Recently I suggested a new definition in Elementary Caclulus.
An introduction to the theory is given in YouTube:
Please take a look, it's short and to the point.
A more detailed version of the theory is available in:
Please let me know if you have any questions or comments, in:
Bear with me, my training is in numerical analysis, so I am not an expert. I am not on par with any of the people I am going to mention, so my explantions and conclusions will be unpolished.
Nice video. Are you familiar with the work of Doron Zeilberger? I read a short paper by him suggesting the replacement of continuous math by discrete math. An approach suggested by Kronecker and others in the past.
He is leading a growing group of mathematicians and physicists. They want to replace derivatives by differences and integrals by sums. In short no more continuous analysis. Topology to be replaced by graph theory. They are heavily influenced by computer algebra systems.
Also upon reading some of David Deutsch's ( a follower of discrete math ) stuff , they want to change what constitutes a proof, to concepts that he calls less rigorous and more practical. You are right in stating that computer scientists have long been replacing continuous concepts with their discrete counterparts. Many algorithms utilize differences in place of derivatives.
Until I read the Zeilberger paper, I had no idea there was any opposition to the way math was being done today. Of course I was aware that Poincare, Kronecker and others were opposed to the modern concepts of both infinity and set theory. As well as continuous math as it is called.
When I saw a lecture that was given to some physicists and mathematicians by a set theorist, I was astounded to see that the physicists and some mathematicians were hostile to it. Hostile to set theory!
As a follower of discrete math, I am sympathetic to the position of Zeilberger, Deutsch, Gosper, Wilf, Pflouffe, Bailey, The Borweins, The Risch group... But not totally convinced they should just dump 200 years of continuous analysis.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.