Abstraction and/or concreteness - What should be emphasized

Question

Alexandar Grothendieck was probably a mathematician focusing on theory developement and abstraction much much more than focusing on concrete examples and/or problems. In his biography, he wrote:

If you think of a theorem to be proved as a nut to be opened, so as to reach “the nourishing flesh protected by the shell”, then the hammer and chisel principle is: “put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks—and you are satisfied”.

I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration...the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it...yet it finally surrounds the resistant substance.

Saharon Shelah remarked that

I have always felt that examples usually just confuse you (though not always), having always specific properties that are traps as they do not hold in general. Note that by ``generality'' I mean I prefer, e.g., to look more at general complete first order theories (possibly uncountable) rather than at simple groups of finite Morley rank.

And there comes Paul Halmos:

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.

Although Grothendieck didn't mentioned explicitly of getting used to generalizations and/or concrete examples, his method of work (and the second and third analogy) usally reflects to extremely focus on the former. Saharon Shelah mentioned of getting used to generalizations and Paul Halmos (speaking nothing of abandoning abstraction) advices to focus on explicit examples.

Given that (probably) Examples build intuition, aid our visual, geometrical and physical instincts and abstraction and/or generalizations cover a broad range of objects, interconnections and big picture; which aspect(s; ?) should one prefer in mathematics and why ?

Answer 1

There is no universal agreement among experts that humans have a moral obligation to do mathematics in a certain way. The variety of mathematical activities present in the modern world is a consequence of the fact that different people do mathematics in different ways for different reasons.

What can be seen in the quotations which you've listed is that though people may do math in different ways for different reasons, they all seem to find some sense of satisfaction in doing it the way they do. This search for satisfaction is perhaps the only "way" of doing math, otherwise you are unlikely to continue doing that which will never satisfy you.

If it is your purpose to seek out a more "concrete" or "practical" reason for doing math one way as opposed to another, then you are unlikely to find the satisfaction you seek, for there are, and will likely always be, contrary opinions on why one way is better than an other (and each opinion is likely to be upheld by a so-called "expert" as you've demonstrated).

Finally, if you find satisfaction in the mathematics, then you are likely to share it with others.