''The Story of Maths'' is a four-part British
television
Television, sometimes shortened to TV, is a telecommunication medium for transmitting moving images and sound. The term can refer to a television set, or the medium of television transmission. Television is a mass medium for advertisin ...
series outlining aspects of the
history 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 ...
. It was a co-production between the
Open University
The Open University (OU) is a British public research university and the largest university in the United Kingdom by number of students. The majority of the OU's undergraduate students are based in the United Kingdom and principally study off- ...
and the
BBC #REDIRECT BBC #REDIRECT BBC #REDIRECT BBC
Here i going to introduce about the best teacher of my life b BALAJI sir. He is the precious gift that I got befor 2yrs . How has helped and thought all the concept and made my success in the 10th board ex ...
and aired in October 2008 on
BBC Four
BBC Four is a British free-to-air public broadcast television channel owned and operated by the BBC. It was launched on 2 March 2002 . The material was written and presented by
University of Oxford
, mottoeng = The Lord is my light
, established =
, endowment = £6.1 billion (including colleges) (2019)
, budget = £2.145 billion (2019–20)
, chancellor ...
professor
Marcus du Sautoy
Marcus Peter Francis du Sautoy (; born 26 August 1965) is a British mathematician, Simonyi Professor for the Public Understanding of Science at the University of Oxford, Fellow of New College, Oxford and author of popular mathematics and popu ...
. The consultants were the Open University academics
Robin Wilson, professor
Jeremy Gray
Jeremy John Gray (born 25 April 1947) is an English mathematician primarily interested in the history of mathematics.
Biography
Gray studied mathematics at Oxford University from 1966 to 1969, and then at Warwick University, obtaining his Ph.D ...
and June Barrow-Green. Kim Duke is credited as series producer.
[''To Infinity and Beyond'' 27 October 2008 21:00 BBC Four]
The series comprised four programmes respectively titled: ''The Language of the Universe''; ''The Genius of the East''; ''The Frontiers of Space''; and'' To Infinity and Beyond''. Du Sautoy documents the development of mathematics covering subjects such as the invention of zero and the unproven
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
, a 150-year-old problem for whose solution the
Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation (nonprofit), foundation dedicated to increasing and disseminating mathematics, mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address i ...
has offered a $1,000,000 prize. He escorts viewers through the subject's history and geography. He examines the development of key mathematical ideas and shows how mathematical ideas underpin the world's science, technology, and culture.
He starts his journey in
ancient Egypt and finishes it by looking at current mathematics. Between he travels through
Babylon
''Bābili(m)''
* sux, 𒆍𒀭𒊏𒆠
* arc, 𐡁𐡁𐡋 ''Bāḇel''
* syc, ܒܒܠ ''Bāḇel''
* grc-gre, Βαβυλών ''Babylṓn''
* he, בָּבֶל ''Bāvel''
* peo, 𐎲𐎠𐎲𐎡𐎽𐎢 ''Bābiru''
* elx, 𒀸𒁀𒉿𒇷 ''Babi ...
,
Greece
Greece,, or , romanized: ', officially the Hellenic Republic, is a country in Southeast Europe. It is situated on the southern tip of the Balkans, and is located at the crossroads of Europe, Asia, and Africa. Greece shares land borders with ...
,
India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
,
China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ...
, and the
medieval Middle East. He also looks at mathematics in Europe and then in America and takes the viewers inside the lives of many of the greatest mathematicians.
"The Language of the Universe"
In this opening programme Marcus du Sautoy looks at how important and fundamental mathematics is to our lives before looking at the mathematics of
ancient Egypt,
Mesopotamia
Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...
, and
Greece
Greece,, or , romanized: ', officially the Hellenic Republic, is a country in Southeast Europe. It is situated on the southern tip of the Balkans, and is located at the crossroads of Europe, Asia, and Africa. Greece shares land borders with ...
.
Du Sautoy commences in
Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Mediter ...
where recording the patterns of the seasons and in particular the flooding of the
Nile
The Nile, , Bohairic , lg, Kiira , Nobiin language, Nobiin: Áman Dawū is a major north-flowing river in northeastern Africa. It flows into the Mediterranean Sea. The Nile is the longest river in Africa and has historically been considered ...
was essential to their economy. There was a need to solve practical problems such as land area for taxation purposes.
[BBC Four; ''The Language of the Universe;'' 9:00 pm 6 October 2008] Du Sautoy discovers the use of a decimal system based on the fingers on the hands, the unusual method for multiplication and division. He examines the
Rhind Papyrus
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...
, the
Moscow Papyrus
The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...
and explores their understanding of binary numbers, fractions and solid shapes.
He then travels to
Babylon
''Bābili(m)''
* sux, 𒆍𒀭𒊏𒆠
* arc, 𐡁𐡁𐡋 ''Bāḇel''
* syc, ܒܒܠ ''Bāḇel''
* grc-gre, Βαβυλών ''Babylṓn''
* he, בָּבֶל ''Bāvel''
* peo, 𐎲𐎠𐎲𐎡𐎽𐎢 ''Bābiru''
* elx, 𒀸𒁀𒉿𒇷 ''Babi ...
and discovered that the way we tell the time today is based on the
Babylonian 60 base number system. So because of the Babylonians we have 60 seconds in a minute, and 60 minutes in an hour. He then shows how the Babylonians used
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s to measure their land. He deals briefly with
Plimpton 322
Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a table ...
.
In Greece, the home of ancient
Greek mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
, he looks at the contributions of some of its greatest and well known mathematicians including
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
,
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
,
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, and
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, who are some of the people who are credited with beginning the transformation of mathematics from a tool for counting into the analytical subject we know today. A controversial figure, Pythagoras' teachings were considered suspect and his followers seen as social outcasts and a little be strange and not in the norm. There is a legend going around that one of his followers,
Hippasus
Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes c ...
, was drowned when he announced his discovery of
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s. As well as his work on the properties of right angled triangles, Pythagoras developed another important theory after observing musical instruments. He discovered that the intervals between harmonious musical notes are always in whole number intervals. It deals briefly with
Hypatia of Alexandria
Hypatia, Koine pronunciation (born 350–370; died 415 AD) was a neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire. She was a prominent thinker in Alexandria where ...
.
"The Genius of the East"
With the decline of ancient Greece, the development of maths stagnated in Europe. However the progress of mathematics continued in the East. Du Sautoy describes both the
Chinese use of maths in
engineering projects
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
and their belief in the mystical powers of numbers. He mentions
Qin Jiushao
Qin Jiushao (, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gaug ...
.
He describes
Indian mathematicians
chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India.
Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...
’ invention of
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
; their introduction of a symbol for the number
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
and their contribution to the new concepts of
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
and
negative number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s. It shows
Gwalior Fort
The Gwalior Fort commonly known as the ''Gwāliiyar Qila'', is a hill fort near Gwalior, Madhya Pradesh, India. The fort has existed at least since the 10th century, and the inscriptions and monuments found within what is now the fort campus ind ...
where zero is inscribed on its walls. It mentions the work of
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
and
Bhāskara II
Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroman ...
on the subject of zero. He mentions
Madhava of Sangamagrama
Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He is ...
and
Aryabhata
Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
and illustrates the - historically first exact -
formula for calculating the π (pi).
[BBC documentary "The Story of Maths", second part](_blank)
showing a visualization of the historically first exact formula, starting at 35 min and 20 sec into the second part of the documentary.
Du Sautoy then considers
the Middle East
The Middle East ( ar, الشرق الأوسط, ISO 233: ) is a geopolitical region commonly encompassing Arabia (including the Arabian Peninsula and Bahrain), Asia Minor (Asian part of Turkey except Hatay Province), East Thrace (Europea ...
: the invention of the new language of
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
and the evolution of a solution to
cubic equations
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
. He talks about the
House of Wisdom
The House of Wisdom ( ar, بيت الحكمة, Bayt al-Ḥikmah), also known as the Grand Library of Baghdad, refers to either a major Abbasid public academy and intellectual center in Baghdad or to a large private library belonging to the Abba ...
with
Muhammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...
and he visits
University of Al-Karaouine
The University of al-Qarawiyyin ( ar, جامعة القرويين; ber, ⵜⴰⵙⴷⴰⵡⵉⵜ ⵏ ⵍⵇⴰⵕⴰⵡⵉⵢⵉⵏ; french: Université Al Quaraouiyine), also written Al-Karaouine or Al Quaraouiyine, is a university located in ...
. He mentions
Omar Khayyám
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...
.
Finally he examines the spread of
Eastern knowledge to the West through mathematicians such as
Leonardo Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
, famous for the
Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
. He mentions
Niccolò Fontana Tartaglia
Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Republi ...
.
"The Frontiers of Space"
From the seventeenth century, Europe replaced the Middle East as the engine house of mathematical ideas. Du Sautoy visits
Urbino
Urbino ( ; ; Romagnol: ''Urbìn'') is a walled city in the Marche region of Italy, south-west of Pesaro, a World Heritage Site notable for a remarkable historical legacy of independent Renaissance culture, especially under the patronage of ...
to introduce
perspective using mathematician and artist,
Piero della Francesca
Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca i ...
's ''
The Flagellation of Christ''.
[''The Frontiers of Space'' 20 October 2008 21:00 BBC Four]
Du Sautoy proceeds to describes
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
realisation that it was possible to describe curved lines as equations and thus link algebra and geometry. He talks with
Henk J. M. Bos
Hendrik Jan Maarten "Henk" Bos (born 17 July 1940, Enschede) is a Dutch historian of mathematics.
Career
Hendrik was a student of Hans Freudenthal and Jerome Ravetz at Utrecht University and in 1973 wrote a thesis "Differentials, higher order dif ...
about Descartes. He shows how one of
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
's theorems is now the basis for the codes that protect credit card transactions on the internet. He describes Isaac Newton’s development of math and physics crucial to understanding the behaviour of moving objects in engineering. He covers the
Leibniz and Newton calculus controversy
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...
and the
Bernoulli family
The Bernoulli family () of Basel was a patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physics during the early modern period.
...
. He further covers
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, the father of topology, and
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
' invention of a new way of handling equations, modular arithmetic. He mentions
János Bolyai
János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consisten ...
.
The further contribution of Gauss to our understanding of how
prime numbers
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
are distributed is covered thus providing the platform for
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
's theories on prime numbers. In addition Riemann worked on the properties of objects, which he saw as manifolds that could exist in multi-dimensional space.
OpenLearn
OpenLearn is an educational website. It is the UK's Open University's contribution to the open educational resources (OER) project and the home of free, open learning from The Open University. The original project was part-funded by the Will ...
The Frontiers of Space
retrieved 12 March 2014
"To Infinity and Beyond"
Hilbert's first problem
The final episode considers the great unsolved problems that confronted mathematicians in the 20th century. On 8 August 1900
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
gave a historic talk at the
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...
in Paris. Hilbert posed
twenty-three then unsolved problems in mathematics which he believed were of the most immediate importance. Hilbert succeeded in setting the agenda for 20thC mathematics and the programme commenced with
Hilbert's first problem.
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
considered the infinite set of whole numbers 1, 2, 3 ... ∞ which he compared with the smaller set of numbers 10, 20, 30 ... ∞. Cantor showed that these two infinite sets of numbers actually had the same size as it was possible to pair each number up; 1 - 10, 2 - 20, 3 - 30 ... etc.
If fractions now are considered there are an infinite number of fractions between any of the two whole numbers, suggesting that the infinity of fractions is bigger than the infinity of whole numbers. Yet Cantor was still able to pair each such fraction to a whole number 1 -
1/
1; 2 -
2/
1; 3 -
1/
2 ... etc. through to ∞; i.e. the infinities of both fractions and whole numbers were shown to have the same size.
But when the set of all infinite decimal numbers was considered, Cantor was able to prove that this produced a bigger infinity. This was because, no matter how one tried to construct such a list, Cantor was able to provide a new decimal number that was missing from that list. Thus he showed that there were different infinities, some bigger than others.
However, there was a problem that Cantor was unable to solve: Is there an infinity sitting between the smaller infinity of all the fractions and the larger infinity of the decimals? Cantor believed, in what became known as the
Continuum Hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
, that there is no such set. This would be the first problem listed by Hilbert.
[
]
Poincaré conjecture
Next Marcus discusses Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
's work on the discipline of 'Bendy geometry'. If two shapes can be moulded or morphed to each other's shape then they have the same topology. Poincaré was able to identify all possible two-dimensional topological surfaces; however in 1904 he came up with a topological problem, the Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
, that he could not solve; namely what are all the possible shapes for a 3D universe.[
According to the programme, the question was solved in 2002 by ]Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
who linked the problem to a different area of mathematics. Perelman looked at the dynamics of the way things can flow over the shape. This enabled him to find all the ways that 3D space could be wrapped up in higher dimensions.[
]
David Hilbert
The achievements of David Hilbert were now considered. In addition to Hilbert's problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...
, Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, Hilbert Classification and the Hilbert Inequality, du Sautoy highlights Hilbert's early work on equations as marking him out as a mathematician able to think in new ways. Hilbert showed that, while there were an infinity of equations, these equations could be constructed from a finite number of building block like sets. Hilbert could not construct that list of sets; he simply proved that it existed. In effect Hilbert had created a new more abstract style of Mathematics.[
]
Hilbert's second problem
For 30 years Hilbert believed that mathematics was a universal language powerful enough to unlock all the truths and solve each of his 23 Problems. Yet, even as Hilbert was stating ''We must know, we will know'', Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
had shattered this belief; he had formulated the Incompleteness Theorem
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
based on his study of Hilbert's second problem
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his Hilbert's problems, 23 problems. It asks for a proof that the arithmetic is consistency proof, consistent – free of any internal contradictions. Hilber ...
:
:''This statement cannot be proved''
Using a code based on prime numbers, Gödel was able to transform the above into a pure statement of arithmetic. Logically, the above cannot be false and hence Gödel had discovered the existence of mathematical statements that were true but were incapable of being proved.[
]
Hilbert's first problem revisited
In 1950s American mathematician Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
took up the challenge of Cantor's Continuum Hypothesis which asks "is there is or isn't there an infinite set of number bigger than the set of whole numbers but smaller than the set of all decimals". Cohen found that there existed two equally consistent mathematical worlds. In one world the Hypothesis was true and there did not exist such a set. Yet there existed a mutually exclusive but equally consistent mathematical proof that Hypothesis was false and there was such a set. Cohen would subsequently work on Hilbert's eighth problem, the Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
, although without the success of his earlier work.[
]
Hilbert's tenth problem
Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equat ...
asked if there was some universal method that could tell whether any equation had whole number solutions or not. The growing belief was that no so such method was possible yet the question remained, how could you prove that, no matter how ingenious you were, you would never come up with such a method. He mentions Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
. To answer this Julia Robinson
Julia Hall Bowman Robinson (December 8, 1919July 30, 1985) was an American mathematician noted for her contributions to the fields of computability theory and computational complexity theory—most notably in decision problems. Her work on Hilbe ...
, who created the Robinson Hypothesis which stated that to show that there was no such method all you had to do was cook up one equation whose solutions were a very specific set of numbers: The set of numbers needed to grow exponentially yet still be captured by the equations at the heart of Hilbert's problem. Robinson was unable to find this set. This part of the solution fell to Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich, (russian: Ю́рий Влади́мирович Матиясе́вич; born 2 March 1947 in Leningrad) is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's t ...
who saw how to capture the Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
using the equations at the heart of Hilbert's tenth.[
]
Algebraic geometry
The final section briefly covers algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
had refined a new language for mathematics. Galois believed mathematics should be the study of structure as opposed to number and shape. Galois had discovered new techniques to tell whether certain equations could have solutions or not. The symmetry of certain geometric objects was the key. Galois' work was picked up by André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
who built algebraic geometry, a whole new language. Weil's work connected number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, algebra, topology and geometry.
Finally du Sautoy mentions Weil's part in the creation of the fictional mathematician Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
and another contributor to Bourbaki's output - Alexander Grothendieck.[
]
See also
* '' Chemistry: A Volatile History''
* '' The Story of Science: Power, Proof and Passion''
References
External links
*
*
OU on the BBC: The Story of Maths - About the series
at OpenLearn
OpenLearn is an educational website. It is the UK's Open University's contribution to the open educational resources (OER) project and the home of free, open learning from The Open University. The original project was part-funded by the Will ...
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