Thoughts about Thinking

‘The average student is unable to study from a book, unless the book is dealt out to him in small sections. In order to become proficient in mathematics, or in any other subject, he must realize that most topics involve only a small number of basic ideas, which, once grasped, give easy access to the mass of details with which they are inevitably surrounded. Reading a book, or a paper of any length, should not mean crawling along its outer circumference, but, by whatever method one finds best suited to his own temperament, aiming straight at the centre, from which the clearest view may be found of the whole panorama.’

André Weil, The Mathematics Curriculum, Collected Papers vol II.

‘It may sometimes happen that a truth, an insight, which you have slowly and laboriously puzzled out by thinking for yourself could easily have been found already written in a book; but it is a hundred times more valuable if you have arrived at it by thinking for yourself. For only then will it enter your thought-system as an integral part and living member, be perfectly and firmly consistent with it and in accord with all its other consequences and conclusions, bear the hue, colour and stamp of your whole manner of thinking, and have arrived at just the moment it was needed; it will thus stand firmly and forever lodged in your mind. ‘

‘For the man who thinks for himself becomes acquainted with the authorities on his opinions only after he has acquired them and merely as confirmation of them, while the book-philosopher starts with his authorities, in that he constructs his opinions by collecting together the opinions of others: his mind then compares with that of the former as an automaton compares with a living man.’

Arthur Schopenhauer, On Thinking for Yourself, translated by J. R. Hollingdale.

‘I never am really satisfied that I understand anything; because, understand it well as I may, my comprehension can only be an infinitesimal fraction of all I want to understand about the many connections and relations which occur to me, how the matter in question was first thought of or arrived at, etc.’

Ada Lovelace (apparently as quoted in Harry Henderson’s Modern Mathematicians).


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