The Philosophy of Mathematics Education

Paul Ernest

• Publisher: Routledge
• Year: 2002
• ISBN:
  9781135387532
• Edition: 1
• Topic: Education
• Subtopic: Education General

Although many agree that all teaching rests on a theory of knowledge, there has been no in-depth exploration of the implications of the philosophy of mathematics for education. This is Paul Ernest's aim. Building on the work of Lakatos and Wittgenstein it challenges the prevalent notion that mathematical knowledge is certain, absolute and neutral, and offers instead an account of mathematics as a social construction. This has profound educational implications for social issues, including gender, race and multiculturalism; for pedagogy, including investigations and problem solving; and challenges hierarchical views of mathematics, learning and ability. Beyond this, the book offers a well-grounded model of five educational ideologies, each with its own epistemology, values, aims and social group of adherents. An analysis of the impact of these groups on the National Curriculum results in a powerful critique, revealing the questionable assumptions, values and interests upon which it rests. The book finishes on an optimistic note, arguing that pedagogy, left unspecified by the National Curriculum, is the way to achieve the radical aims of educating confident problem posers and solvers who are able to critically evaluate the social uses of mathematics.

PART 1
The Philosophy of Mathematics

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1

A Critique of Absolutist Philosophies of Mathematics

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1. Introduction

The main purpose of this chapter is to expound and criticize the dominant epistemological perspective of mathematics. This is the absolutist view that mathematical truth is absolutely certain, that mathematics is the one and perhaps the only realm of certain, unquestionable and objective knowledge. This is to be contrasted with the opposing fallibilist view that mathematical truth is corrigible, and can never be regarded as being above revision and correction.

Much is made of the absolutist-fallibilist distinction because, as is shown subsequently, the choice of which of these two philosophical perspectives is adopted is perhaps the most important epistemological factor underlying the teaching of mathematics.

2. The Philosophy of Mathematics

The philosophy of mathematics is the branch of philosophy whose task is to reflect on, and account for the nature of mathematics. This is a special case of the task of epistemology which is to account for human knowledge in general. The philosophy of mathematics addresses such questions as: What is the basis for mathematical knowledge? What is the nature of mathematical truth? What characterises the truths of mathematics? What is the justification for their assertion? Why are the truths of mathematics necessary truths?

A widely adopted approach to epistemology, is to assume that knowledge in any field is represented by a set of propositions, together with a set of procedures for verifying them, or providing a warrant for their assertion. On this basis, mathematical knowledge consists of a set of propositions together with their proofs. Since mathematical proofs are based on reason alone, without recourse to empirical data, mathematical knowledge is understood to be the most certain of all knowledge. Traditionally the philosophy of mathematics has seen its task as providing a foundation for the certainty of mathematical knowledge. That is, providing a system into which mathematical knowledge can be cast to systematically establish its truth. This depends on an assumption, which is widely adopted, implicitly if not explicitly.

Assumption

The role of the philosophy of mathematics is to provide a systematic and absolutely secure foundation for mathematical knowledge, that is for mathematical truth.¹

This assumption is the basis of foundationism, the doctrine that the function of the philosophy of mathematics is to provide certain foundations for mathematical knowledge. Foundationism is bound up with the absolutist view of mathematical knowledge, for it regards the task of justifying this view to be central to the philosophy of mathematics.

3. The Nature of Mathematical Knowledge

Traditionally, mathematics has been viewed as the paradigm of certain knowledge. Euclid erected a magnificent logical structure nearly 2,500 years ago in his Elements, which until the end of the nineteenth century was taken as the paradigm for establishing truth and certainty. Newton used the form of the Elements in his Principia, and Spinoza in his Ethics, to strengthen their claims to systematically expound truth. Thus mathematics has long been taken as the source of the most certain knowledge known to humankind.

Before inquiring into the nature of mathematical knowledge, it is first necessary to consider the nature of knowledge in general. Thus we begin by asking, what is knowledge? The question of what constitutes knowledge lies at the heart of philosophy, and mathematical knowledge plays a special part. The standard philosophical answer to this question is that knowledge is justified belief. More precisely, that propositional knowledge consists of propositions which are accepted (i.e., believed), provided there are adequate grounds available for asserting them (Sheffler, 1965; Chisholm, 1966; Woozley, 1949).

Knowledge is classified on the basis of the grounds for its assertion. A priori knowledge consists of propositions which are asserted on the basis of reason alone, without recourse to observations of the world. Here reason consists of the use of deductive logic and the meanings of terms, typically to be found in definitions. In contrast, empirical or a posteriori knowledge consists of propositions asserted on the basis of experience, that is, based on the observations of the world (Woozley, 1949).

Mathematical knowledge is classified as a priori knowledge, since it consists of propositions asserted on the basis of reason alone. Reason includes deductive logic and definitions which are used, in conjunction with an assumed set of mathematical axioms or postulates, as a basis from which to infer mathematical knowledge. Thus the foundation of mathematical knowledge, that is the grounds for asserting the truth of mathematical propositions, consists of deductive proof.

The proof of a mathematical proposition is a finite sequence of statements ending in the proposition, which satisfies the following property. Each statement is an axiom drawn from a previously stipulated set of axioms, or is derived by a rule of inference from one or more statements occurring earlier in the sequence. The term ‘set of axioms’ is conceived broadly, to include whatever statements are admitted into a proof without demonstration, including axioms, postulates and definitions.

An example is provided by the following proof of the statement ‘1 + 1= 2’ in the axiomatic system of Peano Arithmetic. For this proof we need the definitions and axioms s0 = 1, s1 = 2, x + 0 = x, x + sy = s(x + y) from Peano Arithmetic, and the logical rules of inference P(r), r = t ⇒ P(t); P(v) ⇒ P(c) (where r, t; v; c; and P(t) range over terms; variables; constants; and propositions in the term t, respectively, and ‘ ⇒’ signifies logical implication).² The following is a proof of 1 + 1= 2: x+sy=s(x+y), 1+sy=s(1+y), 1+s0=s(1+0), x+0=x, 1+0=1, 1+s0=s1, s0 =1, 1+1= s1, sl— 2, 1+1=2.

An explanation of this proof is as follows. s0 =1 [D1] and s1= 2 [D2] are definitions of the constants 1 and 2, respectively, in Peano Arithmetic. x + 0 = x [A1] and x + sy = s(x + [A2] are axioms of Peano Arithmetic. P(r), r = t ⇒P(t) [R1] and P(v) ⇒ P(c) [R2], with the symbols as explained above, are logical rules of inference. The justification of the proof, statement by statement as shown in Table 1.1.

Table 1.1 : Proof of 1 + 1= 2 with justification

Step	Statement	Justification of Statement
S1	x+sy=s(x+y)	A2
S2	1 + sy = s(1 + y)	R2 applied to S1, using v= x, c = 1
S3	1 + s0= s(1+0)	R2 applied to S2, using v= y, c = 0
S4	x+0=x	A1
S5	1 + 0= 1	R2 applied to S4, using v= x, c = 1
S6	1 + s0= s1	R1 applied to S3 and S5, using r=1 + 0, t = 1
S7	s0=1	D1
S8	1 + 1 = s1	R1 applied to S6 & S7, using r = s0, t= 1
S9	s1 =2	D2
S10	1+ 1= 2	R1 applied to S8 & S9, using r= s1, t= 2

This proof establishes ‘1 + 1= 2’ as an item of mathematical knowledge or truth, according to the previous analysis, since the deductive proof provides a legitimate warrant for asserting the statement. Furthermore it is a priori knowledge, since it is asserted on the basis of reason alone.

However, what has not been made clear are the grounds for the assumptions made in the proof. The assumptions made are of two types: mathematical and logical assumptions. The mathematical assumptions used are the definitions (Dl and D2) and the axioms (A1 and A2). The logical assumptions are the rules of inference used (R1 and R2), which are part of the underlying proof theory, and the underlying syntax of the formal language.

We consider first the mathematical assumptions. The definitions, being explicit definitions, are unproblematic, since they are eliminable in principle. Every occurrence of the defined terms 1 and 2 can be replaced by what it abbreviates (s0 and ss0, respectively). The result of eliminating these definitions is the abbreviated proof: x+sy=s(x+y), s0+sy=s(s0+y), s0+s0=s(s0+0), x+0=x, s0+0=s0, s0 + s0 = ss0; proving s0 + s0 = ss0’, which represents ‘1 + 1= 2’. Although explicit definitions are eliminable in principle, it remains an undoubted convenience, not to mention an aid to thought, to retain them. However, in the present context we are concerned to reduce assumptions to their minimum, to reveal the irreducible assumptions on which mathematical knowledge and its justification rests.

If the definitions had not been explicit, such as in Peano’s original inductive definition of addition (Heijenoort, 1967), which is assumed above as an axiom, and not as a definition, then the definitions would not be eliminable in principle. In this case the problem of the basis of a definition, that is the assumption on which it rests, is analogous to that of an axiom.

The axioms in the proof are not eliminable. They must be assumed either as self-evident axiomatic truths, or simply retain the status of unjustified, tentative assumptions, adopted to permit the development of the mathematical theory under consideration. We will return to this point.

The logical assumptions, that is the rules of inference (part of the overall proof theory) and the logical syntax, are assumed as part of the underlying logic, and are part of the mechanism needed for the application of reason. Thus logic is assumed as an unproblematic foundation for the justification of knowledge.

In summary, the elementary mathematical truth ‘1 + 1= 2’, depends for its justification on a mathematical proof. This in turn depends on assuming a number of basic mathematical statements (axioms), as well as on the underlying logic. In general, mathematical knowledge consists of statements justified by proofs, which depend on mathematical axioms (and an underlying logic).

This account of mathematical knowledge is essentially that which has been accepted for almost 2,500 years. Early presentations of mathematical knowledge, such as Euclid’s Elements, differ from the above account only by degree. In Euclid, as above, mathematical knowledge is established by the logical deduction of theorems from axioms and postulates (which we include among the axioms). The underlying logic is left unspecified (other than the statement of some axioms concerning the equality relation). The axioms are not regarded as temporarily adopted assumptions, held only for the construction of the theory under consideration. The axioms are considered to be basic truths which needed no justification, beyond their own self evidence (Blanche, 1966).³ Because of this, the account claims to provide certain grounds for mathematical knowledge. For since logical proof preserves truth and the assumed axioms are self-evident truths, then any theorems derived from them must also be truths (this reasoning is implicit, not explicit in Euclid). However, this claim is no longer accepted because Euclid’s axioms and postulates are not considered to be basic and incontrovertible truths, none of which can be negated or denied without resulting in contradiction. In fact, the denial of some of them, most notably the Parallel Postulate, merely leads to other bodies of geometric knowledge (non-euclidean

Beyond Euclid, modern mathematical knowledge includes many branches which depend on the assumption of sets of axioms which cannot be claimed to be basic universal truths, for example, the axioms of group theory, or of set theory (Maddy, 1984).

4. The Absolutist View of Mathematical Knowledge

The absolutist view of mathematical knowledge is that it consists of certain and unchallengeable truths. According to this view, mathematical knowledge is made up of absolute truths, and represents the unique realm of certain knowledge, apart from logic and statements true by virtue of the meanings of terms, such as ‘All bachelors are unmarried’.

Many philosophers, both modern and traditional, hold absolutist views of mathematical knowledge. Thus according to Hempel:

the validity of mathematics derives from the stipulations which determine the meaning of the mathematical concepts, and that the propositions of mathematics are therefore essentially ‘true by definition’.

(Feigl and Sellars, 1949, page 225)

Another proponent of the certainty of mathematics is A. J. Ayer who claims the following.

Whereas a scientific generalisation is readily admitted to be fallible, the truths of mathematics and logic appear to everyone to be necessary and certain.

The truths of logic and mathematics are analytic propositions or tautologies.

The certainty of a priori propositions depends on the fact that they are tautologies. A proposition [is] a tautology if it is analytic. A proposition is analytic if it is true solely in the virtue of the meaning of its consistituent symbols, and cannot therefore be either confirmed or refuted by any fact of experience.

(Ayer, 1946, pages 72, 77 and 16, respectively).

The deductive method provides the warrant for the assertion of mathematical knowledge. The grounds for claiming that mathematics (and logic) provide absolutely certain knowledge, that is truth, are therefore as follows. First of all, the basic statements used in proofs are taken to be true. Mathematical axioms are assumed to be true, for the purposes of developing that system under consideration, mathematical definitions are true by fiat, and logical axioms are accepted as true. Secondly, the logical rules of inference preserve truth, that is they allow nothing but truths to be deduced from truths. On the basis of these two facts, every statement in a deductive proof, including its conclusion, is true. Thus, since mathematical theorems are all established by means of deductive proofs, they are all certain truths. This constitutes the basis of the claim of many philosophers that mathematical truths are certain truths.

This absolutist view of mathematical knowledge is based on two types of assumptions: those of mathematics, concerning the assumption of axioms and definitions, and those of logic concerning the assumption of axioms, rules of inference and the formal language and its syntax. These are local or micro-assumptions. There is also the possibility of global or macro-assumptions, such as whether logical deduction suffices to establish all mathematical truths. I shall subsequently argue that each of these assumptions weakens the claim of certainty for mathematical knowledge.

The absolutist view of mathematical knowledge encountered problems at the beginning of the twentieth century when a number of antinomies and contradictions were derived in mathematics (Kli...