[EM] Mathematical equivalence (was Re: Democracy Chronicles, introductions)

[Kristofer Munsterhjelm]

On 04/24/2012 08:37 PM, Richard Fobes wrote:
> In the non-mathematical world the word "equivalent" means "having
> similar or identical effects" which allows for not _always_ being
> _identical_ in _all_ respects. That is the context for usage in the
> Democracy Chronicles article.

A context which is overriden by prefixing the word "equivalent" with 
"mathematically".

> Even in a rigorous academic mathematical context, "equivalent" means
> "having virtually identical or corresponding parts." In this context
> VoteFair popularity ranking is "virtually identical" to the
> Condorcet-Kemeny method because the word "virtually" allows for the
> _extremely_ _rare_ cases in which there are more than six candidates in
> the Smith set (which can possibly cause a difference in which candidate
> is declared the winner), and allows for an election involving, say, 30
> candidates that _can_ (but may not) result in different full rankings
> between the two methods.

Consider two functions f and g defined on the integers.

f(x, y) = x + y,

g(x, y) = x + y when |x-y| > 2
         = x * y otherwise.

Is f(x,y) mathematically equivalent to g(x, y)? I doubt it, even though 
the range of (x,y) pairs for which f(x, y) = g(x, y) is enormously 
greater than the range of (x,y) pairs for which that is not true*. But, 
of course, I should find something more weighty than my opinion and Andy's.

The paper at http://arxiv.org/abs/1009.3068 has the title "two 
mathematically equivalent versions of Maxwell's Equations". The synopsis 
then contains: "Here, we first show that there are two versions of 
Maxwell's equations...", i.e. that there are two ways of formulating the 
same equations.

Furthermore, the Wikipedia page on Gaussian gravity ( 
https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity ) uses the term 
"mathematically equivalent" multiple times -- to mean "mathematically 
the same" in each case, and http://cvxmod.net/builtinatoms.html states 
its Euclidean norm function is equivalent to sqrt(sum(square(expr))) - 
i.e. square root of sum of squares, which again is an exact correspondence.

I have not, however, found any pages or papers where "mathematically 
equivalent" is used to mean "virtually identical". The closest I could 
get was http://www.mathworks.se/help/toolbox/nnet/ref/tansig.html which 
says, quoting: "[The function] is mathematically equivalent to tanh(N). 
It differs in that it runs faster than the MATLAB implementation of 
tanh, but the results can have very small numerical differences". Still, 
here the text says that while the function does, in a mathematical 
sense, give identical results to tanh(N), roundoff errors can lead to 
different results in practice. In other words, "mathematical equivalence 
does not imply computational equivalence". That differs from the 
VoteFair case, because in the latter case, the algorithm itself gives a 
different result, even if you were to run it on rational number 
arithmetic rather than floating point.

Finally, in logic, two statements are equivalent in the case that they 
mutually imply each other. "X is greater than Y" is equivalent to "Y is 
less than X". While logic is not mathematics, the latter is relatively 
close to the former, and it would be more surprising if the definition 
of equivalence was changed than if it was not.

-

* If you state that it doesn't count, since there are infinite integer 
pairs for which g(x, y) will give x+y and for which g(x, y) will give 
x*y, then the argument works just as well if you restrict f and g to the 
integers between, say, +1 billion and -1 billion.