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Having read several undergraduate textbooks in complex analysis (Stein-Shakarchi, Gamelin, etc.), I find that some "hand-waving" arguments are frequently used. An example (the proof of the Cauchy integral formula in Stein-Shakarchi) is given in the appendix regarding what I mean for "hand-waving".

Hand-waving arguments in students' homework may be sometimes (usually?) due to a certain degree of "dishonesty": one wants to gloss over a part that one does not fully understand to get credits. If one is brutally honest to oneself when doing homework, then one might not use such arguments at all or one would explicitly admit that one is using such arguments so that the readers should be cautious in advance.

On the other hand, textbooks written by masters are another story.

Questions:

  • What are possible reasons for making hand-waving arguments in textbooks of undergraduate analysis? (Some are deliberately left as helpful exercises for the readers but some are not. I should be referring to those non-exercise ones.) A hand-waving argument in algebra is very different from one in analysis and I would like to restrict my question in the context of analysis here.

  • In the views of students, how should one deal with such arguments for learning?

[An example of hand-waving arguments in Stein-Shakarchi.]

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    $\begingroup$ What would you need to make this rigorous? Probably a digression on the convergence of the values of integrals when the integrands are uniformly convergent. That's a fairly large distraction for a fairly minor point in the proof and also something a typical reader very well might have already seen. $\endgroup$
    – Adam
    Oct 29, 2017 at 19:22
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    $\begingroup$ @Adam If the reader is new to complex analysis it should be fair to assume that such digression will likely distract him and leave him confused as to why it is true. Lacking the mathematical sophistication to carry out the calculations in full detail, he may feel discouraged and unable to tell if this is a fairly minor point in the proof. Constraints of time may also limit his ability to reach these conclusions in appropriate time to hand in assignments or do well on tests. $\endgroup$
    – Mark Fantini
    Oct 29, 2017 at 23:55
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    $\begingroup$ My guess is that the author found that including a sketchy, terse appendix on complex analysis was better than not including complex analysis at all. I personally find this okay as long as the author is being clear and explicit about the terseness (telling students not to imitate the style when they're learning proofs). $\endgroup$
    – darij grinberg
    Oct 30, 2017 at 6:22
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    $\begingroup$ Complex analysis is directly relevant to e.g. engineers in a way that real analysis courses are not. The typical undergraduate text in complex analysis doesn't target just math majors, which puts restrictions on how rigorous it can be. $\endgroup$
    – John Coleman
    Nov 5, 2017 at 19:53
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    $\begingroup$ The "example" dies not show any hand waving. The proof is rigorous and filling in the gaps is routine. I understand by hand waving and argument that is not entirely precise, maybe ignores some cases, that sort of thing. Being terse does not qualify. $\endgroup$
    – Andrés E. Caicedo
    Mar 10, 2019 at 1:37

2 Answers 2

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A proof is meant to convince a reader of the truth of some statement. When a mathematician is communicating an argument to another mathematician, you only include the level of rigor that you need so that the other mathematician is convinced that they could (in principle) give a fully rigorous argument. Even that isn't quite accurate because no one (short of people producing fully formal computer verified proofs) actually produces totally formal arguments ever.

In your example, while you could write down equations and inequalities for this keyhole shape, doing so would take up a large amount of room and distract from the clarity of the argument.

If you are not currently at a level where the argument is clear, then it is your job to fill in the details to a level of rigor you feel comfortable with. This is a nice additional exercise. In fact, every textbook (even one which is very rigorous) will have frequent small gaps in logic which you need to fill in on your own. Reading a math book is a creative endeavor in which you create the mathematics, and refine your own arguments, with a lot of assistance and guidance from the book.

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    $\begingroup$ Right to the point. To any reader feeling they are being much more rigorous, think about the following example. At which level do you use freely, without proof, that the two definitions of limit $\forall \varepsilon>0, \exists N\in\mathbb{N}, \forall n>N, |u_n-\ell|\le \varepsilon$ and $\forall \varepsilon>0, \exists N\in\mathbb{N}, \forall n\ge N, |u_n-\ell| < \varepsilon$ are equivalent? At which level do you think your students could provide a correct proof? $\endgroup$
    – Benoît Kloeckner
    Oct 30, 2017 at 12:17
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    $\begingroup$ I don't think my questions are about journal articles or monographs ("When a mathematician is communicating an argument to another mathematician"), the arguments of which are usually suppose to be terse and should be fundamentally different from an undergraduate course textbook. $\endgroup$
    – user5897
    Nov 5, 2017 at 13:06
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    $\begingroup$ @Jack I do not think they are fundamentally different. Whenever you are communicating mathematics to another human, you have to decide on what level of rigor. You can never be 100% except to a computer. See Benoit's comment above for instance. This also means that reading any proof requires some degree of creative input from the reader, which is a good thing. $\endgroup$
    – Steven Gubkin
    Nov 5, 2017 at 13:19
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    $\begingroup$ I guess my point is that it is a matter of taste. Which parts of which arguments do you include more rigor? Which do you include less? Which are you satisfied with a "big picture idea", and where do you want ot dig down into details? These are choices. As the instructor, you may have a different aesthetic than the author of the book in some cases. Great! Now you have some material for your lecture. Or if you are self studying, you now have something to think about deeply. This is healthy. $\endgroup$
    – Steven Gubkin
    Nov 5, 2017 at 13:49
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    $\begingroup$ @user3813 I would personally prefer to carve the domain into two halves, both of which have zero integrals along the boundary, and sum these to obtain the needed cancellation. Then you don't need to be finicky with a limiting argument. But I don't think the original text was bad. It is actually interesting, and provides a good opportunity for discussion. $\endgroup$
    – Steven Gubkin
    Feb 28, 2019 at 13:02
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One reason for not dismissing the complaint that graduate math textbooks do too much handwaving is that some of the authors of graduate textbooks have the same complaint. For example, in his introduction to Stochastic Integration Theory, Peter Medvegyev writes: "[The] books concentrating on the general theory were, for me, . . . a bit sketchy. I very often had quite serious problems trying to decode what the ideas of the authors were, and it took me a long time, sometimes days and weeks, to understand some basic ideas of the theory. I was nearly always able to understand the main arguments but, looking back, I think some simple notes and hints could have made my suffering shorter."

Mathematicians who would publish their "notes and hints" for "textbooks written by masters" as some sort of systematic commentary would be doing a great service to the profession.

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    $\begingroup$ In fact, this service is offered by the Math Stackexchange. Being able to ask experts about textbooks is such a privilege of our modern age. It used to be you had to be somewhere like a university to have hope for such questions. More and more, with the decline of scholarship at many institutes of higher learning, the role of the "university" will decline. Those who thirst for knowledge do so without regard for certification (degree)... I digress. $\endgroup$
    – James S. Cook
    Feb 28, 2019 at 19:50

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