Economics is widely recognized as an independent branch of knowledge, but this was not the case before Adam Smith’s The Wealth of Nations publication in 1776. Since the Sixteenth Century, Mercantilism had been the dominant school of thought among economists. The mercantilistic perspective views economy, which is already under governmental control, as a means at the […]
Before we begin, be aware that talking about electromagnetism in any meaningful way means talking about vector calculus. Please don’t be intimidated, even if you don’t have the first idea what any of the symbols or terms mean. Vector calculus is hard but its core ideas are intuitive and I will explain everything as we […]
The Goldbach conjecture is one of the most captivating unsolved mysteries in mathematics. In this article, I will take you on a journey through time and mathematics. I will present some other ways of looking at the conjecture than the original definition. Both visually and algebraically. We will prove an equivalence and investigate it a […]
Hypothesis testing has some similarities with one of the simplest methods of proof from mathematical logic: Proof by contradiction. Hypothesis testing can be thought of as being the probabilistic counterpart of proof by contradiction. Let us refresh our memories first about this method. How does proof by contradiction work? Let’s say we want to prove a […]
A theorem by two mathematical greats, which anybody, no maths experience required, can understand! An amazing and beautiful result accompanied by some pretty hand-coded visualisations! Regardless of your experience with math, this theorem can be understood, and the elegance of the solution appreciated. Throughout I will both use diagrams to explain concepts where possible. Where I use […]
Hilbert The Hardy space has three equivalent definitions. Each of them emphasizes a different aspect of the space and all three of these aspects… G.H. Hardy (1877–1947) and David Hilbert (1862–1943), whom Hardy- and Hilbert spaces are named after The Hardy space has three equivalent definitions. Each of them emphasizes a different aspect of the […]
Arguably the most important unsolved problem in pure mathematics today is to prove (or disprove) the Riemann hypothesis, which is intimately connected to the distribution of prime numbers. One of the basic techniques needed to understand the problem is called analytic continuation, which is the topic of this article. Analytic continuation is a technique from a branch of mathematics […]
In mathematics, especially when it comes to number theory, there are many basic things that we don’t know. Simple questions that we can’t answer. This can be realized by considering a few of the famous unanswered questions that we mathematicians would like to know the answer to. A few of them are Are there infinitely many […]
Kronecker Kronecker, God and the Integers Natural numbers were created by God, everything else is the work of men — Kronecker (1823–1891). Number Theory Perfect Numbers An ancient riddle Prime Numbers The Goldbach Conjecture An Equivalent Formulation of an Unsolved Mystery Number Theory The Euler Totient Function A Probabilistic Approach Number Theory Transcendental Numbers Into […]
What happens when logic goes wrong? Let’s see why the Law of the Excluded Middle is wrong (sometimes), discover foundational problems at the heart of Mathematics, before finally trying to resolve what’s going wrong. You’ll also learn how to make sense of the negative root of -1, delve briefly into analysis, as well as proving […]
Computational neuroscience, broadly defined, is the mathematical and physical modeling of neural processes at a chosen scale, from molecular and cellular to systems, for the purpose of understanding how the brain represents and processes information. The ultimate objective is to provide an understanding of how an organism takes in sensory information, how information is integrated […]
Proof Theory Methods of Proof for Every Occasion Three essential proof techniques, followed by many not-so-essential Topology Theorema Egregium and Pizza Eating How can Gauss improve your pizza eating technique? Statistics Maximum Likelihood Estimate Part 2: Mixture Models Probability Theory Maximum Likelihood Estimate Part 1: How to best fit a Gaussian Chaos Theory A Gentle […]
Most people may think that mathematics is about doing hard sums with large numbers. Whilst it is true that you can do seriously hard computations all day long (try computing the integral of the square root of tan), the majority of mathematics is based on proofs. Here I will outline some of the most common […]
Open Problem “Mathematics may not be ready for such problems.” The Collatz Conjecture was posed in 1937 by Lothar Collatz, two years after he received his Ph.D. It remains open to this day. “Mathematics may not be ready for such problems.” — Paul Erdős. “This is an extraordinarily difficult problem, completely out of reach of present day […]