Critical Thinking Toolkit: The Fallacy of Equivocation

This is one of many brief articles I am writing about how to avoid fallacious patterns of thinking. Here, we briefly discuss the fallacy of equivocation. Before I try to define it, it will be helpful to see an example of the fallacy in action. (I take this example out of the Wikipedia page forContinue reading “Critical Thinking Toolkit: The Fallacy of Equivocation”

Critical Thinking Toolkit: External and Internal Contradictions

When developing critical thinking skills, learning to recognize falsehood is as important as learning to recognize truth. This is important for many reasons. Recognizing falsehood helps you realize when you are making mistakes, when others might be making mistakes or using confusing language, and can help you find out the truth by process of elimination.Continue reading “Critical Thinking Toolkit: External and Internal Contradictions”

Critical Thinking Toolkit: Valid and Sound Arguments

This is the first post I am making in what will hopefully become a long and detailed series of posts about how to think more clearly about difficult questions. Since we ought to try to think clearly in every domain of life, we must begin the discussion at the broadest level, with the most importantContinue reading “Critical Thinking Toolkit: Valid and Sound Arguments”

Proof by Contrapositive

Sometimes when you are trying to solve a problem, you realize you really don’t have a lot of information to start with. One piece of good advice in problem-solving is to try to work backwards. That is, sometimes if you know what you want your solution to look like, you can backtrack to learn somethingContinue reading “Proof by Contrapositive”

Proof by Contradiction

The proof method that we will talk about here is quite different than many others. In his famous book A Mathematician’s Apology, the great mathematician G.H. Hardy made an analogy between this proof style, which we call a proof by contradiction, to a gambit in chess. So before I try to analyze what this proofContinue reading “Proof by Contradiction”

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”

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”

What Is a Proof?

I’ve talked a fair amount in some of the earlier posts about the idea of a proof. Now that we’ve developed a conceptual underpinning of what that means, we can see one in action. I hope my readers enjoy this as much as I do, as what we will discuss here is among my favorites.Continue reading “What Is a Proof?”