Talk:Infinitesimal

Independent discovery of derivative by Bhaskara and Sharaf in the 12th century

Has someone checked the sources for such a claim? It is currently sourced in footnotes 2, 3, and 4. Tkuvho (talk) 19:50, 7 May 2011 (UTC)

According to the mathscinet review of the 1984 article, Bhaskara is credited with computing the differential of sine. Claims of his knowledge of the derivative are unsourced. Tkuvho (talk) 11:32, 11 May 2011 (UTC)

A more reliable source on Sharaf is the paper by Hogendijk, Jan P.: Sharaf al-Dīn al-Ṭūsī on the number of positive roots of cubic equations. Historia Math. 16 (1989), no. 1, 69–85. Hogendijk explains that Sharaf exploited ancient and medieval methods rather than 17th century methods, and explained his motivations. Tkuvho (talk) 11:50, 11 May 2011 (UTC)

I don't know about Sharaf, but the one about Bhaskara is an old claim. For a refutation see Footnote 4 by Kim Plofker. Fowler&fowler«Talk» 12:11, 11 May 2011 (UTC)

OK, I already trimmed down the Bhaskara claim to sine. Is the current version accurate in your opinion? Tkuvho (talk) 13:29, 11 May 2011 (UTC)

I think according to Plofker it is not accurate to use the word "differential." Better to say, that Bhaskara found a geometric technique for expressing changes in the Sine by means of the Cosine. (And you could footnote Plofker there.) Fowler&fowler«Talk» 14:44, 11 May 2011 (UTC)

I replaced "differential" by "change". Let me know if this is better. Tkuvho (talk) 14:48, 11 May 2011 (UTC)

The review of the article by Shukla seems to mention an term in the original language which is claimed to mean something like "infinitesimal". Is there any truth to this claim? Tkuvho (talk) 14:56, 11 May 2011 (UTC)

More precisely, the reviewer claims that "Manjula, Āryabhata II and Bhāskara II used the expression Tatkālika-gati (instantaneous motion) to denote differentials". The reviewer is Brij Mohan. I can't say I am familiar with the term. Tkuvho (talk) 15:00, 11 May 2011 (UTC)

Plofker's emphasis is that the Indians "remained within that specific trigonometric context. The differential “principle” was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here". I don't read this necessarily as implying that the Indians may not have possessed, in the trigonometric context, a notion of a differential. Do you? Tkuvho (talk) 15:11, 11 May 2011 (UTC)

Ghostbusting of departed quantities

There is an attempt to rewrite history going on at Ghosts of departed quantities. Please comment at Wikipedia_talk:WikiProject_Mathematics#Ghosts_of_departed_quantities . Tkuvho (talk) 04:15, 26 May 2011 (UTC)

John Gabriel's Aryan mathematics

Contents

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lede

The lede was recently shortened in a drastic way. Wiki policy allows for the statutory 4 paragraphs. Is there any reason to make the lede much shorter than provided by policy? Tkuvho (talk) 16:18, 8 January 2012 (UTC)

n extensions of R

Hello We would like to contibute to the infinitesimals article. It the next article n- extensions of R are constructed, each n-extension has cardinality $\aleph_n$ℵn, so every time happens that the next extension has more numbers than before, so every time we have more holes than numbers in the real line.

Sélem Avila, Elías Proper $n-extensions of ${}^\ast \bold R$∗R with cardinalities $\aleph_n$ℵn. (Spanish) XXIX National Congress of the Mexican Mathematical Society (Spanish) (San Luis Potosí, 1996), 13–24, Aportaciones Mat. Comun., 20, Soc. Mat. Mexicana, México, 1997.

I have translated this article in order that you can read it and disccus it. You can find the translation here: https://docs.google.com/open?id=0B1yg2_0X9n2tNTY4ZmUxNDgtYmY2YS00ZjI2LTlkYTYtNWM0NzU5NjZjY2Fj

I'm (Nselem (talk) 14:38, 25 January 2012 (UTC)) and my father is the author of the article, I'm helping him with typing and translations, so please be patiente with us, we really want to discuss this subject and colaborate if possible.

I'm afraid I don't see the benefits to Wikipedia of the chain of hyperreal fields. Assuming the Generalized Continuum Hypothesis, then, using either the compactness theorem or the ultrapower construction, given any real-closed field *R, there is an extension **R, with infinitesimals over *R, and any specified (non-limit) cardinality greater than or equal to the cardinality of *R. (I'm not sure it needs to be a non-limit cardinal, and I'm pretty sure the compactness theorem method doesn't require it, but the ultrapower method does require that it be the cardinal of a power set.) But I also don't see how to work that into the article. — Arthur Rubin (talk) 15:14, 25 January 2012 (UTC)

What might be of interest is the construction of a maximal hyperreal field that contains all of the above, recently developed by Philip Ehrlich, to appear in BFL, and available at his homepage. The field (which is a class) is isomorphic to the maximal surreals that he describes there. Tkuvho (talk) 15:23, 25 January 2012 (UTC)

(Nselem (talk) 03:36, 26 January 2012 (UTC)) For Rubin : The compacity theorem does not aplply to jump from *R to **R, because *N is not numerable (required condition), it is neither usable for any other construction eçwith a greater cardinality. About the second option that it is mentioned (ultrapowers),even when it is true what is said (It is what it is done in the article) the fact is that it does not work any ultrafilter that contains the fréchet filter (isomorphic extentions to *R are obtained) and this is the usual way to construct *R starting with R; in this way, the extentions are "existentials". For the explicit construction it is required an ultrafilter that contains the filter of the co-bounded sets (with bounded complement ) about *N, as is done in the article; then it is possible to do all the proper extentions *R, **R,..... ****...***R, with incressing cardinals aleph-2, aleph-3, ... aleph-n; with infinitesimals every time smaller, limitless (and each time bigger infinit numbers, limitless). This ultrafilter co bounded over R, works for extend R to *R. And the concept of infinitesimal becomes relative in each extention. This article was reviewed in Current Mathematical Publications, American Mathematical Society,Number 4, March 19, 1999; and Zentralblatt MATH, European Mathematical Society, FIZ Karlsruhe & Springer-VErlag, 0945.03097

You may be right about the ultrafilter construction: *(*R) appears to be quasi-isomorphic to *R, and I can't see immediately whether it has infinitesimals over the embedded *R. Still, the compactness argument will produce, for any field X, a larger field Y with infinitesimals over X, with cardinality at least α as follows:

Define constant symbols c_x for each x in X, constant symbols d_β for each β < α, and constant symbol ε with axioms:

c_x + c_y = c_{x + y} and $c_x \cdot c_y = c_{x \cdot y}$ for x, y in X

c_x < c_y for x < y in X

$0 < \epsilon$

$\epsilon < c_x$ for x > 0 in X

$d_\beta \neq d_\gamma$ for β < γ < α

— Arthur Rubin (talk) 06:58, 26 January 2012 (UTC)