>I do not make any metaphysical claims about space, time nor numbers,
>but notice that the mathematics used in practice is always finite
>...I argue that the so-called real numbers are not really real.
>More precisely, I argue that the mathematical real numbers
>are not physically real, by which I mean that they do not represent anything physical
This paper -- although I'm an outsider to physics, and I'm way more comfortable evaluating mathematics and computer science research -- seems like it has an interesting and novel approach to a theoretical problem in physics. I don't understand why the author felt the need to couple this with the given title and stress this point so much. It almost felt like click-bait, until I understood generally what was being done.
On further deliberation, I have come to the following conclusion: there is a difference between assuming "only physical objects exist, and they are finite" and making an argument of that. The author is trying to say he is not making an argument for the validity of the axiom but implicitly says he just assumes it, to guide the focus of the reader to the main argument he is making.
I acknowledge that you definitely do not need to make that assumption, but as I said, you have to start somewhere. I'd say the metaphysical assumption captures a substantial subset of all physics, hence his ideas are worthy of consideration.