Mathematics as a Language: Wittgenstein’s Influence on the Philosophy of Mathematics

The philosophy of mathematics has long been a vibrant area of philosophical inquiry, raising fundamental questions about the nature of mathematical objects, the epistemology of mathematical knowledge, and the role of mathematical language in conveying truths. Ludwig Wittgenstein, known for his exploration of the limits of language and the relationship between language and the world, brought fresh and transformative perspectives to these enduring questions. Wittgenstein’s influence has been nothing short of profound, challenging conventional views and encouraging a more language-centric understanding of mathematics.

Photo by Waldemar on Unsplash

Wittgenstein’s Linguistic Turn

Wittgenstein’s early work in the “Tractatus Logico-Philosophicus” sought to establish a logical structure underlying language, reducing complex propositions to simpler atomic facts. This phase of his philosophy was characterized by a highly formal and systematic approach. However, he soon became dissatisfied with the “Tractatus” and started to reevaluate his earlier positions.

Wittgenstein’s later philosophy, as articulated in the “Philosophical Investigations,” marked a departure from the reductionist approach of the “Tractatus.” He shifted his focus to the idea that language and meaning are deeply intertwined. He famously asserted, “The limits of my language mean the limits of my world,” emphasizing that language is not just a tool for expressing pre-existing thoughts but plays a fundamental role in shaping our understanding of reality.

A central aspect of Wittgenstein’s linguistic turn is the concept of “language games.” Wittgenstein proposed that language games are rule-based activities, and meaning arises from the use of language within these games. He described various language games, such as giving orders, making requests, describing objects, and more. The meaning of words and sentences in these games is contextual and depends on the rules governing the particular language game being played.

Wittgenstein also introduced the compelling concept that mathematical propositions and operations can be likened to language games. These language games have specific rules and conventions, and the meaning of mathematical statements is derived from their use within these linguistic practices. Wittgenstein’s view challenges the traditional Platonist notion that mathematical objects exist independently of human thought and language.

A Challenge to Formalism

Formalism is one of the big three of “ism” in the philosophy of mathematics. According to formalism, mathematics is primarily concerned with the manipulation of symbols and the derivation of theorems. It asserts that mathematical entities are abstract, symbolic constructs with no inherent connection to the real world. The focus in formalism is on the syntactic aspects of mathematics, emphasizing the internal consistency of mathematical systems.

Wittgenstein’s philosophy, particularly his later ideas in the “Philosophical Investigations,” challenges the formalist perspective in several ways:

1. The Language Game Perspective: Wittgenstein’s notion of language games emphasizes the contextual and rule-based nature of language. Mathematical language, according to Wittgenstein, is a particular language game with its own rules and conventions. This perspective challenges formalism’s notion of an entirely abstract and detached mathematical language. Wittgenstein’s approach suggests that mathematical symbols and statements gain their meaning and significance from their use within mathematical language games and from their relation to real-world practices.
2. The Social and Linguistic Nature of Mathematics: Wittgenstein highlights the communal and linguistic aspects of mathematical practices. According to his view, mathematical knowledge is not just a matter of manipulating symbols but is deeply tied to the social and linguistic conventions of mathematical communities. This perspective contrasts with the formalist notion that mathematics is a solitary and purely symbolic endeavor.
3. Certainty and Proof: Wittgenstein challenges the formalist emphasis on rigorous formal proofs as the primary source of mathematical certainty. He suggests that mathematical certainty is related to the rules and conventions of a particular language game. This challenges the idea that mathematical certainty can be solely derived from abstract symbol manipulation.

Contemporary Debates

Wittgenstein’s philosophy continues to be highly relevant in contemporary discussions within the philosophy of mathematics. It informs ongoing debates about mathematical intuition, the nature of mathematical objects, and the significance of mathematical practice in shaping mathematical concepts. The exploration of his ideas enriches and deepens our understanding of the philosophy of mathematics in the 21st century, challenging scholars to consider new perspectives and refine existing positions.

1. Platonism vs. Nominalism: Wittgenstein’s ideas have provoked fresh debates concerning the perennial philosophical conflict between mathematical Platonism and nominalism. Platonists argue for the objective existence of mathematical objects, while nominalists maintain that they are human constructs. Wittgenstein’s emphasis on language and practice adds complexity to this debate, raising questions about how different philosophical positions relate to mathematical language games.
2. The Nature of Mathematical Objects: The debate about the nature of mathematical objects persists in light of Wittgenstein’s linguistic turn. Scholars examine whether mathematical objects have an existence independent of human language and cognition or if they are linguistic entities embedded in particular language games. This debate has implications for how we conceptualize the foundations of mathematics.
3. Formal Proofs and Certainty: Wittgenstein’s challenge to the role of formal proofs in establishing mathematical certainty sparks debates about the reliability of formalism in contemporary mathematics. Some argue that formal proofs continue to be indispensable for ensuring mathematical rigor, while others contend that other forms of validation and reasoning are equally valid in the mathematical practice.

Ludwig Wittgenstein’s philosophy has left an indelible mark on the philosophy of mathematics. By challenging traditional views and emphasizing the linguistic and social dimensions of mathematical understanding, Wittgenstein’s influence expands our comprehension of this abstract and intriguing discipline. As we continue to explore the implications of his ideas, we gain new insights into the language of mathematics and the ways in which it shapes our understanding of the world. Wittgenstein’s enduring influence stands as a testament to the transformative power of philosophy in shaping our perspectives on the world and the nature of mathematical truth. Wittgenstein’s legacy continues to inspire, provoke, and guide the ongoing exploration of mathematics as a language.

Further Readings:

1. “Philosophical Investigations” by Ludwig Wittgenstein
2. “Wittgenstein on Rules and Private Language” by Saul Kripke