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| The '''language of mathematics''' is the system used by [[mathematician]]s to communicate [[Mathematics|mathematical]] ideas among themselves. This language consists of a [[substratum (linguistics)|substrate]] of some [[natural language]] (for example [[English language|English]]) using [[technical term]]s and grammatical conventions that are peculiar to mathematical discourse (see [[Mathematical jargon]]), supplemented by a highly specialized symbolic notation for [[mathematical formula]]s.
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| Like natural languages in general, discourse using the language of mathematics can employ a scala of [[register (sociolinguistics)|register]]s. [[Research article]]s in [[academic journals]] use a more formal tone than oral exchanges over a scribbled-upon napkin in the university cafeteria.
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| ==What is a language?==
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| Here are some definitions of [[language]]:
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| * ''a systematic means of communicating by the use of sounds or conventional symbols'' [http://www.cogsci.princeton.edu/~wn/ WordNet]
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| * ''a system of words used in a particular discipline'' [http://www.cogsci.princeton.edu/~wn/ WordNet]
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| * ''the code we all use to express ourselves and communicate to others'' [http://www.nlg.nhs.uk/SpeechLanguage/glossary.htm Speech & Language Therapy Glossary of Terms]
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| * ''a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements'' [[Noam Chomsky]].
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| These definitions describe language in terms of the following components:
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| * A [[vocabulary]] of symbols or words
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| * A [[grammar]] consisting of rules of how these symbols may be used
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| * A [[community]] of people who use and understand these symbols
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| * A range of [[Meaning (semiotics)|meanings]] that can be communicated with these symbols
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| Each of these components is also found in the language of mathematics.
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| ==The vocabulary of mathematics==
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| [[Mathematical notation]] has assimilated [[table of mathematical symbols|symbols]] from many different [[alphabet]]s and [[typeface]]s. It also includes symbols that are specific to mathematics, such as
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| :<math>\forall \ \exists \ \nabla \ \wedge \ \infty.</math>
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| Mathematical notation is central to the power of modern mathematics. Though the [[algebra]] of [[Al-Khwārizmī]] did not use such symbols, it solved equations using many more rules than are used today with symbolic notation, and had great difficulty working with multiple variables (which using symbolic notation can simply be called <math>x, y, z</math>, etc.). Sometimes formulas cannot be understood without a written or spoken explanation, but often they are sufficient by themselves, and sometimes they are difficult to read aloud or information is lost in the translation to words, as when several parenthetical factors are involved or when a complex structure like a [[matrix (mathematics)|matrix]] is manipulated.
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| Like any other profession, mathematics also has its own brand of [[technical terminology]]. In some cases, a word in general usage has a different and specific meaning within mathematics—examples are [[group (mathematics)|group]], [[ring (mathematics)|ring]], [[field (mathematics)|field]], [[category theory|category]], [[term (mathematics)|term]], and [[factor (mathematics)|factor]]. For more examples, see [[:Category:Mathematical terminology]].
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| In other cases, specialist terms have been created which do not exist outside of mathematics—examples are [[tensor]], [[fractal]], [[functor]]. Mathematical statements have their own moderately complex taxonomy, being divided into [[axiom]]s, [[conjecture]]s, [[theorem]]s, [[lemma (mathematics)|lemma]]s and [[corollary|corollaries]]. And there are stock phrases in mathematics, used with specific meanings, such as "''[[if and only if]]''", "''[[necessary and sufficient]]''" and "''[[without loss of generality]]''". Such phrases are known as [[mathematical jargon]].
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| When mathematicians communicate with each other informally, they use phrases that help to convey ideas. Examples of some of the more idiomatic phrases are "kill this term"{{further explanation needed|date=November 2012}}, "vanish this interval"{{further explanation needed|date=November 2012}} and "grow this variable"{{further explanation needed|date=November 2012}}.
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| The vocabulary of mathematics also has visual elements. Diagrams are used informally on blackboards, as well as more formally in published work. When used appropriately, diagrams display schematic information more easily. Diagrams also help visually and aid intuitive calculations. Sometimes, as in a [[visual proof]], a diagram even serves as complete justification for a proposition. A system of diagram conventions may evolve into a mathematical notation – for example, the [[Penrose graphical notation]] for tensor products.
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| ==The grammar of mathematics==
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| The [[grammar]] used for mathematical discourse is essentially the grammar of the natural language used as substrate, but with several mathematics-specific peculiarities.
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| Most notably, the mathematical notation used for formulas has its own grammar, not dependent on a specific natural language, but shared internationally by mathematicians regardless of their mother tongues. This includes the conventions that the formulas are written predominantly [[left to right]], also when the writing system of the substrate language is right-to-left, and that the [[Latin alphabet]] is commonly used for simple [[variable (mathematics)|variable]]s and [[parameter]]s. A formula such as
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| :<math>\sin x + a\cos 2 x \ge 0\,</math>
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| is understood by Chinese and Israeli mathematicians alike.
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| Such mathematical formulas can be a [[part of speech]] in a natural-language phrase, or even assume the role of a full-fledged sentence. For example, the formula above, an [[equation]], can be considered a sentence or sentential phrase in which the [[greater than or equal to]] symbol has the role of a [[verb]]. In careful speech, this can be made clear by pronouncing "≥" as "is greater than or equal to", but in an informal context mathematicians may shorten this to "greater or equal" and yet handle this grammatically like a verb. A good example is the book title ''Why does {{nowrap|1=E = mc<sup>2</sup>}}?'';<ref>{{cite book |title=Why does {{nowrap|1=E = mc<sup>2</sup>}}? (and why should we care?) |author=Brian Cox |authorlink=Brian Cox (physicist) |author2=Jeff Forshaw |publisher=Da Capo Press |year=2010 |isbn=978-0-306-81876-9}}</ref> here, the [[equals sign]] has the role of an [[infinitive]].
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| Mathematical formulas can be ''[[Speech production|vocalized]]'' (spoken aloud). The vocalization system for formulas has to be learned, and is dependent on the underlying natural language. For example, when using English, the expression "''ƒ'' (''x'')" is conventionally pronounced "eff of eks", where the insertion of the preposition "of" is not suggested by the notation per se. The expression "<math>\tfrac{dy}{dx}</math>", on the other hand, is vocalized like "dee-why-dee-eks", with complete omission of the [[fraction bar]], in other contexts often pronounced "over". The book title ''Why does {{nowrap|1=E = mc<sup>2</sup>}}?'' is said aloud as ''Why does ee equal em see-squared?''.
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| Characteristic for mathematical discourse – both formal and informal – is the use of the [[Clusivity|inclusive]] [[Grammatical person|first person]] [[plural]] "we" to mean: "the audience (or reader) together with the speaker (or author)".
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| ==The language community of mathematics==
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| Mathematics is used by [[list of mathematicians|mathematicians]], who form a global community composed of speakers of many languages. It is also used by students of mathematics. As mathematics is a part of primary education in almost all countries, almost all educated people have some exposure to pure mathematics. There are very few cultural dependencies or barriers in modern mathematics. There are international mathematics competitions, such as the [[International Mathematical Olympiad]], and international co-operation between professional mathematicians is commonplace.
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| ==The meanings of mathematics==
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| Mathematics is used to communicate information about a wide range of different subjects. Here are three broad categories:
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| * '''Mathematics describes the real world''': many areas of mathematics originated with attempts to describe and solve real world phenomena - from measuring farms ([[geometry]]) to falling apples ([[calculus]]) to gambling ([[probability]]). Mathematics is widely used in modern [[mathematical physics|physics]] and [[engineering]], and has been hugely successful in helping us to understand more about the universe around us from its largest scales ([[physical cosmology]]) to its smallest ([[quantum mechanics]]). Indeed, the very success of mathematics in this respect has been a source of puzzlement for some philosophers (see [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]] by [[Eugene Wigner]]).
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| * '''Mathematics describes abstract structures''': on the other hand, there are areas of pure mathematics which deal with [[abstract structure]]s, which have no known physical counterparts at all. However, it is difficult to give any categorical examples here, as even the most abstract structures can be co-opted as models in some branch of physics (see [[Calabi-Yau spaces]] and [[string theory]]).
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| * '''Mathematics describes mathematics''': mathematics can be used reflexively to describe itself—this is an area of mathematics called [[metamathematics]].
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| Mathematics can communicate a range of meanings that is as wide as (although different from) that of a natural language. As [[England|English]] mathematician [[Rolph Ludwig Edward Schwarzenberger|R.L.E. Schwarzenberger]] says:
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| :''My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the [[Fundamental Theorem of Algebra]] using the English language.''
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| ==Alternative views==
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| Some definitions of language, such as early versions of [[Charles Hockett]]'s "design features" definition, emphasize the spoken nature of language. Mathematics would not qualify as a language under these definitions, as it is primarily a written form of communication (to see why, try reading [[Maxwell's equations]] out loud). However, these definitions would also disqualify [[sign language]]s, which are now recognized as languages in their own right, independent of spoken language.
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| Other linguists believe no valid comparison can be made between mathematics and language, because they are simply too different:
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| : ''Mathematics would appear to be both more and less than a language for while being limited in its linguistic capabilities it also seems to involve a form of thinking that has something in common with art and music.'' - Ford & Peat (1988)
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| ==See also==
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| * [[Linguistics]]
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| * [[Philosophy of language]]
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| ==References==
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| <references /> | |
| * {{cite book
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| | last = Knight
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| | first = Isabel F.
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| | year = 1968
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| | title = The Geometric Spirit: The Abbe de Condillac and the French Enlightenment
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| | publisher = Yale University Press
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| | location = New Haven
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| }}
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| * R. L. E. Schwarzenberger (2000), ''The Language of Geometry'', published in ''A Mathematical Spectrum Miscellany'', Applied Probability Trust.
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| * Alan Ford & F. David Peat (1988), ''The Role of Language in Science'', Foundations of Physics Vol 18.
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| * Kay O'Halloran, ''Mathematical Discourse: Language, Symbolism and Visual Images'', Continuum, 2004. ISBN 0826468578
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| ==External links==
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| *[http://www.aucegypt.edu/academic/anth/anth352/what_is_language.htm What is Language]
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| *[http://www.fdavidpeat.com/bibliography/essays/maths.htm ''Mathematics and the Language of Nature''] - essay by F. David Peat.
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| *[http://www.economics.soton.ac.uk/staff/aldrich/Mathematical%20Words.htm Mathematical Words: Origins and Sources] (John Aldrich, University of Southampton)
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| *[http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics Communicating in the Language of Mathematics] by Dr. David Moursund
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| *[http://www.abstractmath.org/Handbook/handbook.pdf Handbook of Mathematical Discourse] by Charles Wells.
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| [[Category:Philosophy of mathematics]]
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| [[Category:Language]]
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Were you inside a relationship with a man with who we had ardentfervent emotions for and he broke up with you? Do you state to the self all time "I desire my spouse back"? If this really is the case then read on to locate out what you can do to get ex back.
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The basal metabolic rate offers a wise baseline for minimal calories. Obviously, the proper foods and exercise are crucial to the success. A diet of sugary foods and/or an exercise system consisting of endless strolling on a treadmill usually make weight loss difficult. But should you utilize the BMR as a beginning point, you will understand not to go below which level plus add food plus exercise accordingly to create a calorie deficit.
Then the more you exercise, the greater gets a metabolism. And with a higher metabolism you'll burn calories more efficiently. As for a person with a low metabolism rate, he/she usually burn fewer calories within the same amount of food plus shop the calories that have been not 'burned' into fats.
Your daily bmr and the calories we burn for sport plus different escapades create the total amount of calories you burn a day. This really is exactly why BMR is so important - it assists you plan a fat reduction or weight gain program.
Once you have established the daily calorie needs, we then need to decide how several carbs you are going to eat daily. Estimates range from 40% to 60% of calories coming from carbs for what is described as a "healthy diet." If, but, we are eating 60% of your diet because carbs, how several calories are left for the proteins and fats which are imperative for life? Only 40% and which 40% should be distributed between proteins and fats. Why does the American Diabetic Association recommend a diet higher in carbs, that create the blood sugar diabetics are supposed to be controlling, than proteins and fats combined?
So you're suggesting that when a metabolic style workout I could eat any I want? Absolutely not! The types of foods you eat are as important because the sort of workout you do. If you eat is only because important as what we eat. In the upcoming few blogs I might be discussing foods to eat and those to avoid, as well the appropriate times to consume for the many successful fat loss. Be on the lookout for the brand-new N-Motion Fitness recipe section on nmotionfitnessllc.com.