Things Take as Long as They Take

My friend Darryl has been a meditation teacher for about ten years (besides being a medical doctor), and he told me a lovely story recently. A version of this story follows, which I have copied from here: A Buddhist monk approached his teacher and asked the Zen Master, “If I meditate very diligently, how long … Read more

Blue Snow

This past winter I noticed a curious phenomenon while I was shovelling snow. I had seen the same thing once the previous winter as well: What appears to be blue snow! The following photograph was taken on 25 December 2020 just before 11 am local time in St. Catharines, Ontario, where I currently live. I … Read more

On Brainwashing

Humans are typically very easy to brainwash, as experience proves. All you have to do is repeat the same lie often enough, and many people will believe you. If you are a beloved parent or other beloved authority figure, and you start repeating your lies when your children are young enough, then you can make … Read more

Ghosts of Departed Quantities

Calculus was developed by many workers, and their incremental progress was independently systematized by Newton and Leibnitz in the late 1600s. At that time the concept of limit had not been devised yet, and even the concept of a function was still in development, and there was not yet a precise definition of a function. … Read more

Humanity is in Trouble

Just about everyone nowadays realizes that humanity faces multiple crises, but a new article makes the point that even experts underestimate just how bad things are now and how precipitously our situation is likely to deteriorate. The short article is Underestimating the Challenges of Avoiding a Ghastly Future, by Bradshaw et al and it is … Read more

The Dunning-Kruger Effect and Learning

People tend to be biased when assessing their own abilities; this bias is called the Dunning-Kruger effect. Studies show that the poorer the ability, the greater the difference between one’s overestimate of one’s ability and one’s actual ability. One manifestation of this phenomenon is that people who know nothing about a subject and then learn … Read more

Linear Algebra is Scary, Even for Future Mathematicians

The Calculus of Friendship: What a Teacher and a Student Learned about Live While Corresponding about Math (2009) is a delightful book by Steven Strogatz. Strogatz is Jacob Gould Schurman Professor of Applied Mathematics at Cornell University, and he writes beautifully. The book is about his decades-long correspondence with one of his high-school mathematics teachers, … Read more

What is Infinity?

One of the common errors made by calculus learners is to consider infinity as a number. True, there are number systems such as the extended real number system in which infinity is successfully treated as a number, and you can ponder on them if you wish, but for our purposes at this level of learning … Read more

What is Calculus?

What is calculus and what is it used for? Calculus includes an enormous number of ideas, methods, and applications, and this post is an attempt to provide an overview. Most of the interesting phenomena that are analyzed scientifically involve change. The flow of wind and water, the orbits of the planets, the path of a … Read more