UPDATE (3 May): To clarify, when I write
"places where we are used to using arithmetic/harmonic means", I mean more
than just the positive reals. For example, you may want to consider also the
example of S being the complex numbers in the first and fourth
quadrants.
For anything other than the positive reals, I do not require, however, any proof of existence for the value you define. In general, if F is a field and S is this field restricted in such a way that
Consider the following three definitions:
In mathematics, we often try to define things using the most restricted set of tools possible, in order to make them most generally applicable. For example: let S be a metric space and let x^{1} be an involution over S. We refer to the function x^{1} as "the reciprocal" in S. (Explanation: a metric space is a set over which a function d is defined that provides a distance between any two elements, the distance being a nonnegative real number. Here, a second function, x^{1}, is also defined on the set. This function maps elements of S back into elements of S in such a way that for any x in S, (x^{1})^{1}=x. This is what being an "involution" means.) I claim that this is enough to define both the arithmetic and harmonic means. As a first shot, we can try to redefine the arithmetic mean of a and b as "the unique element x in S for which d(a,x)=d(b,x) and d(a,x) is minimal". This will work if S is, for example, the reals, but over a general S nothing guarantees us that the element x defined in this way really is unique, or, in fact, exists at all. (Readers are welcome to come up with examples where it is not unique or does not exist.) We therefore refine the definition as follows:
and further define the harmonic mean by
Things to note about this style of definition:

List of solvers:Yu Gao (1 May 18:45)RaduAlexandru Todor (2 May 05:51) Dan Dima (2 May 18:45) Michael Blaszczyk (2 May 20:26) Claudio Baiocchi (3 May 17:54) Joseph DeVincentis (4 May 04:52) Lorenzo Gianferrari Pini (6 May 16:15) Xiaoyi Cao (9 May 18:24) Rui Viana (12 May 02:30) Kujou Miu (12 May 17:50) Mengxiao Zhang (18 May 11:35) Zilin Jiang (19 May 12:39) Itsik Horovitz (23 May 08:31) Harald Bögeholz (29 May 22:09) Oscar Volpatti (31 May 14:57) 
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Enjoy!