As a mathematical structure, I have no problem with the hyperreals. But I came across the following from Keisler's book "Elementary Calculus: An Infinitesimal Approach".
"We have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."
We get real number answers to physical problems in distance, from integers to transcendental numbers. However, we never get a nonzero number $w$ s.t. $w \lt 1/k, \forall k \in \mathbb N$ coming from a physical calculation, theoretical or applied. Isn't this what makes physical distances real and not hyperreal? Is it really possible that our space was never (locally) Euclidean all this time? Where are these infinitesmals and why are they hiding?
Secondly, we say the real line, by construction from completing the rationals, has "no holes". Yet, the reals are a proper subfield of the hyperreals. Where do these infinitesmals "fit" on the real line to make a hyperreal line when there is no room for them to fit? In other words, if we begin by assuming a physical line segment is a hyperreal line segment and then (mathematically) remove all the infinitesmals, we get a hyperreal line segment with "holes" in the form of missing hyperreal points, but this just gives a real line segment, which has no holes. There seems to be problems in assuming physical lines can be hyperreal lines.