I was just wondering if someone would be kind enough to tell me in what order (I know that there is no real "best order") one would most profitably study these subjects/books:

(edited to conform with suggested order of study)

  • Pre-Algebra
  • Algebra 1
  • Geometry
  • Algebra 2
  • Pre-Calculus/Trig.
  • Calculus 1
  • Calculus 2
  • Combinatorics: Topics, Techniques, Algorithms - Cameron, Peter J.
  • Lectures on Probability Theory and Mathematical Statistics - Taboga, Marco
  • Classical Mathematical Logic - Epstein, Richard L.
  • Calculus, 4th edition - Spivak, Michael
  • Linear Algebra - Shilov, G. E.
  • Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus - Spivak, Michael
  • Naive Set Theory - Halmos, Paul R.
  • Elementary Real and Complex Analysis - Shilov, Georgi E.
  • Linear Algebra Done Right - Axler, Sheldon
  • Ordinary Differential Equations - Tenenbaum, Morris
  • Partial Differential Equations: Second Edition - Evans, Lawrence C.
  • Abstract Algebra - Dummit & Foote
  • Topology (2nd Economy Edition) - Munkres, James
  • Introduction to Set Theory, Third Edition, Revised and Expanded - Hrbacek & Jech (only suggested for those with great interest in Set Theory)

Also, are there any books/subjects missing from a fairly well rounded advanced mathematics education?

Any help would be greatly appreciated.

(edited in) P.S.

I think that I mistook Combinatorics for Discrete Mathematics. Can someone enlighten me on the difference and maybe suggest a good book for discrete mathematics (perhaps a supplement to Cameron's Combinatorics)?

  • $\begingroup$ I think, based on what little is known of your background with more sophisticated mathematics, that Naive Set Theory or Calculus (Spivak) are most likely to be the books accessible to you. Several of the others in your list are appropriate for someone who has a greater level of "mathematical maturity" and is prepared for some more sophisticated and abstract material. The combinatorics book might also be accessible, though I am not familiar with it, as is perhaps the Aleksandrov book (again, unfamiliar). $\endgroup$ – izœc Apr 16 '14 at 6:40
  • 1
    $\begingroup$ (1) You are missing an actual linear-algebra text, not a linear-algebra-for-functional-analysts. Dummit/Foote might be somewhat difficult otherwise. There are various threads on m.se about alternatives. (2) Abstract algebra should come before topology unless the topology is really very basic. (3) Calculus on Manifolds should certainly not come before a good linear algebra course. (4) Combinatorics requires very little and gives a lot familiarity with mathematics; it could easily be taken to the front. $\endgroup$ – darij grinberg Apr 17 '14 at 4:53
  • 2
    $\begingroup$ (5) I'm somewhat biased here as a constructivist, but I'd say set theory (as in Hrbacek & Jech, not as in Halmos) is a graduate subject for those interested; it looks out of place in the "general education" laundry list you designated. You might have mistaken it for the very basic set-theoretic language taught in undergraduate classes; it is a different thing. (6) Graham-Knuth-Patashnik's Concrete Mathematics is a very accessible and motivated introduction into ideas which are nowadays universal and will meet you in various places in mathematics. $\endgroup$ – darij grinberg Apr 17 '14 at 4:57
  • $\begingroup$ I can't help much with probability. I think the difference between combinatorics and discrete mathematics is a matter of culture (judging from the Wikipedia, in the US it is a vague umbrella term for combinatorics, theoretical computer science, logic, cryptography and finite field theory -- aka the part of mathematics needed for CS majors). $\endgroup$ – darij grinberg Apr 18 '14 at 0:14
  • $\begingroup$ From a brief look, William Chen's lecture notes on LA look good maths.mq.edu.au/~wchen/ln.html (I got the link from math.stackexchange.com/questions/160056/… but I am not vowing for all of the other suggestions there; some are really just handouts accompanying a class.) $\endgroup$ – darij grinberg Apr 18 '14 at 0:19

I suggest you not try to stick to some specific order. If something interests you one day then read it. Forcing yourself to stick to a schedule will only cause you to be unmotivated and as a result not get as much out of it. Also, if your going to self study, try and still talk with people about what your doing or make videos in which your try to explain what you've learned. If you really want a challenge test your progress by trying to do problems here on MSE. I hope this helps and good luck. Also, search around for an ebook which shows you some elementary proof techniques because mathematics at this level requires a lot more time and understanding.

  • $\begingroup$ I've thought about this. But I believe that I need structure. Perhaps not a schedule, but a structured order. I love the idea of making videos or sharing my progress. I know from language learning that this helps keep you motivated to do more. What do you mean, though, by "do problems here on MSE"? Do you mean to post questions? Or do you mean to have others post questions for me? Or do you mean to search through the boards for problems posted by others? Or do you mean that there are some exercises somewhere on this site? $\endgroup$ – Dubby Apr 17 '14 at 1:52
  • $\begingroup$ Go through the various tags and try to find problems and do them. $\endgroup$ – user96137 Apr 17 '14 at 2:57
  • $\begingroup$ Thank you very much! I think that that is a great idea. $\endgroup$ – Dubby Apr 17 '14 at 4:26

I believe the highly structured approach of modern day high school education hides the true nature of mathematics research: it is messy, chaotic and vast. The following is a typical way mathematics progresses at higher levels:

  1. Distinct topics A and B are studied heavily, and a set of mature tools and techniques developed,

  2. Certain problems in topic A is rewritten as a problem in problem B. Now the techniques of topic B can be applied to topic A,

  3. A new topic BA is developed, which allows problems of A to be solved by techniques of B, and vice versa,

  4. Generalising ideas encountered from AB leads to a topic C which appears as topic A when viewed in a certain manner, and B in another.

This and more intermingling of techniques keep on happening between different fields constantly, so by the time you read a certain topic- it might involve ideas and techniques inspired and abstracted from other areas. So, structuring topics in a rigid manner can be rather detrimental to the study of mathematics.

The way I see it is- study a topic, understand its techniques, read something else, and when possible- try to review the older topic from this new angle. A definition or a proof often makes a lot more sense when viewed from many angles, and this ability to switch your angle of approach is what makes for a good understanding of mathematical ideas.


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