A delta for your epsilon

Time to fact-check Barsenas again!

As a matter of fact, a great deal of Berkeley's works were dirct [sic] assaults against the science of his time. In his essay "The Analyst" he attempted to undercut calculus by attacking its very logical foundations, particularly regarding fluxion or infinitesimal change (referenced here). An attack that mathematicians did not fully recover from until Abraham Robinson's 1966 book Non-standard Analysis.

This is partially correct. When Newton and Leibniz developed their calculus, it was all built upon the notion of "fluxions", or very tiny quantities. Newton came up with so many cool results that worked so well that he neglected to actually prove his results in a rigorous fashion. So yes, calculus was initially built on very shaky foundations, and Newton didn't really care.

However, plenty of mathematicians found this unsatisfactory, not just this Berkeley character. Cauchy started the main drive to make calculus a rigorous field (which is now called "analysis"). Weierstrass improved Cauchy's work. However, both of these mathematicians worked in the 19th century, not the 20th.

For a more enlightening look at this topic, check out this article.