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I am writing an essay for my calculus class and one of the requirements to meet within the essay is to demonstrate an understanding of integration by explaining a metaphor that uses integration.

This is the passage that I think meets that requirement but I am not sure if I should expand more on integration just to be sure:

To a person familiar with integration attempting to relate the metaphor back to math, this statement likely brings to mind images of their first calculus instructor drawing rectangles below a function when showing the class how to calculate the area under a curve. The reason Tolstoy’s statement conjures this reminiscent math memory to is because the two concepts being discussed are abstractly identical. Just as the wills of man that direct the compass of history are innumerable, so are the number of rectangles that are required to be summed to get an exact measurement of area under a curve. Despite the impossibility of calculating an infinite amount of something we must still calculate some amount of it if we wish to obtain the valuable information an approximation can provide.

For reference, here is the metaphor I am writing about:

"The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history. . . . Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history"

Could anyone provide some feedback? thanks!

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What is that Tolstoy statement you're referring to? – t.b. Jul 8 '11 at 23:51… The passage right below july in that paper – Matt Jul 8 '11 at 23:52
Oy... Thanks for the clarification. Maybe you should include that in your post because otherwise it is a bit difficult to follow what you're saying. One small stylistic remark: In two consecutive sentences you're using the construction "... statement likely brings ... to mind". Also: can a memory be reminiscent? – t.b. Jul 9 '11 at 0:00
I think this is better suited for English stack exchange, as I personally do not see a mathematics question here. – Eric Naslund Jul 9 '11 at 0:12
I'd include the quotation before the excerpt of your essay. The "conjures" sentence has a "to" too much (I think) and I'd drop "reminiscent". I'm not entirely convinced by your last sentence either. To be honest, I think you're not saying a lot but you're using many words - but maybe that's what people do in essays... I'm no native speaker, so I'm not used to reading such things and leave your question to those people here. – t.b. Jul 9 '11 at 0:23
up vote 8 down vote accepted

In my opinion, if this is a serious assignment, then it would be a very difficult one for most students. In order to write something really solid, one needs to (i) have strong general essay-writing skills (this is an unusually difficult topic), (ii) have a very solid theoretical grasp of calculus in order to be able to compare metaphors with theorems and (iii) be able to merge the humanities stuff in (i) with the math stuff in (ii) in a coherent and plausible way. It's a lot to ask!

Since you have found Tolstoy's integration metaphor, I should probably mention that Stephen T. Ahearn wrote a 2005 article in the American Mathematical Monthly on this topic. (His article is freely available here.) Ahearn's article is quite thorough: I for instance would have a tough time trying to write a piece on this topic going beyond what he has already written. (And the fact that I've never read War and Piece is not exactly helping either...) If the assignment is "sufficiently serious", I would recommend that you pick some other integration metaphor to explain. (Exactly how one comes across "integration metaphors" is already not so clear to me, but the internet can do many magical things, probably including this...)

I should say though that in the United States at least it would be a very unusual calculus class that would require a student to complete such an assignment and be really serious about it, as above. (A part of me would really like to assign such an essay in my calculus class, but I think the results would be...disappointing.) If as you say the goal is to demonstrate knowledge of integration, then you should indeed concentrate on that. As ever, it couldn't hurt to talk to your instructor and get more specific information about this assignment: e.g. what is the suggested length of the essay? What sort of places does s/he have in mind for finding such a metaphor? Could you create your own metaphor? And so on.

In summary, if you put this question to us (at present the majority of the "answerers" are advanced mathematics students or math researchers) I fear you're setting yourself up to get picked on. It's probably best to clarify exactly what you need to do: it may not be so much, and it might just be worth taking a crack at it (as you've done) and seeing if that will be sufficient for the instructor.

P.S.: I have read some of Tolstoy's other works (especially Anna Karenina) and nothing math-related springs to mind. However, Dostoyevsky's Notes from Underground has some fun mathy material, although maybe not integration per se. I could imagine writing an ironic piece on whether integration (specifically, explicitly finding anti-derivatives) is as hard-scientific and deterministic as Dostoyevsky's view of mathematics is in this book, or whether the "art of finding antiderivatives" is messy and uncertain like the human condition. But, you know, this could be a failing essay!

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+1: Very nice answer, Pete! It is indeed the kind of assignment I'd love to incorporate in a calculus class! Thanks to for the link to the article you reference. – amWhy Jul 9 '11 at 1:18
Thanks for taking the time to answer! It is an extra credit assignment that only has to be one page in length, so it isn't really all that serious haha. I am taking it semi serious though because I find it interesting. requirement 1 was to demonstrate an understanding of integration by explaining Tolstoy's metaphor and requirement 2 is to discuss whether mathematics is a reasonable tool for analyzing non-numerical concepts. I think I am going to include this comic somehow: :P – Matt Jul 9 '11 at 1:26
@mwnj: Okay, you sound quite on top of things then. The comic that you link to is on one of the office doors in my department. I'm sure many others could say the same... – Pete L. Clark Jul 9 '11 at 2:39

I think that this is a really great assignment, and I think that you likely have an unusual and enlightened instructor, from whom I would encourage you to try to learn as much as you can.

If I were to undertake this assignment, however, I would adopt a more critical tone about the strength of the metaphor. Tolstoy is saying that the movement of humanity arises as the continuous sum of each individual's infinitesimal contribution to it, and is continuous as a result, just as the area under a curve can be thought of as the sum of the increasingly large numbers of increasingly thin rectangles below it, each contributing infinitesimally to the area. Perhaps such a perspective would lead one to a morose attitude on the fate of humanity and the ability of of an individual to affect it---after all, if your contribution has only an infinitesimal affect, then you will not significantly change the outcome.

But my personal outlook on life would compel me to resist this perhaps-depressing conclusion. And so for such an assignment as you have described, I would search for mathematical grounds on which to do so. First of all, of course, it seems that the development of humanity is not continuous; it is rather punctuated by singular developments, such as scientific advances and discovery or political events, such as revolution. In addition, from our study of history it often seems that certain individuals can have and often have had a non-infinitesimal affect on human progress. Think of the great inventors and innovators in history, who changed the course of humanity and scientific development. Isaac Newton was not a slender thin rectangle, contributing only infinitesimally to humanity. And neither were the other great thinkers in our history. (And perhaps consider also those who have had a great negative affect.) Although these individuals may not have acted alone, but made their critical actions after the actions of many others that came before, perhaps they stood on the shoulders of giants---and this will be Tolstoy's reply---the counterargument is that nevertheless it was often comparatively small groups that led to outsize developments, and so their contributions could not have been infinitesimal. After all, the sum of finitely many infinitesimals is still infinitesimal, so if a small group has had outsize total affect, the individual contributions must have been non-trivial. Finally, since there are indeed only finitely many people altogether, the effect of each of them will not be an infinitesimal proportion of the total effect.

I often make term-paper writing assignments for my more advanced courses (and almost always in my graduate courses), but I am now inspired by your question to look for ways to have my calculus-level students undertake more writing.

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$@$Joel: +1. If you actually pull off such an assignment in an undergraduate math course, could you please let me know? I would really like to do it, but I need some help being optimistic as to the likely results. – Pete L. Clark Jul 9 '11 at 2:44

the drop and bucket problem and economics. each person contributes nothing indiidually to the ecnomy. but together as a whole they do.

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@Theo Buehler: Thank you for your kind words. – gall Jul 9 '11 at 2:28
Some contribute less than others though. At least in the spell-checking department! :) – Mariano Suárez-Alvarez Jul 9 '11 at 4:23

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