Everything about Mathematic totally explained
Mathematics (colloquially,
maths or
math) is the body of knowledge centered on such concepts as
quantity,
structure,
space, and
change, and also the academic discipline that studies them.
Benjamin Peirce called it "the science that draws necessary conclusions".
Other practitioners of mathematics maintain that mathematics is the science of pattern, and that
mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new
conjectures and establish their truth by
rigorous deduction from appropriately chosen
axioms and
definitions.
Through the use of
abstraction and
logical
reasoning, mathematics evolved from
counting,
calculation,
measurement, and the systematic study of the
shapes and
motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the
ancient Egyptian,
Mesopotamian,
Indian,
Chinese,
Greek and
Islamic worlds.
Rigorous arguments first appeared in
Greek mathematics, most notably in
Euclid's
Elements. The development continued in fitful bursts until the
Renaissance period of the
16th century, when mathematical innovations interacted with new
scientific discoveries, leading to an acceleration in research that continues to the present day.
Today, mathematics is used throughout the world in many fields, including
natural science,
engineering,
medicine, and the
social sciences such as
economics.
Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in
pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later.
Etymology
The word "mathematics" (Greek: μαθηματικά or
mathēmatiká) comes from the
Greek μάθημα (
máthēma), which means
learning,
study,
science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (
mathēmatikós),
related to learning, or
studious, which likewise further came to mean
mathematical. In particular, μαθηματικὴ τέχνη (
mathēmatikḗ tékhnē), in
Latin ars mathematica, meant
the mathematical art.
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, and meaning roughly "all things mathematical". In English, however, the noun
mathematics takes singular verb forms. It is often shortened to
math in English-speaking North America and
maths elsewhere.
History
The evolution of mathematics might be seen as an ever-increasing series of
abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of
numbers. The realization that two apples and two oranges have something in common was a breakthrough in human thought.
In addition to recognizing how to
count physical objects,
prehistoric peoples also recognized how to count
abstract quantities, like
time —
days,
seasons,
years.
Arithmetic (
addition,
subtraction,
multiplication and
division), naturally followed.
Further steps need
writing or some other system for recording numbers such as
tallies or the knotted strings called
quipu used by the
Inca to store numerical data.
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. The
Indus Valley civilization developed the modern
decimal system, including the concept of
zero.
From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to
taxation and
commerce, to understand the relationships among numbers, to
measure land, and to predict
astronomical events. These needs can be roughly related to the broad subdivision of mathematics into the studies of
quantity,
structure,
space, and
change.
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and 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."
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in
commerce,
land measurement and later
astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example,
Richard Feynman invented the
Feynman path integral using a combination of mathematical reasoning and physical insight, and today's
string theory continues to inspire new mathematics. Some mathematics is only relevant 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. The 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 in mathematics. One major distinction is between
pure mathematics and
applied mathematics. 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's often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the
elegance of mathematics, its intrinsic
aesthetics and inner
beauty.
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 expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest
Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs. The popularity of
recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Notation, language, and rigor
16th century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. In the
18th century,
Euler 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. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.
Mathematical
language also is hard for beginners. Words such as
or and
only have more precise meanings than in everyday speech. Also confusing to beginners, words such as
open and
field have been given specialized mathematical meanings.
Mathematical jargon includes technical terms such as
homeomorphism and
integrable. But there's a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
Rigor is fundamentally a matter of
mathematical proof. 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. 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. Today, mathematicians continue to argue among themselves about
computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.
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.
Mathematics as science
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". In the original Latin
Regina Scientiarum, as well as in
German Königin der Wissenschaften, the word corresponding to
science means (field of) knowledge. Indeed, this is also the original meaning in English, and there's no doubt that mathematics is in this sense a science. The specialization restricting the meaning to
natural science is of later date. If one considers
science to be strictly about the physical world, then mathematics, or at least
pure mathematics, isn't a science.
Albert Einstein has stated that
"as far as the laws of mathematics refer to reality, they're not certain; and as far as they're certain, they don't refer to reality."
Many philosophers believe that mathematics isn't experimentally
falsifiable, and thus not a science according to the definition of
Karl Popper. However, in the 1930s important work in mathematical logic showed that mathematics can't be reduced to logic, 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 itself.
An alternative view is that certain scientific fields (such as
theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist,
J. M. Ziman, proposed that science is
public knowledge and thus includes mathematics. In any case, 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, weakening the objection that mathematics doesn't use the
scientific method. In his 2002 book
A New Kind of Science,
Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
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 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 don't 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.
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the
Fields Medal, established in 1936 and now awarded every 4 years. It is often considered, misleadingly, the equivalent of science's
Nobel Prizes. The
Wolf Prize in Mathematics, instituted in 1979, recognizes lifetime achievement, and another major international award, the
Abel Prize, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such 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. Solution of each of these problems carries a $1 million reward, and only one (the
Riemann hypothesis) is duplicated in Hilbert's problems.
Fields of mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict
astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (for example,
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.
Quantity
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, whence such popular results as
Fermat's last theorem. Number theory also holds two widely-considered unsolved problems: the
twin prime conjecture and
Goldbach's conjecture.
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
quarternions and
octonions. Consideration of the natural numbers also leads to the
transfinite numbers, which formalize the concept of counting to infinity. Another area of study is size, which leads to the
cardinal numbers and then to another conception of infinity: the
aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
»
Applied mathematics
Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the
sciences,
business, and other areas. An important field in applied mathematics is
statistics, which uses
probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, don't consider themselves to be mathematicians, but rather part of an allied group.)
Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of
rounding errors or other sources of error in computation.
Image:Gravitation space source.png | Mathematical physics
Image:BernoullisLawDerivationDiagram.svg | Mathematical fluid dynamics
Image:Composite trapezoidal rule illustration small.png | Numerical analysis
Image:Maximum boxed.png | Optimization
Image:Two red dice 01.svg | Probability
Image:Oldfaithful3.png | Statistics
Image:Market Data Index NYA on 20050726 202628 UTC.png | Financial mathematics
Image:Arbitrary-gametree-solved.png | Game theory
Common misconceptions
Mathematics isn't a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.
Mathematics isn't
numerology, nor is it
accountancy; nor is it restricted to
arithmetic.
Pseudomathematics is a form of mathematics-like activity undertaken outside
academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between
pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigor;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
- lack of familiarity with, and therefore underestimation of, the existing literature.
The case of
Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like
astronomy, mathematics owes much to amateur contributors such as
Fermat and
Mersenne.
Mathematics and physical reality
Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it isn't necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while many
axiom systems are derived from our perceptions and experiments, they're not dependent on them.
For example, we could say that the physical concept of two apples may be accurately
modeled by the
natural number 2. On the other hand, we could also say that the natural numbers are
not an accurate model because there's no standard "unit" apple and no two apples are exactly alike. The modeling idea is further complicated by the possibility of
fractional or partial apples. So while it may be instructive to visualize the axiomatic definition of the natural numbers as collections of apples, the definition itself isn't dependent upon nor derived from any actual physical entities.
Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led physicist
Eugene Wigner to write an article titled "
The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
Further Information
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