Is it not effective to learn math top-down?

By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail’s pace, looking up definitions and learning just what you need and occasionally proving basic results. Eventually you’ll get there but is this a bad idea? Is learning each required math area by textbook the better way?

Answer

I think this sort of question, and pursuant discussion/answers/comments, can be very useful for people who’re new in the math biz. Namely, in my considered opinion, neither a “strict” bottom-up approach, nor “strict” top-down approach is optimal (except, in both cases, for a few extreme personality types). This is plausible on general principles, since, after all, the smart money says we should do a good bit of both, as in “hedging”. And this is true, and for more than those general principles, in my opinion.

The way that “do both” is the only sensible route would seem to be that the extremes have lent themselves to highly stylized, almost caricatured, and editorially-pressured extremes. Research papers very often are written in the first place not to inform and help beginners, but to impress experts, etc. Sometimes journals’ editorial pressures push in this direction. Peoples’ understandable professional insecurities push in this direction. Experts’ boredom with beginners’ issues helps drift. And, at the other end, publish-for-profit situations push … for profitability, which in real “textbook markets” means that “new” textbooks will mostly resemble old ones. We are fortunate that some people (Joe Silverman, as Matt E. noted) manage to move things forward within that strangely constrained milieu.

An important further point, in my experience, is exactly that of the question’s “follow the branching graph of references backward to reach the ground…” ‘s hidden, unknown fallacy. That is, (having tried this in all good faith, very many times in my life) peculiar conclusions are reached when/if one does not give up, but pursues things to their actual ends. Namely, some significant fraction of the time, no one ever proved the thing that gradually was back-attributed to them… though it is true, and by now people have figured out how to prove such things. Another is that the “standard reference” is nearly incomprehensible, and only if one knows that subsequent explications were life-savers can a beginner find a readable thing.

And, of course, these graphs going backward branch so rapidly that literally reading everything referred-to is … well, physically impossible for most of us… and even if one tries to approximate it, the effort is … let’s say… “not repaid in kind”. At various moments I did estimates of how many pages I’d need to read to correctly honor all the background. When it hit “more than 10 pages a second, for the next 20 years”, I knew both that I couldn’t do it and that either it was crazy-impossible or … not the real issue. 🙂

Yet, at the same time, exaggerated reliance on “popular” textbooks (excepting some like Silverman’s… hard for the novice to know who to trust, yes, …) really doesn’t lead one forward/upward.

After all the above blather, my operational advice would approximately be that one should try to figure out _what_to_do_. Thus, any source which cannot be fairly interpreted as giving help is instantly secondary. More tricky are the sources that do well-describe the Mount Fuji at a great distance… To see Mt Fuji is a great thing. What should one do?

At this date, it seems to me that on-line notes on topics that otherwise seem to be “standard”, but have become cliched, are far better than most “textbooks”. But, yes, there is some volatility in this, because, as above, Joe S.’s books on elliptic curves are excellent, even while being entirely within all conventions and such.

Summary: run many threads…

Attribution
Source : Link , Question Author : PenAndPaperMathematics , Answer Author : paul garrett

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