Mathematics

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This episode is from a Wikipedia article titled Mathematics.

Mathematics,

From Greek,

Mathema,

Knowledge,

Study,

Learning,

Includes the study of such topics as quantity,

Number theory,

Structure,

Algebra,

Space,

Geometry,

And change,

Mathematical analysis.

It has no generally accepted definition.

Humans seek and use patterns to formulate new conjectures.

They resolve the truths or falsity of such by mathematical proof.

When mathematical structures are good models of real phenomena,

Mathematical reasoning can be used to 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 inquiry.

Rigorous arguments first appeared in Greek mathematics,

Most notably in Euclid's Elements.

Since the pioneering work of Giuseppe Piano,

David Hilbert,

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.

Mathematics is essential in many fields,

Including natural science,

Engineering,

Medicine,

Finance,

And the social sciences.

Modern mathematics has led to entirely new mathematical disciplines,

Such as statistics and game theory.

Mathematicians engage in pure mathematics,

Mathematics for its own sake,

Without having any application in mind,

But practical applications for what began as pure mathematics are often discovered later.

History 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.

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,

Seasons,

Or years.

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 most ancient mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC.

Many early texts mention Pythagorean triples,

And so by inference,

The Pythagorean theorem seems to be the most ancient and widespread mathematical development,

After basic arithmetic and geometry.

It is in Babylonian mathematics that elementary arithmetic,

Addition,

Subtraction,

Multiplication,

And division,

First appear in the archaeological record.

The Babylonians also possessed a place value system and used a sexagesimal numeral system,

Which is still in use today for measuring angles and time.

Beginning in the 6th century BC with the Pythagoreans,

The ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics.

Around 300 BC,

Euclid introduced the axiomatic method,

Still used in mathematics today,

Consisting of definition,

Axiom,

Theorem,

And proof.

His textbook Elements is widely considered the most successful and influential textbook of all time.

The greatest mathematician of antiquity is often held to be Archimedes of Syracuse.

He developed formulas for calculating the surface area and volume of solids of revolution,

And used the method of exhaustion to calculate the area under the arc of parabola,

With the summation of an infinite series,

In a manner not too dissimilar from modern calculus.

Other notable achievements of Greek mathematics are conic sections,

Apollinius or Perga,

3rd century BC,

Trigonometry Harparcus of Nyssaea,

2nd century BC,

And the beginnings of algebra,

Diophantus,

3rd century AD.

The Hindu-Arabic numeral system and the rules for the use of its operations,

In use throughout the world today,

Evolved over the course of the 1st millennium AD in India,

And were transmitted to the Western world via Islamic mathematics.

Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine,

And an early form of infinite series.

During the Golden Age of Islam,

Especially during the 9th and 10th centuries,

Mathematics saw many important innovations building on Greek mathematics.

The most notable achievement of Islamic mathematics was the development of algebra.

Other notable achievements of the Islamic period are advances in spherical trigonometry,

And the addition of the decimal point to the Arabic numeral system.

Many notable mathematicians from this period were Persian,

Such as Al-Khwarisimi,

Omar Qayyam,

And Sharraf al-Din al-Tusi.

During the early modern period,

Mathematics began to develop at an accelerating pace in Western Europe.

The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics.

Leonhard Euler was the most notable mathematician of the 18th century,

Contributing numerous theorems and discoveries.

Perhaps the foremost mathematician of the 19th century was the German mathematician Karl Friedrich Gauss,

Who made numerous contributions to fields such as algebra,

Analysis,

Differential geometry,

Matrix theory,

Number theory,

And statistics.

In the early 20th century,

Kurt Gödel transformed mathematics by publishing his incompleteness theorems,

Which show in part that any consistent axiomatic system,

If powerful enough to describe arithmetic,

Will contain true propositions that cannot be proved.

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.

Sevruk 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,

000 items are added to the database each year.

The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs.

Etymology.

The word mathematics comes from ancient Greek mathema,

Meaning that which has learned,

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 mathematicos,

Meaning related to learning or studious,

Which likewise further came to mean mathematical.

In particular,

Mathematikai techni meant the mathematical art.

Similarly,

One of the two main schools of thought in Pythagoreanism was known as the mathematicoi,

Which at the time meant learners rather than mathematicians in the modern sense.

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,

St.

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 from les mathématiques and the less commonly used singular derivative la mathématique,

Goes back to the Latin neuter plural Mathematica,

Based on the Greek plural ta Mathematica,

Used by Aristotle and meaning roughly all things mathematical.

Although it is plausible that English borrowed only the adjective mathematical 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 North America,

Math.

Definitions of Mathematics Mathematics has no generally accepted definition.

Aristotle defined mathematics as the science of quantity,

And this definition prevailed until the 18th century.

However,

Aristotle also noted a focus on quantity alone may not distinguish mathematics from science like physics.

In his view,

Abstraction and studying quantity as a property separable in thought from real instances set mathematics apart.

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.

A great many professional mathematicians take no interest in a definition of mathematics or consider it undefinable.

There is not even consensus on whether mathematics is an art or a science.

Some just say,

Mathematics is what mathematicians do.

Three Leading Types Three leading types of definition of mathematics today are called logistic,

Intuitionist,

And formalist,

Each reflecting a different philosophical school of thought.

All have severe flaws,

None has widespread acceptance,

And no reconciliation seems possible.

Logistic Definitions An early definition of mathematics in terms of logic was that of Benjamin Pierce,

A science that draws necessary conclusions.

In the Principia Mathematica,

Bertrod Russell and Alfred North Whitehead advanced a 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 logistic definition of mathematics is Russell's 1903 All Mathematics is Symbolic Logic.

Intuitionist Definitions 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,

Mathematics 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.

Intuitionists also reject the law of excluded middle.

While this stance does force them to reject one common version of proof by contradiction as a viable proof method.

Formalist Definitions Formalist definitions identify mathematics with its symbols and the rules for operating on them.

Haskell-Keyre defined mathematics simply as the science of formal systems.

The formal system is a set of symbols or tokens and some rules on how the tokens are to be combined into formulas.

In formal systems,

The word axiom has a special meaning different from the ordinary meaning of a self-evident truth,

And is used to refer to a combination of tokens that is included in a given formal system without needing to be delivered using the rules of the system.