By Mohan Ganesalingam

The Language of arithmetic used to be offered the E.W. Beth Dissertation Prize for extraordinary dissertations within the fields of common sense, language, and data. It innovatively combines ideas from linguistics, philosophy of arithmetic, and computation to provide the 1st wide-ranging research of mathematical language. It focuses fairly on a style for opting for the entire which means of mathematical texts and on resolving technical deficiencies in all usual money owed of the principles of arithmetic. "The thesis does way over is needed for a PhD: it's extra like a lifetime's paintings packed into 3 years, and is a really remarkable achievement." Timothy Gowers

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**Extra info for The Language of Mathematics: A Linguistic and Philosophical Investigation**

**Example text**

Hartshorne, 1977, p. 1 As in this example, one always deﬁnes new entities in terms of entities that can already be described by the language, such as ‘the length of Mp over Sp ’. Thus deﬁnitions always contain enough information to fully specify the semantics of the material being deﬁned. Textual adaptivity may be compared to the way in which technical terms are introduced in texts in other ﬁelds. There are two diﬀerences that are worth emphasising. First, mathematical terms contain no vagueness, and as a result, deﬁnitions are perfect speciﬁcations of the entities they introduce.

However, deﬁnite descriptions like ‘the set of natural numbers’ abound in mathematical language, and the (mathematically important) presuppositions that they carry can be analysed in purely semantic terms. In the next chapter, we will also analyse certain selectional restrictions as being presuppositional in nature, because this analysis copes with the embedding of symbolic material inside text. For example, as neither ‘π is prime’ nor ‘π is not prime’ is felicitous, we will say that ‘prime’ presupposes that its argument is an integer; this analysis is well-suited to cope with phrases like ‘x is prime’, in which the properties of x cannot be compositionally determined.

This convention requires that in running prose each sentence in a text is a logical consequence of previous sentences. Thus arguments ﬂow from beginning to end in linear order; it is extremely rare that an author will, for example, state a fact and then elaborate on the reason why it is true. This convention reﬂects and supports the way in which mathematical texts are normally read; one typically reads a sentence, mentally veriﬁes it, and only then moves to the next sentence. In this respect, stating a fact in one sentence and elaborating it in later sentences is potentially confusing: some readers might attempt to verify the bare fact without looking forwards.