Philosophy of Science

Abstract 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.

Abstract The study of the certainty and inexorability of mathematical theorems is one of the important topics in the philosophy of mathematics. Various schools such as logicism, intuitionism, Platonism and naturalism have tried to present theories on this subject. These schools usually seek the foundation for mathematics in order to justify the certainty of mathematical theorems and logic. This has always been accompanied by numerous failures. In this article, we try to examine this issue from Wittgenstein's point of view. That is, instead of asking "What is the foundation of logic and mathematics?" we ask "Why does mathematics need the foundation?". Therefore, we first examine foundationalism. Then we show that mathematics is a language-game. In this regard, we examine geometry and logic as two language-games in mathematics. Finally, we show that, according to Wittgenstein's philosophy, the certainty and inexorability of a valid inference or a correct calculation comes from the practical procedure of inference and calculation. Keywords Wittgenstein, practice, philosophy of mathematics, language-game, foundationalism Extended Abstract 1. Introduction The question “Where does the certainty of logical and mathematical propositions come from?” has been the central preoccupation of the philosophy of mathematics. Logicism (Frege, Russell) sought to reduce mathematics to logic, thereby grounding its certainty in the apparent indubitability of logical truths. Intuitionism (Brouwer) located the foundation in basic mental constructions, arguing that mathematical objects are products of intuitive temporal processes. Platonism posits a mind‑independent realm of abstract entities, making mathematical truth a matter of discovery rather than invention. Naturalism treats mathematics as continuous with empirical science, justifying its claims through their overall role in successful theories. Despite their differences, these schools share a foundationalist assumption: that the certainty of mathematics must be justified by appealing to a privileged class of basic, self‑evident beliefs or entities. However, as we show, each programme faces insurmountable difficulties—regress problems, circularity, or the inability to account for the actual practices of mathematicians. This paper, therefore, proposes a radical shift: instead of seeking a foundation, we investigate why mathematics is thought to need one. Drawing on Wittgenstein’s later philosophy, we argue that the search for a foundation is itself a grammatical misunderstanding. Mathematics, like any language‑game, is not a structure built upon a base; it is an activity embedded in our form of life. Its certainty is not derived from something external but is exhibited in the way we use its rules. Our aim is to articulate this practice‑based view and to show how it resolves the puzzle of mathematical necessity without metaphysical commitments. 2. Materials and Methods This study is a philosophical analysis, not an empirical experiment. Our method consists of a critical reinterpretation of Wittgenstein’s later works, primarily Philosophical Investigations, Remarks on the Foundations of Mathematics, and On Certainty, combined with a historical and comparative examination of foundationalist projects. We adopt the following conceptual tools: · Language‑game: we treat mathematics as a multiplicity of language‑games, each with its own vocabulary and rules. A language‑game is a rule‑governed activity that is learned and practiced in a community. · Rule‑following: we analyse the notion that obeying a rule is a practice. A rule cannot be a private mental object; it is publicly taught, learned, and applied in repeated situations. · Grammar: we distinguish between empirical propositions (which can be true or false) and grammatical rules (which determine what counts as a meaningful statement and what counts as a valid inference). Questions about the truth of rules are category mistakes. To illustrate our method, we examine two case studies: 1. Geometry as a language‑game: we reconstruct Hilbert’s axiomatic method as a paradigmatic example. Hilbert’s primitive terms (point, line, plane) receive their meaning solely through the axioms, which function as rules for manipulating symbols. By changing a single axiom (the parallel postulate), we obtain new, consistent language‑games (Euclidean vs. hyperbolic geometry). This shows that geometry is not about a pre‑existing spatial reality but about a formal system whose “truth” is replaced by consistency and usability. 2. Logic as a language‑game: we analyse propositional logic, with its vocabulary (sentence letters, connectives) and rules (modus ponens, double negation, non‑contradiction). We argue that logical rules are not discovered a priori but are constitutive of what we call “reasoning”. They are part of the normative framework of our inferential practices. Additionally, we employ comparative anthropological data (Lévy‑Bruhl on the Ipon tribe; counting practices in the Torres Strait) to demonstrate that alternative inferential and computational practices exist, which suggests that our logical and mathematical rules are contingent upon our social forms of life rather than being universally necessary. 3. Discussion and Results Our analysis reveals that foundationalism fails because it mistakes the grammatical status of rules for the propositional status of factual claims. When we ask for the justification of modus ponens or of the principle of non‑contradiction, we are trying to measure our measuring rod. The certainty of these rules is not a property of their correspondence with reality; it is a reflection of their role as standards of correctness within the practice of inference. Wittgenstein explicitly states: “Inference is part of a language‑game. And the person who makes logical inferences follows certain instructions that have been given to him in learning the language‑game” (RFM, p. 378). Thus, the “hardness” of logic is not a metaphysical necessity but a pragmatic one: we call something “valid inference” only if it conforms to these rules. This is analogous to using a metre stick: we cannot question the correctness of the metre stick itself in the same way we question the length of a table, because the metre stick is the standard by which correctness is determined. The two case studies support this conclusion. In geometry, the shift from Euclidean to non‑Euclidean systems shows that no axiom is intrinsically certain; its role is determined by the game we choose to play. The consistency of the system is all that matters for internal validity. In logic, the law of non‑contradiction is not a high‑level empirical generalisation; it is a rule that tells us when to stop playing—it signals that a contradiction has arisen and we cannot continue with the same moves. As Wittgenstein notes, “Contradiction can only occur among the rules of a game” (Notebooks, p. 321). Its force is not ontological but procedural. Our anthropological examples further illustrate that alternative practices are entirely coherent within their own contexts. The Ipon tribe do not infer death from a wound in the way we do; they invoke sorcery. This does not mean they are irrational; they are following a different set of rules for explanation. The Torres Strait counting method is not an inferior version of our arithmetic; it is a different practical technique for enumeration. These examples underscore that what we call “necessary” is only necessary relative to the practices we have adopted. There is no trans‑cultural, trans‑historical foundation for arithmetic or logic; there is only the practical skill and shared training that enable us to count and infer in the ways we do. Consequently, the certainty of a valid inference or a correct calculation derives from our trust in the practice—our confidence, built through repeated successful applications, that following the rules will lead to a determinate outcome. This is not a blind faith but a trained ability. As Wittgenstein puts it: “I examine the proof and then accept its result. That is simply what we do. This is a custom and a practice among us, or a fact of our natural history” (RFM, p. 61). The “inexorability” of mathematics is therefore not a property of the propositions themselves but of our commitment to the practice. 4. Conclusion We conclude that the traditional foundationalist project for mathematics is misconceived. The later Wittgenstein offers a viable alternative: mathematics is a language‑game, and its rules are not in need of external justification because they are constitutive of the activity of reasoning and calculating. The certainty and inexorability that we attribute to mathematical and logical propositions arise from the grammatical role these rules play within the practical procedure (practice) of inference and computation. This certainty is a form of practical trust and skill, not a reflection of metaphysical necessity. Our analysis shows: 1. The demand for a foundation stems from a grammatical error: confusing a rule with a proposition that can be true or false. 2. Hilbert’s axiomatic geometry and classical logic are best understood as language‑games with their own internal standards of correctness; these standards are arbitrary in the sense that they could be different, but they are not arbitrary in the sense that they are freely chosen by individuals—they are embedded in our shared forms of life. 3. The “hardness” of logic and mathematics is the hardness of the practice itself; it is the result of our training and the social consensus that sustains the practice. 4. We do not need a foundation to justify mathematics; we need a clarification of its grammar. The task of philosophy is not to reform or justify mathematics but to describe what mathematicians actually do and how their practices hang together. In sum, the paper provides a robust Wittgensteinian response to the problem of mathematical certainty, showing that the practice‑based approach dissolves the classical foundationalist anxieties without reducing mathematics to mere convention. It invites philosophers and mathematicians to abandon the search for an indubitable ground and instead to appreciate the rich, dynamic, and socially embedded character of mathematical activity.