3 ms·

Both temperatures and dates have affine structure. For dates it's really easy to see, because when you're talking about the calendar you're actually talking ab

by qsort 6y ago

Both temperatures and dates have affine structure. For dates it's really easy to see, because when you're talking about the calendar you're actually talking about two different types of objects, absolute time references (i.e. Thursday January 1, 1970 12:00 AM) and time durations (i.e. 45 minutes). (If you're familiar with Java, Instants and Durations). Time references cannot be added nor subtracted, but you can add a time duration to a time reference (midnight + 45 minutes = 12:45 AM of the same day) and the difference of two absolute time references is well-defined, unique and is a duration (1 AM - 12 AM = 1 hour). Furthermore, durations satisfy the axioms of a vector space (trivially, just pick a canonical unit, e.g. 1 second, and measure everything as a multiple/fraction of that). The same holds with temperatures, with the unfortunate difference that we use the same notation for differences in temperatures (vectors of the affine space) and temperature references (base set of the affine space). For example, you can't add 18 celsius and 20 celsius, but saying that the temperature of a room increased by 2 celsius is meaningful, because "+2 celsius" is actually a vector. You can also add vectors to points, if the temperature increased by 2 more celsius, it would be 22 celsius. This does even have a physical meaning, and is independent of the frame of reference you pick for temperatures: a difference of 2 celsius is the same as a difference of 2 kelvin. (the same reasoning holds when replacing "celsius" with "fahrenheit" and "kelvin" with "rankine"). It is equally easy to verify temperature differences satisfy the axioms of a vector space, with the same argument as before.

ttoinou 6y ago
That's your interpretation but it doesn't change the fact that physically you cannot add temperatures. You can take the numerical value of one temperature, add something to it, and re interpret it as a temperature; But the '+' operation on temperature does not have any physical meaning. Replace "temperature" by "speed in kilometer/hour" in your sentence : "we cannot add 18 km/h and 20 km/h but I can add 2 km/h to 18 km/h because +2 km/h is a vector" it doesn't make sense to me
- qsort 6y ago
  I don't follow your line of reasoning. You can add speeds (vector space), you can't add temperatures (points in an affine space cannot be added), but you can add temperatures and temperature differentials (points and vectors in an affine space). Speeds are vectors in the first place, that's why you get to add them, while "temperature" denotes, depending on context, either a point (not addable) or a vector (addable). This explains why you can't add temperatures, but the average temperature makes 100% sense and has a direct physical meaning.
  - ttoinou 6y ago
    Speed and temperature are intensive properties, they don't add up. You can remove the unit and do whatever you want mathematically and add back the physical unit, but it's not physics. Average temperature has no meaning it's a common fallacy, the only reason I ever got for that meaning is "my teacher told me it was OK !"
    - qsort 6y ago
      I'm not following your train of thought. > Speed and temperature are intensive properties, they don't add up. You can add speeds by their very definition. They are vectors in R^3 (or some other coordinate system, you get the point). The fact that addition of vectors is allowed is literally baked into the axioms of a vector space. > Average temperature has no meaning it's a common fallacy Yes, average temperature does have unambiguous meaning: https://en.wikipedia.org/wiki/Boltzmann_constant https://en.wikipedia.org/wiki/Boltzmann_constant > You can remove the unit and do whatever you want mathematically and add back the physical unit, but it's not physics. You do realize math can handle units of measurements/dimensions just fine, do you?
      - ttoinou 6y ago
        > You do realize math can handle units of measurements/dimensions just fine, do you? Yes that's exactly what I'm talking about. Kelvin don't support the addition operator. You can pretend it's not Kelvin/Celsius, do something with the number, and add back the unit if you like, but it won't have any physical meaning. > Yes, average temperature does have unambiguous meaning: https://en.wikipedia.org/wiki/Boltzmann_constant https://en.wikipedia.org/wiki/Boltzmann_constant ... in an ideal gas model you can have a meaning adding and averaging k*T with k a magical constant that transforms your intensive property into extensive (energy) > You can add speeds by their very definition Definition of time derivation of the position function ? Yeah well [length]/[time] don't add up that's it. You need to first give a meaning to the addition operator of [length]/[time]. I'm talking in one dimension, of course if you can prove this then going 2D or 3D is easy

by qsort 6y ago

ttoinou 6y ago

That's your interpretation but it doesn't change the fact that physically you cannot add temperatures. You can take the numerical value of one temperature, add something to it, and re interpret it as a temperature; But the '+' operation on temperature does not have any physical meaning. Replace "temperature" by "speed in kilometer/hour" in your sentence : "we cannot add 18 km/h and 20 km/h but I can add 2 km/h to 18 km/h because +2 km/h is a vector" it doesn't make sense to me

qsort 6y ago
I don't follow your line of reasoning. You can add speeds (vector space), you can't add temperatures (points in an affine space cannot be added), but you can add temperatures and temperature differentials (points and vectors in an affine space). Speeds are vectors in the first place, that's why you get to add them, while "temperature" denotes, depending on context, either a point (not addable) or a vector (addable). This explains why you can't add temperatures, but the average temperature makes 100% sense and has a direct physical meaning.
- ttoinou 6y ago
  Speed and temperature are intensive properties, they don't add up. You can remove the unit and do whatever you want mathematically and add back the physical unit, but it's not physics. Average temperature has no meaning it's a common fallacy, the only reason I ever got for that meaning is "my teacher told me it was OK !"
  - qsort 6y ago
    I'm not following your train of thought. > Speed and temperature are intensive properties, they don't add up. You can add speeds by their very definition. They are vectors in R^3 (or some other coordinate system, you get the point). The fact that addition of vectors is allowed is literally baked into the axioms of a vector space. > Average temperature has no meaning it's a common fallacy Yes, average temperature does have unambiguous meaning: https://en.wikipedia.org/wiki/Boltzmann_constant https://en.wikipedia.org/wiki/Boltzmann_constant > You can remove the unit and do whatever you want mathematically and add back the physical unit, but it's not physics. You do realize math can handle units of measurements/dimensions just fine, do you?
    - ttoinou 6y ago
      > You do realize math can handle units of measurements/dimensions just fine, do you? Yes that's exactly what I'm talking about. Kelvin don't support the addition operator. You can pretend it's not Kelvin/Celsius, do something with the number, and add back the unit if you like, but it won't have any physical meaning. > Yes, average temperature does have unambiguous meaning: https://en.wikipedia.org/wiki/Boltzmann_constant https://en.wikipedia.org/wiki/Boltzmann_constant ... in an ideal gas model you can have a meaning adding and averaging k*T with k a magical constant that transforms your intensive property into extensive (energy) > You can add speeds by their very definition Definition of time derivation of the position function ? Yeah well [length]/[time] don't add up that's it. You need to first give a meaning to the addition operator of [length]/[time]. I'm talking in one dimension, of course if you can prove this then going 2D or 3D is easy