Mathematics – Cantor's Archive
Philosophy
Is ‘Philosophy of Mathematics’ somehow useful?
What the philosophy of mathematics is useful for? Max Black, the author of The Nature of Mathematics (1933), thought the main task of the foundation of mathematics (and, consequently the main task of any philosophy of mathematics) would be to elucidate “and analyze the notion of integer or natural number”
Series
An Amazing Formula
Generalizing Leibniz’ formula for π, the alternating harmonic series, and the Basel problem
Philosophy
Mathematics as Fiction: A Common Sense Approach
The following essay was written in 2014 when I was an undergraduate student at the University of Florida; it was the winner of the mathematics department’s Robert Long Prize for writing in the history and/or philosophy of mathematics. In the intervening time, my views on ontological/epistemological status
Philosophy
Beginner’s Guide to Mathematical Constructivism
The foundational crisis in mathematics along with roughly four decades following it, was likely the most fertile period in the history of logic and studies in the foundations. After discovering the set-theoretic paradoxes, such as the paradox of the set of all sets, together with the logical ones, like Russell’
Calculus
The Best Numerical Derivative Approximation Formulas
Approximating derivatives is a very important part of any numerical simulation. When it is no longer possible to analytically obtain a value for the derivative, for example when trying to simulate a complicated ODE. It is of much importance though, as getting it wrong can have detrimental effects on the
Geometry
Understanding Perpendicular Lines and Orthogonal Trajectories
When two lines are perpendicular (or orthogonal) with each other, it means they form a 90° angle when they intersect. If you were taught about perpendicular lines in Algebra or for the SAT, you were probably forced to memorize something like “the slope of the perpendicular line is the negative
Series
The Basel Problem: 1+1/2² + 1/3² +… = π²/6
Solving the Basel problem, using an elementary introduction to Fourier series! Part 0: Motivation, some History, and Welcome! Hello and welcome! We are going to be tackling a beautiful problem, the ‘Basel Problem’. It was posed in 1650 by Pietro Mengoli, and it took 84 years to solve it, when
Fractal
(Not Only Frozen) Fractals All Around
A Discussion of Fractals
Field Theory
A Finite Number Of “Numbers”
What exactly is arithmetic on a field of numbers which are merely elements in some set?
Babylonian
A Modern Look at Square Roots in the Babylonian Way
Revisiting the Babylonian method for square roots: why and how does it work?
Philosophy
The Demons of the Mathematicians
There is a rich history of mathematicians, philosophers, and other thinkers invoking demons (real or imaginary) in their reasonings. The tradition goes back at least to the iconic ancient Greek philosopher Socrates.
Mathematics
Cubic Polynomials — Managing the Architecture to Calculate Roots
Rotating cubic roots into managed quadratic segments where the math is easier!