Having started with some thought experiments in the original article, today I want to start to make some concrete progress.

We're starting in the middle of a large map here. Where we are now is the place of knowing about numbers and how we use them in every day life. We can add, multiply, divide, maybe do some trigonometry. In each direction are slight variations on this theme. Removing properties, as we'll do today, or adding in extra features. Some of these directions are no doubt familiar to people, others will be new.

For anybody wanting to follow this series, without having to read any of my other writing, this Atom feed or this web link will contain just the articles in this feed (the Atom feed contains the full text of each article).

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Topics: mathematics/series: deriving

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Another multi-part series of posts (along with storytelling) I want to try here is laying out my thinking about how we can try to understand (and teach?) formal mathematics. Nothing too revolutionary. Everything I'm going to write about has been done before in various places. I'd like to have a shot at putting my spin on the exercise.

My idea is to motivate everything. Start from what we commonly accept as "true" today — and by "we" here, I mean, say, your basic early high-school student. What do we understand about numbers right now? How can we motivate the leaps that mathematicians make to get to abstract things like sets, rings, fields, formal logic, analytic number theory, ... the list is endless.

Given my finite time here on earth, I'm not going to try and cover everything (surprise!). However, I will claim that a lot of what is taught in senior school classes, even in university courses in Pure Mathematics, really isn't that far away from what "everybody" knows. It's partly a matter of notation, but also, if you learn by building mental models for yourself, a matter of seeing the motivation behind particular idea so that you can put it in perspective.

Let's begin with some thought experiments and on Tuesday I'll start working towards answers.

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Topics: mathematics/series: deriving

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