The closest I'm aware of is What Is Mathematics? by Courant, Robbins, and Stewart -- starts off developing the natural numbers, goes onto number theory, analysis, complex numbers, set theory, projective geometry, non-euclidean geometry, topology, calculus, optimization, and some chapters on recent developments (as of its republishing in 1996, book was originally published in 1941). A lesser known one that isn't quite as comprehensive is a little Dover tome by Mendelson: Number Systems and the Foundations of Analysis. It starts off with the (abstract) natural numbers, and from there develops (parts of) real and complex analysis, using a categorical point of view throughout. One of my favorite parts in the latter: “What is our intuitive understanding of the natural numbers? Surely this being the firmest of all our mathematical ideas, should have a definite, transparent meaning. Let us examine a few attempts to make this meaning clear: (1) The natural numbers may be thought of as symbolic expressions:…
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