# Assumptions

## More on Uncountable Irrationals

(An archive question of the week) While I was researching for the post on uncountable sets, I ran across a discussion that didn’t quite fit, but raises interesting questions about how countable and uncountable sets can fit together. How can the rational numbers be countable, but the irrational numbers, which are closely intertwined with them, …

## Uncertain Sequences

We’ve often pointed out that pattern or sequence problems, when nothing is given but a list of numbers, are not really math, in the sense that there is no one correct answer. They are psychology questions: What would a math teacher think is an interesting sequence to ask about? Mathematically, any number could come next, …

As I slow down the site for the summer, I plan to run a couple series of connected posts, one per week, on subjects that have seemed too large to cover in one post. I’m starting with questions about the structure of mathematics, particularly the postulates and theorems that are common in geometry classes. This …

## When Math Doesn’t Make Sense

(Archive Question of the Week) One of my favorite questions, from 2001, asked about how to convince a skeptical friend, when a clear mathematical result goes against their intuition. Why should they believe the math? It led me into thoughts about the relationship of intuition to math, whether (and when) math can be trusted, and …

## What Role Should a Figure Play in a Proof?

Questions about geometric proofs have often been handicapped by the inability to show us the associated figure (until we made that easier to do on this new site). In principle, that should not be a problem, because the statement to be proved should contain all the necessary information. It should never be necessary to refer …

## How Do You Know That Events Are Equally Likely?

(Archive Question of the Week) Having recently discussed a couple different issues that touched on the relationship of math to reality, I was reminded of this old favorite – a  question that is not asked often enough, and reveals some of the “dirty little secrets” behind math. Math is not reality; it is often used …

## Tracking Down the Meaning of a Problem

(Archive question of the week) Sometimes, we get a question that seems familiar, and can look back at past questions to see if we have already answered it, or to get ideas about what it means. This is the story of a problem that seems to float around as folklore, in varied forms. At first …

## How to Find an Inverse Function: Conflicting Approaches

While pondering issues that often give students trouble in algebra, I decided to check what we have said in Ask Dr. Math about inverse functions. I discovered four answers (all, as it happens, written by me – I tend to be attracted to certain topics!) to essentially the same question, spread over 13 years. It …