“How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?” — Albert Einstein Inthe garden, through a microscope, or deep in the depths of the ocean, patterns are all around us if we know how to look for them. Whether it’s humans […]
On August 2. 2020 Cantor’s Paradise published Bruno Campello’s brief critique of Cantor’s approach in transfinite mathematics. The author raises some doubts about Cantor’s debunking Euclid’s 5th principle stating that the whole is greater than the part. Campello gives an interesting argument against Cantor’s reasoning, but it itself raises a number of question marks to say the […]
Back in June, I wrote this article about Noether’s Theorem, the idea that continuous symmetries and conservation laws are intrinsically linked. But, while Noether’s Theorem is one of the most beautiful concepts in physics, symmetries themselves are equally as fundamental, and often similarly beautiful. And further, much of the coal-face of theoretical physics is a consequence of […]
Mathematical modelling of uncertainty stands or falls on our ability to assess probabilities, but how do you know if your assessments are any good and what does that even mean? This article takes a pragmatic approach to answering these questions. We’ll start in the shadows of the valley of frequentism, step out on to the […]
Quantum field theory or QFT is the framework in which the Standard Model of particle physics (which includes the theories of the electroweak and strong nuclear interactions) is formulated. In particular, quantum electrodynamics or QED has predicted the values of physical quantities with unprecedented precision. For example, the magnetic dipole moment of the muon has a measured value of 233 […]
In Albert Einstein’s original formulation of general relativity (his theory of gravity), the fundamental field is the metric tensor g and the theory is covariant in spacetime. Covariance (more precisely general covariance) “consists of the invariance of the form of physical laws under arbitrary coordinate transformations.” The idea is that since coordinates are only human-made labels, physical laws shouldn’t depend on the […]
In 1982, Kahnemann, Slovic and Tversky published “Judgement under uncertainty: Heuristics and biases” and shattered humanity’s collective self-delusion that we had any functional intuition for even the most rudimentary problems in probability theory. This work has seen a renaissance in popularity since the publication of Kahnemann’s rather more accessible “Thinking fast and slow”. Kahnemann is […]
Since existence, humans have been trying to predict the weather. Early methods looked to astrology and the lunar phases. Even the Bible contains references to Jesus deciphering local weather patterns! And it makes sense, understanding the weather would offer immense advantages both in the battlefield and for agriculture. However, like many achievements, it took a […]
Analytic Number Theory The Abel Summation Formula Sums, Integrals, and Series 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 […]
Everything the human brain is capable of is the product of a complex symphony of interactions between many distinct signaling events and myriad of individual computations. The brain’s ability to learn, connect abstract concepts, adapt, and imagine, are all the result of this vast combinatorial computational space. Even elusive properties such as self-awareness and consciousness […]
Gödel Gödel’s Constitutional Quarrel “The examiner was intelligent enough to quickly quieten Gödel and say ‘Oh god, let’s not go into this’ and broke off the examination at this point, greatly to our relief” — Oskar Morgenstern Kurt Gödel (1906–1978) was the greatest logician who ever lived. At the age of 24, he Applied Statistics The […]
You may have heard of this incredibly large number called Graham’s number, but more often than not when this subject is discussed, Graham’s number is rarely given in context. What does it mean? Did mathematicians decide to write down a really big number one Sunday afternoon because they were bored? No. In fact, Graham’s number […]
From elementary mathematics, we are familiar with the four basic mathematical operations associated with numbers — addition, subtraction, multiplication and division. There are many kinds of numbers in mathematics with their own set of unique operations. Here we are concerned about the most commonly used numbers that we deal with everyday, which are technically called […]
Which student hasn’t been there? It’s exam time and the multiple-choice questions are waiting for you. Can’t be that hard to guess yourself to success, can it? An exam has 20 questions. Each question has four choices. Exactly one choice is correct for each question. You need at least 10 correct answers to pass. What […]