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

you need to start off with - logic and set theory. an introduction to proofs (level 0). something on (proofs in) classical geometry. - then linear algebra. (level 1) - group theory (fe Joe Armstrongs book) and an introductory (real, single-variate) analysis course. also probability theory (I'd recommend Meester's book) (level 2) - calculus, rings & galois theory, topology (fe Munkres) (level 3) - complex analysis, (and other stuff I didnt even pass) (level 4)

I'd recommend buying one book at a time and working through the entire thing, all the problems. It can quickly become too difficult if you try paralellize. But it can actually be a good experience to do one thing well.

oh yeah. the payoff of this stuff isnt very good. take it from a guy coding php for 10 euro / hour. so another warning not to do this.

Calculus is usually taught early because physicists need to know it as well but it depends on other things so if you do this early you will not really understand.

It is also a difficult question to answer without some context of your current mathematical understanding. Do you know any calculus? Any linear algebra? If you don't, those would be good places to start as they underpin many areas with applications of mathematics. Bear in mind too that Mathematics is a huge subject in that even if you take to it naturally, you're not going to acquire a breadth and level of understanding without a fair amount of study. Looking back I probably put a lot of hours in my youth into really understanding linear algebra fully for example to the level that I could teach it at a high-ranking university, and that was with the help of people whom I could pester with my questions and the incentive of exams to take.

The other approach is to look at the areas you want to understand and then work out what topics you need to study to fully understand them. Cryptography is worlds away from electromagnetism for example. Looking at cryptography, are you interested in public key cryptography or symmetric key cryptography? If the former, then you need to start learning number theory and if the latter, knowing some statistics is probably more relevant.

The first principle of learning mathematics is that the notation describing idea `M{n}` depends on an understanding of some notation describing idea `M{n-1}. That's why there is some sense in which "first principles" of mathematics makes sense. In the end, learning mathematics is a long haul - the academically elite of the world normally spend twelve years just getting to the point of completing a first calculus course before heading off to university.

Of course, there isn't really an explicit ordering to the notation. This despite our ordering of the school-boy educational system. Out in the adult world, mathematicians, engineers, scientists, etc. just grab whatever notation is convenient for thinking about the problem they are trying to solve. Thus, it is common for separate domains to have wildly different underlying abstractions for a common mathematical concept: ie. two problems which are reducible to each other by manipulating notation using replacement.

What this means is that there's no meaningful reason to derive the domain specific language [notation] of cryptography and antenna design simultaneously from Peano arithmetic...sure there's a formalism, but it's a Turing tarpit equivalent to building Facebook's infrastructure in Brainfuck. Starting from first principles is a task for mathematicians of Russel's and Whitehead's calibers. For a novice, it constitutes a rookie mistake; keeping in mind that the problem Gödel found with Principia Mathematica is foundational to computer science.

The philosopher CS Pierce's criticism of Descartes Meditations can be elevator pitched as: enquiry begins where and when we have the doubt, not later after we have travelled to some starting point. The base case for extending our knowledge is our current knowledge; creating better working conditions and unlearning poor habits of mind are part of the task.

If the enquiry grows out of knowledge in computing, it is impossible to start anywhere but from computing. Getting to the "No! I want to start over here!" place is part of the enquiry and a sham exercise.

All of which is to preface two suggestions:

+ Knuth's Art of Computer Programming presents a lot of mathematics in a context relevant to people with an interest in computing. Volume I starts off with mathematics, Volume II is all about numbers, Volumes III and IV are loaded with geometry and the algebraic equivalents of things we think about geometrically.

+ Iverson's Math for the Layman and other works are useful for introducing the importance of notation and tying it to computing. [Disclaimer: I'm currently in love with J, and posting the following link was where I started this comment]. http://www.cs.trinity.edu/About/The_Courses/cs301/math-for-t...

+ Because notions of computability are implicit in mathematics, automata theory is another vector for linking knowledge of computing to an increased understanding of mathematics.

Good luck.

There are instead, roughly, between 4 and 50 branches of mathematics which each start and "end" in different places with different goals and philosophies and styles.

What makes this all "math" is that almost inexplicably these branches tread the same ground over and over. Which is to say: learning one branch can dramatically improve your ability to understand another branch. Learning several builds your "mathematical intuition" all together.

In order to learn more math you will most likely want to choose one of these branches and study it intensely. You will not want to start from first principles to begin. Nobody does, it's too complex. Instead, you should seek to understand some set of "introductory core ideas" from that branch.

In order to study any branch you will need to learn the language of mathematics: logic, theorems and proofs. Essentially, this is a language you can think in and speak. Without it, you will be incapable of carefully expressing the kind of sophisticated ideas math is founded upon.

Fortunately, programming is an application in logic. If you can program a computer you're between 1/3rd and 2/3rds of the way to understanding mathematical logic well-enough to begin to understand mathematical argument. That said, you will not yet know enough. There are books which teach this language directly (Velleman's How to Prove It, perhaps) and there is an entire field of study of this language. Usually, however, you just learn by doing. Certain branches are more amenable to this learning of the logical language than others.

One thing to note about the logical language that would be told to you by any teacher but is only mentioned in a few books is that it is not much like English in that you can just listen to or read something in the logical language and have it immediately form a cogent picture in your mind. Mathematical language is a language of action---you MUST complete proofs, often on your own, in order to have grasped what was being said. This doesn't mean there isn't value in skimming a math book and reading the results without doing the proofs. Indeed, that's often a great first pass through a book! But think of doing that like reading the Cliff's Notes for a great work of literature. You might be able to talk about it a little bit, but you certainly haven't understood the material.

One final note with respect to learning any branch—where you start is critical. Often, even the simplest reviews of the material of one branch of mathematics will assume "basic, working knowledge" of many other branches. This is done in order to accelerate learning for those who possess that working knowledge—it takes advantage of the frequent crossover properties from one branch of mathematics to another. Finding resources which do this minimally will be important to begin... but you will probably not succeed entirely. Sometimes, you just have to read a math book and walk away from it without being too much the wiser, but recognizing that there was some technique from another field you could learn to unlock a deeper understanding.

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Some major fields of mathematics are:

1. Algebra. This is like and unlike what you may call algebra today. It is the study of how things are built and decomposed. Indeed, it notes that many "things" can be described entirely in terms of how they are built and decomposed. It is often a good place to begin for programmers as it espouses a way of thinking about the world not dissimilar to the way we model domains in while programming. Some books include Algebra: Chapter 0 by Aluffi and Algebra by MacLane.

2. Combinatorics. This is the study of "counting", but counting far more complex than anything meant by that word in normal usage. It is often a first field of study for teaching people how to read and speak proofs and theorems and therefore is well recommended. It is also where the subfield of graph theory (mostly) lies which makes it more readily accessible to programmers with an algorithms background. I can recommend West's Introduction to Graph Theory, but only with the caveat that it is incredibly dry and boring---you will get out of it what you put into practicing the proofs and nothing more.

3. Topology. This is the study of what it means for one thing to be "near" another. Similarly, it is the study of what it means to be "smooth". It's a somewhat more abstract topic than the others, but in modern mathematics it holds a privileged role as its theorems tend to have surprising and powerful consequences elsewhere in mathematics. I don't know any good introductory material here---perhaps Munkres' Topology.

4. Calculus and Analysis. This is the study of "smooth things". It is often the culminating point of American high school mathematics curricula because it has strong relationship with basic physics. Due to this interplay, it's a remarkably well-studied field with applications throughout applied mathematics, physics, and engineering. It is also the first "analyst's" field I've mentioned so far. Essentially, there are two broad styles of reasoning in mathematics, the "algebraicist's" and the "analyst's". Some people find that they love one much more than the other. The best intro book I know is Spivak's Calculus.

5. Set Theory. This is, on its surface, the study of "sets" which are, often, the most basic mathematical structure from which all others arise. You should study it eventually at this level to improve your mathematical fluency---it's a bit like learning colloquial English as compared to just formal English. More deeply, it is a historical account of the philosophical effort to figure out what the absolute basis of mathematics is---a study of foundations. To understand Set theory at this level is far more challenging, but instrumental for understanding some pieces of Logic. This can therefore be a very useful branch of study for the computer scientist investigating mathematics. I don't know a good introductory book, unfortunately.

6. Number Theory. This is, unlike the others above excepting "surface" Set theory, a branch which arises from studying the properties of a single, extremely interesting mathematical object: the integers. Probably the most obvious feature of this field is the idea that numbers can be decomposed into "atomic" pieces called prime numbers. That idea is studied generally in algebra, but the properties of prime numbers escape many of the general techniques. I don't know a good introductory book, unfortunately.

7. Measure Theory and Probability Theory. Measure theory is the study of the "substance" of things. It generalizes notions like length, weight, and volume letting you build and compare them in any circumstance. Furthermore, if you bound your measure, e.g. declare that "all things in the universe, together, weigh exactly 1 unit", then you get probability theory---the basis of statistics and a form of logical reasoning in its own right. I don't know a good introductory book, unfortunately.

8. Linear Algebra. A much more "applied" field than some of the others, but one that's surprisingly deep. It studies the idea of "simple" relationships between "spaces". These are tackled in general in (general) algebra, but linear algebra has vast application in the real world. It's also the most direct place to study matrices which are vastly important algebraic tools. I don't know a good introductory book, unfortunately.

9. Logic. A much more philosophical field at one end and an intensely algebraic field at the other. Logic establishes notions of "reasoning" and "judgement" and attempts to state which are "valid" for use as a mathematical language. Type Theory is closely related and is vital for the development of modern programming languages, so that might be an interesting connection. I don't know a good introductory book, unfortunately.

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Hopefully, some of the ideas above are interesting on their surface. Truly understanding whether one is interesting or not is necessarily an exercise in getting your feet a little wet, though: you will have to dive in just a bit. You should also try to understand your goals of learning mathematics---do you seek beauty, power, or application? Different branches will be appealing based on your goals.

Anticipate studying mathematics forever. All of humankind together appears to be on the path of studying it forever---you personally will never see its end. What this means is that you must either decide to make it a hobby, a profession, or to consciously leave some (many) doors unopened. Mathematics is a universal roach motel for the curious.

But all that said, mathematics is the most beautiful human discovery. It probably always will be. It permeates our world such that the skills learned studying mathematics will eke out and provide value in any logical concern you undertake.

Good luck.