hypotenuse still has no meaning at all. — aletheist
Time and space are continuous in themselves, so any units assigned to them are completely arbitrary, and you cannot measure a worldline except along the two different axes. Apples are discrete and dollars are defined, so comparing them to time and space is like comparing ... apples to oranges. — aletheist
How about nowhere, since it is nonsense.I don't know where to go with this. — TheMadFool
What are you asking? Where the nonsense is? — tim wood
Actually, no. On a Euclidean plane. But spacetime is not a Euclidean plane except as an approximation. What you're referring to is a spacetime diagram.Similarly, if a moving object, p, were to begin at (0, 0) and travel 4 units of distance in 3 units of time, it would be at point (3, 4). In this case 5 would represent the worldline of the object p in spacetime. — TheMadFool
What's the hypotenuse in terms of apples and dollars? That's all I'm asking. — TheMadFool
The square root of the square of the number of apples plus sixteen. An amazing breakthrough in marketing! — jgill
How did you come to the realization that the angle is perpendicular when constructing your mathematical model of the problem domain? What property in the actual domain of application of your model prompted the idea? If you were not concerned with any domain when you constructed the model, how do you come to need to ask semantic questions now?If memory serves, Pythagora's theorem works only for right triangles and yes the axes that I used are perpendicular and yes there's a right triangle (3, 4, 5) formed. — TheMadFool
I don't get your joke. We have two items in our list: apples and dollars, each of them forming a side of a right triangle. What's the hypotenuse in terms of apples and dollars? — TheMadFool
The whole point is that the 2D Cartesian coordinate system is not a picture. It is ascription of coordinates to some plane of points, which points correspond in pairs to vectors, which vectors individually correspond to lengths and in pairs correspond to angles. Ok, the points are angles-apples, but the remaining properties are not automatic. They are not provided by the Cartesian coordinate system for you, mathematically. They are provided by you, originally, so that you can justify the use of Cartesian coordinate system. Otherwise, what you have are just pairs of numbers corresponding to points, and the rest is as real as Tolkien's world — simeonz
The whole point is that the 2D Cartesian coordinate system is not a picture. It is ascription of coordinates to some plane of points, which points correspond in pairs to vectors, which vectors individually correspond to lengths and in pairs correspond to angles. Ok, the points are angles-apples, but the remaining properties are not automatic. They are not provided by the Cartesian coordinate system for you, mathematically. They are provided by you, originally, so that you can justify the use of Cartesian coordinate system. Otherwise, what you have are just pairs of numbers corresponding to points, and the rest is as real as Tolkien's world. — simeonz
My point is that you need to have the concepts of "angles" (so that they can be equal), "directions" (so that you can make the points on your lines aligned, i.e. colinear), "distances" apriori, before resorting to analytic geometry. (And formally, we would call that an affine space today. Although the terminological designation would not be present historically, the ideas would be the same.) It would be backwards thinking if we started with pairs of numbers, declared them to be the Cartesian coordinate system for implicit space of entities and finally tried to infer a sensible explanation of the nature of the metrics of those entities.A right angle (if I remember my high school geometry) is when you have a line intersecting another line and making equal angles on each side. — fishfry
My point is that you need to have the concepts of "angles" (so that they can be equal), "directions" (so that you can make the points on your lines aligned, i.e. colinear), "distances" apriori, before resorting to analytic geometry. (And formally, we would call that an affine space today. Although the terminological designation would not be present historically, the ideas would be the same.) It would be backwards thinking if we started with pairs of numbers, declared them to be the Cartesian coordinate system for implicit space of entities and finally tried to infer a sensible explanation of the nature of the metrics of those entities. — simeonz
(a + b) (a + b) = a ^ 2 + b ^ 2 + 2 a * b
I don't see it that way really. We still come from the geometric perspective, to define angles and distances in one way or another, and only then we have the privilege of calling an n-tuple of points being from a Cartesian coordinate system. Cartesian coordinate systems come with semantics that need to be defined apriori. They are not just mechanical assignment of pairs of numbers to some arbitrary point space.And in modern math we start with a 2-dimensional coordinate system and define the Euclidean distance. — fishfry
I say that the Pythagorean theorem applies to affine spaces over inner product spaces, — simeonz
I don't see it that way really. We still come from the geometric perspective, to define angles and distances in one way or another, and only then we have the privilege of calling an n-tuple of points being from a Cartesian coordinate system. — simeonz
Cartesian coordinate systems come with semantics that need to be defined apriori. They are not just mechanical assignment of pairs of numbers to some arbitrary point space. — simeonz
There is only one sense, in fact, in which I am not correct. And it is that a Cartesian coordinate system might be a applied to the very n-tuples, with vectors being n-tuples, distances and angles computed in the usual way, etc. But then, we couldn't talk about apples and dollars, because since the underlying point space is just a mechanical bonding of numbers, it is unitless. — simeonz
Those are not analytic. They are intuitional. — simeonz
I still want to be certain that you concur with me on the definition of Cartesian coordinate systems.No that's not true. We define R2 as the set of ordered pairs of real numbers. Then we define the usual Euclidean Euclidean distance, and we define the usual dot product. — fishfry
I agree with the fact that we can define the dot product as you specify, but we need inner product as well, or we are just manipulating unitless numbers that don't correspond to anything.I assume you agree. — fishfry
To some extent. But I was saying that there is one more hop (probably) in my mind to how this intuition translates to Cartesian coordinates. We first justify the requirements of the affine spaces with the constructive proofs, such as the properties of the inner product in the inner product space. Then we assign n-tuples to the points in the point space, proving that we preserve the inner product with the dot product. Since, in the OP's question there is no inner product, just dot product, there is nothing to preserve and no Pythagorean semantics to be had. We either have arbitrary assignment of numbers to points somewhere, in some semantic domain, or we work with numbers as our semantic domain, and those numbers have no units.Is that what you're saying, that we need the ancient geometric intuition to ground the modern analytic approach? — fishfry
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