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I currently have an engineering-style education in mathematics. We covered quite a lot of material (e.g. real and complex analysis, some probability theory and graph theory), but more often than not we stated theorems without formal proofs, and, what's worse, there was a significant amount of hand waving (moving limits around without proper justification, and the like).

Now I am coming to the realization that this approach is akin to going on steroids to build your muscles: while it lets you cover a lot of ground in a relatively short time, it creates more problems than it solves in the long term.

What I would like are suggestions to 'convert' my mathematics knowledge into proper knowledge, using a self-study approach complemented with questions on here. What I have in mind are questions like (but not limited to):

  • Where should I start? (set theory?, predicate logic?).
  • What follows? (real analysis? complex analysis?).
  • Should I work through whole books, solving every problem?
  • What are the core topics that any self-respecting mathematician should know well?
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    $\begingroup$ Get How to Prove It by Velleman. $\endgroup$ – Tyler Mar 2 '14 at 15:40
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    $\begingroup$ I wonder what your motivation is? If you're just looking to be a better engineer, it seems like you should ask engineers about that. If you want to learn maths for the love of it, go ahead and do that, and IMO don't let people tell you what to learn, just follow what excites you. $\endgroup$ – Ben Millwood Mar 2 '14 at 15:59
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    $\begingroup$ Look up Kenneth S. Miller's Advanced Real Calculus in a library if you have access to one. It's one of the nicest treatments I know for what I would guess your background is. The book is short, rigorous, and would be a great prerequisite for Rudin's Principles of Mathematical Analysis. Weinberger's A First Course in Partial Differential Equations: with Complex Variables and Transform Methods is also worth a look (learn all about uniform convergence and the like by having it come up "naturally"). $\endgroup$ – Dave L. Renfro Mar 3 '14 at 19:45
  • $\begingroup$ Have you also studied linear algebra? You do not mention it specifically in your post. $\endgroup$ – J W Jun 28 '14 at 6:31
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Two possibilities:

  1. Take what you have learned for granted (yes, there are holes; but what you got explained by vigorous handwaving can be proved rigurously). Dig down to the formal proofs of new material as you need it, perhaps filling in whatever underlying material makes you uneasy.
  2. Start from the beginning, fill in any and all holes.

As a matter of expediency, I'd go with (1), (2) will take you a long time to get to where you want to go. In fact, I have found (by pesonal experience) that trying to learn preemptively tends to lead you astray by not having a goal in mind. Besides, it often happens than when you need something already studied, you'll have forgotten; or, perhaps even worse, you don't ever need it.

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    $\begingroup$ It is not really a matter of expediency in my case. The problem with the "learn as you do" approach is that you frequently get stuck in infinite loops: you try to understand a proof in real analysis, but you can't because you don't know the underlying set theory, etc. I guess a bottom-up approach would be better in my case? $\endgroup$ – user60297 Mar 2 '14 at 15:39
  • $\begingroup$ @user60297 All that you need to know about set theory and other "fundamentals" to fully appreciate real analysis is completely contained in Rudin's PMA chapters 1 and 2. $\endgroup$ – Emily Mar 2 '14 at 15:41
  • $\begingroup$ If you get into an infinite loop, you've found a flaw in math ;-) I'd prefer the top-down approach. First know why something like the intermediate value theorem is crucial, then dig into the proof. Gives a sense of purpose. But people are different... $\endgroup$ – vonbrand Mar 2 '14 at 15:45
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To understand the basics of most of the math used in engineering, I suggest you start with real analysis, then move on to complex analysis, functional analysis and measure theory. Real analysis might be a bit hard at first, if you're not used to finding and writing rigorous proofs, but it'll get easier afterwards.

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  • $\begingroup$ I'm trying to imagine someone with little (no) proof experience picking up Principles of Mathematical Analysis, and I don't imagine it going very well. I think a short proof-course would be a much better place to start. $\endgroup$ – Tyler Mar 2 '14 at 15:42
  • $\begingroup$ @Tyler I completely agree. Rudin's book isn't good for jumping into mathematics. But I found Berberian's 'Real Analysis' perfect for the same purpose. It is one of my favourite books. It doesn't cover a lot of material though. $\endgroup$ – caffeinemachine Mar 2 '14 at 16:14
  • $\begingroup$ @fgp I guess it depends. If you went through an engineering curriculum, chances are that you already understand a large part of real analysis on an intuitive level. I've never used Rudin's book on real analysis, but I have used his book on functional analysis and found it to work quite well for me. $\endgroup$ – fgp Mar 2 '14 at 19:52

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