How would the world look different on each of those hypotheses? (Can you please taboo “fundamental” and “exists,” too?)
Pythagorean theorem clearly seems to be a property of the real world, as does pi, and geometry in general.
You’re confusing the map and the territory here. The Pythagorean theorem and pi are both mathematical features that fall out of a particular model of the world, namely Euclidean geometry, which is an inaccurate model for at least two historically major reasons (the Earth not being flat and relativity).
The Pythagorean theorem and pi are both mathematical features that fall out of a particular model of the world, namely Euclidean geometry, which is an inaccurate model for at least two historically major reasons (the Earth not being flat and relativity).
How does “the Earth not being flat” make Euclidean geometry inaccurate?
If you draw a big enough right triangle on the Earth, it will visibly fail to satisfy the Pythagorean theorem. The geometry of the Earth is approximately spherical geometry, not Euclidean geometry.
The Pythagorean theorem and pi could both accurately be described as predictive scientific hypothesis of observed phenomenon. “In 3 dimensional space, if I measure two sides of a right triangle, the third side will be the square root of the sum of the squares of the other two sides.” That is a scientific hypothesis, and it can be tested; not only that, but you lose none of the meaning of Pythagorean theorem by putting it in those terms. (Yes, if you bring relativity into it, it turns out to be a slightly inaccurate hypothesis because of the curvature of space in a gravitational field, so I suppose that puts it in the same catagory of hypothesis as Newtonian physcis.) It still seems to be an attempt to describe a feature that exists in nature, though.
Ah. I wouldn’t call that claim the Pythagorean theorem. To me, the Pythagorean theorem is a mathematical statement about mathematical objects called Euclidean triangles (or if we want to get really fancy, it’s a statement about vectors in inner product spaces), and there is a separate claim, which is not mathematical, which asserts that a certain model which includes things like Euclidean triangles describes some part of the real world in some way.
In other words, I think it’s sensible to enforce a strong separation between talking about the mathematical details of a mathematical model and the relation of that mathematical model to reality. To me this dissolves what I think your original question is (although I am not sure I have correctly understood what your original question is).
Actually, the pythagorean theorem and pi still apply regardless of what dimension of geometry the world obeys (3-dimensional newtonian physics, 4-dimensional relativistic spacetime, 11-dimensional string theory, etc).
The Pythagorean theorem doesn’t apply to curved space, only to flat space (regardless of number of dimension). And pi is the number 3.14159..., which can be defined in ways that have nothing to do with geometry, so I’d put it as “in convex (concave) space, the ratio of a circumference to its diameter is less (greater) than pi”, not as “in convex (concave) space, pi is less (greater) than 3.14159...)”.
How would the world look different on each of those hypotheses? (Can you please taboo “fundamental” and “exists,” too?)
You’re confusing the map and the territory here. The Pythagorean theorem and pi are both mathematical features that fall out of a particular model of the world, namely Euclidean geometry, which is an inaccurate model for at least two historically major reasons (the Earth not being flat and relativity).
How does “the Earth not being flat” make Euclidean geometry inaccurate?
If you draw a big enough right triangle on the Earth, it will visibly fail to satisfy the Pythagorean theorem. The geometry of the Earth is approximately spherical geometry, not Euclidean geometry.
Euclidean geometry is a set of principles and conclusions for flat space. That Earth is not flat in no way makes Euclidean geometry inaccurate.
The Pythagorean theorem and pi could both accurately be described as predictive scientific hypothesis of observed phenomenon. “In 3 dimensional space, if I measure two sides of a right triangle, the third side will be the square root of the sum of the squares of the other two sides.” That is a scientific hypothesis, and it can be tested; not only that, but you lose none of the meaning of Pythagorean theorem by putting it in those terms. (Yes, if you bring relativity into it, it turns out to be a slightly inaccurate hypothesis because of the curvature of space in a gravitational field, so I suppose that puts it in the same catagory of hypothesis as Newtonian physcis.) It still seems to be an attempt to describe a feature that exists in nature, though.
(minor edits for clarification)
Ah. I wouldn’t call that claim the Pythagorean theorem. To me, the Pythagorean theorem is a mathematical statement about mathematical objects called Euclidean triangles (or if we want to get really fancy, it’s a statement about vectors in inner product spaces), and there is a separate claim, which is not mathematical, which asserts that a certain model which includes things like Euclidean triangles describes some part of the real world in some way.
In other words, I think it’s sensible to enforce a strong separation between talking about the mathematical details of a mathematical model and the relation of that mathematical model to reality. To me this dissolves what I think your original question is (although I am not sure I have correctly understood what your original question is).
Maybe your question is secretly a question about the unreasonable effectiveness of mathematics in the natural sciences?
Actually, the pythagorean theorem and pi still apply regardless of what dimension of geometry the world obeys (3-dimensional newtonian physics, 4-dimensional relativistic spacetime, 11-dimensional string theory, etc).
The Pythagorean theorem doesn’t apply to curved space, only to flat space (regardless of number of dimension). And pi is the number 3.14159..., which can be defined in ways that have nothing to do with geometry, so I’d put it as “in convex (concave) space, the ratio of a circumference to its diameter is less (greater) than pi”, not as “in convex (concave) space, pi is less (greater) than 3.14159...)”.
I don’t know what you mean by that.