Numerical Analysis Deriving the Quadratic Equation From the Roots Up Solving Quadratics using gradients at the roots Numerical Analysis Sextic Polynomials — A Practical Approach At a young age a builder taught me how to mark out a right angle using the 3,4,5 triangle rule with a tape measure. That math practicality has stuck with […]
Physics Another Approach To Understand Accelerated Motion 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 […]
This story is also available on Kindle! “If the dismissal of Jewish scientists means the annihilation of contemporary German science, then we shall do without science for a few years!” — Adolf Hilter Prior to World War II, Germany had led the world in science for more than one hundred and fifty years. Its reputation for excellence in […]
Here we come in our journey through the past across the first substantial error in science that I want to talk about. Aristarchus measured the distance to the Sun to be 20 times more than that from the Earth to the Moon, which was off by the factor of 20. To clarify, however, it did […]
Before terrorist Theodore John Kaczynski (1942-) began sending mail-bombs to faculty members at various American universities, he had a promising career in mathematics. In particular, between 1964–69, he published a total of six single-authored research papers in renowned mathematical journals, including The American Mathematical Monthly and Proceedings of the American Mathematical Society. The young Kaczynski did work in […]
Probability Theory Why Probability Theory is Hard It’s not because you’re stupid or weren’t concentrating in school Probability Theory Transforming Scores Into Probability How to show what subjective scores do and do not tell you about probability of success on the basis of their past performance Pragmatism Probability for Pragmatists What does it mean that […]
Last time, we solved some very simple IBVPs for the heat equation in which there was no forcing and T(0,t)=T(L,t)=0°C. These problems had the form: The equation Tₜ-α²Tₓₓ=0 is called the homogeneous heat equation. From now on, we will use α² for the diffusivity instead of k/ρc. A variable in a subscript means a partial derivative with respect to […]
Logic Formal Logic vs. Material Logic? In a purely logical argument, even if the premises aren’t in any way (semantically) connected to the conclusion, the argument may still be both valid and sound. Professor Edwin D. Mares displays what he sees as a problem with purely formal logic when he offers us the following example Logic Lewis […]
Factorials I am sure you all remember the surprise you had the first time you saw a “5!” or “7!” I am sure you all remember the surprise you had the first time you saw a “5!” or “7!”, that exclamation mark, and the realization that this was formal mathematical notation… We define the factorial of a […]
The brain is an amazingly complex machine for processing information. Whatever your philosophy of mind is, you can’t deny that it depends in large part on the brain’s ability to process incoming signal data and convert that data into higher level understanding. These days brain-inspired techniques have taken over Artificial Intelligence (AI) in a way […]
In math there are a few, select, magical constants that transverse all branches. These constants, steadily discovered throughout our collective history, provide the numerical infrastructure to our everyday lives; like chemical elements in the periodic table, special constants in math are foundational. To name a few, we have zero (0), circular-darling pi (~3.142), square-root of negative one […]
In part one we introduced holomorphic and meromorphic functions along with their interesting properties. In this part, we shall continue down this road, but with a focus on yet another amazing fact about holomorphic functions. Before stating these famous results let’s get a feeling for different representations of functions. Let’s define the function g as the following It […]
Electromagnetism Electricity, Gravity and Statics Electrogravitostatics Physics Is it time to retire Newton’s Gravitational Constant? Mathematical operations that involve Newton’s gravitational constant are extremely counter intuitive at the subatomic level. Physics Black Holes, explained What they are, how they form, types & shapes, and how they die Computer Science About Scientific Theories What amount of […]
Philosophy The Backstage of Mathematical Thinking: How Our Brains Bring About the Abstract A brief survey of recent insights from cognitive science 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 […]