I will attempt to enumerate the most useful, mind-opening books.

The following sequence of books indicate how mathematics should be taught to children (ages 3-10) - no baby "calculus", no unnecessary formalism (e.g. set theory or algebraic structure), just an intricate interplay of a few germinating concepts (complex numbers, inequalities, algebra, coordinate space, plane geometry, convexity), around a few
*
motifs
*
- isoperimetric problems for example. Basic mathematical ingenuity and a physical "intuition" for modes of representation - say, the usage of the geometric mean, or translating geometrical arguments into kinematic ones involving "motion". Most adults (who have been miseducated by blind manipulation of formulae) would find the following quite "challenging":

* Harold Jacobs / Geometry [The 2nd Edition - later editions are shite. There's an interesting two-part book by Kiselev that was used in Russia, that may be used as a sequel to this one.]

* Coxeter / Geometry Revisited

* Posamentier / Challenging Problems in Algebra; Challenging Problems in Geometry

* Gutenmacher, Vaasilyev / Lines and Curves (now translated and augmented from the Russian)

* Niven / Maxima and Minima Without Calculus

* Andreescu / the IMO problems series (Inequalities; Geometric Problems on Maxima and Minima, Complex Numbers from A to Z, Trigonometry) [Notice how most of the books above were inspired by IMO problems - those are the only problems of
*
any
*
interest at this level of knowledge.]

* Kazarinoff / Geometric Inequalities

* Alsina, Nelsen / When Less is More: Visualizing Basic Inequalities

--

Ultimately the metamorphosis between basic and "higher" mathematics is discontinuous, when the first general theory is broached - finite-dimensional linear spaces (linear algebra), the presence of which can be felt in almost all parts of mathematics:

* Axler /
Linear Algebra Done Right
[This should be supplemented by Hoffman & Kunze for the connection to explicit computations - historically the present subject goes back to Grassmann's calculus of extension.]

* Artin / Algebra

* Courant, John / Intro. to Calculus and Analysis [
*
This
*
is how you learn calculus properly - with a minimum of calculation and a maximum of clear ideas - Spivak could be a good companion to this. There is another excellent text, written by Peter D. Lax, which is equivalent to Vol. 1 of Courant-John but perhaps more to-the-point and incisive. Hijab's "unconventional" book is particularly excellent - especially Ch 5., on the gamma function, theta functions and the like, which is amazing.]

* Stein, Shakarchi / Complex Analysis [Arguably, complex function theory is the peak of mathematics.]

* Dieudonne / Foundations of Modern Analysis

* Spivak / Calculus on Manifolds

... and so on. Those are all pedagogical classics. From this point on you "are on a roll", and making your own investigations. (Although, there are some other subjects that I would like to see in the curriculum and yet are not well-known: one would be
functional equations
- a subject cultivated by Lagrange, Gauss, Cauchy, Babbage, and so on, potentially very useful in applications.)

--

No educated person would miss these gems (on physics, probability theory, statistical inference, data analysis):

* Newton / Principia (a good companion has been written by Densmore, that dissects the arguments)

* Faraday / Experimental Researches in Electricity

* Landau, Lifshitz / Mechanics; Statistical Physics (also see the rest of the series)

* Sommerfeld / Lectures on Physics

* Lanczos / The Variational Principles of Mechanics

* Schwartz / Principles of Electrodynamics

* Shankar / Quantum Mechanics [Not perfect, but still the best introduction.]

* Dirac / Prin. of Quantum Mechanics

* Weyl / Space, Time, Matter [By far the best book, even after all these years, on general relativity. Weyl was also inspired by Husserl's phenomenology - although the "philosophical" parts are completely garbled in the English translation.]

* Feller / An Introduction to Probability Theory and Its Applications (2 Vols)

* Thompson / On Growth and Form

--

If your mind has a natural bent in this direction, with a natural sense for the deeper problems (and with an absolute will to establish solid results instead of treating any single person as a "master" or falling into sweeping "ideologies" like Marxism, positivism or whatever "ism"):

* Aristotle / Organon

* Duns Scotus / Opera Omnia

* Ockham / Summa Logicae

* Leibniz / Nachlass (especially the ones in
*
Opuscules et fragments inedits de Leibniz
*
, ed. by Couturat - and speaking of which here is
*
La Logique de Leibniz
*
)

* Kant / Kritik der reinen Vernunft

* Hegel / Wissenschaft der Logik, PhÃ¤nomenologie des Geistes

* Bolzano / Wissenschaftslehre

* Peirce / everything and anything - the best places to start are probably Vol 2 of the
*
Chronological Edition
*
(that contains his early papers), or
*
Reasoning and the Logic of Things
*
(1898 Cambridge Lectures), or Vol. 4 of the
*
Collected Papers
*
(
*
The Simplest Mathematics
*
), or Vol III/1 of
*
The New Elements of Mathematics
*
(the 1903 Lowell Lectures). Or you can start with
this website
(with a free online book).

* Husserl / Husserliana (especially:
*
Logische Untersuchungen
*
, and the "Materialien" subseries of lectures - the four giant system-builders were Aristotle, Kant, Peirce and Husserl)

* Russell / The Principles of Mathematics

* Lesniewski / Collected Works

* Wittgenstein / Tractatus Logico-Philosophicus, The Cambridge Lectures, Philosophische Untersuchungen [See some of the secondary literature - e.g.,
*
Investigating Wittgenstein
*
by Hintikka & Hintikka. And of course, read the review of the Tactatus by Frank P. Ramsey.]

* Ramsey / Foundations of Mathematics; Notes on Philosophy, Probability and Mathematics

* Whitehead / Process and Reality

* Some more recent books of importance, e.g. Hintikka's
*
Prin. of Mathematics Revisited
*
, Mertz's
*
Moderate Realism and Its Logic
*
, Simon's
*
Parts: A Study in Ontology
*

(I will add more later.)