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Part 3: Mathematical Thinking and Rigor, First Steps.

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Title: Part 3: Mathematical Thinking and Rigor, First Steps.
Authors: Moreno-Armella, Luis1 lmorenoa@cinvestav.mx, Brady, Corey2 corey.brady@smu.edu
Source: Mathematics Enthusiast. Oct2025, Vol. 22 Issue 3, p229-256. 28p.
Subject Terms: *Algebra, *Intuition, Ground motion, Mathematical forms, Arithmetic
Abstract: Agustín thus sought to abandon the unsafe use of inductive methods in algebra, as Leonardo had employed them, and to rather recover Euclid's ideal that is crystallized in the Elements of Geometry. Nevertheless, his idea of a limit is grounded in the dynamism of motion or a process: It is an embodied idea. From here, a conceptual exploration of functions and their possible properties begins. The effort of conceptual reorganization that Agustín undertakes is remarkable, and it prepares the ground for the in-depth research suggested by his work. With this arose a tangible need need to review the forms of existence of mathematical objects and what they meant. What was continuity in arithmetic reality? and How can we measure the size of a set? As mathematics pursued these questions, it became almost possible to abandon confidence in intuition completely, and this led to (among other consequences) a profound rupture: the dissociation of continuity from the differentiability of a function. All this awaits us below. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics Enthusiast is the property of Prof. Bharath Sriraman and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Education Research Complete
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  Data: Part 3: Mathematical Thinking and Rigor, First Steps.
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+Enthusiast%22">Mathematics Enthusiast</searchLink>. Oct2025, Vol. 22 Issue 3, p229-256. 28p.
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  Data: Agustín thus sought to abandon the unsafe use of inductive methods in algebra, as Leonardo had employed them, and to rather recover Euclid's ideal that is crystallized in the Elements of Geometry. Nevertheless, his idea of a limit is grounded in the dynamism of motion or a process: It is an embodied idea. From here, a conceptual exploration of functions and their possible properties begins. The effort of conceptual reorganization that Agustín undertakes is remarkable, and it prepares the ground for the in-depth research suggested by his work. With this arose a tangible need need to review the forms of existence of mathematical objects and what they meant. What was continuity in arithmetic reality? and How can we measure the size of a set? As mathematics pursued these questions, it became almost possible to abandon confidence in intuition completely, and this led to (among other consequences) a profound rupture: the dissociation of continuity from the differentiability of a function. All this awaits us below. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematics Enthusiast is the property of Prof. Bharath Sriraman and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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