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# INTRODUCTION

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# INTRODUCTION

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INTRODUCTION book

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## ABSTRACT

The philosopher and mathematician Gottlob Frege (1848-1925) pursued throughout his career a single project. He strove to settle an important question in the philosophy of arithmetic, the science of numbers: what is the source of our knowledge of arithmetic? According to Frege, the truth of arithmetic can be known merely by exercising the faculty of reason:

Frege published four books in his lifetime: Concept Script, a Formula Language of Pure Thought Modelled upon the Formula Language of Arithmetic (Begriffsschrift, eine der Arithmetischen nachgebildete Formelsprache des reinen Denkens, 1879), The Foundations of Arithmetic (Die Grundlagen der Arithmetik, 1884) and Basic Laws of Arithmetic I (Grundgesetze der Arithmetik I, 1893 and II, 1903). In the last two books he sets out to prove the truths of arithmetic from the laws of logic on the basis

of suitable definitions of concepts such as number. Frege’s first book, true to its title, develops a formula language for conducting inferences. Why does Frege first develop such a language? Why is he, in general, concerned with the meaning of language? He himself feels the need to answer this question. In a representative passage he writes: ‘Symbols have the same importance for thought that discovering how to use the wind to sail against the wind had for navigation. Thus, let no one despise symbols! A great deal depends on choosing them properly’ (CN: 84, 49). When we discovered how to sail against the wind, we discovered how to overcome the limitations imposed by an instrument by using this very same instrument. Language is an indispensable instrument of inferential thinking, but it has its limitations. Yet, we can overcome its limitations if we choose the symbols with which we think properly. Or so Frege argues. He attempts to design a language that expresses thoughts without the intermediary of sound. (‘Begriffsschrift’ translates as ‘concept script’ or, better, ‘ideography’. Later Frege remarks that ‘concept script’ is not the best name for his language. Perhaps ‘thought script’ would have been better.)1 In addition, the design of the Begriffsschrift answers to further specific demands arising from Frege’s scientific project. He aimed to give proofs of the laws of arithmetic in which every step is clearly set out. In the Begriffsschrift everything relevant for inference and nothing else is expressed in signs. The Begriffsschrift contains a logic and a theory of judgeable content, that is, a theory of what a statement says or how a judgement represents the world to be. Every Begriffsschrift sentence has as its judgeable content a circumstance, a complex constituted by particulars and properties.