Mathematics (from Greek μάθημα máthēma, "knowledge, study,
learning") is the study of such topics as quantity, structure,
space, and change. It has no generally accepted
Mathematicians seek out patterns and use them to formulate new
conjectures. Mathematicians resolve the truth or falsity of
conjectures by mathematical proof. When mathematical structures are
good models of real phenomena, then mathematical reasoning can provide
insight or predictions about nature. Through the use of abstraction
and logic, mathematics developed from counting, calculation,
measurement, and the systematic study of the shapes and motions of
physical objects. Practical mathematics has been a human activity from
as far back as written records exist. The research required to solve
mathematical problems can take years or even centuries of sustained
Rigorous arguments first appeared in Greek mathematics, most notably
in Euclid's Elements. Since the pioneering work of Giuseppe Peano
David Hilbert (1862–1943), and others on axiomatic
systems in the late 19th century, it has become customary to view
mathematical research as establishing truth by rigorous deduction from
appropriately chosen axioms and definitions.
Mathematics developed at
a relatively slow pace until the Renaissance, when mathematical
innovations interacting with new scientific discoveries led to a rapid
increase in the rate of mathematical discovery that has continued to
the present day.
Galileo Galilei (1564–1642) said, "The universe cannot be read until
we have learned the language and become familiar with the characters
in which it is written. It is written in mathematical language, and
the letters are triangles, circles and other geometrical figures,
without which means it is humanly impossible to comprehend a single
word. Without these, one is wandering about in a dark labyrinth."
Carl Friedrich Gauss
Carl Friedrich Gauss (1777–1855) referred to mathematics as "the
Queen of the Sciences".
Benjamin Peirce (1809–1880) called
mathematics "the science that draws necessary conclusions". David
Hilbert said of mathematics: "We are not speaking here of
arbitrariness in any sense.
Mathematics is not like a game whose tasks
are determined by arbitrarily stipulated rules. Rather, it is a
conceptual system possessing internal necessity that can only be so
and by no means otherwise."
Albert Einstein (1879–1955) stated
that "as far as the laws of mathematics refer to reality, they are not
certain; and as far as they are certain, they do not refer to
Mathematics is essential in many fields, including natural science,
engineering, medicine, finance and the social sciences. Applied
mathematics has led to entirely new mathematical disciplines, such as
statistics and game theory. Mathematicians also engage in pure
mathematics, or mathematics for its own sake, without having any
application in mind. There is no clear line separating pure and
applied mathematics, and practical applications for what began as pure
mathematics are often discovered.
2 Definitions of mathematics
Mathematics as science
3 Inspiration, pure and applied mathematics, and aesthetics
4 Notation, language, and rigor
5 Fields of mathematics
5.1 Foundations and philosophy
5.2 Pure mathematics
5.3 Applied mathematics
Statistics and other decision sciences
5.3.2 Computational mathematics
6 Mathematical awards
7 See also
11 Further reading
12 External links
Main article: History of mathematics
The history of mathematics can be seen as an ever-increasing series of
abstractions. The first abstraction, which is shared by many
animals, was probably that of numbers: the realization that a
collection of two apples and a collection of two oranges (for example)
have something in common, namely quantity of their members.
Pythagoras (c. 570 BC – c. 495 BC), commonly
credited with discovering the Pythagorean theorem
As evidenced by tallies found on bone, in addition to recognizing how
to count physical objects, prehistoric peoples may have also
recognized how to count abstract quantities, like time – days,
Evidence for more complex mathematics does not appear until around
3000 BC, when the Babylonians and Egyptians began using
arithmetic, algebra and geometry for taxation and other financial
calculations, for building and construction, and for astronomy.
The earliest uses of mathematics were in trading, land measurement,
painting and weaving patterns and the recording of time.
In Babylonian mathematics, elementary arithmetic (addition,
subtraction, multiplication and division) first appears in the
Numeracy pre-dated writing and numeral systems
have been many and diverse, with the first known written numerals
created by Egyptians in Middle Kingdom texts such as the Rhind
Mathematical Papyrus.
Between 600 and 300 BC the
Ancient Greeks began a systematic
study of mathematics in its own right with Greek mathematics.
Persian mathematician Al-Khwarizmi (c. 780 – c. 850), the
inventor of algebra.
During the Golden Age of Islam, especially during the 9th and
10th centuries, mathematics saw many important innovations
building on Greek mathematics: most of them include the contributions
from Persian mathematicians such as Al-Khwarismi,
Omar Khayyam and
Sharaf al-Dīn al-Ṭūsī.
Mathematics has since been greatly extended, and there has been a
fruitful interaction between mathematics and science, to the benefit
of both. Mathematical discoveries continue to be made today. According
to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin
of the American Mathematical Society, "The number of papers and books
included in the
Mathematical Reviews database since 1940 (the first
year of operation of MR) is now more than 1.9 million, and more
than 75 thousand items are added to the database each year. The
overwhelming majority of works in this ocean contain new mathematical
theorems and their proofs."
The word mathematics comes from
Ancient Greek μάθημα
(máthēma), meaning "that which is learnt", "what one gets to
know", hence also "study" and "science". The word for "mathematics"
came to have the narrower and more technical meaning "mathematical
study" even in Classical times. Its adjective is
μαθηματικός (mathēmatikós), meaning "related to learning"
or "studious", which likewise further came to mean "mathematical". In
particular, μαθηματικὴ τέχνη (mathēmatikḗ
tékhnē), Latin: ars mathematica, meant "the mathematical art".
Similarly, one of the two main schools of thought in Pythagoreanism
was known as the mathēmatikoi (μαθηματικοί)—which at the
time meant "teachers" rather than "mathematicians" in the modern
In Latin, and in English until around 1700, the term mathematics more
commonly meant "astrology" (or sometimes "astronomy") rather than
"mathematics"; the meaning gradually changed to its present one from
about 1500 to 1800. This has resulted in several mistranslations. For
example, Saint Augustine's warning that Christians should beware of
mathematici, meaning astrologers, is sometimes mistranslated as a
condemnation of mathematicians.
The apparent plural form in English, like the French plural form les
mathématiques (and the less commonly used singular derivative la
mathématique), goes back to the Latin neuter plural mathematica
(Cicero), based on the Greek plural τα μαθηματικά (ta
mathēmatiká), used by
Aristotle (384–322 BC), and meaning
roughly "all things mathematical"; although it is plausible that
English borrowed only the adjective mathematic(al) and formed the noun
mathematics anew, after the pattern of physics and metaphysics, which
were inherited from Greek. In English, the noun mathematics takes
a singular verb. It is often shortened to maths or, in
English-speaking North America, math.
Definitions of mathematics
Main article: Definitions of mathematics
Leonardo Fibonacci, the Italian mathematician who introduced the
Hindu–Arabic numeral system
Hindu–Arabic numeral system invented between the 1st and
4th centuries by Indian mathematicians, to the Western World
Aristotle defined mathematics as "the science of quantity", and this
definition prevailed until the 18th century. Starting in the
19th century, when the study of mathematics increased in rigor
and began to address abstract topics such as group theory and
projective geometry, which have no clear-cut relation to quantity and
measurement, mathematicians and philosophers began to propose a
variety of new definitions. Some of these definitions emphasize
the deductive character of much of mathematics, some emphasize its
abstractness, some emphasize certain topics within mathematics. Today,
no consensus on the definition of mathematics prevails, even among
professionals. There is not even consensus on whether mathematics
is an art or a science. A great many professional mathematicians
take no interest in a definition of mathematics, or consider it
undefinable. Some just say, "
Mathematics is what mathematicians
Three leading types of definition of mathematics are called logicist,
intuitionist, and formalist, each reflecting a different philosophical
school of thought. All have severe problems, none has widespread
acceptance, and no reconciliation seems possible.
An early definition of mathematics in terms of logic was Benjamin
Peirce's "the science that draws necessary conclusions" (1870). In
the Principia Mathematica,
Bertrand Russell and Alfred North Whitehead
advanced the philosophical program known as logicism, and attempted to
prove that all mathematical concepts, statements, and principles can
be defined and proved entirely in terms of symbolic logic. A logicist
definition of mathematics is Russell's "All
Mathematics is Symbolic
Intuitionist definitions, developing from the philosophy of
mathematician L.E.J. Brouwer, identify mathematics with certain mental
phenomena. An example of an intuitionist definition is "
the mental activity which consists in carrying out constructs one
after the other." A peculiarity of intuitionism is that it rejects
some mathematical ideas considered valid according to other
definitions. In particular, while other philosophies of mathematics
allow objects that can be proved to exist even though they cannot be
constructed, intuitionism allows only mathematical objects that one
can actually construct.
Formalist definitions identify mathematics with its symbols and the
rules for operating on them.
Haskell Curry defined mathematics simply
as "the science of formal systems". A formal system is a set of
symbols, or tokens, and some rules telling how the tokens may be
combined into formulas. In formal systems, the word axiom has a
special meaning, different from the ordinary meaning of "a
self-evident truth". In formal systems, an axiom is a combination of
tokens that is included in a given formal system without needing to be
derived using the rules of the system.
Mathematics as science
Carl Friedrich Gauss, known as the prince of mathematicians
The German mathematician
Carl Friedrich Gauss
Carl Friedrich Gauss referred to mathematics
as "the Queen of the Sciences". More recently, Marcus du Sautoy
has called mathematics "the Queen of Science ... the main driving
force behind scientific discovery". In the original Latin Regina
Scientiarum, as well as in German Königin der Wissenschaften, the
word corresponding to science means a "field of knowledge", and this
was the original meaning of "science" in English, also; mathematics is
in this sense a field of knowledge. The specialization restricting the
meaning of "science" to natural science follows the rise of Baconian
science, which contrasted "natural science" to scholasticism, the
Aristotelean method of inquiring from first principles. The role of
empirical experimentation and observation is negligible in
mathematics, compared to natural sciences such as biology, chemistry,
Albert Einstein stated that "as far as the laws of
mathematics refer to reality, they are not certain; and as far as they
are certain, they do not refer to reality."
Many philosophers believe that mathematics is not experimentally
falsifiable, and thus not a science according to the definition of
Karl Popper. However, in the 1930s Gödel's incompleteness
theorems convinced many mathematicians[who?] that mathematics cannot
be reduced to logic alone, and
Karl Popper concluded that "most
mathematical theories are, like those of physics and biology,
hypothetico-deductive: pure mathematics therefore turns out to be much
closer to the natural sciences whose hypotheses are conjectures, than
it seemed even recently." Other thinkers, notably Imre Lakatos,
have applied a version of falsificationism to mathematics
An alternative view is that certain scientific fields (such as
theoretical physics) are mathematics with axioms that are intended to
correspond to reality.
Mathematics shares much in common with many
fields in the physical sciences, notably the exploration of the
logical consequences of assumptions. Intuition and experimentation
also play a role in the formulation of conjectures in both mathematics
and the (other) sciences.
Experimental mathematics continues to grow
in importance within mathematics, and computation and simulation are
playing an increasing role in both the sciences and mathematics.
The opinions of mathematicians on this matter are varied. Many
mathematicians feel that to call their area a science is to
downplay the importance of its aesthetic side, and its history in the
traditional seven liberal arts; others[who?] feel that to ignore its
connection to the sciences is to turn a blind eye to the fact that the
interface between mathematics and its applications in science and
engineering has driven much development in mathematics. One way this
difference of viewpoint plays out is in the philosophical debate as to
whether mathematics is created (as in art) or discovered (as in
science). It is common to see universities divided into sections that
include a division of Science and Mathematics, indicating that the
fields are seen as being allied but that they do not coincide. In
practice, mathematicians are typically grouped with scientists at the
gross level but separated at finer levels. This is one of many issues
considered in the philosophy of mathematics.
Inspiration, pure and applied mathematics, and aesthetics
Main article: Mathematical beauty
Isaac Newton (left) and
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz (right), developers
of infinitesimal calculus
Mathematics arises from many different kinds of problems. At first
these were found in commerce, land measurement, architecture and later
astronomy; today, all sciences suggest problems studied by
mathematicians, and many problems arise within mathematics itself. For
example, the physicist
Richard Feynman invented the path integral
formulation of quantum mechanics using a combination of mathematical
reasoning and physical insight, and today's string theory, a
still-developing scientific theory which attempts to unify the four
fundamental forces of nature, continues to inspire new
Some mathematics is relevant only in the area that inspired it, and is
applied to solve further problems in that area. But often mathematics
inspired by one area proves useful in many areas, and joins the
general stock of mathematical concepts. A distinction is often made
between pure mathematics and applied mathematics. However pure
mathematics topics often turn out to have applications, e.g. number
theory in cryptography. This remarkable fact, that even the "purest"
mathematics often turns out to have practical applications, is what
Eugene Wigner has called "the unreasonable effectiveness of
mathematics". As in most areas of study, the explosion of
knowledge in the scientific age has led to specialization: there are
now hundreds of specialized areas in mathematics and the latest
Mathematics Subject Classification runs to 46 pages. Several
areas of applied mathematics have merged with related traditions
outside of mathematics and become disciplines in their own right,
including statistics, operations research, and computer science.
For those who are mathematically inclined, there is often a definite
aesthetic aspect to much of mathematics. Many mathematicians talk
about the elegance of mathematics, its intrinsic aesthetics and inner
Simplicity and generality are valued. There is beauty in a
simple and elegant proof, such as Euclid's proof that there are
infinitely many prime numbers, and in an elegant numerical method that
speeds calculation, such as the fast Fourier transform.
G.H. Hardy in
A Mathematician's Apology
A Mathematician's Apology expressed the belief that these aesthetic
considerations are, in themselves, sufficient to justify the study of
pure mathematics. He identified criteria such as significance,
unexpectedness, inevitability, and economy as factors that contribute
to a mathematical aesthetic. Mathematicians often strive to find
proofs that are particularly elegant, proofs from "The Book" of God
according to Paul Erdős. The popularity of recreational
mathematics is another sign of the pleasure many find in solving
Notation, language, and rigor
Main article: Mathematical notation
Leonhard Euler, who created and popularized much of the mathematical
notation used today
Most of the mathematical notation in use today was not invented until
the 16th century. Before that, mathematics was written out in
words, limiting mathematical discovery. Euler (1707–1783) was
responsible for many of the notations in use today. Modern notation
makes mathematics much easier for the professional, but beginners
often find it daunting. According to Barbara Oakley, this can be
attributed to the fact that mathematical ideas are both more abstract
and more encrypted than those of natural language. Unlike natural
language, where people can often equate a word (such as cow) with the
physical object it corresponds to, mathematical symbols are abstract,
lacking any physical analog. Mathematical symbols are also more
highly encrypted than regular words, meaning a single symbol can
encode a number of different operations or ideas.
Mathematical language can be difficult to understand for beginners
because even common terms, such as or and only, have a more precise
meaning than they have in everyday speech, and other terms such as
open and field refer to specific mathematical ideas, not covered by
their laymen's meanings. Mathematical language also includes many
technical terms such as homeomorphism and integrable that have no
meaning outside of mathematics. Additionally, shorthand phrases such
as iff for "if and only if" belong to mathematical jargon. There is a
reason for special notation and technical vocabulary: mathematics
requires more precision than everyday speech. Mathematicians refer to
this precision of language and logic as "rigor".
Mathematical proof is fundamentally a matter of rigor. Mathematicians
want their theorems to follow from axioms by means of systematic
reasoning. This is to avoid mistaken "theorems", based on fallible
intuitions, of which many instances have occurred in the history of
the subject.[b] The level of rigor expected in mathematics has varied
over time: the Greeks expected detailed arguments, but at the time of
Isaac Newton the methods employed were less rigorous. Problems
inherent in the definitions used by Newton would lead to a resurgence
of careful analysis and formal proof in the 19th century.
Misunderstanding the rigor is a cause for some of the common
misconceptions of mathematics. Today, mathematicians continue to argue
among themselves about computer-assisted proofs. Since large
computations are hard to verify, such proofs may not be sufficiently
Axioms in traditional thought were "self-evident truths", but that
conception is problematic. At a formal level, an axiom is just a
string of symbols, which has an intrinsic meaning only in the context
of all derivable formulas of an axiomatic system. It was the goal of
Hilbert's program to put all of mathematics on a firm axiomatic basis,
but according to
Gödel's incompleteness theorem every (sufficiently
powerful) axiomatic system has undecidable formulas; and so a final
axiomatization of mathematics is impossible. Nonetheless mathematics
is often imagined to be (as far as its formal content) nothing but set
theory in some axiomatization, in the sense that every mathematical
statement or proof could be cast into formulas within set theory.
Fields of mathematics
Areas of mathematics and Glossary of areas of mathematics
An abacus, a simple calculating tool used since ancient times
Mathematics can, broadly speaking, be subdivided into the study of
quantity, structure, space, and change (i.e. arithmetic, algebra,
geometry, and analysis). In addition to these main concerns, there are
also subdivisions dedicated to exploring links from the heart of
mathematics to other fields: to logic, to set theory (foundations), to
the empirical mathematics of the various sciences (applied
mathematics), and more recently to the rigorous study of uncertainty.
While some areas might seem unrelated, the
Langlands program has found
connections between areas previously thought unconnected, such as
Galois groups, Riemann surfaces and number theory.
Foundations and philosophy
In order to clarify the foundations of mathematics, the fields of
mathematical logic and set theory were developed. Mathematical logic
includes the mathematical study of logic and the applications of
formal logic to other areas of mathematics; set theory is the branch
of mathematics that studies sets or collections of objects. Category
theory, which deals in an abstract way with mathematical structures
and relationships between them, is still in development. The phrase
"crisis of foundations" describes the search for a rigorous foundation
for mathematics that took place from approximately 1900 to 1930.
Some disagreement about the foundations of mathematics continues to
the present day. The crisis of foundations was stimulated by a number
of controversies at the time, including the controversy over Cantor's
set theory and the Brouwer–Hilbert controversy.
Mathematical logic is concerned with setting mathematics within a
rigorous axiomatic framework, and studying the implications of such a
framework. As such, it is home to Gödel's incompleteness theorems
which (informally) imply that any effective formal system that
contains basic arithmetic, if sound (meaning that all theorems that
can be proved are true), is necessarily incomplete (meaning that there
are true theorems which cannot be proved in that system). Whatever
finite collection of number-theoretical axioms is taken as a
foundation, Gödel showed how to construct a formal statement that is
a true number-theoretical fact, but which does not follow from those
axioms. Therefore, no formal system is a complete axiomatization of
full number theory. Modern logic is divided into recursion theory,
model theory, and proof theory, and is closely linked to theoretical
computer science, as well as to category theory. In
the context of recursion theory, the impossibility of a full
axiomatization of number theory can also be formally demonstrated as a
consequence of the MRDP theorem.
Theoretical computer science
Theoretical computer science includes computability theory,
computational complexity theory, and information theory. Computability
theory examines the limitations of various theoretical models of the
computer, including the most well-known model – the Turing
machine. Complexity theory is the study of tractability by computer;
some problems, although theoretically solvable by computer, are so
expensive in terms of time or space that solving them is likely to
remain practically unfeasible, even with the rapid advancement of
computer hardware. A famous problem is the "P = NP?" problem, one of
the Millennium Prize Problems. Finally, information theory is
concerned with the amount of data that can be stored on a given
medium, and hence deals with concepts such as compression and entropy.
displaystyle pRightarrow q
Theory of computation
Main article: Arithmetic
The study of quantity starts with numbers, first the familiar natural
numbers and integers ("whole numbers") and arithmetical operations on
them, which are characterized in arithmetic. The deeper properties of
integers are studied in number theory, from which come such popular
results as Fermat's Last Theorem. The twin prime conjecture and
Goldbach's conjecture are two unsolved problems in number theory.
As the number system is further developed, the integers are recognized
as a subset of the rational numbers ("fractions"). These, in turn, are
contained within the real numbers, which are used to represent
continuous quantities. Real numbers are generalized to complex
numbers. These are the first steps of a hierarchy of numbers that goes
on to include quaternions and octonions. Consideration of the natural
numbers also leads to the transfinite numbers, which formalize the
concept of "infinity". According to the fundamental theorem of algebra
all solutions of equations in one unknown with complex coefficients
are complex numbers, regardless of degree. Another area of study is
the size of sets, which is described with the cardinal numbers. These
include the aleph numbers, which allow meaningful comparison of the
size of infinitely large sets.
displaystyle ldots ,-2,-1,0,1,2,ldots
displaystyle -2, frac 2 3 ,1.21
displaystyle -e, sqrt 2 ,3,pi
displaystyle 2,i,-2+3i,2e^ i frac 4pi 3
Main article: Algebra
Many mathematical objects, such as sets of numbers and functions,
exhibit internal structure as a consequence of operations or relations
that are defined on the set.
Mathematics then studies properties of
those sets that can be expressed in terms of that structure; for
instance number theory studies properties of the set of integers that
can be expressed in terms of arithmetic operations. Moreover, it
frequently happens that different such structured sets (or structures)
exhibit similar properties, which makes it possible, by a further step
of abstraction, to state axioms for a class of structures, and then
study at once the whole class of structures satisfying these axioms.
Thus one can study groups, rings, fields and other abstract systems;
together such studies (for structures defined by algebraic operations)
constitute the domain of abstract algebra.
By its great generality, abstract algebra can often be applied to
seemingly unrelated problems; for instance a number of ancient
problems concerning compass and straightedge constructions were
finally solved using Galois theory, which involves field theory and
group theory. Another example of an algebraic theory is linear
algebra, which is the general study of vector spaces, whose elements
called vectors have both quantity and direction, and can be used to
model (relations between) points in space. This is one example of the
phenomenon that the originally unrelated areas of geometry and algebra
have very strong interactions in modern mathematics. Combinatorics
studies ways of enumerating the number of objects that fit a given
displaystyle begin matrix
Main article: Geometry
The study of space originates with geometry – in particular,
Euclidean geometry, which combines space and numbers, and encompasses
the well-known Pythagorean theorem.
Trigonometry is the branch of
mathematics that deals with relationships between the sides and the
angles of triangles and with the trigonometric functions. The modern
study of space generalizes these ideas to include higher-dimensional
geometry, non-Euclidean geometries (which play a central role in
general relativity) and topology.
Quantity and space both play a role
in analytic geometry, differential geometry, and algebraic geometry.
Convex and discrete geometry were developed to solve problems in
number theory and functional analysis but now are pursued with an eye
on applications in optimization and computer science. Within
differential geometry are the concepts of fiber bundles and calculus
on manifolds, in particular, vector and tensor calculus. Within
algebraic geometry is the description of geometric objects as solution
sets of polynomial equations, combining the concepts of quantity and
space, and also the study of topological groups, which combine
structure and space. Lie groups are used to study space, structure,
Topology in all its many ramifications may have been the
greatest growth area in 20th-century mathematics; it includes
point-set topology, set-theoretic topology, algebraic topology and
differential topology. In particular, instances of modern-day topology
are metrizability theory, axiomatic set theory, homotopy theory, and
Topology also includes the now solved Poincaré
conjecture, and the still unsolved areas of the Hodge conjecture.
Other results in geometry and topology, including the four color
theorem and Kepler conjecture, have been proved only with the help of
Main article: Calculus
Understanding and describing change is a common theme in the natural
sciences, and calculus was developed as a powerful tool to investigate
it. Functions arise here, as a central concept describing a changing
quantity. The rigorous study of real numbers and functions of a real
variable is known as real analysis, with complex analysis the
equivalent field for the complex numbers.
Functional analysis focuses
attention on (typically infinite-dimensional) spaces of functions. One
of many applications of functional analysis is quantum mechanics. Many
problems lead naturally to relationships between a quantity and its
rate of change, and these are studied as differential equations. Many
phenomena in nature can be described by dynamical systems; chaos
theory makes precise the ways in which many of these systems exhibit
unpredictable yet still deterministic behavior.
Main article: Applied mathematics
Applied mathematics concerns itself with mathematical methods that are
typically used in science, engineering, business, and industry. Thus,
"applied mathematics" is a mathematical science with specialized
knowledge. The term applied mathematics also describes the
professional specialty in which mathematicians work on practical
problems; as a profession focused on practical problems, applied
mathematics focuses on the "formulation, study, and use of
mathematical models" in science, engineering, and other areas of
In the past, practical applications have motivated the development of
mathematical theories, which then became the subject of study in pure
mathematics, where mathematics is developed primarily for its own
sake. Thus, the activity of applied mathematics is vitally connected
with research in pure mathematics.
Statistics and other decision sciences
Main article: Statistics
Applied mathematics has significant overlap with the discipline of
statistics, whose theory is formulated mathematically, especially with
probability theory. Statisticians (working as part of a research
project) "create data that makes sense" with random sampling and with
randomized experiments; the design of a statistical sample or
experiment specifies the analysis of the data (before the data be
available). When reconsidering data from experiments and samples or
when analyzing data from observational studies, statisticians "make
sense of the data" using the art of modelling and the theory of
inference – with model selection and estimation; the estimated
models and consequential predictions should be tested on new data.[c]
Statistical theory studies decision problems such as minimizing the
risk (expected loss) of a statistical action, such as using a
procedure in, for example, parameter estimation, hypothesis testing,
and selecting the best. In these traditional areas of mathematical
statistics, a statistical-decision problem is formulated by minimizing
an objective function, like expected loss or cost, under specific
constraints: For example, designing a survey often involves minimizing
the cost of estimating a population mean with a given level of
confidence. Because of its use of optimization, the mathematical
theory of statistics shares concerns with other decision sciences,
such as operations research, control theory, and mathematical
Computational mathematics proposes and studies methods for solving
mathematical problems that are typically too large for human numerical
Numerical analysis studies methods for problems in analysis
using functional analysis and approximation theory; numerical analysis
includes the study of approximation and discretization broadly with
special concern for rounding errors.
Numerical analysis and, more
broadly, scientific computing also study non-analytic topics of
mathematical science, especially algorithmic matrix and graph theory.
Other areas of computational mathematics include computer algebra and
Arguably the most prestigious award in mathematics is the Fields
Medal, established in 1936 and awarded every four years
(except around World War II) to as many as four individuals. The
Fields Medal is often considered a mathematical equivalent to the
The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime
achievement, and another major international award, the Abel Prize,
was instituted in 2003. The
Chern Medal was introduced in 2010 to
recognize lifetime achievement. These accolades are awarded in
recognition of a particular body of work, which may be innovational,
or provide a solution to an outstanding problem in an established
A famous list of 23 open problems, called "Hilbert's problems", was
compiled in 1900 by German mathematician David Hilbert. This list
achieved great celebrity among mathematicians, and at least nine of
the problems have now been solved. A new list of seven important
problems, titled the "Millennium Prize Problems", was published in
2000. A solution to each of these problems carries a $1 million
reward, and only one (the Riemann hypothesis) is duplicated in
Philosophy of mathematics
Lists of mathematics topics
Mathematics and art
National Museum of Mathematics
Relationship between mathematics and physics
Science, Technology, Engineering, and Mathematics
^ No likeness or description of Euclid's physical appearance made
during his lifetime survived antiquity. Therefore, Euclid's depiction
in works of art depends on the artist's imagination (see Euclid).
^ See false proof for simple examples of what can go wrong in a formal
^ Like other mathematical sciences such as physics and computer
science, statistics is an autonomous discipline rather than a branch
of applied mathematics. Like research physicists and computer
scientists, research statisticians are mathematical scientists. Many
statisticians have a degree in mathematics, and some statisticians are
^ a b "mathematics, n.". Oxford English Dictionary. Oxford University
Press. 2012. Retrieved June 16, 2012. The science of space, number,
quantity, and arrangement, whose methods involve logical reasoning and
usually the use of symbolic notation, and which includes geometry,
arithmetic, algebra, and analysis.
^ Kneebone, G.T. (1963). Mathematical
Logic and the Foundations of
Mathematics: An Introductory Survey. Dover. pp. 4.
ISBN 0-486-41712-3. Mathematics ... is simply the study of
abstract structures, or formal patterns of connectedness.
^ LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter,
Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus
Concepts: An Informal Approach to the
Mathematics of Change. Cengage
Learning. p. 2. ISBN 1-4390-4957-2.
Calculus is the study of
change—how things change, and how quickly they change.
^ Ramana (2007). Applied Mathematics. Tata McGraw–Hill Education.
p. 2.10. ISBN 0-07-066753-5. The mathematical study of
change, motion, growth or decay is calculus.
^ Ziegler, Günter M. (2011). "What Is Mathematics?". An Invitation to
Mathematics: From Competitions to Research. Springer. p. 7.
^ a b c d Mura, Roberta (Dec 1993). "Images of
Mathematics Held by
University Teachers of Mathematical Sciences". Educational Studies in
Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907.
^ a b Tobies, Renate & Helmut Neunzert (2012). Iris Runge: A Life
at the Crossroads of Mathematics, Science, and Industry. Springer.
p. 9. ISBN 3-0348-0229-3. [I]t is first necessary to ask
what is meant by mathematics in general. Illustrious scholars have
debated this matter until they were blue in the face, and yet no
consensus has been reached about whether mathematics is a natural
science, a branch of the humanities, or an art form.
^ Steen, L.A. (April 29, 1988). The Science of
Patterns Science, 240:
611–16. And summarized at Association for Supervision and Curriculum
Development Archived October 28, 2010, at the Wayback Machine.,
^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for
Order in Life, Mind and the Universe (Scientific American Paperback
Library) 1996, ISBN 978-0-7167-5047-5
^ Eves, p. 306
^ Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and
Leibniz Archived December 6, 2012, at the Wayback Machine.,
4, September 27, 2010.
^ a b Waltershausen, p. 79
^ Peirce, p. 97.
^ Hilbert, D. (1919–20), Natur und Mathematisches Erkennen:
Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung
von Paul Bernays (Edited and with an English introduction by David E.
Rowe), p. 14, Basel, Birkhäuser (1992).
^ a b Einstein, p. 28. The quote is Einstein's answer to the question:
"How can it be that mathematics, being after all a product of human
thought which is independent of experience, is so admirably
appropriate to the objects of reality?" This question was inspired by
Eugene Wigner's paper "The Unreasonable Effectiveness of Mathematics
in the Natural Sciences".
^ Peterson, p. 12
^ Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (Aug
1998). "Abstract representations of numbers in the animal and human
brain". Trends in Neurosciences. 21 (8): 355–61.
doi:10.1016/S0166-2236(98)01263-6. PMID 9720604.
^ See, for example, Raymond L. Wilder, Evolution of Mathematical
Concepts; an Elementary Study, passim
^ Kline 1990, Chapter 1.
^ "A History of Greek Mathematics: From Thales to
January 8, 2014, at the Wayback Machine.". Thomas Little Heath (1981).
^ Sevryuk 2006, pp. 101–09.
^ "mathematic". Online Etymology Dictionary. Archived from the
original on March 7, 2013.
^ Both senses can be found in Plato. μαθηματική. Liddell,
Henry George; Scott, Robert;
A Greek–English Lexicon at the Perseus
^ Boas, Ralph (1995) . "What Augustine Didn't Say About
Mathematicians". Lion Hunting and Other Mathematical Pursuits: A
Collection of Mathematics, Verse, and Stories by the Late Ralph P.
Boas, Jr. Cambridge University Press. p. 257.
^ The Oxford Dictionary of English Etymology, Oxford English
Dictionary, sub "mathematics", "mathematic", "mathematics"
^ "maths, n." and "math, n.3". Oxford English Dictionary, on-line
^ James Franklin, "Aristotelian Realism" in Philosophy of Mathematics,
ed. A.D. Irvine, p. 104 Archived September 6, 2015, at the Wayback
Machine.. Elsevier (2009).
^ Cajori, Florian (1893). A History of Mathematics. American
Mathematical Society (1991 reprint). pp. 285–86.
^ a b c Snapper, Ernst (September 1979). "The Three Crises in
Mathematics: Logicism, Intuitionism, and Formalism". Mathematics
Magazine. 52 (4): 207–16. doi:10.2307/2689412.
^ Peirce, Benjamin (1882). Linear Associative Algebra. p. 1.
Archived from the original on September 6, 2015.
^ Bertrand Russell, The Principles of Mathematics, p. 5 Archived
September 6, 2015, at the Wayback Machine.. University Press,
^ Curry, Haskell (1951). Outlines of a Formalist Philosophy of
Mathematics. Elsevier. p. 56. ISBN 0-444-53368-0.
^ du Sautoy, Marcus (June 25, 2010). "Nicolas Bourbaki". A Brief
History of Mathematics. Event occurs at min. 12:50.
BBC Radio 4.
Archived from the original on December 16, 2016. Retrieved October 26,
^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds:
The Lives and Discoveries of 15 Great Computer Scientists. Springer.
^ Popper 1995, p. 56
Imre Lakatos (1976), Proofs and Refutations. Cambridge: Cambridge
^ Gábor Kutrovátz, "Imre Lakatos’s Philosophy of Mathematics"
^ See, for example Bertrand Russell's statement "Mathematics, rightly
viewed, possesses not only truth, but supreme beauty ..." in his
History of Western Philosophy
^ Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral
and Feynman's Operational Calculus. Oxford University Press.
^ Wigner, Eugene (1960). "The Unreasonable Effectiveness of
Mathematics in the Natural Sciences". Communications on Pure and
Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W.
doi:10.1002/cpa.3160130102. Archived from the original on February 28,
Mathematics Subject Classification 2010" (PDF). Archived (PDF) from
the original on May 14, 2011. Retrieved November 9, 2010.
^ Hardy, G.H. (1940). A Mathematician's Apology. Cambridge University
Press. ISBN 0-521-42706-1.
^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas:
Mathematics and Philosophy. MAA.
^ Aigner, Martin; Ziegler, Günter M. (2001). Proofs from The
Book. Springer. ISBN 3-540-40460-0.
^ "Earliest Uses of Various Mathematical Symbols". Archived from the
original on February 20, 2016. Retrieved September 14, 2014.
^ Kline, p. 140, on Diophantus; p. 261, on Vieta.
^ Oakley 2014, p. 16: "Focused problem solving in math and science is
often more effortful than focused-mode thinking involving language and
people. This may be because humans haven't evolved over the millennia
to manipulate mathematical ideas, which are frequently more abstractly
encrypted than those of conventional language."
^ Oakley 2014, p. 16: "What do I mean by abstractness? You can point
to a real live cow chewing its cud in a pasture and equate it with the
letters c–o–w on the page. But you can't point to a real live plus
sign that the symbol '+' is modeled after – the idea underlying the
plus sign is more abstract."
^ Oakley 2014, p. 16: "By encryptedness, I mean that one symbol can
stand for a number of different operations or ideas, just as the
multiplication sign symbolizes repeated addition."
^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988,
ISBN 0-7167-1953-3. p. 4 "A few complain that the computer
program can't be verified properly", (in reference to the
Haken–Apple proof of the Four Color Theorem).
^ "The method of 'postulating' what we want has many advantages; they
are the same as the advantages of theft over honest toil." Bertrand
Russell (1919), Introduction to Mathematical Philosophy, New York and
London, p. 71. Archived June 20, 2015, at the Wayback Machine.
^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972,
ISBN 0-486-61630-4. p. 1, "Among the many branches of modern
mathematics set theory occupies a unique place: with a few rare
exceptions the entities which are studied and analyzed in mathematics
may be regarded as certain particular sets or classes of objects."
^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics,
Oxford University Press, 2005.
Mathematics Institute, P=NP, claymath.org
^ Rao, C.R. (1997)
Statistics and Truth: Putting Chance to Work, World
Scientific. ISBN 981-02-3111-3
^ Rao, C.R. (1981). "Foreword". In Arthanari, T.S.; Dodge, Yadolah.
Mathematical programming in statistics. Wiley Series in Probability
and Mathematical Statistics. New York: Wiley. pp. vii–viii.
ISBN 0-471-08073-X. MR 0607328.
^ Whittle (1994, pp. 10–11, 14–18): Whittle, Peter (1994).
"Almost home". In Kelly, F.P. Probability, statistics and
optimisation: A Tribute to Peter Whittle (previously "A realised path:
The Cambridge Statistical Laboratory upto 1993 (revised 2002)" ed.).
Chichester: John Wiley. pp. 1–28. ISBN 0-471-94829-2.
Archived from the original on December 19, 2013.
^ Monastyrsky 2001, p. 1: "The
Fields Medal is now indisputably
the best known and most influential award in mathematics."
^ Riehm 2002, pp. 778–82.
Courant, Richard and H. Robbins, What Is Mathematics? : An
Elementary Approach to Ideas and Methods, Oxford University Press,
USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
du Sautoy, Marcus (25 June 2010). "Nicolas Bourbaki". A Brief History
BBC Radio 4. Retrieved 26 October 2017.
Einstein, Albert (1923). Sidelights on Relativity: I. Ether and
Geometry and experience (translated by G.B. Jeffery,
D.Sc., and W. Perrett, Ph.D). E.P. Dutton & Co., New York.
Eves, Howard, An Introduction to the History of Mathematics, Sixth
Edition, Saunders, 1990, ISBN 0-03-029558-0.
Kline, Morris, Mathematical Thought from Ancient to Modern Times,
Oxford University Press, USA; Paperback edition (March 1, 1990).
Monastyrsky, Michael (2001). "Some Trends in Modern
the Fields Medal" (PDF). Canadian Mathematical Society. Retrieved July
Oakley, Barbara (2014). A Mind For Numbers: How to Excel at Math and
Science (Even If You Flunked Algebra). New York: Penguin Random House.
Oxford English Dictionary, second edition, ed. John Simpson and Edmund
Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
The Oxford Dictionary of English Etymology, 1983 reprint.
Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised
edition (June 1989). ISBN 0-933174-65-9.
Peirce, Benjamin (1881). Peirce, Charles Sanders, ed. "Linear
associative algebra". American Journal of
expanded, and annotated revision with an 1875 paper by B. Peirce
and annotations by his son, C.S. Peirce, of the 1872 lithograph ed.).
Johns Hopkins University. 4 (1–4): 97–229. doi:10.2307/2369153.
JSTOR 2369153. Corrected, expanded, and annotated revision with
an 1875 paper by B. Peirce and annotations by his son,
C. S. Peirce, of the 1872 lithograph ed. Google Eprint and
as an extract, D. Van Nostrand, 1882, Google Eprint. .
Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of
Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
Popper, Karl R. (1995). "On knowledge". In Search of a Better World:
Lectures and Essays from Thirty Years. Routledge.
Riehm, Carl (August 2002). "The Early History of the Fields Medal"
(PDF). Notices of the AMS. AMS. 49 (7): 778–72.
Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of
the American Mathematical Society. 43 (1): 101–09.
doi:10.1090/S0273-0979-05-01069-4. Retrieved June 24, 2006.
Waltershausen, Wolfgang Sartorius von (1965) [first published 1856].
Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend.
Benson, Donald C., The Moment of Proof: Mathematical Epiphanies,
Oxford University Press, USA; New Ed edition (December 14, 2000).
Boyer, Carl B., A History of Mathematics, Wiley; 2nd edition, revised
by Uta C. Merzbach, (March 6, 1991). ISBN 0-471-54397-7. – A
concise history of mathematics from the Concept of Number to
Davis, Philip J. and Hersh, Reuben, The Mathematical Experience.
Mariner Books; Reprint edition (January 14, 1999).
Gullberg, Jan, Mathematics – From the Birth of Numbers. W. W.
Norton & Company; 1st edition (October 1997).
Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer
Academic Publishers 2000. – A translated and expanded version
of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the
most complete and authoritative work available. Also in paperback and
on CD-ROM, and online.
Jourdain, Philip E. B., The Nature of Mathematics, in The World of
Mathematics, James R. Newman, editor, Dover Publications, 2003,
Maier, Annaliese, At the Threshold of Exact Science: Selected Writings
of Annaliese Maier on Late Medieval Natural Philosophy, edited by
Steven Sargent, Philadelphia: University of Pennsylvania Press, 1982.
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