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I read a useful definition of the word understanding. I can't recall it verbatim, but the notion was that understanding is 'data compression': understanding happens when we learn one thing that explains many things.

I've been studying setential and predicate logic in school; it is my first brush with math outside of high-school, where I, regretably, decided not to study it past grade 11. I just didn't suspect how much of everything math can help people understand. Now, I find it at the heart of almost everything: decisions (game theory); music; co-ordinating colours; natural patterns; and, in a way, thinking itself (logic, set theory). More recently, I've been studying to complete a philosophy major, and I've found that a person could restate most of the questions as math questions (and, not only in formal logic).

I wrote that to make two contextual points relevant to the question: The first is that I seem to decide what to study based on what I suspect will help me understand the most; that is, what will help me explain the most with the fewest rules. So it seems I find subjects with a greater 'ratio' the most interesting. Accordingly, that ratio continues to drive most of my decisions about my education. So, the first point is that I've asked this question to help determine what subjects in math I should study (what direction should I head in?). The second is that I don't know much math yet, so in my attempt to express my main question (below), I'll bump around a bit, and probably use some terms imprecisely(*). My apologies.

I've noticed that set theory and formal logic have something in common that seems to underlie both of them. For example, the Wikipedia page on logical connectives depicts each of them with set diagrams. I've also noticed that geometry and logic have something in common. For example, I once saw them depicted using a matrix* that produced triangular patterns; I regard the truth tables for implication and non implication as 'turned-around' relative to one another, and the converses of those as 'upside-down'; I've also seen logical bi-simplexes plotted on a number of different shapes. Finally, there is something common that underlies the logical connectives: we know that a person can effect all the logical connectives from just one of the connectives; truth trees seem to work on the equivalent of disjunctions and conjunctions.

It seems that the different parts of logic (and probably much of math) rest on something common. I'd bet that mathematicians have identified that underlying thing that a person could use to understand disparate areas of logic (or math). If that's true, what do you call that thing? What would a person need to study to know it well?

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The commonality you've noticed between logic and basic set theory can be summarised in the term: boolean algebra. You're probably looking for something deeper, though. –  goblin Dec 17 '13 at 15:55
    
Thanks. I've wondered about that for a while. –  Hal Dec 17 '13 at 15:58
    
You're welcome. Note that if $X$ is a set, then the subsets of $X$ actually form a complete Boolean algebra; however, not every complete Boolean algebra arises in this way. If after googling those terms their meaning still isn't clear, feel free to ask me about them. (Or whatever search engine you use). –  goblin Dec 17 '13 at 16:00
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I suppose Category Theory is relevant here. See this pdf linking Heyting algebras (and hence intuitionistic logic) to topoi. –  Shaun Dec 17 '13 at 16:50
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Although to put @Shaun's comment in perspective, set theory is quite an advanced subject as well. Really, the difference is that we have centuries of experience with set-oriented pedagogy and framing introductory material in terms of "naive" set theory. We only have a few decades of experience with category-oriented pedagogy, and AFAIK "naive" category theory hasn't really seeped into the basics yet. –  Hurkyl Dec 17 '13 at 20:59

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Some comments mentioned category theory. The book Conceptual Mathematics by Lawvere and Shanuel is meant as an introduction to category theory for people with only a little background in mathematics. In this book the authors suggest doing all the exercises. If you don't know much about category theory this is imperative. Hope this helps.

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