Talk:Outline of calculus

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Major rename proposal of certain "lists" to "outlines"[edit]

See Wikipedia:Village pump (proposals)#Major rename proposal of certain "lists" to "outlines".

The Transhumanist 01:17, 12 June 2008 (UTC)

Rename proposal for this page and all the pages of the set this page belongs to[edit]

See the proposal at the Village pump

The Transhumanist 09:09, 4 July 2008 (UTC)

are limits critical?[edit]

The current version states:

"Calculus is a central branch of mathematics, developed from algebra and geometry. It is built around two major complementary ideas, both of which rely critically on the concept of limits"

This may be a misconception, as well as an ahistorical inaccuracy. Katzmik (talk) 18:12, 15 January 2009 (UTC)

Why is it a misconception or a historical inaccuracy? Limits appear early in the work of Newton and implicitly in the work of Leibniz. But aside from history, could you explain why you feel it is a misconception? Thenub314 (talk) 21:15, 15 January 2009 (UTC)
The central notions of calculus are derivative and integral. I would not call limits a "critical" concept, but rather a technical tool in developing the two central notions. Katzmik (talk) 14:45, 18 January 2009 (UTC)
I would disagree. I agree that differentiation and integration are the two most fundamental concepts in calculus, limits play a much more important role then just a technical tool. I think "critical" is a fair assessment. Thenub314 (talk) 10:02, 19 January 2009 (UTC)
From discussions at other pages I am convinced that you are perfectly aware of the fact that calculus developed very nicely thank you for two hundred years before limits were ever formalized. I have seen elsewhere that you consider that Newton already had some sort of a notion of limit, and I assume you are referring to his fluxions. However, fluxions are not limits. To say that limits are a technical tool is not to denigrate them, incidentally. After all, mathematicians are concerned with rigor. However, calling them "critical" seems to imply that you can't understand calculus without them, which is hardly the case (one can do fine with standard parts, for example). Katzmik (talk) 12:58, 19 January 2009 (UTC)

(unindent) I was not thinking simply of fluxions, I was thinking of the paper "Newton and the Notion of Limit" by B. Pourciau in Historia Mathematica vol 28, no 1 (2001). Specifically he quotes Newton as saying "Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...." and goes on to make a good case that Newton's notion of limit was much clearer then typically thought. The notion of limit also explicitly appears before Newton, for example in the work of Wallis, according the the histories I have read his definition was vague, but contained the right idea. You can do with standard parts, and you could also do with epsilons and deltas and avoid the use of the word limit. Either of these just seems to be unraveling the definition. Thenub314 (talk) 15:51, 19 January 2009 (UTC)

As far as the last point you made, I would like to comment as follows: that's precisely my point! One could avoid limits altogether in one way or another, but the definitions will certainly become more awkward. Just think of how a freshman would react to a definition of the derivative as the number such that for all epsilon there exists a delta etc. In other words, limit is a convenient tool in defining the critical notions of calculus such as derivative and integral.
As far as the second point you mentioned, I was not aware of the paper by Pouciau. I must say I am a little sceptical. How influential was this paper? Is it frequently cited? Also, just because Newton used the word, it does not mean that he went any further than that. People before Enstein also used to say that everything is relative, but this does not make then inventors of relativity theory. Katzmik (talk) 16:00, 19 January 2009 (UTC)
Well I can all I can say is I disagree. I don't spend valuable class time covering techniques for finding limits because the only interest is to apply them to the definition of derivative and integrals. They are an essential idea in calculus, and I agree with the use of "critically" here.
As for your other question, I do not know how influential it was. As you probably know, I am not a historian, so it is difficult for me to assess the overall impact of the paper. My understanding is that Historia Mathematica is a respected journal, and I don't think they would publish a paper that whose thesis was, "he used the word!". It point more the second part of the quote where Newton describes what a limit is in terms of approaching to within an given quantity. In the paper he goes on to argue that Newton uses delta epsilon style proofs for some of his lemmas. Thenub314 (talk) 16:32, 19 January 2009 (UTC)
At any rate, that's a rarely encountered point of view. I am not generally in favor of WP rhetoric, but this is a clear case of undue weight, even if everything Pouciau says is correct. Is it really up to us to decide that his view point is the truth at the expense of what most people out there think? As far as "essential idea", what you write is exaggerated in my opinion. We do limits precisely because they are convenient tools in defining derivative and integral. The proof is that we STOP using them, for the most part, once the definitions are out of the way. Katzmik (talk) 16:39, 19 January 2009 (UTC)
We only stop explicitly using limits for well behaved and well understood functions such as polynomials, where we have developed tried and tested shortcuts. Calculating the derivative or integral of a general function or a value based on such an integral (like the Euler–Mascheroni constant, for example) still requires the explicit use of limits. I agree with Thenub314 - limits play a central role in calculus, and wording of the first sentence of the article is fine as it stands. Gandalf61 (talk) 17:00, 19 January 2009 (UTC)
OK, I find your viewpoint interesting though somewhat surprising. That makes three of us. I think this is an intriguing discussion and would like to get some more input. On the other hand, it does not seem so critical :) that it would be necessary to take it to WPM. So I am not sure what to do at this point. Katzmik (talk) 17:20, 19 January 2009 (UTC)
P.S. At any rate, Gandalf61's comment points to the criticality of "limits" in analysis, not calculus the way it is usually taught today, where one rarely encounters the Euler–Mascheroni constant. Katzmik (talk) 17:29, 19 January 2009 (UTC)
Katzmik, we all know where this is going. You want to bang your non-standard calculus drum and assert calculus could be taught without the concept of limits and so they can't be central to calculus. And you could be right - in theory. However, in practice, limits play a central role in the field of calculus as it is taught and used by most mathematicians, and most mathematicians would be happy with the first sentence of this article as it stands, and your contention that this is a misconception is a tiny minority view. Now you may say that is just my opinon. But if you are really interested in what the wider community thinks, then I suggest you go ahead and flag this discussion at WT:WPM. Gandalf61 (talk) 10:23, 20 January 2009 (UTC)
I'd be quite happy to remove the bit about limits. It is more to do with analysis than calculus. I think it would be silly to use non-standard analysis with people who aren't ever going to be interested in things like ultrafilters, it is just asking for trouble. Limits is better for introducing the subject especially for people who want it as a practical tool. Here though it is just unnecessary to mention limits. Dmcq (talk) 14:01, 20 January 2009 (UTC)

It seemed to me that the lead was anyway scanty, so I have expanded it and tried to give some decent perspective on analysis relative to the needs of learning calculus. Charles Matthews (talk) 16:47, 1 February 2009 (UTC)