tl;dr: Mathematicians prefer to start with a structure’s most general form, while physicists prefer specific (applicable) cases.
Here are some brilliant old math textbooks:

…for the Practical Man  J. E. Thompson
 Calculus Richard Feynman used this.
 Trigonometry
 Geometry
 Algebra
 Arithmetic
 Calculus Made Easy  Silvanus P. Thompson
Here are brilliant modern tomes:
 Maths: A Student’s Survival Guide: A SelfHelp Workbook for Science and Engineering Students  Jenny Olive
 Div, Grad, Curl and All That  H.M. Shey Best Vector Calculus book
 Principles of Vector Analysis  Jerry B Marion
German speakers are especially lucky: Papula
As you see, the common thread emerges from engineers' engagement with mathematics. When taught with physics, the traditional mathematical curriculum makes far more sense. Calculus makes more sense when you understand the problems Newton was solving etc.
Click for a video covering historical applications.
Engineering is applied mathematics and science, which explains why books aimed at them are more approchable. Chapters in physics textbooks often start with a phonomena or system, then introduce theory to describe it (you just accept the mathematics.) Modern mathematics instruction Since Bourbaki? prefers the opposite, building the foundations from which you can derive the theory. …as an excercise to the reader.
This is because mathematics is a 2000 year old proofs are still true, unlike 2000 year old medical or alchemical knowledge. formal science with true, false and undecidable statements, while physics is an empirical science, where concepts emerge from describing experiments. From these statements, mathematicians focus on patterns, structure and the relationships between groups of patterns and structures. In other words: while mathematicians prefer a structure’s most general form, physicists prefer specific cases.
But approaching mathematics this way (without fundementals, from physicists' and engineers' books) foregoes the intuition etc. required to engage with mathematical literature. A mathematician’s development sees him doing countless excercises, constantly referring back to theorems and definitions, to appreciate on an intuitive level how the structures function and combine to construct proofs (the fundemental building block of math.) This practice echoes traditional language learning, combined with understanding different primitive sets. This work tells him which definitions are key to what kind of problems and which are incidential. Better texts (bulleted lists) avoid intuition (they don’t mark facts' importance), so our mathematician can discover it himself. Centuries have forged the language and descrioptions of these objects, so wrestling with these dense abstractions takes particular effort. To “read” math, our mathematician, after reading the introduction, skims to find an interesting result and search for the relevant lemmas Lemma = unimportant result, only used to build a more important result (thoerem), but the key part is how they combine, not the final theorem itself. to understand it. Mathematicians develop tools without specific applications, which others find later. Most people, however, are interested in applying mathematical tools, not deriving them, whence the mathematical mindset falls on deaf ears. Math’s frameworks help construct and manipulate abstract structures, like the primitive sets software engineers design
(The graph of facts which rely on which other facts is fertile soil for Good Old Fashioned AI to plow.)
One alternative is to learn both at the same time:
 Road to Reality  Penrose Very hard
 No Bullshit Guide to Math and Physics  Ivan Savov Very approchable
If desirious to get a dose of mathematical intuition, construct proofs etc.
 Mathematical Notation: A Guide for Engineers and Scientists  Edward R. Scheinerman
 Some Modern Mathematics for Physicists and Other Outsiders: An Introduction to Algebra, Topology, and Functional Analysis
 How to Think Like a Mathematician: Companion to Undergraduate Mathematics  Kevin Houston
 Mathematical Methods of Classical Mechanics  V. I.
But it is key to:
 know exactly what you want from the subject (“how to apply x thoerem to…” or “understand y theorem in z paper”)
For programmers:
 Programmer’s Introduction to Mathematics  Jeremy Kun
 Discrete Math and Functional Programming  Thomas Van Drunen
Four groups:
 mathematicians: “What’s true?”
 give me the primitives, with which i will built higher level structures
 definition > theorem > proof Bulleted lists of them count as a textbook
 notation driven by succinctness (short yet clear)
 use 1 char vars, because normal equations, structs etc. are known entities
 physicists: “What works?”
 physical problem > theory > math
 programmers
 name vars (counterpoint: APL) because new functions, structs etc. are created on the fly
 notation driven by “how can we change this when needs change?”
 engineers
A few odds and ends:
Some books teach physics to mathematicians (definition proof theorem style):
 General Relativity  Sasane
The Art of Problem Solving
 Prealgebra
 Intro to Algebra
 Intro to Counting & Probability
 Intro to Geometry
 Intro to Number Theory
 Intermediate Algebra
 Intermediate Counting & Probability
 Precalculus
 Calculus
 Vol 1 & 2 The Art of Problem Solving Everything distilled.
The series focuses on problem solving, eschewing proofs/formalism, yet don’t provide concrete examples of what the techniques are used for.
N.b. Not a book but Math Academy is well reviewed, more succinct than Khan Academy.