To the best of our knowledge, physical reality has both discrete quantities and continuous quantities. Space and time are continuous, for example. Energy, charge, mass - all discrete. And this is not a question of just models - we've tried to create models where time or space are discrete, and they don't work. They fail to predict reality correctly.
And you're still wrong about the nature of irrational numbers. They are not a model for computational irreducibility. They basically separate different types of quantities that are not directly relatable as ratios of one another. The circumference of a circle (or any non-trivial ellipse, for that matter) is a fundamentally different quantity than the length of a straight line: this is what Pi being irrational tells us.
And yet, you can do arithmetic with circle lengths just as much as you can with sides of squares. You can add them up, divide them, everything. If for whatever reason we decided to define 1m as the circumference of a particular circle, and base our geometry around circles rather than straight lines, we'd consider circle lengths to be integer/rational numbers, and then line lengths would be irrational (the radius of a circle with circumference 1 would be 1/(2pi), an irrational number). Similarly, if we decided that 1m is defined as the diagonal length of a particular square, we'd say that the lengths of that square have irrational length, and then you'd claim that no true square has lengths of exactly sqrt(2)/2.
As I said, it's just as correct to think of any physical circle/ellipse as a many-sided polyhedron as it is to think of any physical straight line as a segment of the circumference of a circle/ellipse with a really huge radius (focal length).
In fact, it's more correct physically to think of apparently straight lines as curved than to think of curves as composed of polygons, as straight paths are very rare in physics, any kind of bias will tend to induce a slight curvature. And atoms are more circular in nature than they are "boxy".
Ultimately we can only really measure ratios of things, and we know some things are not an exact ratio of another thing when we measure them precisely enough. Which of the quantities you consider to be represented by a rational number and which you consider to be irrational is purely a choice of definitions.
> And yet, you can do arithmetic with circle lengths just as much as you can with sides of squares.
No, you can't. There's no way to add or to multiply two numbers that don't have a finite expansion in some basis beyond just writing it as a sum outside of a very few special cases where there's a round-about way of finding the answer. 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. You just choose to believe that it will check out somehow. But, even if you get a 2pi, it's still not an answer you want because to figure out what 2pi is, you still need to add a pi to a pi, so, you are back to square one.
> As I said, it's just as correct to think of any physical circle/ellipse as a many-sided polyhedron
Because you prayed hard enough and it was revealed to you in a dream? Based on what do you believe this?
Take a circle with a radius of 1cm. Unroll its perimeter and declare that length is 1 of new unit called a squelk.
You can measure things and build things in squelks just fine, but if you try to take something that is 100 squelks long and measure it in centimeters you will get an irrational number of centimeters because there is no rational conversation from squelks to centimeters.
A given length can be irrational in one unit of measure but not in another.
Of course we do have limits of precision in the real world, so in reality nothing lines up quite right.
> 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).
And you're still wrong about the nature of irrational numbers. They are not a model for computational irreducibility. They basically separate different types of quantities that are not directly relatable as ratios of one another. The circumference of a circle (or any non-trivial ellipse, for that matter) is a fundamentally different quantity than the length of a straight line: this is what Pi being irrational tells us.
And yet, you can do arithmetic with circle lengths just as much as you can with sides of squares. You can add them up, divide them, everything. If for whatever reason we decided to define 1m as the circumference of a particular circle, and base our geometry around circles rather than straight lines, we'd consider circle lengths to be integer/rational numbers, and then line lengths would be irrational (the radius of a circle with circumference 1 would be 1/(2pi), an irrational number). Similarly, if we decided that 1m is defined as the diagonal length of a particular square, we'd say that the lengths of that square have irrational length, and then you'd claim that no true square has lengths of exactly sqrt(2)/2.
As I said, it's just as correct to think of any physical circle/ellipse as a many-sided polyhedron as it is to think of any physical straight line as a segment of the circumference of a circle/ellipse with a really huge radius (focal length).
In fact, it's more correct physically to think of apparently straight lines as curved than to think of curves as composed of polygons, as straight paths are very rare in physics, any kind of bias will tend to induce a slight curvature. And atoms are more circular in nature than they are "boxy".
Ultimately we can only really measure ratios of things, and we know some things are not an exact ratio of another thing when we measure them precisely enough. Which of the quantities you consider to be represented by a rational number and which you consider to be irrational is purely a choice of definitions.