Yep.
Well, that's short answer.
For long answer, we must to remember that in old books of phys and math there are always entire chapters about magnitudes. In those books demonstrates that given the newton's laws, there are a well defined entity called mass, which was a magnitude just because have a correspondence between plus and times operations, given the relationship m=a/F. In geometry I remember Puig Adams books, where there are a chapter called "Magnitudes" (magnitudes in english?) where define geometrically distances, and demonstrate the interception between circle and line without the algebraic theory of reals, adding an axiom to the Euclides ones.
Point is: Valery show us a curve with the property that L1^L2 = k, where L1, L2 are the distance to two fix points. Math programs refuses to consider magnitude^magnitude, meter^meter in this case. But, what about k? Is it a magnitude? Probably, not but must to be something. If you want to see the same properties as those for plus and times, must to prove first a closure property: magnitude^magnitude is a magnitude. And I guess that this isn't true.
For example, any superior limit of oscillation of real up bounded real valued successions are a real number (axioma de completitud o axioma del supremo). This property is a closure property. Other closure properties are Plus is closed in N, but not the inverse, minus, which is closed in Z. Product is closed in Z, but the inverse is closed in Q. Pow is not closed in Q, but closed in C, and with restrictions, in R. They two inverses too. Vectorial spaces have their own closure properties. Probability theory too. So, they are important.
If closure property don't holds, new things appear, with some unusual properties, like the limit for a succession of functions. Dirac's delta is the simplest example: isn't a function, and have some unusual new properties.
Old books each time that introduces a new magnitude, take some pages to prove that they are well defined magnitudes. With the correspondence for plus and times. So, I guess that meter^meter are not a magnitude, but, given that can show a geometrical origin for that, must to be something with sense. So, maybe can't preserve all the magnitudes properties, like "in system evaluation" (if all are in ft, the result is with algebra and in ft). Some thing happend when starting with Peano's axioms, want to preserve properties to the new entities, Z, Q, R, C. Between R and C, for example, lost some importants things, like order: can't preserve all.
Best regards.
Alvaro.
PD: To Werner: "degree of distances" are the words that a no native english speaker, like me and Valery, could use to say measured length.
