By José Ferreirós

ISBN-10: 0691167516

ISBN-13: 9780691167510

This booklet provides a brand new method of the epistemology of arithmetic through viewing arithmetic as a human task whose wisdom is in detail associated with perform. Charting an exhilarating new course within the philosophy of arithmetic, José Ferreirós makes use of the an important notion of a continuum to supply an account of the improvement of mathematical wisdom that displays the particular event of doing math and is sensible of the perceived objectivity of mathematical results.

Describing a traditionally orientated, agent-based philosophy of arithmetic, Ferreirós indicates how the mathematical culture developed from Euclidean geometry to the genuine numbers and set-theoretic buildings. He argues for the necessity to consider a complete net of mathematical and different practices which are discovered and associated through brokers, and whose interaction acts as a constraint. Ferreirós demonstrates how complex arithmetic, faraway from being a priori, relies on hypotheses, unlike straight forward math, which has robust cognitive and sensible roots and consequently enjoys certainty.

Offering a wealth of philosophical and historic insights, Mathematical wisdom and the interaction of Practices demanding situations us to reconsider a few of our most elementary assumptions approximately arithmetic, its objectivity, and its dating to tradition and science.

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Mathematics in this period focused on structures, complex relational systems that can be characterized by morphisms. And yet, in spite of these profound changes which some have called a rebirth of the discipline, despite all the foundational changes and the reshaping of theories, many mathematical practices remained largely the same. Take as clear examples a whole series of rules for the resolution of algebraic equations (regardless of the epochal change in algebra from a theory of equations to the modern theory of structures), for the finding of roots by approximation, or the integration and differentiation of functions (notwithstanding the famous rigorization of the calculus).

The case is very interesting, because the rise of modern mathematics around 1900 (with careful reconstructions of algebra, analysis, and even geometry along the lines first of so-­called “arithmetization” and then of set-­theoretic approaches, achieving “full rigor,” as it was then said, at the price of eliminating intuition) and the subsequent development of foundational studies have both led to an anti-­K antian orientation that reduced visual thinking to a merely heuristic role, emphasizing its lack of rigor.

The goal here is to select those methods that are judged most appropriate and relevant for a given subject matter, and to rework systematically a theoretical corpus so as to develop it in accordance with that approach. , in mathematics, the art of posing questions is more relevant than that of solving them. 18 Value judgments are of the utmost importance in such cases—judgments about the most adequate conceptual framework; about the “purity” of the methods employed with respect to the subject matter; about which methods or ways of proving theorems are more fruitful, more explanatory; and so on (see section 2 for references).

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Mathematical Knowledge and the Interplay of Practices by José Ferreirós

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