The saddle point involved in magnetic reconnection has a field line that can go infinitesimally close to the saddle point. I guess you could say that they're gone at that point, but you get lines going into and out of that point in equal measure, so... it's like they don't end. Is that all you mean? If so, that's a minor technicality in my book, of relevance to actual mathematicians (like the difference between [0,1] and [0,1) ) but not something a more general audience need dwell on. Are there any scenarios you can think of where it's more pronounced than that?
It is, as you almost say, a mathematical fact that magnetic field lines can end. It is also, as you do say, an empirical fact that some physicists don't care or regard that as "a minor technicality in my book".
That becomes a problem when,
as stated by Zilberti:
Many textbooks support the idea that B-field lines are always closed (or, at most, extend to infinity). Among them, we may mention those by Stratton [1941], Morse and Feshbach [1953], Sommerfeld [1964], Weber [1965] (who associates closed lines to any solenoidal vector), Paul and Nasar [1982], and Griffiths [1999]. The same happens in many handbooks of electrical engineering (see for instance Laughton [2003] and Schmitt [2002]).
The idea that magnetic field lines "are always closed (or, at most, extend to infinity)" is not only out-and-out false, but an impediment to understanding the genuine meaning of ∇∙
B=0.
That false explanation of
Gauss's law for magnetism is very common. For example, it shows up in
Wikipedia's current article on the magnetic field:
Wikipedia said:
...magnetic
B-field lines neither start nor end (mathematically,
B is a
solenoidal vector field); a field line may only extend to infinity, or wrap around to form a closed curve, or follow a never-ending (possibly chaotic) path.
[33]
As seen in that quotation, the false idea that magnetic field lines neither start nor end leads immediately to a false understanding of what it means to be a solenoidal vector field, as in
this Wikipedia article:
Wikipedia said:
A common way of expressing this property is to say that the field has no sources or sinks.
[note 1]
Happily, the superscript
"note 1" amends that false statement by saying
Wikipedia said:
This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.
Pragmatically, the confusion caused by these false claims (that magnetic field lines can neither start nor end, are always closed or go to infinity, etc) has been an impediment to understanding some real physical phenomena, such as magnetic reconnection.
Donald Scott may be unusually susceptible to confusion, but he is not the only one who has been confused by this. Consider, for example, a
stackexchange discussion of "Is magnetic reconnection reconcilable with magnetic field lines neither starting nor ending?" The answers given by @DavidHansen are incorrect. The highest rated answer, by
tok3ratOr, is correct, and the second highest rated answer, by
Kyle Kanos, is mostly correct but unnecessarily complex as it brings in matters that are not needed to answer the question that was asked.