The mathematician Dirichlet (1805—1859), around 1830, proposed a definition for “function” that is essentially the modern set-theoretic definition of function, namely: If and If , Then . See a more detailed explanation here. The cool thing about this definition is that we don’t have to think about functions as “formulas” or “black boxes” anymore. It’s… [Read More]

## Root 2 is Irrational & So am I… (Proof & a Little History)

The Pythagoreans would say that the diagonal of a square is incommensurable with its side. That is, if you draw a square, where each side is measured 1, then what is the length of its diagonal? The answer is , an irrational number (one that cannot be written as a ratio, such as ). There… [Read More]

## 1 = 2? How to Destroy the Universe with Shitty Logic – Math 4 Monkeys

Did you know that 1=2? Don’t believe me? Check this out: Square both sides: Since , then: Now if we pull the to the left side, that should be equivalent to: Because . Now factor: Which means that and so therefore ! What’s… [Read More]

## Penetration: Mathematics & Philosophy – Quote: Leibniz

Without mathematics we cannot penetrate deeply into philosophy. Without philosophy we cannot penetrate deeply into mathematics. Without both we cannot penetrate deeply into anything. — Leibniz … he said, “penetrate”. Now go lift something heavy, Nick Horton

## Hagia Sophia & the Mathematicians

I got an email from my Grandmother this morning that included a scanned section of an old book on Byzantine Architecture. She’s been doing some scanning for my Mom, who is an art professor and teaches art history. The scanned section was describing the Basilica of the Hagia Sophia in Constantinople and how it was… [Read More]

## How to Teach Math: Jason Zegestowsky – Quote of the Day

Here is some great advice from a friend of mine who is a math teacher (and a lifter). This is how he likes to introduce his students to the process of math — which is exactly like the process of weightlifting… or the arts… or life: I like to say in the first class of… [Read More]

## The Tuba of Mathematics: M.N. Huxley Quote of the Day

“If mathematics were an orchestra, the exponentials would be the violins. The ρ(t) would be the flutes; they are introduced by the exponentials. The Poisson summation formula would be the tuba: powerful, but ridiculous when used too much” — M.N. Huxley From Area, Lattice Points and Exponential sums, by M.N. Huxley. Now go lift something… [Read More]

## “Behavioral Experiments in 15 Small-Scale Societies” vs Modern Economic (Wo)Man – Game Theory

Modern first-world societies are all FAR more like one another than they are like “small-scale” societies. Culture matters, but rarely in the way you think it does. We have a tendency to hype cultural distinctions (vs differences) that don’t matter — such as whether you grew up eating sushi or hotdogs. Among the cultural differences… [Read More]

## The Soul of Immensity Dwells in Minutia – Jacob Bernoulli Quote & Harmonic Series

“So the soul of immensity dwells in minutia. And in narrowest limits no limits inhere. What joy to discern the minute in infinity! The vast to perceive in the small, what divinity!” — Jacob Bernoulli in his Tractatus de Seriebus Infinitis (Treatise on Infinite Series), 1689 A divergent series is a sum such as Eventually… [Read More]

## Hermann Weyl: Quote of the Day (On the Shaky Foundations of Mathematics)

“The question of the foundations and the ultimate meaning of mathematics remains open; we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. “Mathamatizing” may well be a creative activity of man, like language or music, of primary originality, whose… [Read More]

## Henri Poincare: Quote of the Day (On K.I.S.S… ing?)

“It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.” — Poincare. Henri Poincaré is one of my favorite mathematicians, and one of the leaders in the development of what we now think of as the theories of Relativity. He… [Read More]

## Strength Is To Athlete As Math Is To Scientist

Come join me on a trip down the river mathematica… where biceps rule. The Problem: Too Much Learnin’ There is a TON of advice out there on strength training. But how do you know what advice applies to YOU, and what advice doesn’t? For instance: Exactly how strong a squatter do you NEED to be?… [Read More]

## The Propositional Calculus – How To Argue Like God & Win!

Part 1 – Why The Magic Of Logic Can Save Your Ass In The Eternal Battle With Evil Humans are dumb. Not only do we spend inordinate amounts of time arguing about nothing. But in most of our arguments we are talking right past each other. You are saying one thing I think you are… [Read More]

## The Geometric Heart Of Darkness – Euclid’s FAIL (part 2)

In the last post, I discussed how Euclid made 3 huge mistakes. However, they were so subtle, so under-the-radar that it took over 2,000 years for anyone to take notice! I also explained that filling up these “holes” in Euclid is the foundation of what we now think of as the ‘formalization of mathematics’ or… [Read More]

## Euclid’s 3 Big Screw Ups & How To Build The Language Of Love

“How could anyone hope to approach the concept of beauty without deeply studying the nature of formal patterns and their organizations and relationships to Mind.” – Douglas R. Hofstadter Euclid was wrong. He was also completely right. It is within the details of that pair of statements that a remarkable amount of modern (last 200… [Read More]