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1、Mathematics What is Mathematics,Anyway?DiscussDid you ever study mathematics?What sub-fields have you studied?What is Mathematics?What is Mathematics?Make a definitionTwo definitions attempted in the articlea broad definitiona standard definitionMovie clipThe Division of MathematicsHow to classify m
2、athematical ideas?There are two ways_Sentence understandingWhile it is perhaps easier for the reader to have the material pre-filtered according to application,this hides the fact that the underlying mathematics is really quite similar radioactive decay is essentially the same as inflationary deprec
3、iation of investments,for example.How Many Parts of Mathematics?How Many Parts of Mathematics?Why foundations,algebra,geometry,analysis are grouped together?Keyto decide which books and articles intend to reveal structureofmathematicsitself,and which intend to applymathematicstocloselyalliedareas.Wh
4、at are the examples which may support“applying mathematics to closely allied areas?How Many Parts of Mathematics?Sentence understandingHowever,these themes are sometimes best woven together into areas of study which are not so much research into mathematics as research into the enterprise of mathema
5、tics“epi-mathematic”,perhaps.But Is This Division“Real”?Whats the relationship between pure mathematics and applied mathematics?Exercises(P.6)IIIIIKEY(1)此此处处我我们们对对数数学学进进行行宽宽泛泛地地定定义义。它它包包括括所所有有涉涉及及数数量量、几几何何和和逻逻辑辑的的相相关关领领域域；其其中中包包括括统统计计学学、计计算算机机学学、逻逻辑辑学学、应应用用数数学学、以以及及其其它它通通常常被被认认为为迥迥异异于于数数学学的的领领域域，也也包
6、包括括研研究究数数学学研研究究的的学学科科，如数学史、数学教育等。如数学史、数学教育等。(2)数数学学与与逻逻辑辑学学二二者者的的相相似似性性仅仅此此而而已已，不不应应加加以以高高估估。后后者者可可被被视视为为对对任任一一概概念念的的处处理理，而而前前者者主主要要关关心心数数值值概概念念。尽尽管管在在对对数数量量概概念念进进行行分分析析时时，数数学学可可能能会会介介入入逻逻辑辑，在在从从事事数数学学研研究究时时我我们们也也可可能能要要使使用用逻逻辑辑思思维维能能力力，但但它它们们到到底底分分属属于于不不同同的的兴兴趣趣领领域域。二二者者虽虽有有理性共核，但研究范围殊异，更非互补理性共核，但研究
7、范围殊异，更非互补。The reason for it is that mathematics is a two-stage process.Rather than studying the world directly,mathematicians create so-called models of the world,and study them.This applies even to the simplest mathematics.After the age of four or five we do not study addition by actually combining
8、 groups of objects and counting them.Instead we use an abstract mathematical construction,or model,known as the positive integers(that is,the numbers 1,2,3,4,5 and so on).Similarly,we do not do basic geometry by cutting shapes out of paper,partly because it is not necessary and partly because in any
9、 case the resulting shapes would not be exact squares,triangles or whatever they were supposed to be.Once again,we study a model,a sort of idealized world that contains things that we do not come across in everyday life,such as infinitely thin lines that stretch away to infinity,or absolutely perfec
10、t circles,and does not contain untidy,worldly things like hamburgers,chairs or human beings.Definition in EST DefiningDefining:Making a statement conveying fundamental character.A definition should be both complete and restricted.Those that can be defined include substances,concepts,processes,though
11、ts,etc.How to make a definition?Term(术语)=Class(范畴)+Characteristics(特性)e.g.Chemical energy is potential energy that is stored in gasoline,food,and oil.TERMCLASSCHARACTERISTICSChemical energypotential energythat is stored in gasoline,food,and oil.Definition in EST Extended DefinitionFor a concept that
12、 is extremely complex,we often use an extended definition to make it clear.Other than class and characteristics,extended definition may use such means as exemplification,contrast,description,illustration,etc.e.g.A telescope is an instrument for magnifying distant objects.It has two essential parts:t
13、he objective which collects light from the distant object and forms a real image,and the eyepiece which forms a magnified image of this image.Refracting telescopes use a convex lens as the objective and reflecting telescopes use a curved mirror of large diameter.Definition in EST How to make a defin
14、ition in a right way?1)Identify the class.2)Be precise.3)Beware of circular definitions.e.g.*An agronomist(农艺学家农艺学家)is a person who practices agronomy.4)Avoid obscurity.5)If a negative statement(What isnt it?)goes first,an affirmative one(What is it?)should follow directly.6)Be objective.e.g.*A pizz
15、a is something really good to eat.7)“The”should be excluded from the term to be defined.8)Make a definition as in a dictionary entry.Definition in EST ExercisesThe followings are some poorly written definitions.Determine what is wrong with each definition and discuss how you would improve it.1.A sci
16、entific theory is a theory like Darwins theory.2.An apple is round,red,and about the size of a fist.3.Calculus is a tough subject.4.An ear is auditory appendage of Homo sapiens and other species.5.A unicorn is not a real animal.6.Tornadoes are very dangerous.7.Radium is an element.8.A compass looks
17、like a clock.9.An amphibian is like a frog or turtle.10.A supernova is big nova.Definition in EST ExercisesGive extended definition of each concept by means of generalizations,examples and uses.1.biosphere2.aluminum3.an insectDefinition in EST Supplementary-1.Expressionsdefineas 把把定义为定义为bywe mean 所谓
18、所谓是指是指be defined as 被定义为被定义为byis meant 所谓所谓是指是指refer to 是指是指Motion may be defined as a continuous change in position.运动可被定义为位置的不断变化。运动可被定义为位置的不断变化。Humidity refers to the amount of water vapor in the air.湿度是指空气中的水蒸气含量。湿度是指空气中的水蒸气含量。By conductivity we mean the ability to conduct an electric current.所谓
19、电导率是指传导电流的能力。所谓电导率是指传导电流的能力。By electrical resistance is meant the hindrance to the flow of electrons.所谓电阻是指对电子流动的阻力。所谓电阻是指对电子流动的阻力。Definition in EST Supplementary-2.PatternsWhich-clause-ed-ingfor+-ingwith+NPprep.+Which clauseA pump is a machine which is used for transferring a liquid or a gas from o
20、ne place to another.泵是将液体或气体从一处输送到另一处用的机械。泵是将液体或气体从一处输送到另一处用的机械。A1uminum is a metal produced from bauxite.铝是从铝土矿提炼出的一种金属。铝是从铝土矿提炼出的一种金属。Brass is an alloy containing a large proportion of copper.黄铜是一种含有大量铜的合金。黄铜是一种含有大量铜的合金。Tungsten is a metal with the property of remaining hardness at red heat.钨是一种在赤热状态下仍能保持坚硬性质的金属。钨是一种在赤热状态下仍能保持坚硬性质的金属。The micrometer is an instrument by means of which very accurate measurements can be made.测微计是一种可用来进行精密测量的仪器。测微计是一种可用来进行精密测量的仪器。Thank you!