A few days ago, I was speaking with a good friend of mine who graduated with a degree in philosophy, but had taken courses through "vector calculus" in mathematics. I spent some time trying to explain why his vision of mathematics was incorrect-- but a problem speaks a thousand words. I gave him a problem I saw in Paul Lockhart's "A Mathematician's Lament"
Given two points who rest above a horizontal line, find the shortest distance between them so that the path touches the line only once. It can be found here on SE: An elementary (?) minimization problem
He spent a solid hour on it, intermittently asking questions throughout (kind of a game of "20 questions") and we eventually came to the correct answer.
I want him to perhaps see some more mathematics, but in a more "abstract" or "Advanced setting." He is out of college, and so, has extra time to enjoy mathematics, and I think he has a real knack for it.
In response to the close vote, I will rephrase my main question, I agree it was perhaps too broad.
My main question: I'm requesting a book suitable for someone who has a solid background in logic (via philosophy), and also interested in more abstract/pure mathematics. His interests are rested in solving interesting questions, but also in more general mathematical structure. I was tempted to give something like Paul Zeitz's Art and Craft of Problem Solving but elected against it, since it didn't give any insight regarding the coherence of mathematical concepts.
Thus, I'm asking for a reference that you may have found inspiring, but also elucidated a novel mathematical concept in a natural way. Recommendations for a particular subject in math is also appreciated, I'm not sure where to start him.
I ended up using some of the recommendations:
Mathematician's Delight and Euler's Gem seemed like good starting points.
I really considered Visual Complex Analysis, since my acquaintance had some familiarity with vector calculus, and I thought it may have been an interesting place to start.
I also supplied the introduction (and 1st half of chapter 1) of Mathematics Made Difficult, because I find it to be very funny, but also I thought the Platonic Dialogue in the first chapter is quite captivating. I may also send over some of his discussion of Topology and the "open men," for similar reasons.
I definitely understand your desire to find some books that reveal some unification of mathematical themes. Any books I can think of that might come close are either for people with a fairly solid mathematical background (e.g. Stillwell's Mathematics and its History), or leaning in a direction that's perhaps more historical or "popular" math book style. These are great, but I don't really think they're what you're after.
I'll just recommend books of mathematical problems that can be handled by the "uninitiated" and that, hopefully, have a common theme lurking in the background. After all, I personally like math for the fun things you get to struggle thinking about!
I've been using books from the Discovering the Art of Mathematics series for a "math for liberal arts" course this semester. Some of them are probably more appropriate for your friend than others, but I've liked all the ones I've seen. They're all worth looking at, and each is more or less conceptually unified. However, some of the activities are really geared towards groups of people/discussions.
On one page of the website, they mention Harold Jacob's Mathematics, A Human Endeavor. I wish I could recommend it personally, but I've never taken the time to get my hands on a copy. It seems pretty great, and is definitely investigation/exercise oriented from what I understand. I didn't realize how reasonably priced a used copy is.
One book I can personally recommend is David Farmer's Groups and Symmetry: A Guide to Discovering Mathematics. This is very investigation-based as well, and has a pretty self-explanatory title. I haven't sat down with all of it, but it has some very nice geometry and light group theory from the very beginning. Also very reasonably priced.
Finally, just one more book that isn't quite like the rest. J. H. Conway and friends released a book, Symmetries of Things, that's really quite wonderful. It falls a little more into the story/text end of things (less investigation baked in, but plenty to be found if you're willing to ask the questions yourself), and covers much of the same material as Farmer, plus much more. Gradually the sophistication of the material increases, the latter two-thirds of the book for strong undergrad math majors, the final third beyond that. But it tells a really nice story very well, and shows some interesting unification (in particular, the 17 wallpaper groups are classified using purely topological arguments!). The fascinating story and nice pictures make the latter parts of the book enjoyable, even if most of the math isn't accessible.