After graduating from "Cambridge University, Rowbottom studied under "Howard Jerome Keisler at the "University of Wisconsin–Madison, earning his "Ph.D. degree in 1964, with a thesis entitled Large Cardinals and Small Constructible Sets, under the supervision of "Jerome Keisler. With a recommendation from "Georg Kreisel, he took a position at the "University of Bristol in 1965, where he spent the rest of his professional career.
He published a paper called "Some strong axioms of infinity incompatible with the axiom of constructibility" in the Annals of Mathematical Logic, 3 1971. This paper, together with his thesis, "showed that Ramsey cardinals were weaker than measurable cardinals, and that their existence implied the constructible real continuum was countable; he further proved that this followed also from weaker partition and two cardinal properties." The "large cardinal notion of "Rowbottom cardinals is named after him, as is the notion of a Rowbottom "ultrafilter.
"Keith Devlin studied "set theory under Rowbottom. In 1992 he and a student, Jonathan Chapman, wrote a textbook on "topos theory, Relative Category Theory and Geometric Morphisms: A Logical Approach, published in Oxford Logic Guides, No. 16. Rowbottom retired in 1993 at the age of 55.