12

I find that some quantities share serveral characteristics.

For date:

  • "1st July" + "1 day" = "2nd July"
  • "2nd July" - "1st July" = "1 day"
  • But "20th August" + "29th August" is nonsense.

For temperature:

  • An object can be heated up from 25 degree Celsius by 5 degree Celsius to 30 degree Celsius.
  • The temperature of boiling water is 100 degree Celsius. The temperature of the surrounding is 25 degree Celsius. The difference is 75 degree Celsius.
  • But we cannot add the temperature of a cup of tea to the temperature of a cup coffee.

Also, we cannot do multiplication on date or temperature. Depending on context, their relative values are useful, but their absolute value are not relavent. E.g., we don't count date from the Big Bang; weather forecast does not involve absolute temperature.

How does abstract algebra describe these quantities? Is there an algebraic structure captures their characteristics?

Nicky C
  • 229
  • 4
    Generally, heaps and torsors. – Chris Culter Aug 21 '15 at 07:53
  • Note all limitations go away if you introduce an absolute zero. For temperatures this is 0K, for dates this is an epoch. Adding two temperatures become "how much ~energy is in this system", and adding two dates become "how many seconds are between these dates and the epoch, together?". This is why programmers love Unix Time. – BoppreH Aug 21 '15 at 20:41
  • @BoppreH Well, I am a programmer, but I don't like Unix Time. It does not handle leap seconds very well ^_^ I agree that some limitations go away when a universal origin point is introduced, or when the underlying meaning of the quantity is considered. But it is still interesting to consider the contexts that do not allow these approaches. – Nicky C Aug 22 '15 at 02:16

2 Answers2

18

What you are describing is similar to the concept of an affine space.

An affine space over a vector space $V$ is a space filled with points. You can add a vector to a point to get another point, and you can subtract one point from another to get a vector. But you cannot add two points together.

5xum
  • 123,496
  • 6
  • 128
  • 204
-1

Calendar dates and temperatures in °C are sets where group operations are defined on. You can add and subtract days as differences in days follow the model of integer numbers ($\mathbb{Z}$). The elements of the group of day differences then operate on the "amorphic" set of calendar dates which has no own algebraic structure (calendar dates are not a vector space or a group; you cannot add 2nd July to 1st July as you correctly noted).

You can add and subtract temperature differences in K. Kelvin, because by convention temperature differences are never measured in °C but in K. Here we see clearly that we don't have an algebraic structure on the temperatures themselves (14°C - 25°C is not a temperature in °C but a temperature difference in Kelvin). The group of temperature differences thus operates on the set of temperatures.

Operations like these are quite important in analysing (multidimensional) data: You are not interested in the numbers themselves but on the information they convey. So if this information is not changed by some operation on the data, you have to use an analysis tool that is invariant to this class of operations. There are also examples where a more specific structure than a group operates on the data.

Horst Grünbusch
  • 399
  • "calendar dates are not a vector space; you cannot add 2nd July to 1st July as you correctly noted, so it is not an affine space" That makes no sense. An affine space is precisely a space where zou cannot add the points inside, but you can subtract them to get vectors. – 5xum Aug 21 '15 at 19:38
  • Oops, so myna typos! – Horst Grünbusch Aug 23 '15 at 16:32
  • So instead of answering my point, you decided to point out that I made typos (pore precisely, one typo) in it... – 5xum Aug 23 '15 at 16:41
  • No, I corrected more other typos in my own answer since I noticed I didn't proofread it. Downvote my answer if you're not satisfied. – Horst Grünbusch Aug 23 '15 at 16:43