Part two 6

Copyright © 2011 by William Allan Kritsonis/All Rights Re-served

6

MATHEMATICS

INSIGHTS

1. The uses of ordinary language are largely practi-cal.

2. Mathematics has many uses.3. Mathematical symbolisms are essentially theoreti-

cal.4. Many students and teachers of mathematics never

really understand the subject because they iden-tify it with calculation for practical ends.

5. Mathematics occupies a world of its own.6. Mathematics applications, no matter how useful,

are secondary and incidental to the essential sym-bolic meanings.

7. Complete abstractness makes possible the elabo-rate developments of mathematical systems, yielding in the long run the most practical applica-tions.

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116 PART TWO: FUNDAMENTAL PATTERNS OF MEANING

8. Mathematical communities tend to be specialized and limited rather than inclusive.

9. Mathematics are designed to achieve complete precision in meaning and rigor in reasoning.


10. In mathematics one really knows the subject only if he knows about the subject.

11. It is not enough to teach students of mathematics how to make calculations and demonstrations skillfully and automatically.

12. The student of mathematics can be said to know mathematically only if he understands and can ar-ticulate his reasons for each assertion he makes.

13. The sovereign principle of all mathematical rea-soning is logical consistency.

14. The subject matter of mathematics is formal (ab-stract) symbolic systems within which all possible propositions are consistent with each other.

15. Mathematics only yields conclusions that follow by logical necessity from the premises defining each system.

16. In mathematics theory is the whole body of sym-bolic content of a given postulational system.

17. Technical skill in computation and the ability to use mathematics in scientific investigation, valu-able as they may be, are not evidence of mathe-matical understanding.

18. Mathematical understanding consists in compre-hending the method of complete logical abstrac-tion and of drawing necessary conclusions from basic formal premises.

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In the realms of meaning, mathematics keeps com-pany with the languages. The reason for this classifica-tion is that mathematics, like the ordinary languages, is a collection of arbitrary symbolic systems. It will be a main goal in this chapter to elaborate and explain this assertion.

It was stated in the previous chapter that knowl-edge of ordinary language consists in the ability to use symbols to communicate meanings. While the same statement also holds for mathematics, there are signifi-cant differences in emphasis in the two cases. The use of ordinary language are largely practical. Its symbolic systems exist for the most part to serve the everyday needs of communication. Mathematics is not primarily practical, nor is it created as a major basis for social co-hesion. To be sure, mathematics has many uses, as its wide applications in science and technology demon-strate. But these practical uses are not of the essence of mathematics, as the social uses of ordinary discourse are. Mathematical symbolisms are essentially theoreti-cal. They constitute a purely intellectual discipline, the forms of which are not determined by the urgencies of adjustment to nature and society.

MANY STUDENTS AND TEACHERS OF MATHEMATICS

NEVER REALLY UNDERSTAND THE SUBJECT

Many students and teachers of mathematics never really understand the subject because they identify it with calculation for practical ends. Ordinary language is chiefly concerned with the community’s adaptation to the actual world of things and people. On the other hand, mathematics has no such relation to tangible ac-tuality. Mathematical symbolisms occupy an indepen-dent, self-contained world of thought. Mathematical symbolisms need not stand for actual things or classes of actual things, as the symbols of ordinary language do. Mathematics occupies a world of its own. Its realm is that of “pure” symbolic forms. Mathematics applica-


tions, no matter how useful, are secondary and inciden-tal to the essential symbolic meanings.

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THE ESSENCE OF MATHEMATICS

Another way of expressing the essence of mathe-matics is to say that it is a language of complete ab-straction. Ordinary language is abstract, too, in the sense that its concepts refer to classes or kinds of things and that its conventional patterns are types of expression. But ordinary language is less abstract in the sense that it refers back to actual things, events, per-sons, and relations. Mathematics, having no necessary reference to actuality, is fully abstract. It is purely for-mal, without any necessary anchorage in the actual world. Interestingly, it is just this complete abstractness that makes possible the elaborate developments of mathematical systems, yielding in the long run the most practical applications.

MATHEMATICS DIFFERS FROM ORDINARY LANGUAGE

Mathematics further differs from ordinary lan-guage in the usual nature of its symbolisms. The sym-bol-patterns of common discourse grow naturally out of the experience of the speaking community; for the most part they are not deliberately invented. On the other hand, mathematical symbolisms normally are ar-tificial, in that they are freely and consciously adopted, constituting deliberate inventions or constructions. Any person may adopt arbitrarily and without reference to previous customary usage any symbolism that serves his formal purposes. It is only incumbent on any such innovator, if he wishes to be understood, that he indi-cate clearly the terms in which his symbolism is de-fined.

Mathematical meanings are communicated effec-tively only to those who choose to become familiar with the symbolic constructions within particular mathemati-cal systems. Mathematical communities tend to be spe-cialized and limited rather than inclusive, like the major ordinary language communities. Mathematical lan-guages are artificial dialects understood only by the members of special communities of voluntary initiates.


The natural ordinary languages are meant to express the whole range of common experiences, while the par-ticular artificial symbolisms of mathematics express special and strictly limited conceptual relationships.

The symbolic systems of mathematics are de-signed to achieve complete precision in meaning and rigor in reasoning. Ordinary language, by contrast, growing informally out of the complex experiences of many persons and groups over long periods to time, is relatively vague and ambiguous. Ordinary reasoning is usually full of unexamined commonsense assumptions and inconsistencies. In fact, one of the two main pur-poses of using special symbols in mathematics is to avoid the imprecision of common speech. The other purpose is to provide symbols that can be more readily manipulated in reasoning processes than is possible us-ing the symbols of common language. On the other hand, since mathematics could be done entirely with the symbols of ordinary discourse, with meticulous care in definition of terms, the usual artificial symbolism of mathematics is a convenient expedient and not a nec-essary feature of the discipline.

It was pointed out earlier that knowing ordinary language does not depend on knowing about it. The same does not hold for mathematics. In mathematics one really knows the subject only if he knows about the subject. Specifically he does his mathematics with self-conscious awareness, examining and justifying each step in his reasoning in the light of the canons of rigor-ous proof. This is why it is not enough to teach stu-dents of mathematics how to make calculations and demonstrations skillfully and automatically. Yet, facility in speaking is properly the primary purpose of ordinary language instruction. The student of mathematics can be said to know mathematically only if he understands and can articulate his reasons for each assertion he makes.

MATHEMATICS IS MORE THAN A DESIGNATED LANGUAGE

MATHEMATICS 121

In one crucial respect mathematics is other and more than what is usually designated a language. Cus-tomarily the term “language” refers to a means of ex-pression and communication using written or spoken symbols. Mathematics includes much more than this, specifically, chains of logical reasoning. The subject matter of mathematics includes far more than the for-mal symbol-patterns. It is chiefly concerned with the transformation of the symbols in accordance with cer-tain rules included in the definition of each particular system. The sovereign principle of all mathematical reasoning is logical consistency. The only admissible rules of transformation for mathematical symbols are those that do not entail contradictory propositions within any given system.


Even at the most basic level,in order for a student to understand

math,he/she must first recognize the sym-

bols used and what their functions are.

This is an early step in the develop-ment

of the understanding of math. Soon after this, however, math begins

to develop as a language of its own.

It is necessary for math to become so disconnected from other disciplines,

or could a teacher develop methods to keep math inclusive across the curricu-

lum?

MATHEMATICS 121


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MATHEMATICS 123

THE SUBJECT MATTER OF MATHEMATICS

The subject matter of mathematics is, then, formal (abstract) symbolic systems within which all possible propositions are consistent with each other. Mathemati-cal reasoning consists in the demonstration of relation-ships among the symbols of the system by means of necessary inference: in which each proposition (affirm-ing some relation between symbols) must be shown to be logically entailed by one or more other propositions within the system. Mathematics is more than a lan-guage. Mathematics adds to the patterns of symbolic expression the methods of deductive inference by which logically consistent relationships can be system-atically elaborated.

MATHEMATICS DOES NOT YIELD FACTS—ONLY CONCLUSIONS

Though mathematics is more than language in containing deductive reasoning, it is like language in re-spect to the indefinite plurality of its admissible sym-bolic systems. Contrary to what was once universally believed and is still a common misconception, mathe-matics is not a single system of ideas containing the “truths” of ordinary arithmetic, algebra, Euclidean ge-ometry, the differential and integral calculus, and other subdivisions of the traditional mathematics curriculum. There are many ordinary languages, each with its own patterns for conceptualizing experience and its charac-teristic ways of combining expressive elements into the larger structures of discourse. There are also any num-ber of different mathematical systems that can be con-structed, each with its own pattern of basic elements and characteristic rules of transformation and each con-sistent within itself but independent of every other mathematical system. Accordingly, mathematics does not express “true” propositions in any absolute or em-pirical sense, as a statement of the way things really are, or of what is actually so. It does no more than re-veal the consistency of propositions within any particu-


lar symbolic system. Mathematics does not yield knowl-edge of facts that have to do with the contingent actu-alities of the world as it is. Mathematics only yields con-clusions that follow by logical necessity from the premises defining each system.

MATHEMATICAL PATTERNS

Any number of self-consistent mathematical pat-terns can be defined and deductively elaborated. There are many geometries besides that of Euclid. In fact, the discovery of consistent geometries, such as those of Riemann and Lobachevsky, in which through any point outside of a given straight line there are, respectively, no parallels or any infinity of parallels to the line, was a major step in the development of the modern under-standing about the plurality of mathematical systems generally. Lest it be thought that such geometries are merely mathematical oddities without practical impor-tance, it should be noted that the theory of relativity, that has played such an important role in the revolu-tionizing of modern physics and astronomy, shows that physical space-time and the laws of motion require non-Euclidean geometry for their formulation. Similarly, there are many algebras besides ordinary algebra. For example, it is possible to define consistent algebraic systems in which the relation a • b = b • a does not al-ways hold. Some such “noncommutative” algebras also are of great importance in their scientific applications.

THE METHOD OF MATHEMATICS IS ESSENTIALLY POSTULATIONAL

The method of mathematics is essentially postula-tional. This means that certain postulates, or axioms, are arbitrarily chosen as part of the foundation of a given mathematical system. These postulates are not “self-evident truths,” as, for example, the axioms of Eu-clidean geometry were formerly thought to be. They are assumptions taken as a starting point for the devel-opment of a chain of deductive inferences. All mathe-

MATHEMATICS 125

matical reasoning is of the form “if . . . then,” where the “if” is followed by a postulate (or some necessary infer-ence therefrom) and the “then” is followed by a conclu-sion, or a theorem. Neither the postulates nor the theo-rems deduced from them are either true or false. All that can be said of them is that if the mathematical rea-soning has been done correctly, they are related in the manner of necessary implication.


MATHEMATICAL SYSTEM REQUIRES BASIS

OF UNDEFINED TERMS

Every mathematical system requires some basis of undefined terms. This basis, together with the postu-lates, constitutes what is called the “foundation” of the system. In any language undefined terms are necessary as a basis for defining other terms because the process of definition by reference to other terms cannot pro-ceed indefinitely; somewhere it must come to rest in certain primitive terms that are not themselves defined. From the basis and the postulates, theorems are de-duced. The entire body of undefined terms, definitions, postulates, and theorems comprises a particular sym-bolic system, or a theory. From various bases and ax-ioms various theories may be developed such as, the-ory of groups, theory of numbers, theory of continuous functions, theory of infinite sets, and theory of complex variables, to name only a few.

THE MEANING OF THEORY IN MATHEMATICS

The meaning of “theory” in mathematics differs somewhat from its meaning in the empirical sciences, as the analysis in following chapters will show. In the sciences a theory usually refers to a general explana-tion for a group of related facts and generalizations. For example, in physics the behavior of gases is explained by the kinetic theory, and the facts of paleontology and comparative anatomy may be explained by the theory of evolution. In mathematics, on the other hand, theory is the whole body of symbolic content of a given postu-lational system.

Any mathematical theory can be defined by means of sets. A “set” is simply a class, family, or aggregate of abstract conceptual entities (elements) all of which have some common property or properties specified by the axioms upon which the theory is founded. Two sets, A and B, are said to be equal if they contain the same elements. A set B is called a “subset” of a set A if all of the elements of B are elements of A. The sum (A + B) of

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two sets A and B is defined as the set containing all ele-ments that are either in A or in B. The product (A • B) of two sets A and B is defined as the set containing all ele-ments that are in both A and B. The difference (A – B) of A and B is defined as the set consisting of all elements in A and not in B. The Cartesian product (A X B) of A and B is defined as the set of all ordered pairs of ele-ments in A and B, that is, a set each of whose elements consists of a pair of elements one of which is an ele-ment of A and the other an element of B.

These ideas of elements, sets, equality, sum, prod-uct, and difference comprise basic terms from which all other mathematical concepts can be developed, pro-vided certain basic logical concepts are also pre-sup-posed. These essential logical concepts include the fol-lowing: “is a member of,” “not,” “all, or every,” “such that,” “there exists, or there is,” “if . . . then,” “or,” and “and.” The primitive materials (elements, sets, and their rules of combination) together with the elemental logical concepts constitute the basis for any mathemati-cal theory.

THE MEANING OF RELATION IN MATHEMATICS

Another concept of far-reaching importance in mathematics is that of relation. A “relation” is defined simply as a subset of the Cartesian product of two sets. It is a means of separating out certain pairs of elements from others. For example, if two sets A and B have ele-ments (a1, a2, a3 ) and (b1, b2, b3 ) respectively, the Cartesian product is the set A X B with elements [(a1, b1 ), (a1, b2 ), (a1, b3 ), (a2, b1 ), (a2 , b2 ), . . . , (b1 , a1 ), (b1 , a2 ), . . .] covering all possible pair combinations. Any subset of A X B, such as the set containing only the two elements [(a1, b2 ), (a3, b1 )], is then a particular re-lation on A X B. The three-element subset [(a1, b1 ), (a2, b2), (a3, b3 )] is another and different relation.

THE MEANING OF FUNCTION IN MATHEMATICS


A special case of a relation is a function, that is an-other concept of great importance in mathematics. A “function” is defined as a relation in which one and only one element in one set corresponds to any element in another set. For example, in the above illustration the first of the relations cited is not a function because no element of B is paired with a2. The second relation is a function because a1, a2, and a3 are each uniquely paired with an element of B. On the other hand, the relation [(a1, b2 ), (a1, b3 ), (a2, b2 ), (a3, b1 )] is not a function because a1 is paired with two different elements of B. When the functional relation works both ways, so that to each element of A, a unique element of B corre-sponds and vice versa, the relation is called “one-to-one correspondence.”

Finally, the concept of binary operation on B by A to C is defined by the requirement that to each pair of elements in the Cartesian product A and B a unique ele-ment of C corresponds (i.e., that C is a function of A X B).

SIMILARITY OF MATHEMATICS TO LANGUAGES

Returning to the similarity of mathematics to a language, or better, to a collection of languages, one can compare the undefined terms to the elements of sound and meaning upon which any given language is based. One can compare the various rules of combina-tion (sum, difference, product, relation, function, one-to-one correspondence, and binary operation) to the morphological and syntactic rules by which ordinary discourse is organized into an ordered hierarchy of ex-pressions. The above fundamental combinatorial con-cepts are the grammar of mathematics. They designate the patterns according to which the deductive elabora-tion of any mathematical system (i.e., the drawing of successive inferences from primitive terms, definitions, and axioms) must proceed.1

1 The above outline of the basic concepts in any mathematical theory largely follows the treatment given by R. B. Kershner and

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WHY THE CONCEPT OF SET IS SO IMPORTANT IN MATHEMATICS

The reason why the concept of set is so central in mathematics is that it embodies the principle of ab-straction. Abstraction is the essence of mathematical thinking. A set is specified completely by the properties of the elements composing it. Those properties are ab-stractions since they define elements in these terms: “any entity such that . . . is an element of the set.” The idea of any such that entails that particular things are not under consideration, but only kinds or classes of things. By means of this idea of abstraction the key mathematical concept of variable may be understood. A variable does not refer to something that moves or changes, as it would in ordinary speech. In mathemat-ics a variable, designated, say, by the symbol χ, is such that χ stands for, or in the place of, any element of a specified set. Variables are simply ways of representing the general idea of any or some as contrasted with par-ticular elements. For example, if the variable χ belongs to the set of rational numbers (fractions) between 0 and 1, it represents the idea of any or some rational number between 0 and 1.

It will have been noted that such ideas as number, point, line, distance, and quantity, which in everyday thought are considered typically mathematical, have hardly been mentioned in the preceding account of mathematical knowledge. The reason is that such con-cepts (with the possible exception of number, that in some formulations is taken as primitive) are special and derivative in comparison with the very general and pri-mary concepts used in the above analysis. For example, the integers and the counting process can be defined by means of the theory of finite sets, and the rational and real numbers by the theory of infinite sets. Further-more, Euclidean geometry and common algebra can be

L. R. Wilcox, The Anatomy of Mathematics, The Ronald Press Company, New York, 1960 esp. chaps. 4-5.


shown to be alternative interpretations of an identical theory of sets of real numbers (R). Thus a “point” may be defined as an ordered pair of real numbers, and a “line” as a relation on (i.e., a subset of) the Cartesian product (R X R) of two real number sets. Similarly, the calculus and the theory of functions can be shown to follow directly from a general study of relations on R X R, and the theory of complex numbers can be shown to result from the study of ordered pairs, combined ac-cording to the following rules: (a, b) + (c, d) = [(a + b), (c + d)] and (a, b) (c, d) = [(a • c – b •d), (a •d + b • c)].

MATHEMATICS 131

At younger ages, moststudents are not capable of

grasping the abstract concepts that deal with higher level math. Even if the student recognizes all of the symbols used, their functions must be under-

stood. Knowing that children develop at

different stages, how can a teacher deal with a class containing students

at many different levels of development?


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MATHEMATICS 133

THE IMPORTANCE OF GENERALITY INDEVELOPING MATHEMATICAL IDEAS

Alfred North Whitehead called form, variable, and generality “a sort of mathematical trinity which preside over the whole subject,” and he added that “they all re-ally spring from the same root, namely from the ab-stract nature of the science.”2 We have already dwelt on the formal nature of mathematical systems and on the concept of variable. Some discussion is now needed about generality. The development of mathematical ideas is marked by a progressive increase in generality. For example, the concept of number beginning with the positive integers, may be successively generalized to include zero and the negative integers, rational num-bers (fractions), irrational numbers (like √2), real num-bers (having a one-to-one correspondence with all the points on a line), complex numbers, vectors (directed magnitudes), and infinite (or transfinite) numbers.

In geometry the study of two-dimensional mani-folds (defined by ordered pairs of numbers) can be gen-eralized to three, four, or any higher number of dimen-sions by using ordered triples, quadruples, or generally, n-tuples (where n is any integer). Though such hyper-space geometries cannot be visualized, in the way that two- and three-dimensional configurations can be, they are nonetheless valid systems of geometry, which, inci-dentally, prove to have important applications in the sciences.

One of the ways in which generalization takes place in mathematics is in connection with the transfor-mation of one set into another through what was earlier defined as a binary operation. In any such transforma-tion certain relations remain unchanged, or invariant. For example, Euclidean geometry is concerned with transformations that leave intervals and angles invari-ant, i.e., in which figures may be translated or rotated but not distorted. A more general geometry (projective 2 An Introduction to Mathematics, American rev. ea., Oxford Uni-versity press, Fair Lawn, N.J., 1948, p. 57.


geometry) is concerned with transformations (projec-tions) that may alter intervals and angles but leave un-changed a quantity known as the “cross ratio.” The most general geometry (topology) deals with transfor-mations where the connectivity pattern of the sub-spaces is not changed (e.g., in three-dimensional space, where surfaces may be distorted but not cut or punctured).

SUMMING UP MATHEMATICS

To recapitulate, mathematics is a discipline in which formal symbolic systems are constructed by positing certain undefined terms (elements, sets, rules of combination), elaborating further concepts by defini-tions (conventions), adopting certain postulates (con-cerning both the undefined and the defined terms), and then, using the principles of logic, drawing necessary deductive inferences, resulting in an aggregate of propositions called “theorems.” The propositions of mathematics are formal and abstract in that they do not necessarily refer to the structure of the actual world but comprise a series of purely abstract formalisms all having in common the one rule of logical consistency.

It is well known that mathematics is of great prac-tical value in science and technology. The nature of the subject is misconstrued if it is regarded primarily as a “tool” subject. Technical skill in computation and the ability to use mathematics in scientific investigation, valuable as they may be, are not evidence of mathe-matical understanding. Mathematical understanding consists in comprehending the method of complete log-ical abstraction and of drawing necessary conclusions from basic formal premises.

WAYS OF KNOWING

1. Mathematics is not exclusively practical. What does this mean?

2. If practical use are not the essences of mathe-matics, what is?

MATHEMATICS 135

3. How do many students and teachers of mathe-matics view math?

4. Mathematics occupies a world of its own. What does this mean?

5. How does mathematics differ from ordinary lan-guage?

6. Mathematical communities tend to be special-ized and limited rather than inclusive like the ma-jor ordinary languages communities. Why?

7. What is the purpose(s) of the symbolic systems of mathematics?

8. In mathematics, one really knows the subject only if one knows about the subject. Why is this true?

9. What is the sovereign principle of all mathemat-ical reasoning? Why is this principle important?

10. What is the subject matter of mathematics?11. Mathematics does not yield knowledge of facts.

Mathematics only yields conclusions that follow by logical necessity from the premises defining each system. What does this mean?

12. What is the meaning of “theory” in mathemat-ics?

13. What are some of the similarities of mathemat-ics to languages?

14. Why is the concept of set so important in math-ematics?

15. Generality is important in the development of mathematics. Why?

16. How is the nature of mathematics miscon-strued?

17. What does mathematical understanding actually mean?

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