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 Self-Help 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.

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.