I would like to learn non-standard analysis, at least the basics of it. I will make use of this book: Elementary Calculus: An Infinitesimal Approach (Dover Books on Mathematics), by H. Jerome Keisler. Before anything else, please let me take up some links that DO NOT have a fully fleshed out answer to my question, at least according to me.
Here are the links:
The content on this link 'Is non-standard analysis worth learning?' more or less discusses the use of studying non-standard analysis. The content on this link mostly refers to the link that discusses the use of studying non-standard analysis, 'Is non-standard analysis worth learning?'. This and this does not answer my question.
The problems with the answers to the question up above, is that while they may scratch on the surface and from to time take up the disadvantages of non-standard analysis, they DO NOT purely discuss the disadvantages or/and mathematical disadvantages of non-standard analysis.
'What are the disadvantages of non-standard analysis?' does not rigorously answer that which I want an answer to. For example, let me cite this answer:
I think there are a number of reasons:
Early reviews of Robinson's papers and Keisler's textbook were done by a prejudiced individual, so most mature mathematicians had a poor first impression of it. It appears to have a lot of nasty set theory and model theory in it. Start talking about nonprincipal ultrafilters and see the analysts' eyes glaze over. (This of course is silly: the construction of the hyperreals and the transfer principle is as important to NSA as construction of the reals is for real analysis, and we know how much people love that part of their first analysis course.) There is a substantial set of opinion that because NSA and standard analysis are equivalent, there's no point in learning the former. Often, the bounds created with NSA arguments are a lot weaker than standard analysis bounds. See Terry Tao's discussion here. Lots of mathematicians are still prejudiced by history and culture to instinctively think that anything infinitesimal is somewhere between false and actually sinful, and best left to engineers and physicists. As Stefan Perko mentions in the comments, there are a number of other infinitesimal approaches: smooth infinitesimals, nilpotents, synthetic differential geometry, . . . none of these is a standout candidate for replacement. It's not a widely-studied subject, so using it in papers limits the audience of your work. Most of these reasons are the usual ones about inertia: unless a radical approach to a subject is shown to have distinct advantages over the prevalent one, switching over is seen as more trouble than it's worth. And at the end of the day, mathematics has to be taught by more senior mathematicians, so they are the ones who tend to determine the curriculum.
This is a good start of an answer to the question that I am asking. What I am missing in the answer that you saw, are if there exists any mathematical problems in non-standard analysis. Does it, and if so, which?
I once read - on this forum, at a place that I really can't remember - that there exists some mathematical problems in non-standard analysis. At least some ideas or concepts that weren't, if I remember correctly, likable. The word likable points towards at least one bias. But is it a bias?
Please help me to understand if there are mathematical problems/ problems in certain concepts of non-standard analysis.