When I was at school, mathematics fascinated me. I used to spend weeks with calculating and deriving different formulas, filling notebooks and sheets of papers day and night. Complex numbers were one of my interests. But I haven’t done everything rigorously, and one day, when I showed some notes to my math teacher, she shook […]
Two simple proofs of the infinity of primes Prime numbers are whole numbers that are only divisible by 1 and themselves. Students first learn about them in middle school, around the time when teachers are first introducing pesky equations with weird not-numbers like x and y. Remember these factor trees? All you need to know […]
Poincaré Poincaré’s Philosophy of Mathematics Was his philosophy of mathematics underrated? 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 […]
Samuelson Implications of the Efficient-Market Hypothesis “The baseball announcer has it, of course, conveniently all wrong. Ted Williams is due to hit because he hasn’t hit for seven days. That’s red noise. Ted Williams is hot, he’s sure to hit tomorrow because he’s been hitting for seven days. That’s blue noise. The better description of […]
The Earth and Venus are altos. Mercury sings soprano and Mars, tenor. Jupiter and Saturn together provide the bass. It’s an idea that dates back to the Pythagoreans. The heavenly orbs resonate to produce mathematical music, audible to the spirit, if not to the ears. We may regard Pythagoreans as those mystic mathematicians that brought […]
Firstly we can say that logical identity is reflexive. In other words, everything (or every thing) is identical to itself. In symbols: ∀x (x = x) To translate: For every x (or for every thing), x must equal x (or everything must be identical to itself). That is, everything has the (quasi)-relation of self-identity.. The logic of identity is also symmetrical. In symbols: If a = b, then b = a This can also be expressed […]
Since Leibniz and Newton developed Calculus, the scientific world and all its disciplines have not been the same. From physics and engineering to biology and mathematics, calculus permeates through the modern sciences as an indispensable tool in the search for truth and inventions. However, there are different ways of looking at it. From a physical and practical view, it has […]
Few mathematical theorems are ever covered & publicized to the greater mainstream (aka non-academic) crowd. Sometimes, because, well, math is boring to sane people, but mainly because many of these theorems are brutally convoluted & unnecessarily hard to grasp. The sweet spot between simplicity & significance is rarely found in the general textbook of unsolved […]
How many spatial dimensions do we live in? How many can we directly perceive? Are the two answers the same? If they are different, what effect, if any, do any higher dimensions have on us? What Is a Dimension? We exist in physical space. In this space, every point or location can be labeled by […]
“I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But […]
Algorithm Binary search is ubiquitous. Even if you are not aware of it, you have most likely used some (approximate) version of binary search one way or another 07 Dec 2020 — 14 min read Photo by Florencia Viadana on Unsplash Binary search is ubiquitous. Even if you are not aware of it, you have most likely used […]
Motion of a moving particle is considered to be one of the most instructive and useful physical systems one can study. In a real world case, such systems may exhibit immense complexity and intractability. But if we are lucky, we may be able to isolate the moving particle from unwanted effects coupled to the system, […]
It seems like the most common introduction to the Calculus of Variations is to talk about the Brachistochrone problem. It goes like this. Imagine you have a bead that can slide along a frictionless wire under the influence of gravity only. You want the bead to slide (starting from rest) from point 1 to point […]
The invention of group testing dates back to WWII, as the U.S. army wanted to weed out all syphilitic men called up for enlistment. The presence of syphilis could be identified with a blood test. However, individually testing the blood sample of each new recruit costed lots of time and resources. Thus, the U.S. army […]