Paul Joseph Salomon Benacerraf, a professor of the philosophy of mathematics at Princeton University, passed away in 2025 at the age of 93. By raising a number of fundamental questions, he transformed the philosophy of mathematics from a primarily logical enterprise into a profound epistemological and metaphysical inquiry. His papers are widely regarded as among the most influential works in twentieth-century philosophy of mathematics and played a pivotal role in the emergence and development of mathematical structuralism. This paper examines the two articles that brought him international recognition and briefly reviews the reactions they elicited from other philosophers. Benacerraf argued that numbers lack an intrinsic identity independent of the mathematical structures in which they occur; what matters is the “role” they occupy within a numerical structure. He also formulated a challenge that later became known as the “Benacerraf Problem”: how is it possible for human beings to have knowledge of abstract mathematical objects.
Keywords: numbers; the Benacerraf problem; mathematical epistemology; structuralism.

Extended Abstract
Introduction
Paul Benacerraf is widely recognized as one of the most influential philosophers of mathematics in the twentieth century. His work fundamentally reshaped philosophical debates by shifting attention from the logical foundations of mathematics to deeper epistemological and metaphysical questions. Rather than asking merely what mathematical objects are, Benacerraf challenged philosophers to explain how human beings can possess knowledge of abstract mathematical entities. The present study examines Benacerraf's philosophical contributions through an analytical review of his two landmark papers, “What Numbers Could Not Be (1965)” and “Mathematical Truth (1973)”, which transformed contemporary philosophy of mathematics. The article also reviews the intellectual responses of later philosophers and evaluates Benacerraf's lasting influence on mathematical structuralism. Special attention is devoted to his rejection of the intrinsic identity of numbers and to the epistemological challenge that has become known as the “Benacerraf Problem”. By exploring these ideas, the study demonstrates how Benacerraf redirected philosophical inquiry from the ontology of numbers toward the nature and possibility of mathematical knowledge.

Materials and Methods
This research adopts a qualitative and analytical approach based on documentary and philosophical analysis. The primary sources consist of Benacerraf's seminal papers published in 1965 and 1973, supplemented by his edited volume with Hilary Putnam and major secondary literature written by contemporary philosophers of mathematics, including Stewart Shapiro, Michael Resnik, Hartry Field, Penelope Maddy, Charles Parsons, Mark Colyvan, Øystein Linnebo, Gideon Rosen, and others. The study systematically examines Benacerraf's principal arguments concerning the ontology of numbers, structuralism, and mathematical epistemology. It also compares his views with major philosophical schools, including Platonism, Logicism, Formalism, Intuitionism, Nominalism, and Structuralism. Furthermore, the historical development of his ideas and their influence on subsequent philosophical debates are investigated through a critical review of published scholarly literature and citation records. The research method is therefore interpretative rather than empirical, aiming to reconstruct Benacerraf's arguments, assess their logical coherence, and evaluate their philosophical significance within the broader development of modern philosophy of mathematics.

Discussion and Results
The analysis demonstrates that Benacerraf introduced two revolutionary arguments that continue to shape contemporary philosophy of mathematics. In his “1965 paper”, he criticized the traditional identification of numbers with particular sets. By comparing the set-theoretical constructions proposed by John von Neumann and Ernst Zermelo, he argued that equally valid mathematical systems assign different set-theoretic identities to the same numbers. Since no principled reason exists for preferring one representation over another, numbers cannot possess unique intrinsic identities. Their significance lies solely in the structural positions they occupy within mathematical systems. This conclusion provided one of the strongest philosophical motivations for mathematical structuralism, according to which mathematical objects are defined relationally rather than independently. The second major contribution emerged in his “1973 paper”, where Benacerraf shifted philosophical attention from ontology to epistemology. He argued that an adequate philosophy of mathematics must satisfy both semantic and epistemological requirements. While Platonism successfully explains the objectivity and necessity of mathematical truths, it fails to explain how finite human beings can acquire knowledge of causally inert abstract objects. Conversely, nominalist or formalist approaches often provide more plausible accounts of mathematical knowledge but struggle to explain the apparent objectivity and truth of mathematics. This tension became known as the Benacerraf Problem and remains one of the central unresolved issues in contemporary philosophy. The study further shows that Benacerraf's work generated extensive philosophical debate. Structuralists such as Stewart Shapiro expanded his relational conception of mathematical objects by providing more elaborate theories of mathematical structures. Nominalists, including Hartry Field, attempted to eliminate commitment to abstract mathematical entities altogether, while naturalists such as Penelope Maddy questioned Benacerraf's assumption that mathematical knowledge requires causal interaction. Other philosophers, including Charles Parsons, Mark Colyvan, Gideon Rosen, and Øystein Linnebo, proposed alternative responses addressing either the ontological or epistemological dimensions of the problem. Despite these diverse approaches, none has achieved universal acceptance, confirming the enduring philosophical significance of Benacerraf's challenge. The article also emphasizes Benacerraf's broader intellectual legacy. Although he published relatively few papers, his writings have received thousands of scholarly citations and fundamentally redirected research in philosophy of mathematics, influencing debates in metaphysics, epistemology, philosophy of language, and philosophy of science. His ideas contributed significantly to the emergence of modern mathematical structuralism and continue to serve as reference points in contemporary discussions concerning mathematical realism and abstract objects.

Conclusion
This study concludes that Paul Benacerraf transformed the philosophy of mathematics by demonstrating that questions concerning the identity of mathematical objects cannot be separated from questions concerning mathematical knowledge. His rejection of the intrinsic identity of numbers provided a powerful philosophical foundation for structuralism, while his epistemological challenge exposed significant weaknesses in both realist and anti-realist accounts of mathematics. The Benacerraf Problem continues to resist definitive resolution because no existing philosophical theory simultaneously offers a fully satisfactory account of both the truth and the knowability of mathematics. Consequently, Benacerraf's work remains central to contemporary philosophical inquiry and continues to inspire new approaches to ontology, epistemology, and the foundations of mathematics. His contributions represent a lasting shift from viewing mathematics merely as a formal logical discipline toward understanding it as a profound philosophical investigation into the nature of abstract knowledge and human cognition.