A colleague of mine asked an interesting question reproduced below with his permission.
It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - even if one accepts that, all things being equal, it is an easier language to learn - when the vast majority of the mathematical and scientific community still speak in the "Weierstrassian" language. It is possible to teach them both, but this may come at a certain expense.
When one wonders whether or not to teach calculus a la Keisler, it is not because it is or isn't the right to do, but rather a matter of personal preference. A typical mathematician grew up with Weierstrass and Cauchy, feels at home with standard analysis, and is comfortable teaching it. He knows where the pitfalls are, where to appeal to intuition and where to warn students to be wary, the right examples to grab onto for support, etc.
He doesn't have the same comfort level with infinitesimals, because of how he learned to think about things and look at them, and therefore could not convey it properly to a classroom, certainly not with the same confidence and enthusiasm, that is important in these classes. Given enough time, he could probably gain that same level of comfort with infinitesimals, but it would take a lot more time and effort than he is willing to take on.
So to summarize, is it really to the students' advantage to learn the language of infinitesimals?
What is requested is a reasoned response (based on reliable sources rather than personal opinions) on (1) historical, (2) mathematical, and (3) philosophical aspects of the question.
This question was posed a while ago at MSE where some of the editors felt that ME is more appropriate.