So I'm confused by the following. Let's say someone makes the claim:

Math is also a language game.

I can imagine 2 different kind of responses Wittgenstein might say:

  1. Indeed, it suffices to only provide examples of concepts one uses in language, and math is merely another example.
  2. You have taken 2 concepts and placed it outside their original intent, and this makes the statement meaningless.

Which (or neither) response of Wittgenstein would be likely? What is the correct Wittgenstein analysis of this claim?

I still haven't been able to assimilate this philosopher's ideas.

  • Wittgenstein devoted many many lectures to mathematics... see Wittgenstein’s Philosophy of Mathematics. Commented Feb 22, 2022 at 7:47
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    Wittgenstein analogizes Russell's Principia symbolism to a game in Remarks on the Foundations of Mathematics, and then adds: "But if you say "The point isn't understanding at all. Mathematical propositions are only positions in a game" that too is nonsense! 'Mathematics' is not a sharply delimited concept". In other words, mathematics is at least a collection of interconnected games, and, perhaps, more, vague as it is.
    – Conifold
    Commented Feb 22, 2022 at 7:57

2 Answers 2



There are several definitions of game with one often being this one. However, what is meant by a language game is not a language competition in this sense, but rather a cooperative decision making process in this sense. From WP:

A language-game (German: Sprachspiel) is a philosophical concept developed by Ludwig Wittgenstein, referring to simple examples of language use and the actions into which the language is woven. Wittgenstein argued that a word or even a sentence has meaning only as a result of the "rule" of the "game" being played. Depending on the context, for example, the utterance "Water!" could be an order, the answer to a question, or some other form of communication.

It is in this scenario with multiple agents executing decisions along strategic lines that the notion proceeds with the use of game also similar to how mathematics calls this process game theory. What is the thrust of Wittgenstein's point, however? It's the idea that languages themselves don't contain meaning. Rather, utterances and propositions are tools to achieving ends because much meaning is prelinguistic and not conceptual. Think about how a monkey, who cannot use a proper language is quite capable of acting as an agent.

Why is this significant? Because many philosophers somehow attribute meaning to words by excluding the agent that uses them from the process of language. In the philosophy of language, this is not contemporaneous, and linguists, scientists of language, do not see words as the sole bearer of meaning as any introduction to current pragmatics attests. More recent developments at the end of the twentieth century put meaning where it belongs, in the interpretation of the words by an agent. Doing so resolves many problems with seeing words as having meaning by having meaning be an experience rather than a property of an entity. In this regard, the language of philosophy has moved towards process philosophy in seeing meaning as a construction of a mental state rather as something tangible and disembodied. Cognitive linguistics has recently added empirical evidence to the idea that intelligence must be embodied.

In mathematics, the Wittgensteinian idea is simply this. That mathematical discourse is a community of experiences built around the exchange of symbols that have mathematical meanings to the mathematicians by and for the purposes of those mathematicians. This dovetails nicely with the scientifically palatable notions of mathematical constructivism which asserts that mathematicians are in the business of constructing a language that asserts facts about mathematical experience. This would be in contradistinction to the more traditional Platonic thinking that somehow mathematicians are engaged in discovering 'mathematical objects' such as 'Forms' that preexist independent of human thought. Ironically, most mathematicians reject mathematical empiricism and constructivism and believe that numbers and circles somehow exist independently of human thought.


Math and language is not a game. This is simply due to a misunderstanding of model theory. A game is played by rules and so anything played by rules can be thought of as a game. Alternatively, seeing these rules as constituting a syntax and grammar, we can think of a game as the writing of sentences, and hence as language games.

But this is only one half of what Wittgenstein's logic was about. The other side is the relation to truth, his logical space of facts, those facts that obtain of the world, this is the logical model and so model theory. So a language game is only a game when you merely view its formal side. But Wittgenstein's logic is a game of two halves and both have to be brought into play. When we do that, it is no longer merely a game, but a language talking about facts.

If you want to understand Wittgenstein's theory as it pertains to math, it would be worth looking at model theory.

The answer to your question, in Wittgensteinian terms, is 2 - you've taken the notion of language game outside its context, that is outside of its relation to truth. Have a look at the first few propositions of the Tractatus.

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    Why do you consider math not a game?
    – Jo Wehler
    Commented Feb 22, 2022 at 9:43
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    @JoWehler: I've just written a post explaining why. It's a science not a game. Wittgenstein's model theory was about modelling scientific languages as they pertain to a logical world of facts which are obtained about the world. Commented Feb 22, 2022 at 10:08
  • There is an approach to Model Theory based on Games Commented Feb 22, 2022 at 10:47
  • @MauroALLEGRANZA: I'm not talking about Game Theory, that's a different thing altogether. I'm taljing about languages considered as a game being played by formal rules. This is different. Commented Feb 22, 2022 at 10:50
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    You are advocating the wrong Wittgenstein here. The language game theory and how language gains meaning through performative acts is late Wittgenstein. There are many scholars who think that Tractatus is at odds with that later theory of Philosophical Investigations and On Certainty.
    – Philip Klöcking
    Commented Feb 22, 2022 at 13:44

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