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You might be interested in the book A Primer of Infinitesimal Analysis by John L. Bell, which is another approach to infinitesimals that uses intuitionistic logic.

https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Be...

Any interesting property of this logic and model is that all functions are infinitely differentiable.

Other introductions are An Invitation to Smooth Infinitesimal Analysis by John L. Bell and Synthetic Differential Geometry by Michael Shulman.

http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf

http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...

This kind of approach simply doesn't make sense

It was poorly explained there, but essentially that style of reasoning does work: http://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bel...

And I at least find that approach easier and more useful. (I learned it from the Feynman lectures on physics, where he didn't axiomatize it; the above link does.)

⬐ stiff

I find the logically sound version of the infinitesimals approach to be much more difficult to understand than the approach using limits. For example, you have to introduce hyperreals to make it work (most common approach):

http://en.wikipedia.org/wiki/Hyperreal_number

⬐ abecedarius

The book I linked to uses http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis (a quite different approach). I haven't studied nonstandard analysis.

⬐ nopinsight

How would you explain the continued popularity of 103-year-old "Calculus Made Easy" then? The method used in the book might not be rigorous enough to solve all calculus problems but it jives well with a large number of people.

Most of them could get a better intuitive understanding of Calculus. (We should of course conduct a rigorous study comparing the effects of using different approaches to teach Introductory Calculus.) Then those who need to use the limits approach for other courses could use the intuition to learn it faster. In addition, they would understand Calculus from two different angles and could select the more suitable approach for each problem.

http://en.m.wikipedia.org/wiki/Calculus_Made_Easy