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Well, no, no physical square, no matter how precisely its sides are 1m long has an irrational-lenght diagonal. That is simply impossible in the physical univer

Well, no, no physical square, no matter how precisely its sides are 1m long has an irrational-lenght diagonal. That is simply impossible in the physical universe because the universe is discrete at this level. Irrational numbers are impossible in discrete context. The whole point they were invented is to capture the idea of continuous functions or continuous number line. But this is a "slight of hand", a definition that's made for convenience of solving useful problems, but isn't based in the physical reality. In a very similar way to how sqrt(-1) is not a thing in a physical reality, but it's useful to work with problems that can be described using complex numbers. > There is nothing more special about sqrt(2) than about 1 That fact that you don't understand what's special about it doesn't mean there isn't. It's a very different thing though. > All of our drawings and constructions are approximations Drawings: yes. Constructions: no. > but that doesn't make them naturally be integers or rationals any more than they are irrational. Here you've ventured into the territory you have no idea about... I'm sorry. You sound more like some LLM-generated gibberish here than anything a human with any expertise on the subject would write. Of course some things are naturally integers. We've invented integers to capture those things (in the physical universe). Similarly, rationals. There's nothing in the physical universe that's irrational in the same way how it can be an integer or a rational. Irrational numbers don't describe quantities or passage of time or forces acting on physical objects or the speed etc. because all those things are made up of small indivisible parts, and there's always a finite computable answer to how big something is, how long a process would take, how strong is the force applied to an object etc. Irrational numbers are a mathematical device to deal with different kinds of problems. Similar to how generating functions use "+" to mean a completely different thing from how it's used in algebra, or how it's used in regular languages, so are irrational called "numbers". But they aren't the same kind of thing as integers or rationals. To be honest, it would've been better not to call them "numbers" at all, to avoid this kind of confusion, but mathematics has a lot of old and bad terminology that's used due to tradition. > elementary particles travel in perfectly straight lines and radiate in perfect circles in our models The root of your problem in understanding this is: our models. Particles don't radiate in perfect circles in reality. Physical reality is discrete and cannot create perfect circles. You can imagine, however, a perfect circle and use it to a great effect to estimate the result of some physical process. But, if you truly measure the effect, you will never have an irrational number. There's no physical process of measuring anything that will end up with an irrational answer. That's simply impossible.

tsimionescu 2y ago
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.
- crabbone 2y ago
  > 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?

tsimionescu 2y ago

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.

crabbone 2y ago
> 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?