There are now quite a few excellent ones, but most of these are pitched at fairly sophisticated readers-graduate students or professional mathematicans. The thinking according to such textbooks, of course,is that the readers are very far along in thier mathematical training and are ready to use that mathematics to learn physics at a very high level. Whether or not that's true is debatable. In any event, here's a list to get you started:
Lectures on Quantum Mechanics for Mathematics Students by L. D. Faddeev and O. A. Yakubovskii: Really the only one for undergraduates-a beautiful and justly famous Russian treatment now in English. But even this beginning text requires a good knowledge of differential equations, real and complex analysis, linear algebra and group theory.
Quantum Mechanics for Mathematicians by Leon A. Takhtajan : One of several such texts for second year graduate students, most of which need at least first year graduate courses in analysis,differential geometry and topology. Excellently written,though.
Physics for Mathematicians, Mechanics I by Michael Spivak : Awesome book by the master. Perfect follow up to his 5 volume opus on graduate differential geometry and shows in depth what all that beautiful manifold theory was good for. Let's hope he finishes the projected other 3 volumes.
Quantum Theory for Mathematicians by Brian C. Hall A recent addition to the list I haven't seen. But if it's as good as his Lie groups book, it'll be well worth checking out.
Mathematical Methods in Quantum Mechanics by Gerald Teschl : This book is really more about the methods of quantum theory then the science itself. That being said,it's by a master and presents the material beautifully for second year graduate students.
Quantum Field Theory by Gerald B. Folland: Another terrific textbook by Folland, covering everything about QFT that can be made rigorous at this point,which sadly isn't as much as we'd like. You better have your graduate analysis chops on before tackling this one.
Semi-Riemannian Geometry With Applications to Relativity by Barrett O'Neill : Exactly what the title says it is. A classic,one of the best books on relativity theory for mathematicians. The book to read after Spivak.
Modern Geometric Structures and Fields by S. P. Novikov and I. A. Taimanov: An incredible and completely unorthodox beginning graduate course in differential geometry by one of the premier mathematicians in the world that completely interweaves the careful theory with virtually the entire modern structure of physics-from basic mechanics through relativity theory through the beginnings of string theory and quantum field theory.A good working knowledge of rigorous calculus of one and several variables and linear algebra is all you need to tackle this one. An absolute must have if you're interested in the relationship between mathematics and physics. I DO wish it had more exercises.
UPDATE: I've added a few more in response to some of the other posts here:
Foundations of Mechanics 2nd edition by Ralph Abraham and Jerrold E. Marsden: A remarkable and careful text for both physicists and mathematicians giving one of the first detailed presentations of classical mechanics from the modern differential geometric point of view. Very advanced, really for strong graduate students in mathematics or very strong graduate students in physics. A good follow up to Spivak's mechanics book and a first year graduate mathematics course sequence.
Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems by Jerrold E. Marsden and Tudor S. RatiuL A very mathematical first year graduate course in classical mechanics,again from the moder geometric point of view and emphasizing symmetry. Can act as the preliminary text for the text immediately preceding this one. About the same level as Spivak,but it's not as rigorous and focuses more on advanced physics then geometry. Very good collateral reading to that text.
Symplectic Techniques in Physics by Victor Guillemin and Shlomo Sternberg: I've never seen this book, but I've heard good things about it. It's by 2 masters,so that by itself makes it worth checking out.
Ordinary Differential Equations by V.I. Arnold: A terrific intermediate level course emphasizing the role of linear transformations and manifolds in the study of linear ODEs. A strong course in linear algebra and geometry is needed; this beautiful text is a must for students interested in the geometry of differential equations and physics.
Geometrical Methods in the Theory of Ordinary Differential Equations by V.I. Arnold: Graduate level follow up to the preceding text. An introduction to ordinary differential equations and dynamical systems emphasizing thier role in chaos theory and physics. A very difficult and sometimes mystifying book, but the sheer richness of the ideas and the interplay of math and physics makes it worth the effort.
I'll also mention in passing at the end the wonderful physics textbooks by Walter Griener. Not only are they concise,wide ranging and extremely clear, they have more solved examples then any physics book I've ever seen. They're written by a master. If you can get them in the original German and can read those,that would be even better as some errors creeped in in translation.
Hope that gets you started. Good luck!
You could start with the first chapter of this book, and then with this three-volumes book. The former is a very nice mathematical introduction to finance, from the viewpoint of someone on the mathematical (or physical) side. The latter may seem, and is, a book on interest rates, but it allows you to cover all mathematical techniques used in finance nowadays, and its first volume is the best introduction I have ever seen on mathematical finance ; it has btw a very nice bibliography that will redirect you to central papers in the discipline etc. I am not that fan of this book, even if I started in the field with him, but it could be ok nevertheless for what you are looking for. Finally, there is a book that is not very good on mathematical finance at all, but it is central on FX implied volatility quoting conventions, and is a must have for this.
Last point, previous books are not books on stochastic processes or PDE's or other mathematical subjects that are used in mathematical finance, they are books on mathematical finance roughly covering these subjects, and using and applying them to fianance - essentially pricing and hedging, curve building etc. This means that sometimes you will need to put your nose in a book or another on stochastic processes or even probabilities (note that this book on probabilities and discrete time martingales is a must-have), or PDE's etc. Theses are my complementary advises. I know that this wasn't you primary question, but I don't see myself giving a piece of advise on mathematical finance without mentioning this.
Best Answer
I suppose that a very good choice is A History of Mathematics, by Victor J. Katz.