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If someone never had any experience with mathematical proofs and had only classes like Calc I-III (which he passed, without paying any attention to the proofs present in the textbooks), how long would it take this person to master the Book of Proofs by Richard H. Hammack?

Now a few clarifications: This person would learn this book completely alone, at home, so no other students, professors or any other type of math enthusiasts. Let's say this person would learn 2 hours a day. And by "mastering" I mean this person would solve 90% of all exercises without looking at the solutions and understand all theorems.

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This is a text for an "introduction to proofs" course.

It might not be well-known outside mathematical circles, because mathematics educators don't like to advertise this fact, but, outside of fairly selective universities, most students taking such courses fail to learn the material despite earning a passing grade. The majority of students never learn to develop or organize on their own any but the simplest proofs, or understand any significantly complex proof. This inability persists despite another year or two of coursework beyond the "introduction to proofs" course that develops and uses these skills.

Hence the answer for the average student is forever.

On the other hand, a student who has an excellent natural grasp of logic has essentially no need for this course and will be able to learn all the material in a week or two. The most selective universities tend to not have a course based on a similar text and simply expect their students to intuitively pick up this material while taking more advanced math courses.

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  • $\begingroup$ About that: "expect their students to intuitively" I surely am not questioning your experience, but it could also be that the students learned material from this book in school or at their own. There are a lot of schools in many countries, where Topics from this book are covered. And I like to think that books like this are mostly for 7-12 graders who like math and want to study in later on. Of course you are right, that most good Math Unis dont offer intro to proofs. But imho most math students had a long and thorough exposure to math before applying. $\endgroup$
    – a a
    Jan 8 at 23:50
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    $\begingroup$ Is it possible that your answer comes from a mostly US-centric perspective? In my German university, there was no "introduction to proofs" course in studying mathematics, we were indeed expected to pick this up along the way in our Analysis and Linear Algebra lectures. Let me emphasize that my university was bog-standard. Also, in France I attended somewhat more advanced courses, but I don't think the beginners got an "intro to proofs" there, either. $\endgroup$ Jan 10 at 11:25
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    $\begingroup$ @StephanKolassa: Yes this is a US-centric answer. However, let me point out that, by virtue of the Abitur, every German university is selective. Also, German universities consider it acceptable for there to be significant numbers of students who put in genuine effort but still fail their courses, and US universities do not. $\endgroup$ Jan 10 at 16:46
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    $\begingroup$ While I agree with you, I've found the observation, e.g., "by virtue of the Abitur, every German university is selective" to be a point of severe argumentation between U.S. and European perspectives: academia.stackexchange.com/questions/139725/… $\endgroup$ Jan 27 at 2:11
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(I had tried to add a clarifying comment to @AlexanderWoo's good answer, but it was mysteriously deleted.)

My point was, and is, that it is not constructive to think of "proofs" as a thing separate from normal human discussion of things. Rather, proofs are really explanations, or discussions that persuade. There is no magic formula, and "proof" is not an alien thing.

Yes, many math curricula (perhaps especially in the U.S., these days) in high school and the beginning of college or university have essentially no content that is about explanation or persuasion, but only about inexplicable rules.

I do have some acquaintance with various "introduction to proof" textbooks and courses, and I am not a fan of any of them, for various reasons. Instead, as suggested in Alex W's answer, simply reading relatively easy proofs of real things (rather than contrived) is the way to learn what the point of "proofs" is. It's not conforming to rules in a ritual, it's about careful appraisal of causality and such...

One particular problem in terms of understanding mathematics, is the notion that we should not trust our physical intuition. It is perverse to invalidate beginners' self-confidence and critical judgement by telling them things like this. Mathematics is not about inscrutable stuff behaving wildly contrary to our physical experience, and it is perverse to portray it as such. One iconic example is the intermediate value theorem in calculus. On one hand, it is "obviously true"... but it is non-trivial to prove, because a precise notion of "real number" is necessary, to begin. But that should not make us doubt the fact.

Especially when I hear people say "what should I memorize?" it worries me. Of course, memorization can be a first step toward understanding, but I strongly prefer thinking of "talking about math" as more akin to throwing and catching a ball, or walking across a room. Maybe it can be axiomatized (and that may be of some interest), but it seems that our best understanding of it is achieved by viewing it as "real", as opposed to disconnected from our reality.

So, sure, look at "intro to proofs" books, if they are fun, and do exercises in them, if they're fun, but that's further from real math than "T-ball" (hitting a ball off a "tee", for little kids) is from baseball.

Better to look at relatively elementary texts that address some bit of mathematics itself, rather than claiming to talk about "proofs".

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    $\begingroup$ When the only epistemology a person is capable of mustering is "Find a prophet and believe everything they say", this is all moot. $\endgroup$ Jan 10 at 3:00
  • $\begingroup$ @AlexanderWoo, true, there is a noticeable tendency in some parts of the human population... :) $\endgroup$ Jan 10 at 19:05
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The text says it is designed for a 14-week semester. With an instructor, we could guess that means 3 hours a week of class and 6 hours a week of additional work ( = 126 hours?). I cannot say whether someone without an instructor could do it in that time, or could do it at all. That will vary greatly depending on the student.

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I have that book. It's actually a very easy gentle book. I think easier than the medium difficulty calculus books or ODE books (not Spivak) that I have. I don't think you'd have any issues self studying it. Note, that it only has answers for 50% of the exercises, but the amount of exercises is vast (and like I said, the content is easy). So I think you could get the gist just by working the odds. For the occasional times you need help, can go online. But probably don't need it that often.

I do think you should also try to develop your ability to work through things though when studying calculus or ODEs (or other courses). Often there are proofs or at least derivations, initial examples, etc. It's a good skill to learn to pre-read such texts and be an active learner (working the material, doing the derivations as you go). If the development is progressive enough, often you can do some of the derivations on your own, checking back as needed. It will help you build up your problem solving muscles.

(Added) P.s. Just saw the remarks by Woo and I agree. If you think you can be a non-curious calculus/ODE student, go do this (very easy) book and then run into Rudin, you're crazy. Again, I think following and solving/cosolving the proofs, derivations, etc. in Thomas Finney, Tennanbaum and Pollard, or the like will actually be more helpful. And closer to the type of content you'll have in real analysis, if that's where you're heading. If you can handle it, I would even just go straight to Spivak or a book like that now. After all, you've had calculus, even if you didn't pay attention to the proofs and derivations.

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  • $\begingroup$ Thanks for the answer, I have a lot of free time no and I decided to learn proofs. But I don't really use the book in the title tough, I am currently working trough "Mathematical Proofs A Transition to Advanced Mathematics" by Gary Chartrand. First half of this book is very similar to Book of Proof by Hammack. I am planning to go trough this book two times, first doing odd exercises and getting a feel for what its all about and a second run for even and extra exercises. I think this repetition will help me memorize some things better. $\endgroup$
    – a a
    Jan 9 at 1:48
  • $\begingroup$ I don't really know whether I want to do Real Analysis afte that. I heard Abstract Algebra and basic number theory are easier and are more fun. $\endgroup$
    – a a
    Jan 9 at 1:49
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    $\begingroup$ Makes sense. You should be fine. $\endgroup$
    – guest
    Jan 9 at 5:41

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