mathematics, that is, mathematics is studied in a markedly different way from other languages. If mathematics is a language, it is a different type of language from natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems. Mohan Ganesalingam has analysed mathematical language using tools from formal linguistics. Ganesalingam notes that some features of natural language are not necessary when analysing mathematical language (such as tense), but many of the same analytical tools can be used (such as context-free grammars). One important difference is that mathematical objects have clearly defined types, which can be explicitly defined in a text: "Effectively, we are allowed to introduce a word in one part of a sentence, and declare its part of speech in another; and this operation has no analogue in natural language." Arguments Indispensability argument for realism This argument, associated with Willard Quine and Hilary Putnam, is considered by Stephen Yablo to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets. The form of the argument is as follows. One must have ontological commitments to all entities that are indispensable to the best scientific theories, and to those entities only (commonly referred to as "all and only"). Mathematical entities are indispensable to the best scientific theories. Therefore, One must have ontological commitments to mathematical entities. The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism. Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the nominalist who wishes to exclude the existence of sets and non-Euclidean geometry, but to include the existence of quarks and other