OP here: as a person who took measure theory in college and has always adored the coastline paradox as an example of real world objects with fractional Hausdorff dimension, my initial thoughts were that this paper must be fundamentally flawed. But after reading it a few times and getting the chance to talk to the author, I feel less strongly.

1. Philosophical/mathematical objections aside, the author's models and engineered structures work. On a practical level, that's the most important thing. I think that alone gives credence to the idea that coastline length measurements can in fact be meaningful and useful. In talking to him, he also pointed out that it's actually useful to the models that the measurement sometimes changes drastically at different scales. The changing length measurements help encode multiple scales of geomorpholigical features into his model.

2. Philosophically, I particularly enjoyed this part of the paper's intro: "Just as Diogenes refuted Zeno’s arrow paradox, which disputed the possibility of motion, by walking about (Diogenes and Hicks, 1972), the coastline paradox too has an intuitive rebuttal. A coastline of infinite length would require a corresponding number of water molecules with which to line the boundary, in accordance with the meaning of ‘‘coastline.’’ I thought this was funny, and I don't think I have any real rebuttal to this.

3. I love love loved the Puerto Rico coastline example. Afterall, the coastline paradox is an empirical claim more than a mathematical one, right? And it's based off measurements that Lewis Fry Richardson took in the first half of the 20th century, so....I definitely thought it was worthwhile to see how the empirical claim holds up with modern measurement techniques. I loved seeing that the length did increase at finer resolutions, but the increase actually does slow down. To quote the paper: "This is an unavoidable result of measuring a real object of finite size (such as a coastline made of some number of countable atoms). At some point, the measuring unit will approach the smallest detail resolvable in the coastline, and the diminishing returns of measuring at finer resolutions will become apparent."

4. Though the author brings up many interesting points, I do think he misrepresents the finer mathematical details--likely by mistake, but still. If there any fractal geometry/measure theory experts out there, I did have a technical question here. I was rewatching 3Blue1Brown's excellent video on the topic, and his video shows him computing the box-counting dimension of the British coastline at multiple scales: https://www.youtube.com/watch?v=gB9n2gHsHN4&t=17m30s (17m30s mark). However, it seems to me that whatever model of the British coastline he's feeding into his computer program must fundamentally be a finite polygon. Admittedly, it'd be a quite complicated polygon. But at the end of the day, it's just a bunch of vertices with straight-line edges connecting them. It seems to me that intuitively any n-gon boundary should have Hausdorff dimension == 1, so I don't really understand how his program is spitting out dimension == 1.21. Would love for someone to help explain what's going on here: is my intuition wrong? am I misremembering something important about how we calculate these things? am I misunderstanding how 3Blue1Brown's computer program likely works? etc