I do not think there is a conceivable way for science to undermine its own search for answers.
Science is the constant search for the truth in our natural world and beyond. And as long as we are looking for answers, it will never obsolesce itself; furthermore, there is no looking into its “very nature” to discover a way in which it undermines itself. Where, and what, is the other side of the debate? The only arguments in favor of this statement that I have seen are nothing more than religious preaching.
Considering those who believe science obsolesces itself, who are almost invariably religious, the reason they are not comfortable with science’s progression is because it leaves their holy books, especially the Bible, further and further behind admist the advancement of mankind. New discoveries consistently reduce their credibility by teaching us more about the world, and universe, in which we live. Examples of this include the existence of dinosaurs and the heliocentric theory of the solar system, which I will not get into here. The point is that religious people often find comfort in attacking the nature of science and endlessly searching for some fundamental flaw within it; which, incidentally, they cannot get people of other faiths to agree on.
Science has taught us just about everything we know about life, our world, and the universe. We have learned what gravity is and why it occurs; how are planet was formed; why we orbit a burning star along with other planets; why life has formed on earth and why it cannot form on other planets; how all life on earth today has evolved over millions of years from single-celled organisms; and the list goes on indefinitely. There really is no telling how much we have learned from science, and how much there still is to learn.
By one of its core definitions, “systematic knowledge of the physical or material world gained through observation and experimentation,” is there any possible way for it to become obsolete? I suppose ceasing to gain knowledge of the world through observation and experimentation is the only way, but calling this a far-off dream (as if anyone would dream of such an occurrence) is an understatement. As long as we are alive, human beings will never stop searching for answers. Knowledge is indeed a powerful ambition of our species that will never be broken.