The article by Michael Barany highlights the evolution of calculus, moving from its intuitive beginnings in the 17th century to a more formalized foundation in the 19th century. It traces the silos of professional mathematics until Carl Gauss and Bertrand Russell challenged the notion of abstract mathematics being tied to physical reality. Then, as calculus expanded, mathematicians like Weierstrass reshaped its framework, leading to its widespread acceptance and formalization.
### Transformation with Weierstrass
In 1872, Weierstrass created the ‘Pillars of Calculus’ with his function, γ(x), defined as Σ((-1)^n * cos(x/3^n))/(2^n). This function is continuous everywhere but nowhere differentiable, challenging intuitive notions of change in calculus. This rejection specified rigor led to the development of two school of thought: earlier approaches prioritized intuition with informal arguments, while later attempts relied on precise, formal definitions. Central to these were Cauchy’s Peano axioms, which emphasized continuity, and Weierstrass’s rigor, championed by famous mathematicians like Cantor and Dedekind.
### Shift Toward Abstract Pillars
The author illustrates that calculus pillars—continuity, differentiability, and integrability—state both general and specific. According to Euclidean standards, continuity, the inability to find derivatives at any point, and integrability (ability to calculate areas) are interconnected. This synthesis, common to real numbers, Pythagorean geometry, and elliptical functions, led to the Pillars of Calculus. These pillars form the cornerstones of modern calculus, organizing its abstract concepts.
### The Rearing of the herd
The pillars of calculus, such as continuity, differentiability, and integrability, remain constant while the rigor beneath gives birth to its spreading influence. Weierstrass’s contribution solidified the foundation, despite initial hesitations. The abstraction in calculus, with every noun beyond algebra, became pivotal for its applications in physics, engineering, and beyond, extending to geology and biophysics.
### Conclusion
Calculus, reimagined through its pillars, transcend mere intuition and becomes a universal language of change. Weierstrass’s example, by challenging intuitions, rekindled the spirit of rigorous questioning, leading to a foundation that no longer distinctions between specific curves and the continuous remain. The pillars bear the marks of history, offering a harmonious blend of technical precision and conceptual depth.
