Subject: Introductory Real (Mathematical) Analysis:
Recommendation: Real Mathematical Analysis by Charles Pugh
The three introductory Analysis books I’ve read cover-to-cover are Lang’s, Pugh’s, and Rudin’s.
What makes Pugh’s book stand out is simply that he focuses on building up repeatedly useful machinery and concepts-a broad set of theorems that are clearly motivated and widely applicable to a lot of problems. Pugh’s book is also chock-full of examples, which make understanding the material much faster. And finally, Pugh’s book has a very large number of exercises of varying difficulty-Pugh has more than 500 exercises total.
In contrast, Rudin’s book tends to focus on “magic.” Rudin uses the shortest possible proofs for a given theorem. The problem is that the shortest proofs aren’t necessarily the most instructive-while Baby Rudin is a beautiful work of Math qua Math, it’s not a particularly good book to learn from.
Finally, Lang’s book is frankly subpar. Lang leaves out critical details of some proofs (dismissing one 6 page proof as trivial!), is poorly motivated by examples, and has a number of mistakes.
If you want to really understand Mathematical Analysis and get to the point where you can use the concepts to create proofs and solve problems, Pugh is the best book on the topic. If you want a concise summary of undergraduate analysis to review, pick Rudin’s book.
It’s not that they’re measuring the wrong variables, it’s most likely that those organizations have already made the decisions based on variables they already measure. In the “Function Points” example, I would bet there were a few obvious learnings early on that spread throughout the organizations, and once the culture had changed any further effort didn’t help at all.
Another example: I took statistics on how my friends played games that involved bidding, such as Liar’s Poker. I found that they typically would bid too much. Therefore a measurement of how many times someone had the winning bid was a high predictor of how they would perform in the game-people who bid high would typically lose.
Once I shared this information, behavior changed and people started using a much more rational bidding scheme. And the old measurement of “how often someone bid high” was no longer very predictive. It simply meant that they’d had more opportunities where bidding high made sense. Other variables such as “the player to your left” started becoming much more predictive.