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Lógos and Máthēma 2

Studies in the Philosophy of Logic and Mathematics

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Roman Murawski

The volume consists of thirteen papers devoted to various problems of the philosophy of logic and mathematics. They can be divided into two groups. The first group contains papers devoted to some general problems of the philosophy of mathematics whereas the second group – papers devoted to the history of logic in Poland and to the work of Polish logicians and math-ematicians in the philosophy of mathematics and logic. Among considered problems are: meaning of reverse mathematics, proof in mathematics, the status of Church’s Thesis, phenomenology in the philosophy of mathematics, mathematics vs. theology, the problem of truth, philosophy of logic and mathematics in the interwar Poland.
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On the Distinction Proof–Truth in Mathematics

On the Distinction Proof–Truth in Mathematics

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Concepts of proof and truth are (even in mathematics) ambiguous. It is commonly accepted that proof is the ultimate warrant for a mathematical proposition, that proof is a source of truth in mathematics. One can say that a proposition A is true if it holds in a considered structure or if we can prove it. But what is a proof? And what is truth?

The axiomatic method was considered (since Plato, Aristotle and Euclid) to be the best method to justify and to organize mathematical knowledge. The first mature and most representative example of its usage in mathematics were Elements of Euclid. They established a pattern of a scientific theory and a paradigm in mathematics. Since Euclid till the end of the 19th century, mathematics was developed as an axiomatic (in fact rather a quasi-axiomatic) theory based on axioms and postulates. Proofs of theorems contained several gaps – in fact the lists of axioms and postulates were not complete, one freely used in proofs various “obvious” truths or referred to the intuition. Proofs were informal and intuitive, they were rather demonstrations; and the very concept of a proof was of a psychological (and not of a logical) nature. Note that almost no attention was paid to the precization and specification of the language of theories – in fact the language of theories was simply the unprecise colloquial language. One should also note here that in fact till the end of the 19th century, mathematicians were convinced that axioms and postulates...

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