Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer. 



Meaning of "quantitative result"
Quantitative is typically used as opposed to qualitative. In the geometric context referred to by the OP, the distinction is explained as follows:



> 
> Quantitative geometry and topology refines the qualitative, discrete
>  questions of algebraic and geometric topology into continuous ones.
>  For example, we may see a loop in a space which is homotopically
>  trivial and ask how difficult it is to trivialize. Depending on what
>  we mean by "difficult", we might obtain different notions of
>  isoperimetry; one common choice is the area of a filling disk, which
>  leads to the definition of the Dehn function of a group.
> 
> 
> A priori, such notions usually depend on the choice of a metric on the
>  space; one can then analyze their dependence on the metric
>  (quantitative geometry) or show results, for example asymptotic ones,
>  which are independent of the choice (quantitative topology).
> 
> 
> 


[source](https://web.math.ucsb.edu/~manin//qgt-workshop.html)


