Talk:Arithmetization of analysis

Created this page (obviously still under construction) thinking it would serve as a helpful place to summarize not only the historical facts of the "arithmetization of analysis" but also some current lively discussion of its significance for the "foundations of mathematics," whatever that may turn out to be! LOL Whayes43 (talk) 18:12, 15 April 2014 (CEST)

Current lively discussion, really? I did not know. Where to look for this?

"the definition of the theory of real numbers" --- do you really mean it, or rather, the definition of [the set, or orderd field,... of] real numbers?

Boris Tsirelson (talk) 19:35, 15 April 2014 (CEST)

Yes, thank you, I ought of course to have written "creation" of the theory of real numbers -- as is contained in the footnoted quotation -- or something equivalent.

as for "lively discussion" I found some here: http://www.cs.nyu.edu/pipermail/fom/1998-January/000804.html Perhaps not as lively as I recalled! In any case, Enjoy!

Whayes43 (talk) 04:32, 16 April 2014 (CEST)

Rather lively, I see, thanks. Boris Tsirelson (talk) 08:24, 16 April 2014 (CEST)

But, it seems, you insists on the spelling "pilars" rather than "pillars"? Boris Tsirelson (talk) 19:54, 16 April 2014 (CEST)

Oooops! I didn't see any further comment here, so I pasted in a changed version of the page. Hope I didn't over-write a change that you made, Boris Tsirelson I will look at the history and try to patch up any damage I may have done. I'll be more careful in future! Whayes43 (talk) 16:50, 19 April 2014 (CEST)

Yes, please. Hope you do not hate TeX. It seems, you keep at home a version and just upload... this is indeed not a good idea on a wiki. On the other hand, if needed, you can temporary write "Please do not disturb now; let me work intensively during two days" or something like that. Another (rather good) possibility is: work on your sandbox (there no one will disturb you) and then, when ready, move it to the mainspace. Boris Tsirelson (talk) 19:54, 19 April 2014 (CEST)

In fact, I only added the dollar signs around your formulas. Boris Tsirelson (talk) 16:43, 20 April 2014 (CEST)

I see Pierpoint in the Notes but not in Primary sources nor References. Boris Tsirelson (talk) 08:14, 6 May 2014 (CEST)

My apology . . . and many thanks for your tireless editorial watchfulness! Greatly appreciated. Spelling of his name actually is Pierpont.

"Cauchy then proved that a necessary and sufficient condition that an infinite series converge is that, for a given value of $p$, the magnitude of the difference between $S_n$ and $S_n+p$ tends toward $0$ as $n$ increases indefinitely." — Really? As far as I understand, this claim is wrong. Is it meant "Cauchy believed that he proved"? Or is it meant, not for a given value of $p$, but just the opposite: $p$ is permitted to grow with $n$? In the form written it implies convergence of every series whose terms tend to $0$. Boris Tsirelson (talk) 18:58, 10 September 2014 (CEST)

You are a tireless editor, Mr. Tsirelson, for which I again express my thanks. My laptop malfunctioned during my last edit (just another poor workman blaming his tools!) requiring that I re-enter a number of small amendments. In the process of doing so, I omitted the {__} around the subscripts in the expression $S_{n+p}$, which I believe solves the problem. It must be (somewhat) amusing for you to be monitoring the efforts of "professionals", who, even after working with computers for 47 years, still make amateurish mistakes?!

In any case, I will attend to the matter with as much haste and as little waste as I can manage! --Whayes43 (talk) 14:59, 11 September 2014 (CEST)

No, the braces around the subscripts do not solve the problem. The problem is mathematical, not technical. Consider for example the harmonic series $ 1+\frac12+\frac13+\dots$; here $ S_{n+1}-S_n = \frac1n \to 0 $; also $ S_{n+2}-S_n = \frac1n+\frac1{n+1} \to 0 $; and so on, for every (fixed!) $p$. Nevertheless, the series diverges. Boris Tsirelson (talk) 17:31, 11 September 2014 (CEST)

Ah, yes, of course. (Apparently, it has been known since the 14th century that the harmonic series diverges!) It's interesting, as Grabiner (1981) notes, that one 18th century way of defining convergence of a series is that the nth term goes to zero, even though (as with the harmonic series) there is no finite sum. She writes this:

In eighteenth-century work on series, sometimes a series is said to converge in the way that the hyperbola 'converges' to its asymptote, that is, when its nth term goes to zero; at other times the series is said to converge in our sense, that is, when its partial sums approach a limit, which is then called the sum of the series. Thus a series may converge in the first sense without converging in the second sense -- Cauchy's (and our) sense.

Thank you for pointing out this lacuna. I will correct -- and endeavour to work with more care.

Nice. Not knowing the history but knowing the notion now called Cauchy sequence I guess that Cauchy had good understanding of the importance of $p$ not to be fixed but be permitted to grow with $n$. Boris Tsirelson (talk) 16:24, 12 September 2014 (CEST)

Thanks for your comments. I will remove the following problematic text:

Cauchy then established the necessary and sufficient condition that an infinite series converges:

for a given value of $p$, the magnitude of the difference between $S_n$ and $S_{n+p}$ tends toward $0$ as $n$ increases indefinitely.

Some investigation reveals that Cauchy did not even attempt a proof of the sufficiency of the the Cauchy criterion!

The excerpt below [from Grabiner (1981) p. 102] provides interesting background that may interest you:

Cauchy called attention to the differences between the first and the succesive partial sums, defined by

$S_{n+1} - S_n = u_n$ ::$S_{n+2} - S_n = u_n + u_{n+1}$ ::$S_{n+3} - S_n = u_n + u_{n+1} + u_{n+2}$ :: . . . :: . . . :: . . . Grabiner continues as follows: :For the series to converge, it was known [by Cauchy and others] to be necessary that the first of these, $u_n$, go to zero. But it was also known, as Cauchy pointed out next, that this was not sufficient: ::It is necessary also, for increasing values of $n$, that the different sums $u_n + u_{n+1}$, $u_n + u_{n+1} + u_{n+2}$ ... , that is, the sums of the quantities $u_n$, $u_{n+1}$, $u_{n+2}$, ... taken, from the first, in whatever number we wish, finish by constantly having numerical [that is, absolute] values less than any assignable limit. ''Conversely, when these diverse conditions are fulfilled, the convergence of the series is assured''. I believe that the following [from Boyer p. 566], which I stated Cauchy had established as a necessary and sufficient condition of convergence, was meant [by Boyer] to be a summary of the above (translated) text of Cauchy's: :for a given value of $p$, the magnitude of the difference between $S_n$ and $S_{n+p}$ tends toward $0$ as $n$ increases indefinitely.