There is no evidence that the real world has anything to do with a continuum, real numbers, or R^3. When we look closely at things, we always find combinatorial structures of discrete entities (smooth matter -> atoms -> protons -> quarks). For example, electrical charge is discrete, not continuous, and the unit appears to be 1/3 the charge on the electron (as carried by quarks).
The balance of evidence is that our universe is spatially compact, with time bounded in the past (Big Bang), so it is possible that there are no infinities at the cosmological scale either. The evidence for the temporal future has recently changed from closed (Big Crunch) to open (infinite expansion).
There are various theories about how space and time emerge from discrete quantum mechanical structures. For example, a simple derivation in Loop Quantum Gravity gives a discrete spectrum for area. At the moment, we don't know which theory is correct, but it is likely there is a smallest finite unit of area (and volume).
Another reason for rejecting real numbers in physics, is that each value contains an infinite amount of information. The Holographic Principle is interesting because it rejects the notion of an infinite amount of information in a finite space. In fact, it goes a lot further, and says the total information within a 3D volume does not scale as the volume (!), but as the 2D surface area. Essentially there is one bit per Planck Area, so 10^66 bits/m^2.
It is easy to imagine, but not to describe or prove, a combination of these theories that implies everything is finite, discrete and combinatorial, without an infinity or a real number in sight.
Using the reals as a model makes the math a lot easier than trying to do everything in discrete structures, especially since we don't actually know the scales that these discrete structures might be at. We don't know what the smallest unit of space is, and it's still not clear that there is one. As a result we use the reals as a model, and it does incredibly well.
So in a sense you are helping support the main point of the original article. The fact that the reals are an approximation to what you claim reality might be should be acknowledged, and then we should explore the limits and limitations of that models. The Banach-Tarski theorem does exactly that, showing us at least one place there the model apparently fails, and should certainly be treated with care.
The balance of evidence is that our universe is spatially compact, with time bounded in the past (Big Bang), so it is possible that there are no infinities at the cosmological scale either. The evidence for the temporal future has recently changed from closed (Big Crunch) to open (infinite expansion).
There are various theories about how space and time emerge from discrete quantum mechanical structures. For example, a simple derivation in Loop Quantum Gravity gives a discrete spectrum for area. At the moment, we don't know which theory is correct, but it is likely there is a smallest finite unit of area (and volume).
Another reason for rejecting real numbers in physics, is that each value contains an infinite amount of information. The Holographic Principle is interesting because it rejects the notion of an infinite amount of information in a finite space. In fact, it goes a lot further, and says the total information within a 3D volume does not scale as the volume (!), but as the 2D surface area. Essentially there is one bit per Planck Area, so 10^66 bits/m^2.
It is easy to imagine, but not to describe or prove, a combination of these theories that implies everything is finite, discrete and combinatorial, without an infinity or a real number in sight.