The word induction refers to any thought pattern which moves from specific examples to more general principles. The best known example of induction is probably the scientific method – we collect data by repeating the same specific experiments multiple times, find regularities and make guesses about a potential overarching framework into which the data mightContinue reading “Proof by Induction”

# Author Archives: Will Craig

## How Would We Know COVID-19 is in Decline?

In light of the current COVID-19 pandemic, I have written an article entitled How Does Disease Spread? in which I discuss how mathematicians attempt to understand the way that a virus is spreading in the population. In said article, my primary focus is building up the concepts and thought processes that are needed to understandContinue reading “How Would We Know COVID-19 is in Decline?”

## How Does Disease Spread?

These are difficult times. Because of the spread of the COVID-19 virus, we are all taking very drastic measures to ‘flatten the curve’ by socially distancing, increasing our awareness of hygiene, and many other measures. These, of course, should be done, because lives will be saved. But how, exactly, do we know this? We don’tContinue reading “How Does Disease Spread?”

## Types of Proofs in Math

Previously, I have talked about logic and some of the most important rules of logic. These are quite important and useful in doing mathematics. However, it is necessary to go further, because logic is not specific enough. Mathematics analyzes patterns that involve concepts like shape, number, repetition, and symmetry. Pure logic does not adequately handleContinue reading “Types of Proofs in Math”

## Some Thoughts on Mental Health

One of my closest friends, on hearing I was going to start doing some writing, brought me the following questions she was curious to hear me address: Have you felt that math and the thought processes that you use to solve math problems have aided you in better understanding your mental health? Or made itContinue reading “Some Thoughts on Mental Health”

## The Foundation of Logic

There are a lot of different ways to approach understanding what mathematics truly is in modern times. And these different approaches are fundamentally different. Does one come from the angle of discussing the beauty of patterns? Or perhaps of the power of the human mind to move from specific examples to general conclusions (similar toContinue reading “The Foundation of Logic”

## Why All the Letters in Math?

One thing that many people dislike about math is all of its special symbols. Why does it need to be so specific? Why so many symbols? Why is writing things down in a specific format so important in math? Shouldn’t something be considered “correct” if it has the right ideas, regardless of how it isContinue reading “Why All the Letters in Math?”

## Pythagorean Triples? (Solution #1, Part 2)

In Part 1, we have begun discussing primitive Pythagorean triples, and thought a little bit about them. Now, we want to try to characterize all primitive triples. That is now our goal. Limiting The Possibilities Suppose we are given a primitive triple (a,b,c). Recall that this means that the three positive whole numbers a, b,Continue reading “Pythagorean Triples? (Solution #1, Part 2)”

## Pythagorean Triples? (Solution #1, Part 1)

(If you haven’t read the “Problem” post with the same title, go there first. This will make more sense if you do.) We want to find all the Pythagorean triples (a,b,c). The first thing a mathematician would probably do is to try some small examples, gather some information, and then look for patterns within thatContinue reading “Pythagorean Triples? (Solution #1, Part 1)”

## Pythagorean Triples? (Problem)

I’ve spent some time thinking about how I want to present things I love on this blog, and one format I’ve come up with is a Problem/Solution series. The idea here is to have two posts with the same name – in one of them, I will explain an interesting problem and try to giveContinue reading “Pythagorean Triples? (Problem)”