Education

The position of what’s sometimes called semantic realism is that every (grammatically-acceptable) statement (or proposition) is bivalent (i.e., it’s determinately true or false); as well as being evidence-transcendent (i.e., it’s true or false independently of our means of establishing its truth-value). Thus if we take any given p, we can state this: Proposition p is determinately true (or false) regardless of any proof, evidence, experimental data, […]

On September 26, 1905, Albert Einstein’s revolutionary paper Zur Elektrodynamik bewegter Körper (On the Electrodynamics of Moving Bodies) was published in the scientific journal Annalen der Physik (Annals of Physics). In it, Einstein describes his theory of special relativity, which provides fundamentally new definitions of the concepts of space and time. Special relativity reconciles electromagnetic theory (more specifically, Maxwell’s equations) with mechanics by introducing […]

Here’s a cute way to prove the existence of Transcendental numbers. It requires *only* a keenness for mathematics, and is combined with a quick guide to the infinite and Cantor’s Diagonalisation argument! We’ll also see that there are ‘more’ transcendental numbers than non-transcendental . Below: Cantor’s Diagonalisation Argument in base 2. More on this later! Cantor’s […]

PDE Laplace’s Equation Flows and fields Physics The Wave Equation Bonus: Separation of Variables Physics The Heat Equation: Inhomogeneous Boundary Conditions This is the third article in my series on partial differential equations. Before reading further you might want to read part one (9… Mathematics The Black-Scholes formula, explained Introduction to the most famous equation […]

Canadian mathematician Simon Kochen recalled in his tribute to Kurt Gödel how during his PhD exam, he was asked to name five of Gödel’s theorems. The essence of the question was that each of the theorems either gave birth to a new branch of, or revolutionized, modern mathematical logic. “Proof theory, model theory, recursion theory, […]

Pierre de Fermat was a genius in many fields. He was a lawyer, mathematician, and fluent in six languages. He contributed significantly to analytic geometry, differential calculus, and his best-known works to number theory in mathematics. So, during the study in mathematics, you come across him several times. I still have my notes from the […]

Last week, in a blog post titled Finding Patterns, I introduced a puzzle. I asked my teenager to solve it using only a pen/pencil and paper. The Puzzle Two integers differ by 22. Each, when multiplied by its successor, yields an eight-digit palindrome. What is the smaller of the two? Using a collaborative effort between human […]

Before we get started on this rather visual investigation into the nature of equations, we need to recall that an equation is a relation and that relation can be written down in more than one way. Throughout this article, it is important that you let go and forget about the old conventions that you learned […]

Computer-Aided Engineering (CAE) is getting more and more important in modern engineering. FEM-programs, simulation tools and huge libraries are offered to the engineer who has to decide what suits his problem best he’s facing. But to make a good decision you have to know at least the basic principles of the algorithms used in the […]

We’re living in the age of the next industrial revolution: the very first three freed most of the humans from hard labor. This one is aiming to take us over the last domain of human dominance on this planet: our intelligence. In this article, we will put aside ethical, political and social effects of such […]

Series The Mind-Boggling Properties of the Alternating Harmonic Series Riemann’s simple algorithm to approximate computationally any real number 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 […]

In my A totally new, very old method for finding square roots article, I described how I discovered an algorithm that actually happened to be very old and well-known: the “Babylonian method.” (See References, at the bottom for more on that topic.) Let’s try to work out how and why this old method works, but “from first principles”, […]

This post is about a beautiful result in the field of dimensionality reduction. It is arguably one of the most impressive and satisfying statements about dimensionality reduction in Euclidean spaces. It is named after the mathematicians who proved it in 1984, William B. Johnson (1944-) and Joram Lindenstrauss (1936–2012). Informally, the lemma says that, given N points with N coordinates each (i.e., these points […]

“Why is surprise the permanent condition of the political and economic elite?” The above (somewhat abbreviated) question is how Nassim Taleb and author Mark Blyth open their essay The Black Swan of Cairo, published in the May/June issue of Foreign Affairs in 2011. The essay takes note of the then-recent turmoil in Egypt, Tunisia and Libya (now know as “the Arab […]