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**1 - 2**of**2**### Non-Standard Models of Arithmetic: a Philosophical and Historical perspective MSc Thesis (Afstudeerscriptie)

, 2010

"... 1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5 ..."

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1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5

### From Lagrange to Frege: Functions and Expressions

, 2011

"... Part 1 I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung] ” of his formal system. It opens with the following claim ([34], § I.1, p. 5; [40], p. 33) 2: When one is concerned with specifying the original reference [Bedeutung] of the word ‘function’ in its mathematical usage, it is e ..."

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Part 1 I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung] ” of his formal system. It opens with the following claim ([34], § I.1, p. 5; [40], p. 33) 2: When one is concerned with specifying the original reference [Bedeutung] of the word ‘function’ in its mathematical usage, it is easy to fall into calling function of x an expression [Ausdruck] formed from ‘x ’ and particular numbers by means of the notations [Bezeichnungen] for sum, product, power, difference, and so on. This is inappropriate [unzutreffend] because in this way a function is depicted [hingestellt] as an expression—that is, as a concatenation of signs [Verbindung von Zeichen]—not as what is designated [Bezeichnete] thereby. Hence, instead of ‘expression’, one should say ‘reference of an expression’. Frege does not explicitly ascribe this inappropriateness to anyone, though he could have ascribed it to many 3. One is Lagrange, which, a little less than one century later, defined functions as follows, both in the Théorie des fonctions analytiques and in the Leçons sur le calcul des fonctions ([58], § 1, p. 1;