Susan M. C. Mac Donald: Euclid Transmogrified

Euclid Transmogrified: A National Scandal

The listed axioms do not produce plane geometry and the listed theorems cannot be proved from them; thus they constitute a bogus axiomatic system. As a consequence, the draft [subsequently introduced] course in geometry is substandard, incompetent mathematics and the quality is so badly defective that it constitutes a national scandal. It is all the more deplorable since the [Department of Education] Syllabus Committee has produced this after receiving [a] submission [from university mathematicians] detailing similar defects in the present syllabus. The members of the Syllabus Committee have evidently encountered strands of three quite separate treatments of plane geometry due to Euclid (c.300 BC), G. D. Birkhoff (1932) and G. Papy (1967), but what they have produced is as little a course in geometry as combining the second act of Hamlet, the first of Julius Caesar and the second of The Merchant of Venice, would result in a play.

— Report of a Royal Irish Academy commission of professors of mathematics

Preface (extract)

This book has been written to tell the extraordinary story of how in the mid twentieth century, in a well-developed European country, a mathematically solid system of geometry was altered to such an extent that more than one million students were taught foundationally flawed geometry over more than a quarter of a century. The almost unbelievable situation came about in Ireland in the wake of early postcolonial developments that gave rise to an impressionable and unqualified civil service in a position of power over a nation’s educational development. This is a multifaceted tale of a perfect storm within mathematics development in Ireland whose result was so destructive that its ramifications continue to be felt into the twenty-first century, forty years after a report of a Royal Irish Academy commission of professors of mathematics described the matter as a national scandal. That this should happen in Ireland is all the more extraordinary given that Ireland has produced mathematicians of the calibre of Berkeley, Casey, Hamilton and Salmon, all graduates of Trinity College Dublin with distinguished academic careers in mathematics (with Casey writing an internationally acclaimed and enduring sequel to the first six books of Euclid’s Elements), and, indeed, Oliver Byrne, the finest modern-day illustrator of the original Elements of Euclid of c.300 BCE.

The events leading to this mathematical fiasco occurred in two phases, here termed Phase One and Phase Two, and involved three sequential mathematics syllabuses, here termed Syllabus I, Syllabus II and Syllabus III.

The geometry of Syllabus I, the Intermediate Certificate mathematics syllabus introduced in 1939, differed almost inconsequentially from that of all previous geometry syllabuses in Ireland (and, indeed, the world) and from that of Euclid more than 2,000 years earlier (despite logical imperfections and the discovery of other geometries in the nineteenth century). A combination of political, economic and social reforms, together with international developments in propaedeutical mathematics in the mid twentieth century, led to major changes to this near status quo.

Phase One of Ireland’s reaction to these revolutionary international developments in mathematics, du jour in the late 1950s and 1960s, brought about the geometry of Syllabus II, the Intermediate Certificate mathematics syllabus introduced in 1966. This materialized following a series of OECD conferences to reform mathematics that were attended by a small number of Irish civil servants, schoolteachers and academics. The conferences and their resultant publications were dominated by Belgian mathematician Georges Papy, an ardent promoter of his own mathematics. Many who attended Papy’s lectures in Ireland and abroad fell under his spell. The result was the geometry of Syllabus II, a mathematically flawed hybrid of the geometry of Euclid and the set-based transformation geometry promoted at the conferences and in Papy’s lectures. Then, for six years, the civil service inspectorate progressively modified Syllabus II examination questions so that the amount of transformation geometry and mathematical incompatibility increased.

Phase Two of Ireland’s reaction to continuing international developments in propaedeutical mathematics led to Syllabus III, the Intermediate Certificate mathematics syllabus introduced in 1973 (last examined in 1989). This far more damaging syllabus, having undeniably sui generis development, content and form, emerged from Ireland’s singular brand of mathematical revolution. Astonishingly, the civil service inspectorate sidelined relevant academic parties and created its own mathematics, seemingly without any awareness that, in general, original mathematics is unlikely to be produced by amateurs. The resulting geometry was a mathematically illogical, foundationally flawed, irreconcilable composite formed from the works of the inspectorate, Papy and Euclid. This could happen only because the civil service inspectorate had supreme control over the nation’s education: not even logical argument, made by academics for many years, could overpower it. Moreover, much of Syllabus III remained in subsequent syllabuses so that the total number of students affected came to be more than two million.

This book is in four parts: Setting, Phase One, Phase Two and Aftermath. Setting gives an account of the national position and the international position in preparation for the substance of the book, Phase One and Phase Two. Aftermath consists of an account of the overall outcome and the eventual denouement. A set of appendices, including a chronology, and a glossary conclude the book.

A case as unparalleled as this warrants a record: this book attempts to supply one.

Contents (extract)

SETTING

1. Two Phases, Three Syllabuses

2. The Endurance of the Geometry of Euclid

3. The Place of Geometry in Mathematics Education in Ireland up to the Time of Syllabus I

4. Syllabus I, Geometry in the Intermediate Certificate Mathematics Syllabus to 1966

4.1 Syllabus I

4.2 The Geometry Contained in Textbooks Associated with Syllabus I

4.3 The Examination of the Geometry of Syllabus I

4.4 Survey of the Academic Year 1959/1960

5. The International Drive to Revolutionize Second-Level Geometry

5.1 Mathematical Developments in the United States: The Evolution of New Maths

5.2 Mathematical Developments in Continental Europe: The Impact of University Mathematicians

5.3 Mathematical Developments in the United Kingdom: The Influence of Schoolteachers

5.4 Axiomatic Systems after Euclid

5.5 Euclid Reconditioned

5.6 Euclid Replaced

5.7 Theoretical Second-Level Alternatives to Euclid

5.8 New Axiom Systems for Geometry in Second-Level Textbooks

5.9 Practical Second-Level alternatives to Euclid

PHASE ONE

Ireland Reacts: The Re-Formation of Second-Level Geometry in Ireland

6. The Background to Syllabus II: Educational and Mathematical Developments in Ireland, 1959–1969

6.1 General Developments in Education in Ireland, 1959–1969

6.2 Developments in Irish Second-Level Mathematics Education Preceding the Introduction of Syllabus II

6.3 The Development of Syllabus II

7. The Geometry of Syllabus II, the Intermediate Certificate Mathematics Syllabus Introduced in 1966

7.1 The Geometry of Syllabus II

7.2 Study of the Geometry of Syllabus II

7.3 Study of the Examination of the Geometry of Syllabus II

7.4 Reaction to the Geometry of Syllabus II

PHASE TWO

Ireland Reacts Again: The Further Re-Formation of Second-Level Geometry in Ireland

8. The Background To Syllabus III: Educational and Mathematical Developments in Ireland, 1969–1973

8.1 General Developments in Education in Ireland, 1969–1973

8.2 Developments in Irish Second-Level Mathematics Education Preceding the Introduction of Syllabus III

8.3 The Development of Syllabus III

9. The Geometry of Syllabus III, the Intermediate Certificate Mathematics Syllabus Introduced in 1973

9.1 The Geometry of Syllabus III

9.2 Study of the Geometry of Syllabus III

9.3 Study of the Examination of the Geometry of Syllabus III

9.4 Reaction to the Geometry of Syllabus III

AFTERMATH

10. Aftermath

10.1 Influence in Ireland

10.2 Broad Impact

10.3 Enduring Impact

10.4 In Closing