Los Angeles (UCLA) in 1996 and became a full professor in 1999, the youngest person to achieve this rank at UCLA.Tao is known for his collaborative approach to research, having worked with over 30 co-authors by 2006. He collaborated extensively with Ben J. Green, with whom he proved the Green–Tao theorem, demonstrating the existence of arbitrarily long arithmetic progressions of prime numbers. His work has garnered attention in various fields, including harmonic analysis, partial differential equations, and analytic number theory.Throughout his career, Tao has received numerous accolades, including the Fields Medal in 2006, making him the first Australian and one of the youngest recipients of this prestigious award. He has also been recognized as a Fellow of several scientific societies and has published over three hundred research papers and sixteen books. Tao's contributions to mathematics have made him a prominent figure in the field, with a reputation for solving complex problems and advancing mathematical knowledge.Tao is married to Laura Tao, an electrical engineer at NASA's Jet Propulsion Laboratory, and they have two children. He speaks Cantonese but cannot write Chinese.