Two kinds of mathematics
Mathematics, at almost any level, can be either practical or ideal. Problems arise when the two kinds are confused.
Greece vs. Bablylon
explanation
Junk mail
Making things easy
New technology often makes a task much easier to do. That doesn’t necessarily mean it’s done better.
School’s out
Exactly useless
Stages of learning a foreign language
Our consultants are mostly concerned with teaching science and mathematics, claiming no expertise in the very different skill of teaching language. They have, however, learned several, which gives some insight into the process.
The ancestral line
Genealogy and history
Some people put tremendous effort into extending and filling in their family trees. In the end, what purpose does it serve?
Ideal lies
Why learn something that isn’t true?
As we progress in school, the lessons get more difficult and complicated. Sometimes they’re difficult because we have to unlearn things.
The tools of a professional
A ladder is often essential
What a photographer needs to do the job is sometimes far from obvious.
Proving nothing
Narrowing it down
Proving that something doesn’t exist is hard, though it can be done. More often, scientists work out more and more restrictions on the characteristics something can have, until the idea has no place left to hide.
Indirect arguments
Simple is not always true
A common feature of paradoxers is a confusion between a simple argument and a correct one.
