# Essays in the Philosophy and History of Logic and by Roman Murawski

By Roman Murawski

The booklet is a suite of the author's chosen works within the philosophy and historical past of common sense and arithmetic. Papers partly I comprise either basic surveys of latest philosophy of arithmetic in addition to reviews dedicated to really expert subject matters, like Cantor's philosophy of set idea, the Church thesis and its epistemological prestige, the background of the philosophical historical past of the idea that of quantity, the structuralist epistemology of arithmetic and the phenomenological philosophy of arithmetic. half II includes essays within the heritage of common sense and arithmetic. They deal with such matters because the philosophical historical past of the improvement of symbolism in mathematical common sense, Giuseppe Peano and his function within the production of up to date logical symbolism, Emil L. Post's works in mathematical common sense and recursion conception, the formalist tuition within the foundations of arithmetic and the algebra of good judgment in England within the nineteenth century. The historical past of arithmetic and common sense in Poland is additionally thought of. This quantity is of curiosity to historians and philosophers of technological know-how and arithmetic in addition to to logicians and mathematicians drawn to the philosophy and heritage in their fields

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Note also that Hilbert’s views were changing over the years, but always took a formalist direction. 2. , using the axiomatic method. He viewed the latter as holding the key to a systematic organization of any sufﬁciently developed subject. This idea was very well stated already in a letter of 29th December 1899 to G. Frege where Truth vs. Provability 43 Hilbert explained his motives of axiomatizing the geometry and wrote (cf. Frege 1976, p. 2 In “Axiomatisches Denken” (1918, p. 405) Hilbert wrote: When we put together the facts of a given more or less comprehensive ﬁeld of our knowledge, then we notice soon that those facts can be ordered.

To be able to study seriously mathematics and mathematical proofs one should ﬁrst of all deﬁne accurately the notion of a proof. In fact, the concept of a proof used in mathematical practice is intuitive, loose and vague, it has clearly a subjective character. This does not cause much trouble in practice. On the other hand if one wants to study mathematics as a science – as Hilbert did – then one needs a precise notion of proof. This was provided by mathematical logic. In works of G. Frege and B.

And in the footnote b to (1972) Gödel added: What Hilbert means by ‘Anschauung’ is substantially Kant’s space-time intuition conﬁned, however, to conﬁgurations of a ﬁnite number of discrete objects. Note that it is Hilbert’s insistence on concrete knowledge that makes ﬁnitary mathematics so surprisingly weak and excludes many things that are just as incontrovertibly evident to everybody as ﬁnitary number theory. , while any primitive recursive deﬁnition is ﬁnitary, the general principle of primitive recursive deﬁnition is not a ﬁnitary proposition, because it contains the abstract concept of function.