Attached is a criticism (perhaps, a little fanatic) of the nature of
mathematical research, written by a certain [sic] Prof. Kline in 1970s
(minimum editing has been done to reduce the length of the article).
Can somebody please come forward with a good defense?
What needs to be defended? Almost everything that is said in the article can equally well be said of research in (say) Physics or Computer Science or Biology or, for that matter, (modern) Art or (classical/modern) Music.
To play the devil's advocate  if someone is willing to pay some people to do Mathematics and these people (the mathematicians) enjoy doing it then that is reason enough. When people decide that they do not need so many research mathematicians they will not pay for them and there will be less of them!
So perhaps the question for mathematicians is "How do we convince people to continue to pay us?"^{1} The applied mathematician's answer is "We should give people something they can use." The pure mathematician's answer is "We should give people something they can enjoy."
However, as long as the "Big Divide" between the mathematical ability of the "common man" and the "research mathematician" remains (and continues to widen) there is no real hope of either of these noble aims succeeding. Even the "common scientist/engineer" finds it difficult enough to understand/enjoy/apply mathematics that is more than 150 years old.
So I think that the real answer is "We should teach (young) people to enjoy and apply mathematics."
I have found that trying to explain "elementary" mathematics to mathematically immature audiences sharpens one's own understanding of it and leads to interesting research problems as well.

This applies only if we want mathematicians in the future to be paid. Our own livelihood does not really depend on this answer. ↩