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

This is a beautiful post! I'm loving what I am reading and the suggestions seem quite insightful :]

I am not sure if any of the following books were recommendation already:

- Carl B. Boyer - A History of Mathematics - William Dunham - Journey Through Genius - Philip J. Davi & Reuben Hersh - The Mathematical Experience - Martin Aigner & Günter M. Ziegler - Proofs from the Book - Imre Lakatos - Proofs and Refutations - Robert M. Young - Excursions in Calculus: An Interplay of the Continuous and the Discrete - Courant & Robbins - What is Mathematics? - George Pólya - How to Solve It - Morris Kline - Kline’s Mathematics: The Loss of Certainty

More or less a general deep understanding of Mathematics but will definitely give you a boost in a direction that you will favor.

The difference is the emphasis: mathematicians focus on proofs, and therefore have to write proofs of theorems that ensure their validity in all cases. Many of these cases are what physicists would call "pathological", and so they basically ignore them when introducing calculus as a tool for students of physics. If you read the table of contents of virtually any book that professes to introduce the "mathematical methods of physics", you will see how different the emphasis is. Since the physicist doesn't care about proofs that involve concepts beyond basic plane geometry and high school algebra, s/he spends a bunch of time introducing a large number of techniques that use no more than algebraic manipulation and limits. You'll find a laundry list of topics, including: tricks of differentiation and integration, the basics of vectors and multivariate calculus (more or less in order to introduce Maxwell's equations), specific examples of Taylor expansions and infinite series (in order to provide asymptotic approximations of functions--usually solutions to differential equations which represent the physics problem you are seeking to solve--which otherwise are unwieldy or impossible to carry out algebraically), some concepts and formulas from "complex variables" (which is a very deep and beautiful subject when studied more systematically, but is also so useful to physicists because of its connection to infinite series, and hence to differential equations), and "special" functions (which may be thought of as important classes of series solutions to differential equation), and most importantly, a little bit of linear algebra theory, which more or less places the majority of these computational tools in a unified framework. (Later on, physicists make use of the an analog of linear algebra, attempting to carry out similar computations in "infinite dimensional" spaces (Hilbert spaces instead of vector spaces), in a subject that mathematicians call functional analysis. Understanding the computational parts of functional analysis is essential for physicists who desire to take their knowledge of problem solving from the domains of mechanics and E&M, and apply them to problems in quantum mechanics, which is founded on functional analysis, and is in large part responsible for the extent to which the subject itself was developed).

The reason the physicist is able to succeed in so completely developing these tools (without even proving any of them) is because s/he is constantly testing their efficacy by trying them out of physics problems. The life of a physics student is more or less an endless game of trying difficult problems, and then invariably ending up resorting to finding approximations in almost all cases.

You might say, then, that where the mathematician spends his or her time looking for watertight proofs of pure concepts about space, logic, and number, from first principles, the physicist spends an equal amount of time thinking about how to approximate differential equations. It's not too surprising that, while in theory there is a great deal of overlap, in practice, the styles of thinking are very different. Part of this is because mathematics is such a vast subject, with more branches than one can care to count, whereas the number of branches used by physicists can more or less be counted on one hand.

In order to go back to a time when mathematics and physics had not yet diverged, you probably have to return to the days of Euler, or even Newton. If you read William Dunham's excellent book, "The Calculus Gallery", in the very first chapter, you will find a discussion of Newton's very basic work on binomial series--something that physicists use almost every day. However, as that book progresses (it goes in chronological order), once you get to Cauchy, who worked in the early 19th century, you begin to see mathematicians turn their focus to questions that no longer address computational questions that could possibly find direct use by physicists, but instead are more or less exist only in the minds of mathematicians.

There certainly is a branch of real analysis, called "classical analysis", which tends to focus more on concrete examples of infinite series, which of course have roots in basic calculus and physics. You can in fact prove a great deal of interesting things about specific infinite series, but you will not invent algebraic geometry or abstract algebra, or be able to fully appreciate the scope of modern 20th century mathematics, if you confine yourself to studying the properties of just one concrete object. If you do, though, you will eventually find yourself studying complex analysis. Somebody who takes this route--that is, to reject the abstract flavor of 20th mathematics--can learn a great deal about mathematics that also happens to be very useful to physicists. On the other hand, you would be missing out on a great deal of fascinating connections between the abstract mathematics of the 20th century, and applications to problems in computer science.

On the other hand, if you really want to study pure mathematics, using proofs, you'll have to be somewhat patient before you can see applications to physics. These applications will come sporatically. Right away, linear algebra is an example of a very basic pure mathematics subject which is absolutely essential to physics. At about the same level is calculus and infinite series. Then, in mechanics, you will probably encounter what physicists call "analytical mechanics", which is an application of a pure mathematics subject called the "calculus of variations". Your understanding of this subject doesn't have to be very deep to start using it in physics, though. At a slightly higher level is complex analysis, which vastly improves your understanding of infinite series. The next application I can think of is quite a bit more advanced; it concerns general relativity, and could be thought of as an advanced setting for multivariable calculus, but which is inspired by the beauty of Euclidian geometry. I am talking about what used to be called "advanced calculus", but now has many different names. The key object of study is what is called a "manifold". A mathematician would call the subject "differential geometry", whereas a physicist would emphasize the use of objects called tensors. At this point, you will begin to see non-trivial mathematics being used in physics, but with the annoyance that physicists continue to rely on computations rather than complete proofs.

Interestingly enough, Michael Spivak, the author of the classic pure mathematics book "Calculus" (mentioned in this thread, by myself and others), is in fact somebody with an interest in mechanics, and happens to be one of the key expositors of differential geometry (see his 5-volume treatise on the subject). To this effect, he has also written an introductory book on mechanics (and is in the process of writing a sequel on the subject of E&M), but with an emphasis and style unlike any mechanics book written for physicists. The book is called "Physics for Mathematicians: Mechanics". It looks deceptively simple in terms of the amount of formula used; however, it will really only be appreciated by mathematicians who have studied some amount of differential geometry (I believe he says in the preface that it would be idea for the read to have read some subset of his 5-volume treatise on the subject). Back in the `60s, Spivak also wrote what, for years, seems to have been definitive text on multivariable analysis and differential geometry for pure mathematicians, in a book called "Calculus on Manifolds". This book is quite difficult to read, though, and today, there are more friendly introductions that presuppose less mathematical maturity, and are less terse (although the book is still a classic).

Those are great books, every single one of them.

Anybody who remotely takes category theory seriously should have Aluffi's book, even if s/he chooses to learn from another book. There isn't really anything that compares in terms of its comprehensive coverage of abstraction at the basic level, and you'll eventually need something like it.

I simply love Moore & Merten's "The Nature of Computation"; it's probably the only book for the absolute (but mathematically-minded) beginner about the theory of computation that manages to be incredibly captivating and lucid (while also covering a significant chunk of material).

Hubbard & Hubbard is a fantastic book; for the right audience, it may be all that is needed. I'd say that every mathematically minded engineer or physicist certainly should have it. It does a fantastic job of exposing the fundamental ideas of linear algebra and multivariable calculus--and in a way that doesn't require really any prerequisites beyond basic calculus. Reading this book is like being in a lecture with an experienced practitioner of both pure and applied mathematics simultaneously (well, this is actually true of Hubbard, but most texts are not written in a way that the author is so detailed and clear so as to seem present); the notes in the margins and the extensive, direct, and clear explanations are absolutely lovely.

That said, somebody interested in building a foundation for pure mathematics, and not so much motivated by the ability to solve problems outside of mathematics, would probably be better served by reading a standard text on real analysis.

On the other hand, I myself have turned to the notes in the margins of Hubbard & Hubbard, even when studying real analysis from the purest point-of-view, because the little tidbits are just so insightful.

Somebody who really took Hubbard & Hubbard seriously, though, could come away with a monster understanding of applied mathematics, while still having learned the craft in a way that is correct enough to lead to further study in pure mathematics as well. Nobody can really go wrong having this book on his or her shelf (although it is a bit expensive).

I'd like to mention one more thing, which is very important to keep in mind when studying pure mathematics: if you feel like you don't quite understand something, do NOT, under any circumstance, "pressure" or force yourself into believing it. While the saying "practice makes perfect" is perfectly applicable to computation, you have to be very careful when studying pure mathematics to fight the instinct to try and convince yourself that you understand something. It is perfectly okay not to understand something. The chances are, your best shot at understanding something that makes you uneasy is to simply admit your present situation doesn't yield any direct lines of attack toward a better understanding, and to tuck the problem in the back of your mind, until a later time, when further study (possibly in an unrelated area) unexpectedly leads you to the missing piece. You also have to be on guard against allowing trying to form an intuitive understanding of something, before you've fully understood how the idea follows from the definitions and proofs alone. Forming an intuitive understanding of ideas in mathematics is always about understanding the connections between abstract ideas, but the abstract ideas themselves should have no intuitive basis when considered by themselves. For very simple things, it is probably okay to rely on intuition in a pinch (e.g., you can safely think of the derivative of a function at a point as the slope of the line tangent to the graph of the function at the point).

To use a crude analogy, you could think of mathematics as one giant puzzle, but with the pieces coming in slowly, one at a time. Any two pieces have a fairly small chance of fitting together, but since you only have the ability to focus on one or two pieces, you need the memory of old puzzle pieces which previously did not fit anywhere in the back of your mind, so that when you do stumble upon the fitting piece, you can go back to it.

Another thing to keep in mind is that the best truths in mathematics are the most general. Every time you consider a specific example, you should always have some amount of innate desire to see a more all-encompassing idea which handles the details of the specific example as a special case. It will usually not be possible for you to come up with the right generalizations yourself, though; mathematics has evolved gradually over the last couple millennia, and it has taken the trial and error of many brilliant people before the "right" abstractions were found. Your best hope is that your teacher (or author) is leading you on a path that will eventually allow you to see how the things you've come to accept as true can be thought of as existing within a broader framework. The fact that applied mathematics books generally do not do this at all is the reason why one cannot simply try to learn about a topic in applied mathematics, and then try to learn the pure setting as an afterthought. If you go straight to the applications and computations, the chances are that you will be leaving out the conceptual legwork that will be needed to understand the subject in a way that can allow you to potentially create new theory.

To take the puzzle analogy further (to the point of breaking it), you can try to view the generalizations as pieces that connect to MANY puzzle pieces simultaneously (which is obviously not how an actual puzzle works, since each piece only connects to adjacent pieces). As you progress in mathematics, you start to see that the subject is composed of a sort of hidden hierarchy, in which you later learn that your past findings are subsumed by more general theories. Without this, the subject would be unwieldy, since no normal human is capable of committing to memory a perfectly interlocking body of thought that is only made of mostly isolated ideas. Inevitably, you will need some governing ideas, which form the root of a sort of conceptual hierarchy. However, this conceptual hierarchy is more or less impossible to convey pedagogically (c.f. all the complaints about "New Math" back in the `60s), without first understanding all the pieces involved.

(For reasons discussed in the above paragraph, you should be prepared to accumulate a very large number of books and documents, should you begin to more broadly become interested in mathematics).

One way to increase the probability that you'll find interlocking pieces in the same span of attention is to be guided by an excellent teacher (and in some cases, an excellent author). Otherwise, your best shot at exploring the space of possible directions to take is to follow this advice of Paul Halmos: "A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one."

All that said, it is true that any successful student of mathematics will eventually reach a point in his or her studies in which the writing of proofs has become natural enough that, when given a theorem that has a straightforward proof, it the student will probably be able to find it 80% of the time without too much stress or outside help. Getting to that point is important; therefore, a significant chunk of the value of studying a book like Spivak's or Pugh's is to increase your ability to write proofs. This will be a gradual process, so don't be too discouraged when you get frustrated. If you feel like you need to improve your proof writing skills, though, it would certainly help to take a break from the analysis text, and read an elementary book on proofs (just search Amazon or a university library) until you feel like you've done a good job of building up this skill. The ability to write proofs with ease is as important in pure mathematics as algebraic manipulations is important in applied mathematics.

One last warning. If you truly become skilled at pure mathematics, be aware that it can be addictive. Research mathematicians spend their entire lives on this stuff, and are most often quite happy to give up a great deal of things which non-mathematicians value (e.g., a career in industry).

Of course, this really depends on how addictive a personality you have, or if you are unfortunate enough to be a creative genius.