Ask HN: How or where to begin learning mathematics from first principles?

I would be very careful before unleashing a beginner on this book. It would be too easy, IMO, for the reader to end up with the wrong idea that mechanical proofs like the ones in this book are all that are needed in mathematics, since it's possible to get as far as the real numbers (or complex numbers) with so little geometric intuition. Furthermore, the real numbers are the most concrete, familiar setting to do analysis in, but it is not healthy to spend so much focus on the concrete details of the real line so early on: a student of mathematics needs variation to keep alive her or his curiosity. Pugh does an excellent job of explaining the simple geometric essence of Dedekind cuts. In principle, one might learn something about proof writing by reading the Landau book. However, it is much, much better, IMO, to defer detailed study of something so specific, until after first surveying the setting in which the results of Landau's book are used. In most real analysis books, the reader is asked to prove a few of the results covered in Landau.

Mathematics is foremost a conceptual subject rather than a mechanical one, and it is immaterial that the reader have firsthand experience that all the theorems are proven. As one learns mathematics, it soon becomes apparent that there will always be gaps in her or his knowledge, and that is therefore best to skip steps that s/he believes could be done in principle.

You're certainly welcome!

I should probably mention one thing. In your post, you mentioned electronics, which does not so much require an understanding of mathematics, but rather a competence in solving differential equations in physics. If you are only interested in topics that are under the umbrella of electrical engineering, then you do not need to study mathematics at all. Rather, you should be studying physics, which, except at the highest theoretical levels, is more or less the practice of solving differential equations (without the kind of abstract proofs that would satisfy a mathematician). Pure mathematics is only about proofs, but with the assurance that this understanding will allow you to apply whatever problem solving techniques you may have to new domains (including situations where it is far from obvious that they apply--this is the value of mathematics).

In short, I would say that computer science is a good segue from pure mathematics, but if your goals is electronics, the kind of thinking you will need comes from learning physics--no more, no less. On the other hand, to truly understand cryptography, a background in pure mathematics is required (specifically, you should understand number theory and abstract algebra).

One thing to keep in mind: do not be mislead by similar notation between mathematics and physics: they are very, very different subjects. Certainly, physics uses equations; in addition, many theoretical results in mathematics explain (at a very high level) why certain problem solving techniques work in physics. However (at least until you reach research level physics), the overlap ends there, with the exception of linear algebra.

In fact, if there was one subject I could recommend to you (besides basic calculus) which is pretty much universally used, from optimization, to physics (all branches), to machine learning and statistics, it would be linear algebra, no contest. (I should admit here that I've contradicted my early remark in the first paragraph that one needn't study mathematics at all to understand physics, since linear algebra and calculus are indeed mathematics. However, while learning these two subjects rigorously in the spirit of mathematics will certainly aid you conceptually when you attempt to apply them to physics, it is also true that in physics you do not need to know how to prove the results in order to use them.)

One way to approach your wishes to learn "mathematics from first principles" is to see modern mathematics as the study and application of formal systems:

http://en.wikipedia.org/wiki/Formal_system

A formal system has several components:

An alphabet of symbols from which sequences or strings of symbols are constructed. Some of these strings of symbols can be well-formed according to some formal grammar, which is the next component.

Next we have a collection of basic assumptions, called axioms, which are supposed to reflect the obvious truths about whatever we want to formalize in the formal system.

And then we have some rules of inference. They allow us to derive conclusions from premises. An example would be the rule of modus ponens: If we have "If A then B" and "A", we can conclude "B".

An example of a formal system is ZFC set theory which can be regarded as a formalization of one concept, the concept of a set:

We take "classical predicate logic" as a background formal system, it already has logical symbols, like symbols for AND, OR and "IF ... THEN ..." and quantifiers "FOR ALL ..." and "THERE EXISTS ...".

We enhance this logic with one non-logical symbol, the binary element-of-symbol ∈. With it we want to express the idea that something is an element of something, for example x ∈ y is supposed to mean that x is an element of y.

Of course this is a bit simplified, but now we can build expressions (with symbols from the alphabet, according to the grammar for logical formulas plus the element symbol) which talk about the element-of-relationship between individuals.

Next, we sit together at a round table and discuss which properties about sets and element-of or membership of a set we see as self-evident - there is room for discussion and there can be many different intuitions.

For example, as in ZFC set theory, we may want to have some existence axioms. They guarantee us that in this formal system certain objects do exist. An example is the axiom of the empty set: There exists a set which has no elements. This statement can be written in our formal language.

Other axioms may have a more constructive meaning. Instead of telling us that something exists, they say that given the existence of some objects we know the existence of further objects. An example would be the axiom of set unions: Given some arbitrary sets A and B, there exists a set C, which contains all the members of A and all the members of B as its elements. Another axiom asserts the existence of an unordered pair of any two given sets, from this we can define the concept of an ordered pair, which is very important.

ZFC is one example of a set theory, there are many different set theories. You could exchange classical logic with intuitionistic logic and arrive at some formal system for intuitionistic or constructive set theory. You can drop certain axioms, because maybe they do not appear as self-evident to you (for example the axiom of choice, which contributes the "C" in ZFC, is not accepted by some people). You may add further axioms to arrive at a possibly stronger theory.

One interesting aspect about set theory is that the concept of set is very powerful and expressive, because many concepts from modern mathematics can be build up from sets: natural numbers 0,1,2,3,... can be constructed from the empty set, functions can be represented through ordered pairs of sets. Sometimes set theory is regarded as "the foundation of all mathematics", but feel free to disagree! Just because natural numbers can be modelled as sets it is not certain that natural numbers are indeed sets.

The basic pattern above is the formalization of an intuitive or natural concept, something from everyday life. We try to capture the essentials of this concept within a formal system. And then we can use the deductive power of the formal system to arrive at new and hopefully interesting conclusions about whatever we wanted to formalize. These conclusions are theorems. Not all theorems are interesting, some are even confusing, paradox and disppointing. Formalization is used to arrive at new insights about the original concept. Interesting in this context is Carnap and his idea of explication of inexact prescientific concepts:

http://en.wikipedia.org/wiki/Explication

What I want to express with this is that it is really possible to start your journey into mathematics at a beginning.

@bobsadinook Thx for the plug!