Very few occasions can surpass the delight and excitement of reading the thoughts of a great thinker and finding in them an echo of your own thoughts and feelings. Something similar happened to me while I was reading Bertrand Russell’s essay The Study of Mathematics.
In this essay Russell has presented his views on how mathematics should be taught and what end it is to serve.
He starts by saying:
In regard to every form of human activity it is necessary that the question should be asked from time to time, What is its purpose and ideal? In what way does it contribute to the beauty of human existence?
He then goes on to comment on the methods of mathematical instruction. In many ways, the following passage is a retelling of countless stories of frustration – individual or collective:
Dry pedants possess themselves of the privilege of instilling this knowledge: they forget that it is to serve but as a key to open the doors of the temple; though they spend their lives on the steps leading up to those sacred doors, they turn their backs upon the temple so resolutely that its very existence is forgotten, and the eager youth, who would press forward to be initiated to its domes and arches, is bidden to turn back and count the steps.
When one reads this essay one should have in mind a great thinker and logician who himself has worked on the foundational questions of Mathematics. Russell has a very personal, but informed view of the dark and deep dungeons of mathematical rigour (or a lack thereof). Russell’s work on the Foundations of Mathematics is very well known. It is in this context that we must read the following passage.
Mathematics, rightly viewed, possesses not only truth, but supreme beauty–a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
Later in the essay, he observes that the contemporary method of instruction fails to achieve one very important goal – belief in the power of reason.
One of the chief ends served by mathematics, when rightly taught, is to awaken the learner’s belief in reason, his confidence in the truth of what has been demonstrated, and in the value of demonstration. This purpose is not served by existing instruction; but it is easy to see ways in which it might be served.
What follows is an all too familiar example. At this point we must remind ourselves that this essay was written in 1902 and the very fact that we still find this example relatable is shameful and profoundly disturbing.
At present, in what concerns arithmetic, the boy or girl is given a set of rules, which present themselves as neither true nor false, but as merely the will of the teacher, the way in which, for some unfathomable reason, the teacher prefers to have the game played.
Next paragraph can be considered a summary of all mathematical education in most classrooms even today.Russell continues:
In the beginning of algebra, even the most intelligent child finds, as a rule, very great difficulty. The use of letters is a mystery, which seems to have no purpose except mystification. It is almost impossible, at first, not to think that every letter stands for some particular number, if only the teacher would reveal _what_ number it stands for. The fact is, that in algebra the mind is first taught to consider general truths, truths which are not asserted to hold only of this or that particular thing, but of any one of a whole group of things.[…] But how little, as a rule, is the teacher of algebra able to explain the chasm which divides it from arithmetic, and how little is the learner assisted in his groping efforts at comprehension! Usually the method that has been adopted in arithmetic is continued: rules are set forth, with no adequate explanation of their grounds; the pupil learns to use the rules blindly, and presently, when he is able to obtain the answer that the teacher desires, he feels that he has mastered the difficulties of the subject. But of inner comprehension of the processes employed he has probably acquired almost nothing.
Following this paragraph Russell deals with some of the ‘technical issues’ in the Philosophy of Mathematics, namely the doctrine of infinity. He mentions Cantor’s work which at last established the concept of infinity on a logically coherent and secure footing. He also mentions Symbolic Logic and how it relates to mathematics.
The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole. […] Until symbolic logic had acquired its present development, the principles upon which mathematics depends were always supposed to be philosophical, and discoverable only by the uncertain, unprogressive methods hitherto employed by philosophers. So long as this was thought, mathematics seemed to be not autonomous, but dependent upon a study which had quite other methods than its own.
Russell concludes the essay with a very important reminder:
Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of mathematics.
This essay, I think, is a must read for all teachers and students. I’d love to have it as first reading assignment in my next Mathematics course.
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