Glasersfeld E. von (1992) Philosophy of mathematics (Review of Paul Ernest). Zentralblatt für Didaktik der Mathematik 24(2): 46. Available at http://cepa.info/1438

Review of Paul Ernest: The Philosophy of Mathematics Education. London/New York/Philadelphia: The Falmers Press, 1991. XIV + 329 p. – ISBN 1-85000-667-9 (pbk).

Thanks to Euclid and Plato, mathematics teachers have for a long time enjoyed the unique luxury of teaching what everyone considered to be unquestionable. Not only was the subject matter of mathematics held to be “true” for all time, but it was also held to be irreplaceable. Teachers of other subjects had to be ready to go with the times, and not infrequently this meant that they had to change some very basic ideas. In contrast, mathematical knowledge was thought to be eternal – it could grow, but once acquired it was indubitable.

Inevitably, this tradition generated a certain pride – some might call it complacency – and the educational establishment was, and still is, reluctant to relinquish its olympian attitude in the face of criticism. Serious doubt, however, was initiated when it was shown, in the middle of the last century, that Euclidean geometry was not unique and could be accompanied, if not substituted, by equally powerful systems built on radically different premises. The subversion spread and has by now affected practically all branches of mathematics. If education is to be an honest business, it can no longer trivialize the upheaval as the appearance of negligible oddities.

In the first chapter of Part I of the book, Paul Ernest provides a complete summary of the diverse theoretical developments that have led to the demise of “certainty” in the main areas of mathematical knowledge. With the help of well-chosen quotations from the leaders in the quest for firm “foundations of mathematics”, he shows some of the drama and the disappointments in the turbulent struggle that began when Bertrand Russell wrote his fateful letter to Gottlob Frege, showing that the conceptual edifice this author had built up in his monumental work, contained a paradox.

Next, the author develops the position he calls “social constructivism” as a “descriptive as opposed to a prescriptive philosophy of mathematics”. Sharing Lakatos’ view that mathematical knowledge is neither a priori nor wholly experiential, but must be considered “quasiempirical”, he characterizes social construction on three grounds (p.42):

Math. knowledge is linguistic knowledge, conventions and rules;Subjective knowledge, after publication, becomes accepted and objective;Objectivity is understood to be social.

In the rest of Part I (chapters 3,4,5), Ernest substantiates this view in detailed discussions of the function of language, the generation of subjective knowledge, its conversion to “objective” knowledge through social interaction, and the delicate relation between psychology and the philosophy of mathematics.

The meticulous presentation and the many apt quotations drawn from the most reputable pertinent sources will make his arguments difficult to reject for all those who tend to react negatively to any attempt to modify the status quo in the field of mathematics. If they are not swayed, they will at least be stimulated to reconsider their positions. For radical constructivists, there is a welcome expansion of arguments with which they are very much in sympathy, and the attempt to specify mechanisms of social interaction that are pertinent to the construction of logical and mathematical knowledge is of great interest. From the radical point of view, of course, the locus of knowledge can be nowhere but in the individual knower. Hence expressions such as “shared ideas” and the path “from subjective knowledge ... via publication to objective knowledge” (p.4243) have to be taken metaphorically until a further analysis of the mechanisms involved render them plausible as parts of the constructivist model.

In one respect the author shows what, to me, is unwarranted timidity. On p.45, he says: “Thus there is the danger of social constructivism straying into the provinces of history, sociology or psychology.” This seems an oddly defensive statement to me. Once one specifies that mathematical knowledge will be described “as a social construction”, that its basis is “linguistic knowledge, conventions and rules” (which themselves are a “social construction”), that “objectivity itself will be understood to be social”, and that “objective knowledge is internalized and reconstructed by individuals”, one can no longer stray into the domains of sociology and psychology, because one is already in them. And to my mind, this does not require an excuse. The traditional philosophy of mathematics and, consequently, the teaching of mathematics have forever presented their subject as though it existed ready-made in some foreign, non-human space towards which students had to grope their way. We are much closer today to the realization that mathematics, before it becomes a social property, must be produced by a mind – and the way human minds produce things is (or should be) the domain of psychology.

Part II forms the remaining two thirds of the book and begins with a description of “ideological” positions (chapters 6,7,8,9). These positions then serve as a scaffolding in the discussion of educators’ diverse views on epistemology, the philosophy of mathematics, theories of the child and of society, of learning and teaching, mathematical ability, aims of mathematics education, and other sub-topics. Descriptions of ideologies are always in danger of slipping into the idiosyncratic, but Ernest is careful to cite and quote a great number of other authors (the book contains a full 20 pages of references!), which makes these chapters a most useful compendium of educational ideas and attitudes.

There is the refreshingly candid statement:

“Education is an intentional activity, and the statement of the underlying intentions constitutes the aims of education. ... Different social groups have different educational aims relating to their underlying ideologies and interests.” (p.123).

In other words, education is a political activity, and the author covers the actual and possible connections between social politics and mathematics all the way to the feminist perspective.

For a social constructivist, this is as it should be and it provides an wide-ranging overview of the present situation as well as some of its historical roots.

From my more radical constructivist point of view, the book is an excellent survey of the philosophies underlying educational practice and makes a solid case against the absolutist notions that in many places and in many ways still govern mathematics education. On the other hand, it may be the very focus on the social that leads to the neglect of what, in my view, is an important aspect of mathematics: the satisfaction mathematical operating yields to the individual thinking mind. However, this book should do much to change attitudes towards the teaching of mathematics, and once the field is cleared, I have no doubt that teachers will find ways and means of showing students that, beyond practical and social necessity, there is personal pleasure, if not fun, to be found by learning to manage numbers.

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