> I.e. if you try to do pi + pi, well, you may get a 2pi, if you pray hard enough and your faith is strong enough, but really, there's no proof that even that is true.

Sure there is. pi + pi = 2pi <=> (pi + pi) / pi = 2pi / pi <=> pi/pi + pi/pi = 2 <=> 1 + 1 = 2, which we know is true. QED.

This is in fact exactly what I'm saying about the circle and its radius. We can't get rid of the irrationality when calculating the ratio between the circumference and the radius of a circle. But it's arbitrary which one we call rational and which we call irrational: a circle with a rational circumference will have an irrational radius, and vice versa.

> Because you prayed hard enough and it was revealed to you in a dream? Based on what do you believe this?

I don't know what exactly you are responding to here.

It's your claim that in the real physical world all "circles" have a rational circumference (perimeter), which is equivalent to saying that the "circle" is really a very very many-sided polyhedron (since only a polyhedron can have a rational perimeter if the sides are of a rational length and all angles are constructible). I don't need to pray (?!?) to see this.

And if you were responding to my full quote, that this comparison is equivalent to saying that all physical "squares" are in fact rounded-corner ovoid shapes (and so their actual side lengths are some multiple of pi, or at least some other irrational number that we don't even have a name for) then that follows from the observation above, that you can arbitrarily decide to call the circumference of a circle "2 pi" or the radius "1/2pi".

It also follows from how trajectories work in physics - if a particle is moving in a straight line and then some force starts acting on it in some direction other than directly in front or behind, its trajectory will become circular, not go at a straight angle. So if an electron in a perfectly isolated environment would follow a perfectly straight line, an electron in a real environment where there are electrical fields everywhere will follow a line that's curvy all around. In contrast, it's in fact impossible to create a trajectory for an electron that has any kind of angles, even in an ideally isolated environment - it's impossible for a physical object to turn on the spot like an ideal angle.

So, again, curves (and their associated irrational numbers) are in fact closer to physical reality, we just chose to approximate them using straight lines because its easier.

And as a final thought, related to the reality of the continuum. In all of the models that we have of physics that actually work, if I fire two particles away from each other arbitrarily in space, the distance between them will cover every real number in some interval [minDist, maxDist]. And any model that requires a minimum unit of distance to exist (so that the distances would be minDist + n*FundamentalMinimum, with n = 1, 2, 3...) doesn't work with special relativity, that says that lengths contract in the direction of movement (because if two particles are at a distance of FundamentalMinimum as measured by one observer, they will be at a distance of gamma*FundamentalMinimum to another observer moving at some speed relative to the first one, with gamma < 1, thus breaking the assumption that all lengths are > FundamentalMinimum).