George Boole, Global Hero

[A.Borovik, Talk at the opening of the The Life and Legacy of George Boole exhibition in Lincoln, 16 July 2015.]

I am privileged to take part in this celebration and I am honored to represent the London Mathematical Society.

The LMS was founded 150 years ago by Augustus De Morgan, a colleague and close friend of George Boole, just a year after Boole’s untimely death. The Society continues the work started by mathematicians of George Boole’s circle.

Some people say that mathematicians are remote from everyday life.

George Boole was not.

Here, in Lincoln, he taught at the Mechanics Institute, fought for the improvement of working conditions of shop workers, founded a building society.

His famous book An Investigation of the Laws of Thought was very down-to-earth, it was a textbook of practical thinking. It was written for humans, not for machines—after all, computers remained non-existent for another century.

Let us take a look at his famous definition of the universe of discourse – a concept that you will immediately recognise as obvious, everyone-knows-it kind of things – but which was new, fresh, and perhaps paradoxical in Boole’s time:

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined.

The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself.

But more usually we confine ourselves to a less spacious field.

Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation.

Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse.

In short,

  • The laws of thought are global; but
  • they are applied locally, for example, at a board meeting of a building society.

Please notice George Boole’s words:

we imply (without expressing the limitation)  …

This is his warning against undeclared assumptions that can poison the discourse, his warning against

  • hidden bias,
  • hidden prejudice,
  • hidden phobia,
  • hidden hatred.

Boole’s time was the era of tectonic shifts in technology, in economy, and in social life.

The need for practical logic for everyday use, logic freed from medieval scholasticism, logic accessible to everyman—was in the air of the epoch.

The great contemporary of George Boole, Abraham Lincoln, used in his political writings and speeches the implicit logic of the Euclidean geometry:

One would start with confidence that he could convince any sane child that the simpler propositions of Euclid are true; but, nevertheless, he would fail, utterly, with one who should deny the definitions and axioms.

The principles of Jefferson are the definitions and axioms of free society.

And yet they are denied, and evaded, with no small show of success.
One dashingly calls them `glittering generalities'; another bluntly calls them `self-evident lies'; and still others insidiously argue that they apply only `to superior races’.

From these two quotes, it is hard to avoid the impression that both Boole and Lincoln were thinking in terms of what we now call “human rights”.

It is also difficult to avoid the feeling that for Boole and Lincoln, Logic was the Logic for the Masses; it was

  • Logic for Personal Empowerment,
  • Logic for Social Advancement,
  • Logic for Liberation.

Abraham Lincoln re-used mathematical thinking of classical geometry dated 2 millennia back in time.

But George Boole took an audacious step into the future. He created a new logic and a new mathematical symbolism which supported it.

He extracted the most basic and fundamental laws of thought, so simple that they are now used by computers. Everyone in this room has a mobile phone; in every mobile phone, microchips contain millions of logical gates carrying out millions of Boolean operations per second.

By discovering the algebra of thought – now implemented in computers and electronic devices all around us – Boole changed the course of human civilization.

George Boole is a global hero.

But he wouldn’t become a global hero, if he was not a local hero here – in Lincoln.

His life and work are the best justification of the dictum:

Think globally – act locally!


Acknowledgements. I use this opportunity to say my thanks to everyone involved in setting-up of the two consecutive exhibitions in Lincoln, in the University of Lincoln and in the glorious Lincoln Cathedral. My special thanks go to Ian Slowley, Mark Hocknull, Dave Kenyon, and Eugene Khukhro.

Disclaimer: The author writes in his personal capacity and the views expressed do not necessarily represent position of his employer or any other person, organisation or institution.

Homeschooling in England

An important legal case reported by the BBC on 16 July 2015:

Council drops home education case

Paul Ernest on Douglas Quadling

[See also: Douglas Quadling]

In 1979 I joined the staff at Homerton College, Cambridge as a temporary replacement for Stuart Plunkett on study leave. This was my first job after school teaching as a teacher educator. I worked alongside Richard Light, Tim Rowland, Bob Burn and Hilary Shuard. Alan Bishop at the university department of education (a separate body then) organised a masters course in mathematics education which our students as well as his attended. I also sat in when I could as an introduction to the fledgling science of mathematics education. Douglas Quadling was around, possibly teaching at the 3rd body, the Institute of Education (where Angela Walsh worked too). I remember most vividly the 2 seminars he gave to the masters course. He was a modest but immensely knowledgeable man who described very clearly and with great insight the development of the mathematics curriculum of the previous 50 years or more, and the great growth of textbook schemes in the 60s and 70s, including, most notably, the SMP series. His seminars were deceptively chatty, but rich in content and atmosphere. He was very active in the Mathematical Association. He published many texts and I especially remember his insightful 1969 book, The same, but different : a survey of the notion of equivalence in the context of school mathematics / by D. A. Quadling (published by Bell for MA), an early acknowledgement of a critical notion in school mathematics. I’m sure many others have further deep and affectionate memories of this man and his contribution.



Douglas Quadling

Douglas Quadling, who was one of the four inspirational drivers behind the School Mathematics Project (SMP) in the 1960s and 70s, and a fine mathematician,  schoolmaster, and author, died on Wednesday 25th March 2015.
His funeral is in Emmanuel College Cambridge on Thursday 9 April at 2pm.

Yagmur Denizhan: Response to Comments

Anonymous on 4 January 2015 at 22:49 said in response to my post:

Anon: My comments are on a few themes which appear within the paper. They are stand-alone, selected on the basis of curiosity, and do not necessarily present a coherent over-arching argument.
On Games:
…the winning strategies in such games were typically based on identifying the underlying algorithm instead of being “misled” by the story.
This is very true, and I view it as a result of the human tendency to simplify or reduce puzzles to their essence.

YD: I would prefer to say the “pragmatically relevant essence”… Yet there is also another tendency that is unfortunately systematically suppressed by the system that I am criticising: The tendency to comprehend and delve into the essence of anything/everything.

Continue reading

David Singerman: X + Y the movie

[To appear in the LMS Newsletter]

There are now an increasing number of movies where mathematics plays an important role. Usually we are let down by the parts featuring the maths because the makers of the film have little knowledge about our subject. So it is a real pleasure to review x+y a beautiful film where the mathematics is carefully done but not in a way that will put off a non-mathematical audience. The director is Morgan Matthews who also made the BBC4 documentary Beautiful Young Minds about the Mathematical Olympiad and the film is clearly based on this documentary. This documentary can be seen on Youtube.

The main character is Nathan. From the BBC synopsis

Preferring to hide in the safety of his own private world, Nathan struggles to connect with people, often pushing away those who want to be closest to him, including his mother, Julie. Without the ability to understand love or affection, Nathan finds the comfort and security he needs in numbers and mathematics.

Even though there are similarities between this film and the documentary, the main story line is totally fictitious. Near the beginning, Nathan, who has Asperger’s syndrome, is involved in a car crash which kills his father to whom he was very close. He is then mentored by his maths teacher Martin Humphreys, who when young had taken part in the Mathematics Olympiad. He was diagnosed with multiple sclerosis but also has other problems to do with self worth and soft drugs and ended up being a secondary school teacher.

Humphreys recognizes Nathan’s abilities and persuades him to enter for the Olympiad. He goes to the preliminaries in Taipei.

One of the scenes where there is actual maths is when Nathan is brought to the board to explain how to solve a problem. This involves playing cards which can be face up or face down.

Nathan’s solution is to model this with binary arithmetic involving 0s and 1s and he then turns the problem into an arithmetic one which is easy to solve.

In Taipei he meets Zhang Mei, a girl on the Chinese team. The film concentrates on two relationships. One between Nathan and Zhang Mei and the other between Martin Humphreys and Julie.

The scene moves from Taipei to Cambridge where the Maths Olympiad takes place.

There is real pathos in the final scenes. One where Nathan finally opens himself up to his Mother, and another when Nathan and Zhang Mei while travelling back from Cambridge by train see a rainbow and the viewer feels that their relationship will last. At last, Nathan feels and understands love and affection. Some critics have thought that this ending is too soapy, but if you see the documentary on which this film is based, the rainbow really was there!

One should also mention the excellent cast. Nathan was played by Asa Butterfield, Martin by Ralf Spall, Julie by Sally Hawkins and Zang Mei by Jo Yang. A lovely film where mathematics plays a central role.

Book Review: “What the Best College Teachers Do” by Ken Bain, 2004

Book review by Richard Elwes:

Open a typical book on the theory of pedagogy, and all too often one is confronted by a morass of impenetrable and, one often suspects, unnecessary jargon. So it is a particular pleasure to read Ken Bain’s “What the Best College Teachers Do”. The book is the outcome of a fifteen year study in which Bain and colleagues identified and analysed around a hundred excellent teachers at US Colleges and Universities. Through extensive observations, discussions, and interviews with the teachers and their students, Bain arrives at a range of conclusions regarding the practice of good teaching. His findings are laid bare in a series of straightforwardly entitled chapters: “How do they conduct class?”, “How do they treat their students?”, and so on.

Few of his discoveries come as complete surprises, yet many are genuinely enlightening. For instance, the best teachers “have an unusually keen sense of the histories of their disciplines, including the controversies that have swirled within them, and that understanding seems to help them reflect deeply on the nature of thinking within their fields”.

Many of the insights within this book derive from the removal of extraneous and superficial aspects of education. How do good teachers speak to their students? Obviously, there are countless possible answers. But what do these approaches have in common? “Perhaps the most significant skill the teachers in our study displayed in the classroom… was the ability to communicate orally in ways that stimulated thought.”

The author often allows his educators to speak for themselves, and as one might expect, they are a thoughtful and frequently amusing group. Thus we read the Harvard political theorist Michael Sandel opining that teaching is “above all… about commanding attention and holding it… Our task… is not unlike that of a commercial for a soft drink”. On the other hand, Jeanette Norden, professor of cell biology at Vanderbilt University, “told us that before she begins the first class in any semester, she thinks about the awe and excitement she felt the first time anyone explained the brain to her, and she considers how she can help her students achieve that same feeling.”

The teachers analysed come from a wide range of Colleges and academic disciplines; some teach only elite students, others specialise in assisting strugglers; while several are eminent researchers, a few have no research publications at all; they deploy a variety of educational techniques. Among this diversity, the conclusions that Bain avoids are as interesting as those he draws. “[P]ersonality played little or no role in successful teaching. We encountered both the bashful and the bold, the restrained and the histrionic…. We found no pattern in instructors’ sartorial habits, or in what students and professors called each other. In some classrooms first names were common; in others, only titles and surnames prevailed.”

All the same, some common traits are apparent. “Exceptional teachers treat their lectures… and other elements of teaching as serious intellectual endeavors, as intellectually demanding and important as their research and scholarship.”

Particularly important, Bain argues, is the fostering of a “natural critical learning environment”. This is the closest the book ever comes to jargon, but that judgement would be unfair: “‘natural’ because students encounter the skills, habits, attitudes, and information they are trying to learn embedded in questions and tasks they find fascinating… ‘critical’ because students learn to… reason from evidence, to examine the quality of their reasoning… and to ask probing and insightful questions about the thinking of other people”.

At this stage, the reader might worry that this catalogue of heroic deeds could be dispiriting to the rest of us. Not so. Whilst Bain is full of admiration for his teachers, he by no means deifies them. “Even the best teachers have bad days… they are not immune to frustrations, lapses in judgement, worry, or failure.” On the contrary, their ability to confront their own shortcomings is one thing which sets the best teachers apart from those others who “never saw any problems with their own teaching, or they believed they could do little to correct deficiencies”. Good teachers show humility and willingness to improve.

In comparison, the teachers identified as the “worst” by their students often appear to carry the attitude, as one of Bain’s subjects puts it, that only “smart men can possibly comprehend this material and that if you can’t understand what I’m saying, that must mean I’m a lot smarter than you are”. As the biologist Craig Nelson says “The trouble with most of us… is that we teach like we were god.” Contrast this to the view of Dudley Herrschbach, another of the teachers in the study (as well as being a Nobel Prize-winning Chemist) that “You have to be confused… before you can reach a new level of understanding anything.”

In summary, this short book is far more readable and entertaining than a text on educational theory has any right to be. It offers every Higher Education teacher an invaluable opportunity: to learn from the best.

Ken Bain, What the Best College Teachers Do, Harvard University Press, 2004. ISBN-10: 0674013255. ISBN-13: 978-0674013254.