Meaning: The quote "Well, I think Barkley and I were the only two who were working with Church for a Ph.D." by Stephen Kleene, a mathematician, refers to a significant period in the history of computer science and mathematics. The quote sheds light on the collaboration between Alonzo Church, a prominent mathematician and logician, and his students, Stephen Kleene and Barkley Rosser, in the pursuit of a Ph.D. degree. This collaboration played a pivotal role in the development of foundational concepts in the field of theoretical computer science, particularly in relation to computability and formal languages.

Alonzo Church, an American mathematician, is best known for his development of the lambda calculus, a formal system in mathematical logic and computer science. His work laid the groundwork for the understanding of computability and the theoretical limits of computation. Church's collaboration with his students, including Stephen Kleene, was instrumental in advancing the understanding of computability and formal systems, making significant contributions to the field of theoretical computer science.

Stephen Kleene, a prominent figure in mathematical logic and theoretical computer science, made substantial contributions to the development of recursive function theory and the foundations of computability. Kleene's work, including his collaboration with Alonzo Church, significantly influenced the understanding of computability and formal languages, shaping the theoretical underpinnings of modern computer science.

The collaboration between Alonzo Church, Stephen Kleene, and Barkley Rosser for their Ph.D. studies reflects a pivotal period in the history of theoretical computer science. Their work contributed to the development of foundational concepts in the field, including the Church-Turing thesis, which formalized the notion of computability and laid the groundwork for the theory of computation.

The quote also highlights the close-knit academic environment in which these influential mathematicians and computer scientists operated. The collaboration between Church, Kleene, and Rosser exemplifies the mentorship and collaborative spirit that characterized the early development of theoretical computer science, with students working closely with their advisors to advance the understanding of fundamental concepts in the field.

In summary, the quote by Stephen Kleene provides insight into the collaborative efforts of Alonzo Church and his students, including Kleene himself, in the pursuit of Ph.D. degrees. Their collaboration was instrumental in advancing the understanding of computability, formal languages, and the theoretical foundations of computer science. The work of these mathematicians and computer scientists continues to have a profound impact on the field of theoretical computer science, shaping the way we understand the fundamental limits and capabilities of computation.