116 PART TWO: FUNDAMENTAL PATTERNS OF MEANING10. 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 mathematical- ly only if he understands and can articulate his reasons for each assertion he makes.13. The sovereign principle of all mathematical reasoning is logi- cal consistency.14. The subject matter of mathematics is formal (abstract) sym- bolic systems within which all possible propositions are consis- tent with each other.15. Mathematics only yields conclusions that follow by logical ne- cessity from the premises defining each system.16. In mathematics theory is the whole body of symbolic content of a given postulational system.17. Technical skill in computation and the ability to use mathemat- ics in scientific investigation, valuable as they may be, are not evidence of mathematical understanding.18. Mathematical understanding consists in comprehending the method of complete logical abstraction and of drawing neces- sary conclusions from basic formal premises. ____________________
MATHEMATICS 117In the realms of meaning, mathematics keeps company with the lan-guages. The reason for this classification is that mathematics, likethe ordinary languages, is a collection of arbitrary symbolic sys-tems. It will be a main goal in this chapter to elaborate and explainthis assertion. It was stated in the previous chapter that knowledge of ordi-nary language consists in the ability to use symbols to communicatemeanings. While the same statement also holds for mathematics,there are significant differences in emphasis in the two cases. The useof ordinary language are largely practical. Its symbolic systems ex-ist for the most part to serve the everyday needs of communication.Mathematics is not primarily practical, nor is it created as a majorbasis for social cohesion. To be sure, mathematics has many uses, asits wide applications in science and technology demonstrate. But thesepractical uses are not of the essence of mathematics, as the socialuses of ordinary discourse are. Mathematical symbolisms are essen-tially theoretical. They constitute a purely intellectual discipline,the forms of which are not determined by the urgencies of adjustmentto nature and society. MANY STUDENTS AND TEACHERS OF MATHEMATICS NEVER REALLY UNDERSTAND THE SUBJECT Many students and teachers of mathematics never really un-derstand the subject because they identify it with calculation forpractical ends. Ordinary language is chiefly concerned with the com-munity’s adaptation to the actual world of things and people. On theother hand, mathematics has no such relation to tangible actuality.Mathematical symbolisms occupy an independent, self-contained worldof thought. Mathematical symbolisms need not stand for actualthings or classes of actual things, as the symbols of ordinary lan-guage do. Mathematics occupies a world of its own. Its realm is thatof “pure” symbolic forms. Mathematics applications, no matter howuseful, are secondary and incidental to the essential symbolic mean-ings.
118 PART TWO: FUNDAMENTAL PATTERNS OF MEANING THE ESSENCE OF MATHEMATICS Another way of expressing the essence of mathematics is to saythat it is a language of complete abstraction. Ordinary languageis abstract, too, in the sense that its concepts refer to classes orkinds of things and that its conventional patterns are types of ex-pression. But ordinary language is less abstract in the sense that itrefers back to actual things, events, persons, and relations. Mathe-matics, having no necessary reference to actuality, is fully ab-stract. It is purely formal, without any necessary anchorage in theactual world. Interestingly, it is just this complete abstractnessthat makes possible the elaborate developments of mathematical sys-tems, yielding in the long run the most practical applications. MATHEMATICS DIFFERS FROM ORDINARY LANGUAGE Mathematics further differs from ordinary language in the usu-al nature of its symbolisms. The symbol-patterns of common discoursegrow naturally out of the experience of the speaking community; forthe most part they are not deliberately invented. On the other hand,mathematical symbolisms normally are artificial, in that they arefreely and consciously adopted, constituting deliberate inventions orconstructions. Any person may adopt arbitrarily and without refer-ence to previous customary usage any symbolism that serves his for-mal purposes. It is only incumbent on any such innovator, if he wishesto be understood, that he indicate clearly the terms in which his sym-bolism is defined. Mathematical meanings are communicated effectively only tothose who choose to become familiar with the symbolic constructionswithin particular mathematical systems. Mathematical communitiestend to be specialized and limited rather than inclusive, like the majorordinary language communities. Mathematical languages are artifi-cial dialects understood only by the members of special communities ofvoluntary initiates. The natural ordinary languages are meant toexpress the whole range of common experiences, while the particularartificial symbolisms of mathematics express special and strictly lim-ited conceptual relationships. The symbolic systems of mathematics are designed to achievecomplete precision in meaning and rigor in reasoning. Ordinary lan-guage, by contrast, growing informally out of the complex experi-ences of many persons and groups over long periods to time, is rela-tively vague and ambiguous. Ordinary reasoning is usually full of un-examined commonsense assumptions and inconsistencies. In fact, one ofthe two main purposes of using special symbols in mathematics is toavoid the imprecision of common speech. The other purpose is to providesymbols that can be more readily manipulated in reasoning processesthan is possible using the symbols of common language. On the otherhand, since mathematics could be done entirely with the symbols ofordinary discourse, with meticulous care in definition of terms, theusual artificial symbolism of mathematics is a convenient expedientand not a necessary feature of the discipline. It was pointed out earlier that knowing ordinary language doesnot depend on knowing about it. The same does not hold for mathemat-ics. In mathematics one really knows the subject only if he knowsabout the subject. Specifically he does his mathematics with self-con-scious awareness, examining and justifying each step in his reasoningin the light of the canons of rigorous proof. This is why it is not
MATHEMATICS 119enough to teach students of mathematics how to make calculationsand demonstrations skillfully and automatically. Yet, facility inspeaking is properly the primary purpose of ordinary language in-struction. The student of mathematics can be said to know mathemati-cally only if he understands and can articulate his reasons for eachassertion he makes. MATHEMATICS IS MORE THAN A DESIGNATED LANGUAGE In one crucial respect mathematics is other and more than whatis usually designated a language. Customarily the term “language”refers to a means of expression and communication using written orspoken symbols. Mathematics includes much more than this, specifical-ly, chains of logical reasoning. The subject matter of mathematicsincludes far more than the formal symbol-patterns. It is chiefly con-cerned with the transformation of the symbols in accordance withcertain rules included in the definition of each particular system. Thesovereign principle of all mathematical reasoning is logical consis-tency. The only admissible rules of transformation for mathematicalsymbols are those that do not entail contradictory propositions with-in any given system.
120 PART TWO: FUNDAMENTAL PATTERNS OF MEANING Even at the most basic level, in order for a student to understand math, he/she must first recognize the symbols used and what their functions are. This is an early step in the development 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 curriculum?
122 PART TWO: FUNDAMENTAL PATTERNS OF MEANING THE SUBJECT MATTER OF MATHEMATICS The subject matter of mathematics is, then, formal (ab-stract) symbolic systems within which all possible proposi-tions are consistent with each other. Mathematical reasoningconsists in the demonstration of relationships among the symbols ofthe system by means of necessary inference: in which each proposi-tion (affirming some relation between symbols) must be shown to belogically entailed by one or more other propositions within the sys-tem. Mathematics is more than a language. Mathematics adds to thepatterns of symbolic expression the methods of deductive inference bywhich logically consistent relationships can be systematically elabo-rated. MATHEMATICS DOES NOT YIELD FACTS—ONLY CONCLUSIONS Though mathematics is more than language in containing deduc-tive reasoning, it is like language in respect to the indefinite pluralityof its admissible symbolic systems. Contrary to what was once univer-sally believed and is still a common misconception, mathematics is nota single system of ideas containing the “truths” of ordinary arith-metic, algebra, Euclidean geometry, the differential and integralcalculus, and other subdivisions of the traditional mathematics cur-riculum. There are many ordinary languages, each with its own pat-terns for conceptualizing experience and its characteristic ways ofcombining expressive elements into the larger structures of discourse.There are also any number of different mathematical systems thatcan be constructed, each with its own pattern of basic elements andcharacteristic rules of transformation and each consistent within it-self but independent of every other mathematical system. According-ly, mathematics does not express “true” propositions in any absoluteor empirical sense, as a statement of the way things really are, orof what is actually so. It does no more than reveal the consistency ofpropositions within any particular symbolic system. Mathematics doesnot yield knowledge of facts that have to do with the contingent ac-tualities of the world as it is. Mathematics only yields conclusionsthat follow by logical necessity from the premises defining each sys-tem. MATHEMATICAL PATTERNS Any number of self-consistent mathematical patterns can bedefined and deductively elaborated. There are many geometries besidesthat of Euclid. In fact, the discovery of consistent geometries, such asthose of Riemann and Lobachevsky, in which through any point outsideof a given straight line there are, respectively, no parallels or anyinfinity of parallels to the line, was a major step in the developmentof the modern understanding about the plurality of mathematical sys-tems generally. Lest it be thought that such geometries are merelymathematical oddities without practical importance, it should be notedthat the theory of relativity, that has played such an important rolein the revolutionizing of modern physics and astronomy, shows thatphysical space-time and the laws of motion require non-Euclidean ge-ometry for their formulation. Similarly, there are many algebras be-sides ordinary algebra. For example, it is possible to define consistentalgebraic systems in which the relation a • b = b • a does not al-
MATHEMATICS 123ways hold. Some such “noncommutative” algebras also are of greatimportance in their scientific applications. THE METHOD OF MATHEMATICS IS ESSENTIALLY POSTULATIONAL The method of mathematics is essentially postulational. Thismeans that certain postulates, or axioms, are arbitrarily chosen aspart of the foundation of a given mathematical system. These postu-lates are not “self-evident truths,” as, for example, the axioms ofEuclidean geometry were formerly thought to be. They are assump-tions taken as a starting point for the development of a chain of de-ductive inferences. All mathematical reasoning is of the form “if . . .then,” where the “if” is followed by a postulate (or some necessaryinference therefrom) and the “then” is followed by a conclusion, or atheorem. Neither the postulates nor the theorems deduced from themare either true or false. All that can be said of them is that if themathematical reasoning has been done correctly, they are related inthe manner of necessary implication.
124 PART TWO: FUNDAMENTAL PATTERNS OF MEANING MATHEMATICAL SYSTEM REQUIRES BASIS OF UNDEFINED TERMS Every mathematical system requires some basis of undefinedterms. This basis, together with the postulates, constitutes what iscalled the “foundation” of the system. In any language undefinedterms are necessary as a basis for defining other terms because theprocess of definition by reference to other terms cannot proceed indef-initely; somewhere it must come to rest in certain primitive terms thatare not themselves defined. From the basis and the postulates, theo-rems are deduced. The entire body of undefined terms, definitions, pos-tulates, and theorems comprises a particular symbolic system, or atheory. From various bases and axioms various theories may be devel-oped such as, theory of groups, theory of numbers, theory ofcontinuous functions, theory of infinite sets, and theory ofcomplex variables, to name only a few. THE MEANING OF THEORY IN MATHEMATICS The meaning of “theory” in mathematics differs somewhat fromits meaning in the empirical sciences, as the analysis in followingchapters will show. In the sciences a theory usually refers to a gen-eral explanation for a group of related facts and generalizations.For example, in physics the behavior of gases is explained by the kinet-ic theory, and the facts of paleontology and comparative anatomymay be explained by the theory of evolution. In mathematics, on theother hand, theory is the whole body of symbolic content of a givenpostulational system. Any mathematical theory can be defined by means of sets. A“set” is simply a class, family, or aggregate of abstract conceptualentities (elements) all of which have some common property or prop-erties specified by the axioms upon which the theory is founded. Twosets, 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 areelements of A. The sum (A + B) of two sets A and B is defined as theset containing all elements 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 Bis defined as the set consisting of all elements in A and not in B. TheCartesian product (A X B) of A and B is defined as the set of allordered pairs of elements in A and B, that is, a set each of whose el-ements consists of a pair of elements one of which is an element of Aand the other an element of B. These ideas of elements, sets, equality, sum, product, and dif-ference comprise basic terms from which all other mathematical con-cepts can be developed, provided certain basic logical concepts are
MATHEMATICS 125also pre-supposed. These essential logical concepts include the fol-lowing: “is a member of,” “not,” “all, or every,” “such that,” “thereexists, or there is,” “if . . . then,” “or,” and “and.” The primitive mate-rials (elements, sets, and their rules of combination) together withthe elemental logical concepts constitute the basis for any mathe-matical theory. THE MEANING OF RELATION IN MATHEMATICS Another concept of far-reaching importance in mathematics isthat of relation. A “relation” is defined simply as a subset of theCartesian product of two sets. It is a means of separating out cer-tain pairs of elements from others. For example, if two sets A and Bhave elements (a1, a2, a3 ) and (b1, b2, b3 ) respectively, the Carte-sian product is the set A X B with elements [(a1, b1 ), (a1, b2 ), (a1,b3 ), (a2, b1 ), (a2 , b2 ), . . . , (b1 , a1 ), (b1 , a2 ), . . .] coveringall possible pair combinations. Any subset of A X B, such as the setcontaining only the two elements [(a1, b2 ), (a3, b1 )], is then a par-ticular relation 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 anotherconcept of great importance in mathematics. A “function” is definedas a relation in which one and only one element in one set correspondsto any element in another set. For example, in the above illustrationthe first of the relations cited is not a function because no element ofB 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 otherhand, the relation [(a1, b2 ), (a1, b3 ), (a2, b2 ), (a3, b1 )] is not afunction because a1 is paired with two different elements of B. Whenthe functional relation works both ways, so that to each element ofA, a unique element of B corresponds and vice versa, the relation iscalled “one-to-one correspondence.” Finally, the concept of binary operation on B by A to C is de-fined by the requirement that to each pair of elements in the Carte-sian product A and B a unique element of C corresponds (i.e., that Cis a function of A X B). SIMILARITY OF MATHEMATICS TO LANGUAGES Returning to the similarity of mathematics to a language, orbetter, to a collection of languages, one can compare the undefinedterms to the elements of sound and meaning upon which any given lan-guage is based. One can compare the various rules of combination(sum, difference, product, relation, function, one-to-one correspon-dence, and binary operation) to the morphological and syntacticrules by which ordinary discourse is organized into an ordered hierar-
126 PART TWO: FUNDAMENTAL PATTERNS OF MEANINGchy of expressions. The above fundamental combinatorial conceptsare the grammar of mathematics. They designate the patterns ac-cording to which the deductive elaboration of any mathematical sys-tem (i.e., the drawing of successive inferences from primitive terms,definitions, and axioms) must proceed.1 WHY THE CONCEPT OF SET IS SO IMPORTANT IN MATHEMATICS The reason why the concept of set is so central in mathematicsis that it embodies the principle of abstraction. Abstraction is theessence of mathematical thinking. A set is specified completely by theproperties of the elements composing it. Those properties are abstrac-tions since they define elements in these terms: “any entity suchthat . . . is an element of the set.” The idea of any such that entailsthat particular things are not under consideration, but only kinds orclasses of things. By means of this idea of abstraction the key math-ematical concept of variable may be understood. A variable does notrefer to something that moves or changes, as it would in ordinaryspeech. In mathematics a variable, designated, say, by the symbol χ,is such that χ stands for, or in the place of, any element of a speci-fied set. Variables are simply ways of representing the general ideaof any or some as contrasted with particular 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 rationalnumber 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 typi-cally mathematical, have hardly been mentioned in the preceding ac-count of mathematical knowledge. The reason is that such concepts(with the possible exception of number, that in some formulations istaken as primitive) are special and derivative in comparison with thevery general and primary concepts used in the above analysis. Forexample, the integers and the counting process can be defined bymeans of the theory of finite sets, and the rational and real numbersby the theory of infinite sets. Furthermore, Euclidean geometry andcommon algebra can be shown to be alternative interpretations of anidentical theory of sets of real numbers (R). Thus a “point” may bedefined as an ordered pair of real numbers, and a “line” as a relationon (i.e., a subset of) the Cartesian product (R X R) of two real num-ber sets. Similarly, the calculus and the theory of functions can beshown 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 thestudy of ordered pairs, combined according to the following rules: (a,1 The above outline of the basic concepts in any mathematical theory largelyfollows the treatment given by R. B. Kershner and L. R. Wilcox, The Anatomyof Mathematics, The Ronald Press Company, New York, 1960 esp. chaps. 4-5.
MATHEMATICS 127b) + (c, d) = [(a + b), (c + d)] and (a, b) (c, d) = [(a • c – b •d),(a •d + b • c)].
128 PART TWO: FUNDAMENTAL PATTERNS OF MEANING At younger ages, most students are not capable of grasping the abstract concepts that deal with higher level math. Even if thestudent recognizes all of the symbols used, their functions must be understood. Knowing that children develop at different stages, how can a teacher deal with a class containing students at many different levels of development?
130 PART TWO: FUNDAMENTAL PATTERNS OF MEANING THE IMPORTANCE OF GENERALITY IN DEVELOPING MATHEMATICAL IDEAS Alfred North Whitehead called form, variable, and generali-ty “a sort of mathematical trinity which preside over the whole sub-ject,” and he added that “they all really spring from the same root,namely from the abstract nature of the science.”2 We have alreadydwelt on the formal nature of mathematical systems and on the con-cept of variable. Some discussion is now needed about generality. Thedevelopment of mathematical ideas is marked by a progressive in-crease in generality. For example, the concept of number beginningwith the positive integers, may be successively generalized to includezero and the negative integers, rational numbers (fractions), irra-tional numbers (like √2), real numbers (having a one-to-one corre-spondence with all the points on a line), complex numbers, vectors (di-rected magnitudes), and infinite (or transfinite) numbers. In geometry the study of two-dimensional manifolds (defined byordered pairs of numbers) can be generalized to three, four, or anyhigher number of dimensions by using ordered triples, quadruples, orgenerally, n-tuples (where n is any integer). Though such hyperspacegeometries cannot be visualized, in the way that two- and three-di-mensional configurations can be, they are nonetheless valid systemsof geometry, which, incidentally, prove to have important applica-tions in the sciences. One of the ways in which generalization takes place in mathe-matics is in connection with the transformation of one set into an-other through what was earlier defined as a binary operation. In anysuch transformation certain relations remain unchanged, or invari-ant. For example, Euclidean geometry is concerned with transforma-tions that leave intervals and angles invariant, i.e., in which figuresmay be translated or rotated but not distorted. A more general ge-ometry (projective geometry) is concerned with transformations(projections) that may alter intervals and angles but leave un-changed a quantity known as the “cross ratio.” The most general ge-ometry (topology) deals with transformations where the connectivitypattern of the subspaces is not changed (e.g., in three-dimensionalspace, where surfaces may be distorted but not cut or punctured). SUMMING UP MATHEMATICS To recapitulate, mathematics is a discipline in which formalsymbolic systems are constructed by positing certain undefined terms(elements, sets, rules of combination), elaborating further conceptsby definitions (conventions), adopting certain postulates (concerningboth the undefined and the defined terms), and then, using the princi-ples of logic, drawing necessary deductive inferences, resulting in anaggregate of propositions called “theorems.” The propositions ofmathematics are formal and abstract in that they do not necessarilyrefer to the structure of the actual world but comprise a series ofpurely abstract formalisms all having in common the one rule of log-ical consistency.2 An Introduction to Mathematics, American rev. ea., Oxford University press,Fair Lawn, N.J., 1948, p. 57.
MATHEMATICS 131 It is well known that mathematics is of great practical valuein science and technology. The nature of the subject is misconstrued ifit is regarded primarily as a “tool” subject. Technical skill in compu-tation and the ability to use mathematics in scientific investigation,valuable as they may be, are not evidence of mathematical under-standing. Mathematical understanding consists in comprehending themethod of complete logical abstraction and of drawing necessaryconclusions from basic formal premises. WAYS OF KNOWING1. Mathematics is not exclusively practical. What does this mean?2. If practical use are not the essences of mathematics, what is?3. How do many students and teachers of mathematics view math?4. Mathematics occupies a world of its own. What does this mean?5. How does mathematics differ from ordinary language?6. Mathematical communities tend to be specialized and limited rather than inclusive like the major ordinary languages com- munities. 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 mathematical 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 mathematics?13. What are some of the similarities of mathematics to languages?14. Why is the concept of set so important in mathematics?15. Generality is important in the development of mathematics. Why?16. How is the nature of mathematics misconstrued?17. What does mathematical understanding actually mean?