TITLE: What is(are) the reason(s) for defining things in the following way?
QUESTION [31 upvotes]: In this answer it is written that,

In modern mathematics, there's a tendency to define things in terms of what they do rather than in terms of what they are.

My questions are,


What is(are) the philosophical and mathematical reason(s) for doing so?

Has there been any criticism of this approach?

REPLY [2 votes]: They are the consequence of formalism, what is the core of the modern mathematics and it giant developing from principles of XX century.
The point is that it is irrelevant to ask about "what they are" because the answer is "nothing" or "anything".
For science exists a maxima that says: the form is the function, i.e. there is no difference between what they "are" and what they "do".
Notice that the question about "what they are" is metaphysical, variable and subjective. 
By the other hand the pragmatism is the core of any science and development, i.e. pragmatism is the requirement for objectivity, from the older Rome empire to today: you know what something is seeing what it does, what are their consequences.
The history of development of humanity is the history to transform "things" to "no-things". The reason is that when something "is" then it is restricted to what it is... but when something "is not" it is free to be used as you want.
One example of this transformation was the change of written languages based on symbols (by example ideograms or hieroglyphs) to languages based on signs (letters, syllables), with the transcendental apparition of the alphabet of the ancient Greek.