CHARACTERISTICS OF MATHEMATICS

 

The characteristics of mathematics that makes it unique among other subjects are:

      Logical sequence

      Structure

      Precision and accuracy

      Abstractness

      Mathematical Language and Symbolism

      Applicability

      Generalisation and classification

      Mathematical systems

      Rigor and logic

      Simplicity and complexity

  Logical Sequence

The modern characteristics of logical derivability and axiomatic arrangement are inherited from the ancient Greek tradition of Thales and Pythagoras and are epitomized in the presentation of Geometry by Euclid (The Elements).

It has not always been this way. The earliest mathematics was firmly empirical, rooted in man’s perception of number (quantity), space (configuration), time, and change (transformation). But by a gradual process of experience, abstraction, and generalization, concepts developed that finally separated mathematics from an empirical science to an abstract science, culminating in the axiomatic science that it is today. It is this evolution from empirical science to axiomatic science that has established derivability as the basis for mathematics.

This does not mean that there is no connection with empirical reality, quite the contrary. But it does mean that mathematics is, today, built upon abstract concepts whose relationship with real experiences is useful but not essential. These abstractions mean that mathematical fact is now established without reference to empirical reality. It may certainly be influenced by this reality, as it often is, but it is not considered mathematical fact until it is established according to the logical requirements of modern mathematics.

Thus logical sequence becomes a main characteristic of mathematics. To put in simple words, the study of mathematics begins with few well – known uncomplicated definitions and postulates, and proceeds, step by step, to quite elaborate steps. Mathematics learning always proceeds from simple to complex and from concrete to abstract. It is a subject in which the previous knowledge has a greater influence. For example, algebra depends on arithmetic, the calculus depends on algebra, dynamics depends on the calculus, analytical geometry depends on algebra and elementary geometry and so on. Even within these topics, the same dependence is found.

Structure

Generally speaking, a structure denotes ‘ the formation, arrangement and articulation of parts in anything build up by nature or art’. Based on this definition mathematical structure should be some sort of arrangement, formation, or result of putting together of parts.

For example, we take as the fundamental building units of a structure the members a,b,c,…… of a non-empty set S. We hold together these building units by using one or more operations.

The familiar operations of addition denoted by +, and multiplication denoted by x, of natural numbes are operations on set N of natural numbers. Subtraction is not an operation on the set of natural numbers since the difference of two natural numbers may not be a natural number (Example: ). But subtraction is an operation on a set I of all integers.

If ‘S’ is a non-empty set on which one or more operations have been uniquely defined with respect to an equivalence relation, then the set S together with the operation or operations is called a mathematical system. We will denote such a system consisting of a set S and an operation O by <S;O>. A system consisting of  asset S and two operations O₁ and O₂ will be denoted by <S; O₁,O₂>. A mathematical structure is a mathematical system with one or more explicitly recognized(mathematical) properties. We may create a structure from a mathematical system by making specific recognition of one or more of the commutative, associative pr distributive properties that the system may have.

A structure which consists of a mathematical system <S;O> with one operation, in which the operation O is associative is called a semi-group. To illustrate, the set of positive integers under addition is a semi group, as is also the set of positive integers under multiplication.

A structure which consists of a mathematical system <S; O₁,O₂> with two operations in which each of the operations O₁ and O₂ is commutative and associative, and in which one of the operations is distributive with respect to the other is called a number system. A number is a member of the set S in a number system. Among the familiar number systems are the following three: the natural number system <N;+, >, the non negative integral number systems <N_o; +, >(here N_o denotes the set of non-negative integers) and the integral number system <I; +, >.

A semi- group with an identity number is called a monoid. The semi group of positive integers under multiplication is a monoid, for I is an identity number of N under multiplication. On the contrary, the semi-group of positive integers under addition is not a monoid, for the set N has no identity number under addition.

In ReXRe where Re is the set of real numbers, plane analytic geometry may be considered as a superstructure based upon the structure known as real number system. Thus mathematics has got definite logical structures. These structures ensure the beauty and order of mathematics.

The capacity to group and order the environment in a logical way is to be gradually developed  in the students, The teacher should demonstrate the beauty of mathematical structures for  a greater understanding of the various mathematical structures and their interrelationships. Such an approach will help the students in appreciating the order and beauty of mathematics.

Precision and accuracy

Mathematics is known as an ‘exact’ science because of its precision. It is perhaps the only subject which can claim certainty of results. In mathematics, the results are either right or wrong, accepted or rejected. There is no midway possible between the right and wrong. Mathematics can decide whether or not its conclusions are right. Mathematicians can verify the validity of the results and convince others of its validity with consistency and objectivity. This holds not only for the expert, but also for anyone who uses mathematics at any level.

Even when there is a new emphasis o approximation, mathematical results can have any degree of accuracy required. Although precision and accuracy are distinctly different as criteria for the measures of approximation, they can be most effectively discussed when contrasted with each other. The most effective measures of both precision and accuracy are in terms of errors(positive or negative) involved. The precision of the measure or a computation is evaluated in terms of the apparent error. The accuracy of a measure or a computation is evaluated in terms of the relative error or percent of error made.

It is the teacher’s job to help the students in taking decisions regarding the degree of accuracy which are most appropriate for a measurement or calculation. This is possible by encouraging the students to observe critically, perceive relationships, analyse the data and arrive at precise conclusions/ inferences to the level of accuracy required.

Mathematical culture is that what you say should be correct. What you say should have a definition. You should know the definition and limits of what you are saying, stating, or claiming. The distinction is between mathematics being developed informally and mathematics being done more formally, with necessary and sufficient conditions stated up front and restricting the discussion to a particular class of objects.

Thus, the modern mathematical culture of precision arises because:

  mathematics has developed a precise, highly symbolic language,

  mathematical concepts have developed in a dialectic manner that allows for the adaptation, adjustment and cumulative refinement of concepts based on experiences, and

  mathematical reasoning is expected to be correct.

Abstractness

Mathematics is abstract in the sense that mathematics does not deal with actual objects in much the same way as physics. But, in fact, mathematicsl questions, as a rule, cannot be settles by direct appeal to experiment. For example, Euclid’s lines are supposed to have no width and his points no size. No such objects can be found in the physical world. Euclid’s geometry describes an imaginary world whch resembles the actual world sufficiently for it is a useful study for surveyors, carpenters and engineers.

Infinity is something that we can never experience and yet it is a central concept of mathematics. Our whole thinking is based on the assumption that there are infinitely many numbers, so that counting need never stop; that there are infinitely many fractions between 0 and 1, that there are infinitely many points on the circumference of a circle etc. We have no way of knowing and justifying that it is so because we cannot observe and count all these. Infinitely, then, it is not a concept corresponding to any object that we have seen or likely to see. It is an abstract concept.

Again someone whose thinking was essentially physical might refuse to believe in negative numbers on the ground that you cannot have a quantity less than nothing. Still more, such a person would refuse to believe in the square root of minus one.

Children form concepts out of experience and lead to certain structures. Furthermore, the same structure should, if possible, be met in different situations. Children eventually learn that there is something about four beats, four chairs, four chocolates, four friends and eventually they extract the notion four. This process of abstraction comes from their experiences of dealing with discrete objects.

However, all mathematical concepts cannot be learned through experiences with concrete objects. Some concepts can be learned only through their definition and they may not have concrete counterparts to be extracted from. Most of the mathematical concepts are such concepts without concretization and hence they are abstract. The concept of prime numbers, concept of probability, concept of function, the concept of limit, the concept of continuous functions are all abstract in the sense they can be learnt only through their definitions and it is not possible to provide concrete objects to correspond to such concepts. Even those concepts which one argues to be concrete are also abstract. For example, one could argue that concepts such as point, a line, a ray, a diagonal, a circle etc., can be learnt through observation of concrete instances and therefore, they are concrete. But a line drawn on a board, or a dot (point), a figure of a circle, are all mere representations of the concepts and they are not objects themselves. Moreover, a student learning a concept by mere observation of such instances can form wrong concepts. Fr example, a student can identify a figure which is not quite a circle as a circle. If a child has learned the concept by its definition as ‘closed curve on which every point is equidistant from a fixed point called center’, the child looks for the correct conditions for a curve to be a circle. Wherever possible, it is always advisable to provide the suitable concrete experiences which will lead to a generalization forming an abstract concept.  

Mathematical Language and Symbolism

Another most important characteristic of mathematics which distinguishes it from many other subjects is its peculiar language and symbolism.  Lindsay says, “Mathematics is the language of physical sicneces and certainly no more marvelous language was ever created by the mind of man”.  Man has the ability to assign symbols for objects and ideas.  Mathematical language and symbols cut short the lengthy statements and help the expression of ideas or thing sin the exact form.  Mathematical language is free form verbosity and helps into the point, clear and exact expression of facts.

Over the course of the past three thousand years, mankind has developed sophisticated spoken and written natural languages that are highly effective for expressing a variety of moods, motives, and meanings. The language in which Mathematics is done has developed no less, and, when mastered, provides a highly efficient and powerful tool for mathematical expression, exploration, reconstruction after exploration, and communication. Its power (when used well) comes from simultaneously being precise (unambiguous) and yet concise (no superfluities, nothing unnecessary). But the language of mathematics is no exception to being used poorly. Just as any language, it can be used well or poorly.

The language for communication of mathematical ideas is largely in terms of symbols and words which everybody cannot understand. There is no popular terminology for talking about mathematics.  For example, the distinction between a number and a numeral could head the list. A number is a property of a set; that property tells how many elements there are in a set. A numeral is a name or a symbol used to represent a number. Essentially, to distinguish between a number and a numeral is to distinguish between a thing and the name of a thing. If the things considered are physical entities, there seems t be little difficulty in making distinctions. But if things are abstract entities such as those who deal within mathematics, it becomes considerably more difficult to make the distinction between the name of the thing and its referent, the things itself.

Since numbers are abstractions and cannot be perceived by any of the five senses, they are often confused with their names. A teacher ought to be very careful t use correct terms, since this helps children to learn and think better. It is important that a student understands the distinction between a number and a numeral so that he may realise the difference between actually operating with numbers and merely manipulating symbols representing those numbers. This is only one item in regard to precision of language. There are many others, such as distinguishing between the line and picture of a line, a point and the dot used to represent a point, to list a few.

In earlier times, mathematics was in fact, fully verbal. Now, after the dramatic advances in symbolism that occurred in the mercantile period (1500s), mathematics can be practiced in an apparent symbolic shorthand, without really the need for very many words. This, however, is only shorthand. The symbols themselves require very careful and precise definition and characterization in order for them to be used, computed with, and allows the results to be correct.

Understanding mathematics is realizing what symbolism corresponds to the structure that has been abstracted. It is not enough for children to understand mathematics; it is necessary for them to speak mathematics; in other words to handle symbols. This corresponds to speaking a language as opposed to understanding a language. The process of speaking of the mathematical language runs as follows: an abstraction process, followed by a symbolization process, followed again by the learning of the use of the symbols.

There are great dangers in a purely symbolic treatment. By purely symbolic, it is meant that no reference is being made to the entities symbolized while the symbols are being used. The danger is that the symbols have not acquired the same firm meanings as other word symbols currently used in the language used by the children, The amount of experience which lies behind the vocabulary used in their own languages is enormous. It is not possible to hope to put the same amount of experience behind mathematical symbols. Symbolic or even verbal statements of a concept are meaningless unless the symbolism is related to something real and concrete. Thus, if children are presented with symbols before they have abstracted the concepts that the symbols represent – the only way they can deal with them is associatively. They treat them as nonsense and learn by rote.

In arithmetic and algebra students deal not with facts, but with symbols. The child who is poor in mathematics is unable to see what concepts the symbols stand for, what the concepts themselves are abstracted from, and hence what the symbols communicate.

The symbols of mathematics constitute a language which is gradually developed by and for the pupil. This language must be acquired just like any other languages and there is need for translating this language into one’s own mother tongue. Long periods of training with patience and endurance are needed to make the students feel at home with this language. The training that mathematics provides in the use of symbols is an excellent preparation for other sciences.

The use of symbols makes the mathematics language more elegant and precise than any other language. For example, the commutative law of addition and multiplication in real number system can be stated in the verbal form as: ‘the addition and multiplication of two real numbers is independent of the order in which they are combined’.

This can be stated in concise form as: a+b = b+a, and axb=bxa,  .

Almost all mathematical statements, relations, operations are expressed using mathematical symbols such as  and so on. It is highly impossible to prepare a comprehensive list of all the mathematical symbols. Anyone, who wants to read and communicate effectively in the mathematical language, has to be well versed in the mathematical symbols and their definite uses.

  Applicability

Knowledge is power only when it is applied. The study of mathematics requires the learner to apply the skills acquired to new situations. The knowledge acquired by students is greatly used for solving problems. The students can always verify the validity of the mathematical rules and relationships by applying and verifying the mathematical ideas. The knowledge and its application, wherever possible, should be related to daily life situations. Concepts and principles become more functional and meaningful only when they are related to actual practical applications. Such a practice will make the learning of mathematics more meaningful and significant.

General applicability is a recurring characteristic of mathematics: mathematical truth turns out to be applicable in very distinct areas of application in phenomena from across the universe to across the street. Mathematics is widely useful because the five phenomena that it studies are ubiquitous in nature and in the natural instincts of man to seek explanation, to generalize, and to attempt to improve the organization of his knowledge. As Mathematics has progressively advanced and abstracted its natural concepts, it has increased the host of subjects to which these concepts can be fruitfully applied.

 Generalisation and classification

Mathematics gives exercises in widening and generalizing conceptions, in combining various results under one head, in making schematic arrangements and classifications. It is easy to find instances of successive generalizations.  For example, the number concept itself enlarged from that of the whole number to include successively fractional numbers, irrational numbers, negative numbers and imaginary numbers. One of the important aspects of algebra is its generalized treatment of the processes of arithmetic. In geometry also, there are repeated occasions for grouping and generating results. When the pupil evolves his own definitions, concepts and theorems, he is making generalizations.            The generalization and classification of mathematics are very simple and obvious in comparison with those of others domains of thought and activity. However, the mathematics teachers should take care to see that the final generations into a rule should always be deferred untilit is almost spontaneously suggested by the pupils themselves. The children should realize that there is always a justification for a rule; the mathematics is logic. If a mechanical rule is given prematurately, the need for any understanding is dispensed with.

  Mathematical systems

As it has been shown earlier, mathematics includes many components which are inn themselves mathematical structures or mathematical systems. A typical mathematical system has the following four parts: undefined terms, defined terms, axioms and theorems.

  Undefined terms

In geometry or in any other mathematical system, we have to start with some terms which are taken as undefined terms or other terms of the system are defined in terms of undefined terms. The choice of the undefined terms is completely arbitrary and generally to facilitate the development of the structure (Example: point, line, set, variable, plane etc)

  Defined terms

We define the other terms in the system in terms of the undefined terms. For example, a triangle having three equal sides is an equilateral triangle. Thus to define an equilateral triangle, one should have learnt the terms ‘triangle’, ‘equal’, or ‘sides’.

  Axioms

Axioms or postulates are statements in a mathematical system which we take for granted and describe the relationships existing among the undefined terms of the system. They are self evident truths.

Example:

         There can be one and only straight line joining two points

         Two lines meet at a point

         A line has one and only one mid-point

         The above postulates describe the relationship existing among undefined terms ‘line’ and ‘point’.

  Theorems

In daily life, we generally use a form of argument called the rule of implication. The rule of implication states that (1) the statement p implies the statement q and (2) if the statement p is true, then the statement q will be true. When we apply the rule of application to the axioms we generate new statements. Again we may apply this rule to these new statements. A statement that we arrive at by successive application of the rule of implication to the axioms and statements previously arrived at is called a theorem.

Example:

         The angles in a semicircle are right angles.

         In a triangle the greater side has the greater angle opposite to it.

         If two parallel lines are cut by a transversal, then alternate angles are equal.

  Rigor and logic

It goes without saying that logic is an important factor in mathematics; it governs the pattern of deductive proof through which mathematics is developed. Of course, logic was used in mathematics centuries ago. During the last few decades there has been great emphasis on the analysis of the logical structure of mathematics as a whole.

The presentation of mathematics in rigorous form is ill – advised. Mathematics must be understood intitutively in physical or geometrical terms. This is the primary pedagogical objective. When this is achieved it is proper to formulate concepts and reasoning in as rigorous a form as students can take. However, rigorous presentation is secondary in importance. AS Roger Bacon, “Argument concludes a question, but it does not make us feel certain, or acquiesce in the contemplation of truth, except the truth also be found to be so by experience”

 Simplicity (Search for a Single Exposition), Complexity (Dense Exposition)

For the outsider looking in, it is hard to believe that simplicity is a characteristic of mathematics. Yet, for the practitioner of mathematics, simplicity is a strong part of the culture. Simplicity in what respect? The mathematician desires the simplest possible single exposition. Through greater abstraction, a single exposition is possible at the price of additional terminology and machinery to allow all of the various particularities to be subsumed into the exposition at the higher level.

This is significant: although the mathematician may indeed have found his desired single exposition (for which reason he claims also that simplicity has been achieved), the reader often bears the burden of correctly and conscientiously exploring the quite significant terrain that lies beneath the abstract language of the higher-level exposition.

Thus, it is the mathematician’s desire for a single exposition that leads to the attendant complexity of mathematics, especially in contemporary mathematics.