Mathematics is such an elusive subject that to devote one hour into regular reading could result in complete intellectual withdrawal. As the renouned (and late) Chinese educationist Sir Ye shengtao once put it simply, one should not try to understand everything in a book, lest it becomes a venue for releasing his exhaustive power, rather than the more profound inspirational resource. When I first read that quote I found it somewhat obvious, almost like an intellectual condescension from the great masters of the old days. But that ultimately justified the deep impression of this seemingly plain quote in my constantly stuffed head (which tend to bulge in the fore these days).
Well the thing is, it’s not so obvious. To the vast majority of largely unmotivated students in society today, the foremost ecucational slogan for them is to try to read between the lines of a book, in a manner that can be dissociated with half-hearted negligence. That is perhaps the most urgent prescription to the current lacking of scholarly aspiration. But the same sort of perfunctoriness could equally well be applied to someone as stuck up as a seasoned research student like myself, who tends to view everything in the domain of knowledge as accessible as the food from the refrigerator. Not the case my good sir. One must work towards such ideal situation, and the working itself is a show of talent in the handling of priority list and time management. It’s a constant race against time as we all have experienced, whatever the field is we are engaged with. But the sheer mental strength of staying cool from a subject in a hot-headed state of existence is far from ploebian. Indeed one needs to acquire an ego of performing artist, to whom the elapse of time factors in as an important, perhaps most important parameter in the output of his synergism. Just as with a good recipe, someone like me who is so eager to learn and open my third eye to the divine edifice of truth, a great textbook or research article can be unfathomably satiating. The sad thing turns out that these books are not as perfectly scribed as some of the more familiar religious texts, whose wording shall never be altered upon occasional convenience or abuse of crafty reasoning by demonable humans. They serve merely as approximation to the truth, so to speak, and can often lead to a misrepresentation thereof or even academic demagoguery (which is of course not as devastating as politcal ones). What’s the cure? As I discovered not long ago, the removal of mysticism or authority of these texts is key. Imagine yourself reading a book that is about to be burned forever, i.e., not a copy of it could be recovered after that irreversible chemical reaction. Then your reaction would be to extract as much relevant information from it as you could within the given time frame, relevant in the sense of what you most wanted to get out of it. Thus a passive pressure of aritificial selection takes place, resulting in profit maximizing learning attitude. But unless you are the emperor of 200 BC China, you are most likely not going to put that into real practice. Instead a more environment friendly procedure must be devised to reach the same capitalistic goal. So the upshot of my long digression is, go look in a mathematical forum where students exchange brief ideas concerning their momentary interest or mental blocks in the pursuit thereof. These are usually coarsely written pre-latex tea time discussion log, but nevertheless contains about the same amount of truth as is usually present in the first one or two chapters of a typical math textbooks. They probably won’t be as efficient in conveying complicated analysis ideas which can often only be visualized through formula chasing; nonetheless they are extremely powerful overviews and save time from going over routine definitions in those texts written by well-respected professors or that spent on looking them up over wikipedia or something if they happen to be written by a burgeoning new star. One often does not remember the detail if his aim is merely to have a generic feel of the subject. Only the divinely favored few like von Neumann, equipped with his marvelous memory, could handle things not only with breadth, but also absolute technical clarity. But even a privileged soul like his couldn’t help whisper "one does not understand mathematics, only gets used to it". Now contrary to most other professions or trades, the best way to get used to some part of mathematics is not to strip it naked of its inner mechanism, but rather to explore its deeper philosophical impact and how each piece of the gear was actually inspired by some lofty intelligence design so to speak. Only by going through such seemingly unaccomplishing ordeals does one secure a true share of the inexhaustible wealth of mathematical knowledge explored by past explorers.