The scope of logic:

The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable intimations concerning the nature and constitution of the human mind” [2, p. 1]. Logic is normative, not descriptive. For, the laws of thought do not “manifest their presence otherwise than by merely prescribing the conditions of formal inference” [2, p. 419]. They are, “properly speaking, the laws of right reasoning only” [2, p. 408]. So they “form but a part of the system of laws by which the actual processes of reasoning, whether right or wrong, are governed” [2, p. 409]. Boole’s idea that logic is the study of the laws of thought was taken over by Hilbert. According to him logic is “a discipline which expresses the structure of all our thought” [31, p. 125]. This is possible because our understanding “does not exercise any mysterious art, on the contrary, it proceeds only according to certain well-defined rules” [29, p. 9]. Such rules “form a closed system that can be discovered and definitively stated”. They are captured by the formal systems of proof theory, where the “formula game is carried out according to certain definite rules in which the technique of our thinking is expressed”. The fundamental idea of proof theory is “none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds”. Making such a protocol is possible because “thinking, it so happens, parallels speaking and writing: we form statements and place them one behind another” [30, p. 475]. In the formal systems of proof theory each symbol denotes “an object of our thought” [28, p. 131]. Moreover, “the axioms and provable propositions, that is, the formulas resulting from this procedure, are copies of the thoughts constituting customary mathematics as it has developed till now” [30, p. 465]. Hilbert assumes that there is a strict correspondence between our thinking and the formal systems of proof theory. This agrees with a logical tradition where Leibniz maintains that what he calls ‘mathematical logic’ is “an art of forming and ordering