The Quadratic Formula (Solution)

Having discussed the motivation of why something like a “quadratic formula” is a useful thing to discover and understand, I’d like to work through some of the ideas that might lead one to discover a quadratic formula. Reducing the Number of Unknowns Remember that the equation we care about is , with (so that thisContinue reading “The Quadratic Formula (Solution)”

The Quadratic Formula (Problem)

This problem is something that is pretty well known from our high school education – the quadratic formula. In a follow-up post, I will present the solution through a shortened version of the process by which a mathematician might have gone about finding a solution. Equations are among the most important objects in mathematics. InContinue reading “The Quadratic Formula (Problem)”

Evaluating a Premise: Don’t be a “Mere Skeptic”

This is the last of the initial series of discussions I am putting forward about evaluating premises in an argument – or more colloquially, evaluating anything that somebody tells you is true. We’ve gone through a variety of nuances so far – about properly evaluating probabilities, when people do and do not need to justifyContinue reading “Evaluating a Premise: Don’t be a “Mere Skeptic””

Current Happenings and Future Plans for the Blog

Life has been quite hectic lately. It’s increasingly difficult to find time to write on the blog. I should have more time to write a few weeks from now, and once that time comes I should be able to make progress on a lot of the ideas I’ve been wanting to develop here. For theContinue reading “Current Happenings and Future Plans for the Blog”

Evaluating a Premise: Get the Categories Right

This is another brief, but important, note within our discussion of evaluating claims that people make while trying to prove a point. The shortest way to say this is that there is more than one type of claim. You should evaluate claims based upon the category into which they fit. I could see this summaryContinue reading “Evaluating a Premise: Get the Categories Right”

Evaluating a Premise: More on Basic Beliefs

In a previous post in this series, I discussed what are called properly basic beliefs. Because this is a frequently misunderstood idea in my experience thus far, I thought I’d add a dedicated but brief exposition of the idea and when it is and is not appropriate. A really great example of this goes backContinue reading “Evaluating a Premise: More on Basic Beliefs”

Evaluating a Premise: Weighing Evidence with Bayes’ Theorem

When we are involved in important discussions, it is important to not take pivotal claims of others – or ourselves – merely at face value. Instead, we debate the various available ideas. That is, we bring forth evidence that we have thought about and that we believe supports our position, and we listen to theContinue reading “Evaluating a Premise: Weighing Evidence with Bayes’ Theorem”

Evaluating a Premise: What Does it Mean to “Know”?

One of the most common and comical aspects of the lives of very young children that I find quite joyful can be summarized in one word… “Why?” Well, it is more like a sequence of these “Why” questions over and over again until your head is about to explode. Even though there is a sillinessContinue reading “Evaluating a Premise: What Does it Mean to “Know”?”

Evaluating a Premise: The Burden of Proof

One important aspect of philosophical arguments is whether or not there exist any “default positions.” That is, if I am trying to convince someone about X and that person is trying to convince me of the opposite of X, is either of us in some kind of default position? When is an idea innocent untilContinue reading “Evaluating a Premise: The Burden of Proof”

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 “Valid and Sound Arguments”