Abstract
Abstract The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert’s Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic.
Highlights
For centuries mathematical proofs have been seen as special, different from any other kind of argument
As late as the 19th century, it turned out that some implicit assumptions were used and that a more complete treatment was needed in order to achieve the goal of having the system of geometry that is purely logical and does not depend on intuitive visualization
The underlying assumption, called sometimes Hilbert’s Thesis or the Frege-Hilbert Thesis, is as follows: Every real mathematical proof can be converted into a formal proof in the appropriate axiomatic theory
Summary
For centuries mathematical proofs have been seen as special, different from any other kind of argument. Hilbert’s famous remark that the objects of his system of geometry can be anything, for instance “tables, chairs, and beer mugs,” as long as they satisfy all the axioms This approach made possible a new variant of the axiomatic method; it slowly emerged in the 19th century. The underlying assumption, called sometimes Hilbert’s Thesis or the Frege-Hilbert Thesis, is as follows: Every real mathematical proof can be converted into a formal proof in the appropriate axiomatic theory. This attractive hypothesis has been, rejected by more and more philosophers of mathematics since at least the 1960s
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