I just watched this youtube video: http://www.youtube.com/watch?v=K4eAyn-oK4M
He lays out his objections against the $ϵ,δ$ definition around 14 min.
Here is the discription of the video:
In this video we aim to give a precise and simpler definition for what it means to say that: a rational polynumber on-sequence p(n) has a limit A, for some rational number A. Our definition is both much simpler and more logical than the usual epsilon -delta definition found in calculus texts. What is required is that we need to find two natural numbers: k called the scale, and m called the start that allow us to bound in a pretty simple way the difference between p(n) and A.
The epsilon-delta definition of a limit is usually considered a high point of logical rigour. Not so. It is also considered too logically involving to be taken seriously as a pedagogical pillar for most undergrads. Hence students may be told about the definition, but are not required to seriously understand it, or be able to use it--unless they are prospective maths majors.
There is a subtle ambiguity in the definition: given an epsilon we are supposed to demonstrate there is a delta (with certain properties) but how are we to do this, since an potential infinity of epsilons are involved? In practice what is required is a correspondence (function/relation etc) between epsilon and delta but the nature of this required correspondence is not clear. We return to our familiar conundrum of using the work``function'' without a proper definition of it.
The key point that makes our simpler more intuitive notion of limit of a sequence work is that we are dealing with very particular and clearly defined on-sequences: those generated by a rationl polynumber. A good example of the benefits of being careful rather than casual when dealing with the foundations of analysis!
My question is: Is this an opinion shared by more mathematicians ? I kind a feel like that this Professor of the University of New South Wales is standing completely alone as it comes to this. I don't really undestand his objections, but I don't think I'm skilled enough to understand if his objections are legit.