Arithmetic

History
It is the method known as the "Method of the Indians" or in Latin "Modus Indoram" that has become our arithmetic today. Prior to this, basic arithmetic operations were highly complicated affairs. Seventh century Syriac Bishop Severus Sebhokt mentioned this method and stated that the method of the Indians is beyond description. Indian arithmetic was much simpler than the Greek arithmetic simply due to the simplicity of the Indian number system which had a zero and place value notation. Arabs learned this new method and called it "Hesab" or "Hindu Science". Fibonacci or Leonardo of Pisa is one of the first European mathematicians who introduced the "Method of the Indians" to Europe. In his famous book "Liber Abaci" Fibonacci says that compared to this new method all other methods were mistakes.


The prehistory of arithmetic is limited by a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango Bone from Africa, dating from 18,000 BC.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic circa 1850 BC, although historians can only infer the methods utilized to generate the arithmetical results (see Plimpton 322). Likewise, definitive addition, subtraction, multiplication, and division facts are used within the unit fraction system, which can be found in the Rhind Mathematical Papyrus dating from Ancient Egypt circa 1650 BC, copied from 1850 BC (Mathematathe septem liberales artes (seven liberal arts).

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Hindu-Arabic numerals and decimal place notation for numbers. Hindu- Arabic numeral based arithmetic was developed by great Indian mathematicians Aryabhatta, Brahmagupta and Bhaskara. Aryabhatta tried different place value notations and Brahmagupta added zero to the Indian number system. Brahmagupta developed modern multiplication, division, addition and subtraction based on Hindu-Arabic numerals. Although it is now considered elementary, its simplicity is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.




Further information: Introduction to Arithmetic

[edit] Decimal arithmetic
Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6 hundredths (10-2). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of 0 as a number comparable to the other basic digits.

Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {...,102,10,1,10-1,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.


[edit] Arithmetic operations
The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero), and wherein these four operations obey the usual laws, is called a field.


[edit] Addition (+)
Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum.

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting.

Addition is commutative and associative so the order in which the terms are added does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number will yield that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) = 0.


[edit] Subtraction (−)
Main article: Subtraction
Subtraction is essentially the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero.

Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.


[edit] Multiplication (× or ·)
Main article: Multiplication
Multiplication is in essence repeated addition, or the sum of a list of identical numbers. Multiplication finds the product of two numbers, the multiplier and the multiplicand, sometimes both are simply called factors.

Multiplication, as it is really repeated addition, is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse is the reciprocal of any number, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity, 1.


[edit] Division (÷ or /)
Main article: Division (mathematics)
Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers and negative one). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it will obey all the properties of multiplication.

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Analytic number theory

Analytic number theory is the branch of number theory that uses methods from mathematical analysis. Its first major success was Dirichlet's application of analysis to prove Dirichlet's theorem on arithmetic progressions, stating the existence of infinitely many primes in arithmetic progressions of the form a + nb, where a and b are relatively prime. The proofs of the prime number theorem based on the Riemann zeta function are another milestone.

The outline of the subject remains similar to the heyday of the subject in the 1930s. Multiplicative number theory deals with the distribution of the prime numbers, applying Dirichlet series as generating functions. It is assumed that the methods will eventually apply to the general L-function, though that theory is still largely conjectural. Additive number theory has as typical problems Goldbach's conjecture and Waring's problem.

Methods have changed somewhat. The circle method of

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Algebraic number theory

Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. An algebraic number field is any finite (and therefore algebraic) field extension of the rational numbers. These domains contain elements analogous to the integers, the so-called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed — Galois theory, group cohomology, class field theory, group representations and L-functions — is that it allows one to recover that order partly for this new class of numbers.

Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite fields). This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory.
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Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.

Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. (See the list of number theory topics).

The term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system. Mathematicians working in the field of number theory are called number theorists.


elementary number theory

In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function, Euler's φ function, integer sequences, factorials and Fibonacci numbers all also fall into this area.

Many questions in number theory can be stated in elementary number theoretic terms, but they may require very deep consideration and new approaches outside the realm of elementary number theory to solve. Examples include:

* The Goldbach conjecture concerning the expression of even numbers as sums of two primes.
* Catalan's conjecture (now Mihăilescu's theorem) regarding successive integer powers.
* The twin prime conjecture about the infinitude of prime pairs.
* The Collatz conjecture concerning a simple iteration.
* Fermat's last theorem (stated in 1637, but not proved until 1994) concerning the impossibility of finding nonzero integers x, y, z such that xn + yn = zn for some integer n greater than 2.

The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem).

Analytic number theory

Analytic number theory employs the machinery of calculus and complex analysis to tackle questions about integers. The prime number theorem (PNT) and the related Riemann hypothesis are examples. Waring's problem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well. Proofs of the transcendence of mathematical constants, such as π or e, are also classified as analytical number theory. While statements about transcendental numbers may seem to be removed from the study of integers, they really study the possible values of polynomials with integer coefficients evaluated at, say, e; they are also closely linked to the field of Diophantine approximation, where one investigates "how well" a given real number may be approximated by a rational one.

Algebraic number theory

In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so-called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed—Galois theory, group cohomology, class field theory, group representations and L-functions—is that it allows to recover that order partly for this new class of numbers.

Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite fields). This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory.

Geometric number theory

Geometric number theory (traditionally called geometry of numbers) incorporates all forms of geometry. It starts with Minkowski's theorem about lattice points in convex sets and investigations of sphere packings.

Combinatorial number theory

Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field.

Computational number theory

Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography.

History

Vedic number theory

Mathematicians in India were interested in finding integral solutions of Diophantine equations since the Vedic era. The earliest geometric use of Diophantine equations can be traced back to the Sulba Sutras, which were written between the 8th and 6th centuries BC. Baudhayana (c. 800 BC) found two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also used simultaneous Diophantine equations with up to four unknowns. Apastamba (c. 600 BC) used simultaneous Diophantine equations with up to five unknow

Jaina number theory

In India, Jaina mathematicians developed the earliest systematic theory of numbers from the 4th century BC to the 2nd century CE. The Jaina text Surya Prajinapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. Each of these was further subdivided into three orders:

* Enumerable: lowest, intermediate and highest.
* Innumerable: nearly innumerable, truly innumerable and innumerably innumerable.
* Infinite: nearly infinite, truly infinite, infinitely infinite.

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized five different types of infinity: infinite in one and two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions).

The highest enumerable number N of the Jains corresponds to the modern concept of aleph-null \aleph_0 (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of transfinite cardinal numbers, of which \aleph_0 is the smallest.

In the Jaina work on the theory of sets, two basic types of transfinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.

Hellenistic number theory

Number theory was a favorite study among the Hellenistic mathematicians of Alexandria, Egypt from the 3rd century CE, who were aware of the Diophantine equation concept in numerous special cases. The first Hellenistic mathematician to study these equations was Diophantus.

Diophantus also looked for a method of finding integer solutions to linear indeterminate equations, equations that lack sufficient information to produce a single discrete set of answers. The equation x + y = 5 is such an equation. Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of answers is known even though a specific answer is not.

Classical Indian number theory

Diophantine equations were extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for the determination of integral solutions of Diophantine equations. Aryabhata (499) gave the first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c, which occurs in his text Aryabhatiya. This kuttaka algorithm is considered to be one of the most significant contributions of Aryabhata in pure mathematics, which found solutions to Diophantine equations by means of continued fractions. The technique was applied by Aryabhata to give integral solutions of simulataneous linear Diophantine equations, a problem with important applications in astronomy. He also found the general solution to the indeterminate linear equation using this method.

Brahmagupta in 628 handled more difficult Diophantine equations. He used the chakravala method to solve quadratic Diophantine equations, including forms of Pell's equation, such as 61x2 + 1 = y2. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. The equation 61x2 + 1 = y2 was later posed as a problem in 1657 by the French mathematician Pierre de Fermat. The general solution to this particular form of Pell's equation was found over 70 years later by Leonhard Euler, while the general solution to Pell's equation was found over 100 years later by Joseph Louis Lagrange in 1767. Meanwhile, many centuries ago, the general solution to Pell's equation was recorded by Bhaskara II in 1150, using a modified version of Brahmagupta's chakravala method, which he also used to find the general solution to other indeterminate quadratic equations and quadratic Diophantine equations. Bhaskara's chakravala method for finding the general solution to Pell's equation was much simpler than the method used by Lagrange over 600 years later. Bhaskara also found solutions to other indeterminate quadratic, cubic, quartic and higher-order polynomial equations. Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations.

Islamic number theory

From the 9th century, Islamic mathematicians had a keen interest in number theory. The first of these mathematicians was the Arab mathematician Thabit ibn Qurra, who discovered a theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. In the 10th century, Al-Baghdadi looked at a slight variant of Thabit ibn Qurra's theorem.

In the 10th century, al-Haitham seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k − 1(2k − 1) where 2k − 1 is prime. Al-Haytham is also the first person to state Wilson's theorem, namely that if p is prime then 1 + (p − 1)! is divisible by p. It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Edward Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771.

Amicable numbers played a large role in Islamic mathematics. In the 13th century, Persian mathematician Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. In the 17th century, Muhammad Baqir Yazdi gave the pair of amicable numbers 9,363,584 and 9,437,056 still many years before Euler's contribution.

Early European number theory

Number theory began in Europe in the 16th and 17th centuries, with François Viète, Bachet de Meziriac, and especially Fermat, whose infinite descent method was the first general proof of diophantine questions. Fermat's last theorem was posed as a problem in 1637, a proof of which wasn't found until 1994. Fermat also posed the equation 61x2 + 1 = y2 as a problem in 1657.

In the eighteenth century, Euler and Lagrange made important contributions to number theory. Euler did some work on analytic number theory, and found a general solution to the equation 61x2 + 1 = y2, which Fermat posed as a problem. Lagrange found a solution to the more general Pell's equation. Euler and Lagrange solved these Pell equations by means of continued fractions, though this was more difficult than the Indian chakravala method.

Beginnings of modern number theory

Around the beginning of the nineteenth century books of Legendre (1798), and Gauss put together the first systematic theories in Europe. Gauss's Disquisitiones Arithmeticae (1801) may be said to begin the modern theory of numbers.

The formulation of the theory of congruences starts with Gauss's Disquisitiones. He introduced the symbolism

a \equiv b \pmod c,

and explored most of the field. Chebyshev published in 1847 a work in Russian on the subject, and in France Serret popularised it.

Besides summarizing previous work, Legendre stated the law of quadratic reciprocity. This law, discovered by induction and enunciated by Euler, was first proved by Legendre in his Théorie des Nombres (1798) for special cases. Independently of Euler and Legendre, Gauss discovered the law about 1795, and was the first to give a general proof. The following have also contributed to the subject: Cauchy; Dirichlet whose Vorlesungen über Zahlentheorie is a classic; Jacobi, who introduced the Jacobi symbol; Liouville, Zeller(?), Eisenstein, Kummer, and Kronecker. The theory extends to include cubic and biquadratic reciprocity, (Gauss, Jacobi who first proved the law of cubic reciprocity, and Kummer).

To Gauss is also due the representation of numbers by binary quadratic forms.

Prime number theory

A recurring and productive theme in number theory is the study of the distribution of prime numbers. Carl Friedrich Gauss conjectured the limit of the number of primes not exceeding a given number (the prime number theorem) as a teenager.

Chebyshev (1850) gave useful bounds for the number of primes between two given limits. Riemann introduced complex analysis into the theory of the Riemann zeta function. This led to a relation between the zeros of the zeta function and the distribution of primes, eventually leading to a proof of prime number theorem independently by Hadamard and de la Vallée Poussin in 1896. However, an elementary proof was given later by Paul Erdős and Atle Selberg in 1949+. Here elementary means that it does not use techniques of complex analysis; however, the proof is still very ingenious and difficult. The Riemann hypothesis, which would give much more accurate information, is still an open question.

Nineteenth-century developments

Cauchy, Poinsot (1845), Lebesgue(?) (1859, 1868), and notably Hermite have added to the subject. In the theory of ternary forms Eisenstein has been a leader, and to him and H. J. S. Smith is also due a noteworthy advance in the theory of forms in general. Smith gave a complete classification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. The investigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and the theory was completed by Smith.

Dirichlet was the first to lecture upon the subject in a German university. Among his contributions is the extension of Fermat's last theorem:

x^n+y^n \neq z^n, (x,y,z \neq 0, n > 2)

which Euler and Legendre had proven for n = 3,4 (and therefore by implication, all multiples of 3 and 4), Dirichlet showing that x^5+y^5 \neq az^5. Among the later French writers are Borel; Poincaré, whose memoirs are numerous and valuable; Tannery, and Stieltjes. Among the leading contributors in Germany were Kronecker, Kummer, Schering, Bachmann, and Dedekind. In Austria Stolz's Vorlesungen über allgemeine Arithmetik (1885-86), and in England Mathews' Theory of Numbers (Part I, 1892) were scholarly general works. Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory.

Twentieth-century developments

Major figures in twentieth-century number theory include Paul Erdős, Gerd Faltings, G. H. Hardy, Edmund Landau, John Edensor Littlewood, Srinivasa Ramanujan and André Weil.

Milestones in twentieth-century number theory include the proof of Fermat's Last Theorem by Andrew Wiles in 1994 and the proof of the related Taniyama–Shimura conjecture in 1999.

Quotations

* Mathematics is the queen of the sciences and number theory is the queen of mathematics. — Gauss
* God invented the integers; all else is the work of man. — Kronecker
* I know numbers are beautiful. If they aren't beautiful, nothing is. — Erdős


References

* Apostol, T. M. (1986). Introduction to Analytic Number Theory. Springer-Verlag. ISBN 0-387-90163-9.
* Dedekind, Richard (1963). Essays on the Theory of Numbers. Cambridge University Press. ISBN 0-486-21010-3.
* Davenport, Harold (1999). The Higher Arithmetic: An Introduction to the Theory of Numbers (7th ed.). Cambridge University Press. ISBN 0-521-63446-6.
* Guy, Richard K. (1981). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-90593-6.
* Hardy, G. H. and Wright, E. M. (1980). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press. ISBN 0-19-853171-0.
* Niven, Ivans Herbert S. Zuckermans and Hugh L. Montgomery (1991). An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books. ISBN 0-471-62546-9.
* Ore, Oystein (1948). Number Theory and Its History. Dover Publications, Inc.. ISBN 0-486-65620-9.
* Smith, David. History of Modern Mathematics (1906) (adapted public domain text)
* Dutta, Amartya Kumar (2002). 'Diophantine equations: The Kuttaka', Resonance - Journal of Science Education.
* O'Connor, John J. and Robertson, Edmund F. (2004). 'Arabic/Islamic mathematics', MacTutor History of Mathematics archive.
* O'Connor, John J. and Robertson, Edmund F. (2004). 'Index of Ancient Indian mathematics', MacTutor History of Mathematics archive.
* O'Connor, John J. and Robertson, Edmund F. (2004). 'Numbers and Number Theory Index', MacTutor History of Mathematics archive.
* Important publications in number theory

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Foundations of mathematics

Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. The search for foundations of mathematics is however also a central question of the philosophy of mathematics: On what ultimate basis can mathematical statements be called true?


The foundational philosophy of Platonist mathematical realism, as exemplified by mathematician Kurt Gödel, proposes the existence of a world of mathematical objects independent of humans; the truths about these objects are discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. The obvious question, then, is: how do we access this world? (cf Anglin 1991 p. 218)

The foundational philosophy of formalism, as exemplified by David Hilbert, is based on axiomatic set theory and formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic (cf Anglin 1991 p. 218).

Formalism does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why "true" mathematical statements (e.g., the laws of arithmetic) appear to be true in the physical world, and so on. Formalistic truth could also turn out to be rather pointless: it is possible that all statements could be derived from the axioms of set theory. Moreover, as a consequence of Gödel's second incompleteness theorem, we can never be sure that this is not the case.

The foundational philosophy of intuitionism or constructivism, as exemplified in the extreme by Brouwer and more coherently by Stephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of non-existence. For example, as a consequence of this the form of proof known as reductio ad absurdum is suspect (cf Anglin 1991 p. 218).

Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.
Foundational crisis

The foundational crisis of mathematics (in German: Grundlagenkrise der Mathematik) was early 20th century's term for the search for proper foundations of mathematics.

After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.

One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell's paradox) and to be inconsistent: an undesirable situation in which every mathematical statement that can be formulated in a proposed system (such as 2 + 2 = 5) can also be proved in the system.

Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a formal system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols [citation needed]. The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic – a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means. Meanwhile, the intuitionistic school had failed to attract adherents among working mathematicians, and foundered due to the difficulties of doing mathematics under the constraint of constructivism.

In a sense, the crisis has not been resolved, but faded away: most mathematicians do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided.

Platonism

”Platonists, such as Kurt Gödel, hold that numbers are abstract, necessarily existing objects, independent of the human mind” (Anglin (1994) p. 218)

Formalism

“Formalists, such as David Hilbert (1862-1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...” (Anglin (1994) p. 218)

Intuitionism

”Intuitionists, such as L. E. J. Brouwer (1882-1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them." (Anglin (1994) p. 218)

References

* W. S. Anglin, Mathematics: A Concise history and Philosophy, Springer-Verlag, New York, 1994. Chapter 39 Foundations contains concise descriptions, for the 20th century, of Platonism (with respect to Gödel), Formalism (with respect to Hilbert), and Intuitionism (with respect to Brouwer).
* Goodman, N.D. (1979), "Mathematics as an Objective Science", in Tymoczko (ed., 1986).
* Hart, W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
* Hersh, R. (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986).
* Hilbert, D. (1922), "Neubegründung der Mathematik. Erste Mitteilung", Hamburger Mathematische Seminarabhandlungen 1, 157–177. Translated, "The New Grounding of Mathematics. First Report", in (Mancosu 1998).
* Kleene, Stephen C. [1952] (1991). Introduction to Meta-Mathematics, Tenth impression 1991, Amsterdam NY: North-Holland Pub. Co. ISBN 0-7204-2103-9.

In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.

* Mancosu, P. (ed., 1998), From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, Oxford, UK.
* Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5–22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).
* Putnam, Hilary (1975), "What is Mathematical Truth?", in Tymoczko (ed., 1986).
* A. S. Troelstra (no date but later than 1987), "A History of Constructivism in the 20th Century", http://staff.science.uva.nl/~anne/hhhist.pdf, A detailed survey for specialists: §1 Introduction, §2 Finitism & §2.2 Actualism, §3 Predicativism and Semi-Intuitionism, §4 Brouwerian Intuitionism, §5 Intuitionistic Logic and Arithmetic, §6 Intuitionistic Analysis and Stronger Theories, §7 Constructive Recursive Mathematics, §8 Bishop's Constructivism, §9 Concluding Remarks. Approximately 80 references.
* Tymoczko, T. (1986), "Challenging Foundations", in Tymoczko (ed., 1986).
* Tymoczko, T. (ed., 1986), New Directions in the Philosophy of Mathematics, 1986. Revised edition, 1998.
* Weyl, H. (1921), "Über die neue Grundlagenkrise der Mathematik", Mathematische Zeitschrift 10, 39–79. Translated, "On the New Foundational Crisis of Mathematics", in (Mancosu 1998).
* Wilder, Raymond L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.
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20th century

The profession of mathematician became much more important in the twentieth century. Every year, hundreds of new Ph.D.s in mathematics are awarded, and jobs are available both in teaching and industry. Mathematical development has grown at an exponential rate, with too many new developments to even touch on any but a few of the most profound.


In the 1910s, Srinivasa Aiyangar Ramanujan (1887-1920) developed over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major breakthroughs and
discoveries in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
Famous theorems of the past yielded to new and more powerful techniques. Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem. Andrew Wiles, working alone in his office for years, proved Fermat's last theorem.

Entire new areas of mathematics such as mathematical logic, the mathematics of computers, statistics, and game theory changed the kinds of questions that could be answered by mathematical methods. Bourbaki, a non-existent French mathematician, attempted to bring all of mathematics into a coherent whole.

There were also new investigations of limitations to mathematics. Kurt Gödel proved that in any mathematical system that includes the integers, there are true statements that cannot be proved. Paul Cohen proved the independence of the continuum hypothesis from the standard axioms of set theory.
By the end of the century, mathematics was even finding its way into art, as fractal geometry produced beautiful shapes never seen before.

source:http://en.wikipedia.org/wiki/History_of_mathematics

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19th century

Throughout the 19th century mathematics became increasingly abstract. In this century lived one of the greatest mathematicians of all time, Carl Friedrich Gauss (1777 - 1855). Leaving aside his many contributions to science, in pure mathematics he did revolutionary work on functions of complex variables, in geometry, and on the convergence of series. He gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival the Hungarian mathematician Janos Bolyai both independently of each other discovered non-Euclidean geometry, their non-Euclidean geometry was called hyperbolic geometry and differed from traditional Euclidean geometry in that it rejected Euclid's fifth postulate, a rule of Euclidean geometry that states that parallel lines go on to infinity and never intersect, they replaced this with a postulate that allowed parallel lines to intersect, in hyperbolic geometry because parallel lines can intersect triangles have less than 180 degrees. The initial reaction of the mathematical and scientific communities to Lobachevsky's findings was hostile. It was very courageous of Lobachevsky to publish his findings in the face of this opposition. Lobachevsky's essay about non-Euclidean geometry A concise outline of the foundations of geometry was published by the Kazan Messenger but was rejected by Ostrogodski when the St. Petersberg Academy of Sciences submitted it for publication.

Another non-Euclidean geometry called elliptic geometry was developed later in the nineteenth century by the German mathematician George Friedrich Bernhard Riemmann. In elliptic geometry parallel lines do not exist and there are three dimensional triangles with more than 180 degrees. Despite the fact that the mathematical and scientific communities' initial reaction to non-Euclidean geometries was at first negative these new geometries and in particular elliptic geometry later turned out to be crucial to Albert Einstein's theory of relativity;e=mc squared, which is a theory about geometrical gravitational fields, although the theory of relativity also used Euclidean flat space. Also in the nineteenth century William Rowan Hamilton developed noncommutative algebra.

Lobachevsky also discovered a method for finding the approximations of the roots of algebraic equations which is still called the Lobachevsky method in Russia although it is called the Dandelin-Graffe method in the West after two western mathematicians who discovered it independently both of each other and of Lobachevsky.

In addition to new directions in mathematics, older mathematics were given a stronger logical foundation, especially in the case of calculus, in work by Augustin-Louis Cauchy and Karl Weierstrass.

A new form of algebra was developed in the nineteenth century called Boolean algebra. It was developed by a British mathematician named George Boole. It was a system that contained true and false statements, in it 1 meant true and 0 meant false. Boolean algebra later became important in the twentieth century because it was the math that would be used for computers.

Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four, and other 19th century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.

Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.

The 19th century also saw the founding of the first mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Mathematico di Palermo in 1884, the Edinburgh Mathematical Society in 1864, and the American Mathematical Society in 1888.

Before the 20th century, the number of creative mathematicians in the world at any one time was limited. For the most part, mathematicians were either born to wealth, like Napier, or supported by wealthy patrons, like Gauss. There were a few meager livelihoods to be had teaching at a university, like Fourier. Niels Henrik Abel, unable to obtain a position, died of tuberculosis.

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8th century

Leonhard Euler by Emanuel Handmann.

As we have seen, knowledge of the natural numbers, 1, 2, 3,..., as preserved in monolithic structures, is older than any surviving written text. The earliest civilizations -- in Mesopotamia, Egypt, India and China -- knew arithmetic.

One way to view the development of the various number systems of modern mathematics is to see new numbers studied and investigated to answer questions about arithmetic performed on older numbers. In prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1. In India and China, and much later in Germany, negative numbers were developed to answer the question: what do you get when you subtract a larger number from a smaller. The invention of the zero may have followed from similar question: what do you get when you subtract a number from itself.

Another natural question is: what kind of a number is the square root of two? The Greeks knew that it was not a fraction, and this question may have played a role in the development of continued fractions. But a better answer came with the invention of decimals, developed by John Napier (1550 - 1617) and perfected later by Simon Stevin. Using decimals, and an idea that anticipated the concept of the limit, Napier also studied a new constant, which Leonhard Euler (1707 - 1783) named e.

Euler was very influential in the standardization of other mathematical terms and notations. He named the square root of minus 1 with the symbol i. He also popularized the use of the Greek letter π to stand for the ratio of a circle's circumference to its diameter. He then derived one of the most remarkable identities in all of mathematics:


(see Euler's Identity.)

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17th century

The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, Johannes Kepler, a German, began to work with this data. In part because he wanted to help Kepler in his calculations, Lord Napier, in Scotland, was the first to investigate natural logarithms. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596-1650), a French mathematician and philosopher, allowed those orbits to be plotted on a graph, in Cartesian coordinates. Building on earlier work by many mathematicians, Isaac Newton, an Englishman, discovered the laws of physics explaining Kepler's Laws, and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz, in Germany, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.

In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In a sense this forshadowed the later 18th-19th century development of utility theory.

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European Renaissance mathematics (c. 1200—1600)

In Europe at the dawn of the Renaissance, most of what is now called school mathematics -- addition, subtraction, multiplication, division, and geometry -- was known to educated people, though the notation was cumbersome: Roman numerals and words were used, but no symbols: no plus sign, no equal sign, and no use of x as an unknown. Most of the mathematics now taught at universities was either known only to the mathematical community in India or had yet to be investigated and developed in Europe.

Through Latin translations of Arabic texts, knowledge of the Hindu-Arabic numerals and other important developments of Islamic and Indian mathematics were brought to Europe. Robert of Chester's translation of Al-Khwarizmi's Al-Jabr wa-al-Muqabilah into Latin in the 12th century was particularly important. The earlier works of Aristotle were redeveloped in Europe, first in Arabic and later in Greek. Of particular importance was the rediscovery of a collection of Aristotle's logical writing, compiled in the 1st century, known as the Organon.

The reawakened desire for new knowledge sparked a renewed interest in mathematics. Fibonacci, in the early 13th century, produced the first significant mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. But it was only from the late 16th century that European mathematicians began to make advances without precedent anywhere in the world, so far as is known today.

The first of these was the general solution of cubic equations, generally credited to Scipione del Ferro circa 1510, but first published in Gerolamo Cardano's Ars magna. It was quickly followed by Lodovico Ferrari's solution of the general quartic equation.

From this point on, mathematical developments came swiftly, and combined with advances in science, to their mutual benefit. At this time Treviso arithmetic was first written, and is still regarded as the first mathematics book ever printed. In the landmark year 1543, Copernicus published De revolutionibus, asserting that the Earth traveled around the Sun, and Vesalius published De humani corporis fabrica, treating the human body as a collection of organs.

Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus' table of sines and cosines was published in 1533.[12]

By century's end, thanks to Regiomontanus (1436—1476) and François Vieta (1540—1603), among others, mathematics was written using Hindu-Arabic numerals and in a form not too different from the elegant notation used today.

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