Talk:Infinitesimal/Archive 2

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&amp;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&amp;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. Tkuvho (talk) 04:15, 26 May 2011 (UTC)

Suggestion to remove infinitesimal from Wikipedia
Unless you are able to well-define an infinitesimal (and its plural), you should remove all references to it from every article on Wikipedia. 12.176.152.194 (talk) 19:44, 3 December 2011 (UTC)

I have commented before on this page from 70.120.182.243 — Preceding unsigned comment added by 12.176.152.194 (talk) 19:45, 3 December 2011 (UTC)


 * You have the option of opening an article for deletion process, see WP:AfD. Could you elaborate on the reasons for your proposal? Tkuvho (talk) 22:09, 3 December 2011 (UTC)

They are not well-defined. Infinitesimals do not exist in theory or reality. The article falsely states that Archimedes used 1/inf. However, 1/inf is not a number. It is also not an infinitesimal because it is ill-defined. Robinson's theory using ultra-filters (set theory) has also been shown to be false. 12.176.152.194 (talk) 03:12, 4 December 2011 (UTC)


 * Wikipedia has many pages on things that "do not exist in theory or reality" - take the Flying Spaghetti Monster as an example. As one of our core policies says, the threshold for inclusion in Wikipedia os verifiability, not truth. In other words, if something has been discussed in multiple reliable independent sources, then we can have a Wikipedia article on it, regardles of whether it is "real" or not. Infinitesimals certainly meet this threshold test. Gandalf61 (talk) 09:21, 4 December 2011 (UTC)


 * Thanks for your comment. I would be interested, though, in clarifying the IP's position on this.  First, the article does not claim that Archimedes used 1/infinity.  Rather, it was Wallis who used 1/infinity.  I am also interested in the IP's position on ultrafilters, as well as on the axiom of choice.  Also, do you have some details on the alleged refutation of Abraham Robinson's theory?  Was this done in the framework of ZFC?  Tkuvho (talk) 12:19, 4 December 2011 (UTC)


 * The article states that Archimedes exploited infinitesimals which in my opinion is the even stronger than saying he used infinitesimals. I have already discussed my opinions on ultra-filters in my dialogue with Rubin.


 * Gandalf61: In mathematics, it is of great importance that concepts are well-defined. You can have as many discussions as you like, but you cannot write articles that masquerade as encyclopedic when ill-defined terms such as infinitesimal/s are used. Furthermore, you cannot say anyone used infinitesimals when it is known that the same do not exist either in theory or in practice. 12.176.152.194 (talk) 03:07, 11 December 2011 (UTC)


 * There is no policy in Wikipedia which says that a concept has to be "well defined" (in any sense) or that it has to exist (in any sense) before it deserves an article. We have many excellent and encyclopedic articles on topics that are not at all well defined, such as intelligence, democracy and socialism. You are entitled to your opinions on infinitesimals, but I think you need to become more familiar with Wikipedia's policies before you can have useful views on which articles should and should not be written. Gandalf61 (talk) 10:38, 11 December 2011 (UTC)


 * Again, an article on mathematics is not the same as an article on intelligence or democracy. In fact, one can argue for these also, but I won't except to state that regardless of Wikipedia policy or not, mathematics articles must contain concepts that are well-defined. If not, then as long as anyone writes an entertaining article, then it should not be deleted, yes? However, such an article would be deleted because it is not factual. This article contains no facts about infinitesimals; only speculation and ill-defined concepts. As for useful views, I would certainly not rely on your opinion. 12.176.152.194 (talk) 03:35, 13 December 2011 (UTC)


 * I am not asking you to rely on my opinion. I am asking you to read and understand Wikipedia's core policies and accept that those policies determine what is suitable for inclusion in Wikipedia. Until you do this, there is no point continuing this discussion. Gandalf61 (talk) 08:58, 13 December 2011 (UTC)


 * You are correct. I have no wish to continue this discussion with you. Perhaps you should cease to respond to my comments. This would be the right thing to do, yes? Bye, bye. 12.176.152.194 (talk) 17:00, 13 December 2011 (UTC)


 * Thanks for your comments. I am actually interested in what the IP has to say about this.  It seems to me that the talkpage can accomodate this type of discussion.  Question to the IP: you seem to feel that an infinitesimal is not a well-defined concept.  How would you react if I gave you a specific well-defined infinitesimal, such as 1 - 0.999... ?  Tkuvho (talk) 15:20, 11 December 2011 (UTC)


 * I do not agree that 1-0.999... is well-defined. Wikipedia has an article that states 0.999... = 1, therefore in this case the difference is 0, which is not an infinitesimal but a well-defined concept. If you are trying to express the magnitude that succeeds zero, it is still not well-defined. But even if you suppose that 1-0.999... is well-defined, then what would be the next infinitesimal? More precisely, where do infinitesimals begin and most importantly, where do these end? 12.176.152.194 (talk) 03:40, 13 December 2011 (UTC)
 * The article you mentioned specifically contains a section 0.999 explaining how to define 1-"0.999..." in such a way as to make it a well-defined concept. I am interested in your contention that infinitesimals are ill-defined but it has to be backed up by sources before it can be implemented at wiki.  Specifically, what kind of framework do you accept as well-defined?  Is it ZFC?  Tkuvho (talk) 11:57, 13 December 2011 (UTC)


 * Well, one must be able to derive an object - either theoretically or physically from any concept in order for it to be well-defined. I am a mathematician and do not accept that an infinitesimal/s is/are well-defined. If I cannot understand the nonsense in this article, I have no doubt that any lay person or beginner or even a graduate of mathematics will be able to understand it. What I am saying is, this article and all other articles that mention infinitesimals are not factual. I put it to you that anyone who reads this does not understand because it makes no sense. Now if you say 1-0.999... is infinitesimal, call it phi. So show me, what can you do with phi? 12.176.152.194 (talk) 17:00, 13 December 2011 (UTC)
 * See Special:Contributions/70.120.182.243 for his/her similar comments in 2007. The ones in this talk page are presently in the first 4 sections of the talk page.  I agree that his/her contentions there make little sense. I tend to agree that the comments here about 1/inf and 1 - 0.999999... are not particularly helpful, either, but the anon's contentions were refuted in 2007. — Arthur Rubin  (talk) 17:36, 13 December 2011 (UTC)


 * That my objections "were refuted" is false.
 * If I recall correctly Mr. Rubin, you could not substantiate any of the claims you made in those arguments. By reading that discussion carefully, one will see that it was you who was refuted.


 * I have no intention of having the same discussion with you again. In fact, I think you should draw attention to your biographical article on Wikipedia which is not notable and has survived so many attempts at deletion. Mr. Rubin, you are a poor excuse for a mathematician. 12.176.152.194 (talk) 05:52, 14 December 2011 (UTC)
 * You do not recall correctly, and you (if you are the author of the web page you refer to below) have no credibility as a mathematician. I have over a dozen published papers in peer-reviewed journals, and (if you're interested in speculation), my analysis of 3-dimensional unitary algebras over the reals was taken from sci.math and published in a real journal.  — Arthur Rubin  (talk) 07:14, 14 December 2011 (UTC)


 * It is irrelevant how many publications you have made. A mathematician is not one who has *credibility* in the eyes of others. He is one who has accomplished great things in mathematics. Your publications are full of errors and serve no purpose whatsoever. This is to be expected because you are exactly the academic that has credibility only in the eyes of his peers. That you have an article on yourself that has survived so many attempts at deletion is proof that you are not a notable mathematician. 12.176.152.194 (talk) 17:25, 16 December 2011 (UTC)


 * I would like to follow up on the IP's suggestion to denote by phi an infinitesimal of the form 1-"0.999...". One needs one further assumption, namely that phi is an element of an ordered field extending the real numbers.  If we are in such a situation, then we can usefully apply phi to develop the calculus.  Thus, we can define the deritive of a function f by forming the quotient f(x+phi)-f(x)/phi.  This quotient in general will not be a real number, but it will be infinitely close to a real number; that real number is the derivative of f at x.  If Leibniz thought that's a good approach to the "differential quotient", perhaps some students may think so, as well.  To pursue this a little bit further: you did not respond to my request to clarify the nature of mathematics that you do find "well-defined and factual".  Does ZFC meet this criterion, in your opinion?  Tkuvho (talk) 18:01, 13 December 2011 (UTC)
 * Well I would like to agree with Arthur Rubin, infinitesimals most certainly have a place in wikipedia. As far as 1-0.999..., I would have to agree that this is not a particularly helpful discussion as the example is quite misleading.  The whole point as I understand it is that is that there is some ambiguity as to what an ellipses means in this setting, and so the quantity might refer to an infinite family of infinitesimals, but it is not a well defined set of symbols.  This is complicated by the fact that in the real numbers $$1-0.\bar 9=0$$ which is of course a necessarily true in the non-standard setting as well, written in this way there is no confusion with the ellipses. Thenub314 (talk) 19:36, 13 December 2011 (UTC)
 * I am familiar with your opinion on this matter. I was hoping to find out the IP's.  Tkuvho (talk) 20:10, 13 December 2011 (UTC)
 * I had forgotten we had ever discussed this particular issue. Thenub314 (talk) 20:25, 13 December 2011 (UTC)
 * Sorry if I misunderstood your position. What is it?  Tkuvho (talk) 12:42, 14 December 2011 (UTC)

Response to Tkuvho
Cauchy's definition of the derivative is a kludge. No "infinitesimals" are used or required in differential calculus. I reject ZF theory because it is not required or particularly useful in mathematics education. My ideas are outlined here: http://thenewcalculus.weebly.com

12.176.152.194 (talk) 06:00, 14 December 2011 (UTC)


 * That's interesting. It could be used elsewhere in Wikipedia if it were published.  It has nothing to do with the fact that infinitesimals are accepted formal mathematics, now that the proper formalism has been introduced, and were accepted mathematics before the formalism was introduced.  It has nothing to do with this article.   — Arthur Rubin  (talk) 07:11, 14 December 2011 (UTC)


 * How can you say it has nothing to do with this article? It has *everything* to do with it. Were it not for Cauchy's unsound work on infinitesimals, the mathematics of infinitesimals would never have existed. Again, the article contains non-factual statements such as "Archimedes exploited infinitesimals". Well, in order for this to be true (which it is not), Archimedes would have had to know what they are. He did not. Archimedes would *never* have used any concept that was ill-defined. We've had this conversation and it's no use rehashing it. According to your Wikipedia rules, information has to be factual. So tell me, what did Archimedes' infinitesimals look like? You can't even tell me what an infinitesimal looks like now. Cauchy's theory leaves much to be desired. He defines it as something infinitely small and proceeds to talk about orders of infinitesimals - also ill-defined. 12.176.152.194 (talk) 16:50, 14 December 2011 (UTC)


 * Arthur, thanks for your comment, with which I agree. John (a.k.a. 12.176.152.194): I read your text on Cauchy with interest.  Unfortunately, there is an error in it.  You propose to choose k and h in such a way that the ratio will be equal exactly to the formula for the derivative that one expects, namely 6x in the example you gave.  However, this procedure will not work in all examples.  Consider, for instance, the function which vanishes for negative x, and is equal to 3x^2 for nonnegative x.  No matter how hard you try, you won't be able to make the quotient at x=0 equal to the value one expects, namely 0.  Good try, though!  Tkuvho (talk) 12:31, 14 December 2011 (UTC)


 * What you have noticed is not an error Tkuvho. It is a *correct* observation. The reason for this is that the cubic is NOT differentiable at x=0 (contrary to popular thought). This has to do with the fact that differentiability in terms of Cauchy's definition is wrong. 12.176.152.194 (talk) 16:44, 14 December 2011 (UTC)
 * Which cubic are you referring to? Tkuvho (talk) 16:50, 14 December 2011 (UTC)


 * You were talking about 3x^2 which is the derivative of x^3. Did I misunderstand what you were saying? Please explain clearly if I did. There are no errors in my New Calculus - I am certain of this. What do you mean by "the function which vanishes for negative x" ?12.176.152.194 (talk) 16:53, 14 December 2011 (UTC)
 * The example you treat in your Cauchy text is 3x^2, whose derivative is 6x. What I am proposing is to consider the function which is equal to zero for negative x, and is equal to 3x^2 for nonnegative x.  If you draw its graph, you will notice that it is smooth at the origin, in the sense that the tangent line exists there.  Therefore one would expect the derivative to exist, as well.  Are you implying that this function is not differentiable in your approach?  That would be a major drawback.  Tkuvho (talk) 17:06, 14 December 2011 (UTC)


 * Okay, I see. This is not a function a then. It is a piecewise "function" which is just another name for two different functions in this case. Of course it would not apply. Calculus applies only to smooth and continuous functions. 12.176.152.194 (talk) 19:12, 14 December 2011 (UTC)


 * Of course Tkuvho's example is a function ! You must have invented a new definition of function to go along with your New Calculus. Fellow editors, I believe we are being trolled. Gandalf61 (talk) 13:35, 15 December 2011 (UTC)


 * Not necessarily. Actually a case can be made in favor of what he says, in the context of intuitionistic mathematics.  The example I proposed is not defined on all of R in that setting.  However, I think it should be possible to construct a counterexample that would satisfy even intuitionistic criteria.  Tkuvho (talk) 13:53, 15 December 2011 (UTC)


 * I feel I would comment here that, even within constructive mathematics, the specific function you mention is fine. Though not all piecewise functions are defined, that one is.  So your correct that even within constructive mathematics.  Mostly thought Gandalf61 is correct.  After the books above are results above regarding new calculus are published in a reliable source they then might be reasonable to include.  But removing references to infinitesimals is not possible.  Even if you disagree with their existence, they are certainly something that is discussed frequently in the wider world and so are suitable for inclusion. Thenub314 (talk) 18:43, 15 December 2011 (UTC)
 * You are in error. For the function to be defined on all of R, we would need the law of trichotomy.  This law is often not assumed in constructive mathematics.  For example, van Dalen's book contains counterexamples based on the violation of the law of trichotomy.  At any rate, as far as the IP's theory is concerned, it turns out that the cubic x^3 is also not differentiable in his sense!  So constructive mathematics is a bit of an overkill.  Tkuvho (talk) 20:31, 15 December 2011 (UTC)
 * I agree with you fully about the law of trichotomy, but that is more a problem with your description of the function then the function itself. For example $$|x|$$ is constructively defined (Techniques of Constructive Analysis Bridges and Simona page 30) as is $$y(x)=(x+|x|)/2$$.  Finally, your function is simply $$3y(x)^2$$. Sums and compositions are fine within the constructive framework, so your function would be fine also.  Thenub314 (talk) 23:10, 15 December 2011 (UTC)
 * Good point, thanks. Tkuvho (talk) 09:05, 16 December 2011 (UTC)
 * The function |x| is a set of conditional functions, that is, f(x) = x when x>0; f(x)=-x when x<0 and f(x)=0 when x=0. By the way, how can you define it  any other way except "constructively"? 64.134.230.145 (talk) 23:40, 15 December 2011 (UTC)

Response to Rubin
Perhaps you can point to a source that shows Archimedes knew what an infinitesimal is? I do not believe there is any such source. 12.176.152.194 (talk) 16:57, 14 December 2011 (UTC)
 * According to our article on The Method, he used infinitesimals, whether or not he knew what they were. — Arthur Rubin  (talk) 17:05, 14 December 2011 (UTC)
 * The article on the Method does not show how Archimedes used infinitesimals; rather it shows how he used the method of exhaustion.

12.176.152.194 (talk) 17:20, 16 December 2011 (UTC)


 * Not only did Archimedes use indivisibles, but also the greatest mathematicians such as Leibniz and Euler did. If you wish to declare them charlatans along with Cauchy, you won't have too many names left to go around.  Tkuvho (talk) 17:07, 14 December 2011 (UTC)


 * So, write down an infinitesimal that Archimedes used?

You cannot say he used them, if you cannot even provide one example. 12.176.152.194 (talk) 19:10, 14 December 2011 (UTC)
 * You got me on this one, thanks. Archimedes used indivisibles, not infinitesimals.  I corrected the page yesterday as soon as I read your message.  It is true that some of our pages may report Archimedes as using infinitesimals, but this is not exactly true, unless of course you extend the meaning of "infinitesimal" to include "indivisible" (in Cavalieri's sense), of course.  Tkuvho (talk) 13:51, 15 December 2011 (UTC)


 * Exactly. Indivisibles are well-defined (I say this with caution...). I invented something called a "positional derivative" which I use in my Average Sum Theorem (similar to Cauchy's different order infinitesimals but well-defined which is something that can't be said for Cauchy's ideas in this regard) and in the proof of the Mean Value Theorem. The positional derivative is the best definition of an indivisible using standard calculus.

At the following link:

http://www.researchgate.net/group/Mathematical_Articles/files/

You will find three articles that might interest you: The Positional Derivative.pdf The definition and proof of the Mean Value Theorem.pdf John Gabriel's Average Sum Theorem.pdf

Just to clarify: An indivisible in my opinion is well represented by the idea of a real number that is part of some interval. As a simple example consider the area between a planar curve and an axis. The integral is given by the product of the interval width and the average value of a function on this interval. The average value which is the average of the infinitely many ordinates in this interval is what caused Cavalieri to think about an indivisible. So, in conclusion an indivisible is represented by some real number in such an interval (analogous to my positional derivative). It (the real number) is rational or "indivisible" when it is not possible to measure it completely. One might say irrational numbers are not completely measurable and think of these as the indivisible points in the interval. The rational numbers make up the rest of the points in the interval. 64.134.230.145 (talk) 17:02, 15 December 2011 (UTC)
 * It is interesting that you are willing to talk about "an average of infinitely many values", but not about "infinitesimals". I think they are actually equivalent. Tkuvho (talk) 10:14, 16 December 2011 (UTC)


 * I meant the average length of infinitely many ordinates. But, one does not actually get to calculate this average by doing the arithmetic in the common way. If you look at "The definition and proof of the mean value theorem", you will understand what I mean. By no means are they equivalent. There is no similarity or correspondence between infinitesimals and an infinite average. However, there is the concept of indivisible which is used as the value of x to find the given length of an ordinate for the infinitely many ordinates in a given interval. 12.176.152.194 (talk) 15:34, 16 December 2011 (UTC)

Response to Gandalf61
I believe you are a troll. A function by definition is composed of *one* rule. Piece-wise functions did not exist until many years after calculus was invented. Any piece-wise function can be written as a set of 2 or more functions (rules). In fact, the phrase "piece-wise" is a misnomer. A better name would have been "conditional functions". Furthermore, it is wrong to say Archimedes used infinitesimals because the only objects he knew and used were rational numbers or approximations to irrational numbers.

Anyway, the reason I suggested you remove infinitesimal is clear - it's an ill-defined concept. In fact, nothing on Wikipedia defines an infinitesimal properly. Cauchy himself started off this section in his Course D'Analys by stating, "...let alpha be an infinitesimal number." even before he defined infinitesimal. Much later he describes an infinitesimal in terms of e (as Rubin has in an archived discussion). However, this is still no different from a real number.


 * Rather than accusing me of being a troll, why don't you try to provide a definition that makes sense? You can't because you don't know.

64.134.230.145 (talk) 23:25, 15 December 2011 (UTC)
 * John, I think editors are reacting the way they do because they feel there are weaknesses in your approach. Incidentally, if you don't accept ZFC as you mentioned above, why don't you request that it be removed from wiki along with infinitesimals?  As far as defining an infinitesimal, I will quote Cauchy.  To translate his terminology into modern language, he takes a null sequence, i.e. a sequence tending to zero.  Then he says that such a null sequence "becomes" an infinitesimal.  In modern terms, one could say that what he is interested in is the asymptotic behavior of the sequence; if one modifies a finite number of terms, this does not affect the resulting infinitesimal.  Thus, it is easy to write down an explicit representative for an infinitesimal: take, for example, the sequence 1, 1/2, 1/3, 1/4, ...  In modern terminology, its equivalence class will be an infinitesimal.  One can even refine the equivalence relation in Cantor's construction of the real numbers in such a way as to obtain an infinitesimal-enriched continuum, but that clearly does not belong in this article.  Tkuvho (talk) 09:11, 16 December 2011 (UTC)
 * ZFC is far more sound than infinitesimal theory and even though I don't think it's important, it is factual for the most part. So I won't debate it. The equivalence class of 1, 1/2, 1/3, ... is *zero*, not an infinitesimal.

If you read my original dialogue with Rubin, you will notice that the infinitesimals are defined as a subset of the interval (0,1) using ultra-filters. This is the short and sweet of it. However, one cannot distinguish between any of the members of this set because they are ill-defined.

The term indivisible came about as a result of Cavalieri. The true area between a curve and an axis is given by the average of the length of infinitely many ordinates. Since a line has no extent (width), each ordinate is thought to be represented by an indivisible line. It is only possible to know this average exactly if a given function has a primitive otherwise one uses numeric integration. The mean value theorem makes it possible to calculate such an infinite average. So in fact, one does not care anymore about the concept of indivisibility; only a means to find the ordinate length for the ordinates in a given interval. This can be well-defined using my new calculus or by use of my positional derivative and standard calculus. So, it is not even necessary to think about indivisibles. Truth is Archimedes did not use indivisibles or infinitesimals. He had thought about indivisibles because he did not know of the mean value theorem. So, it is factually incorrect to say anyone uses indivisibles or infinitesimals. You can say he thought of indivisibles. He did not use them because they are not well-defined. — Preceding unsigned comment added by 12.176.152.194 (talk) 15:10, 16 December 2011 (UTC)


 * John - you "one rule" definition of a function is not only quite different from the standard definition of a function, it is also arbitrary nonsense. You say Tkuvho's example
 * $$f(x) =

\begin{cases} 0 & \text{if }n < 0 \\ 3x^2 & \text{if } n\ge 0 \end{cases}$$
 * is not a function because it involves two rules. But we can rewrite this in a single rule:
 * $$g(x) = \frac{3x(x + \sqrt{x^2})}{2}$$
 * By your definition g is a function but f is not - and yet f and g take the same value for all real values of x. So your definition creates a contradiction. Gandalf61 (talk) 10:59, 16 December 2011 (UTC)


 * Actually, g(x) is not a single function as you claim. Let me explain. You can write g(x) as follows:

$$g(x) = 1.5x(x + |x|)$$

From the fact that |x| appears in the equation we have confirmation that g(x) is a conditional function. There is no contradiction in what I have said.

By the way, I do not wish to discuss other mathematics (*), only the use of infinitesimal. Indivisible is a clearer concept but it is still incorrect in this article.

(*) If we did discuss other mathematics, I would have many different ideas to most mathematicians. For example, I know real numbers are not well-defined; I know all integrals are line integrals; I know that use of the concept of infinity in anything causes it to be ill-defined; I know that radix systems can only represent rational numbers; I know that real numbers are not countable but not for the reasons most mathematicians think; I know Cantor's ideas were mostly wrong.


 * Please discuss this, if on-Wiki, elsewhere. There (appear to be) no mathematicians who support your analysis.  If there were some, you could have named them.  The logical conclusion is that you have made a mistake.  There is sometimes a benefit to doing things no one else believes in, but not on Wikipedia.  — Arthur Rubin  (talk) 15:52, 16 December 2011 (UTC)


 * I agree. See next section called Getting back on topic. That no other mathematicians support my analysis is false. My New Calculus group on Research Gate has over 40 members some of who are PhDs (even though I don't believe that having a PhD makes you a mathematician).

12.176.152.194 (talk) 16:01, 16 December 2011 (UTC)


 * In my opinion, anyone who thinks that what you've written is a constructive is not a mathematician. — Arthur Rubin  (talk) 16:45, 16 December 2011 (UTC)


 * In my opinion, I know that you are not a mathematician. As I have already stated Mr. Rubin, having a PhD does not make you a mathematician. Neither do 1000 publications which have no real significance and are understood only by you. — Preceding unsigned comment added by 12.176.152.194 (talk • contribs)
 * Yes, it is correct to say that you have provided no evidence that any mathematicians support your analysis. If you can provide evidence that someone using mathematical or logical reasoning believes that the "new Calculus" has any mathematical or pedagogical value, you might try supplying such on your web site, or pointing to a reliable source to that effect on Wikipedia. — Arthur Rubin  (talk) 17:52, 27 December 2011 (UTC)

Getting back to the topic.
I think after this has been read, it would be a good idea to remove these last few edits and take the discussion back to the original topic: this article is nonsense because the infinitesimal is not well-defined.

If you must keep it, then at least provide a definition even if you have to use Cauchy's definition (that is, as an element of R(((e)))). Ironically, once you do this, then it will become evident that your claim Archimedes used infinitesimals or indivisibles is sheer nonsense.

If you are concerned about the article being too difficult for a lay person to understand, you needn't worry because it's incomprehensible even as it stands. 12.176.152.194 (talk) 16:37, 17 December 2011 (UTC)


 * There is a universal consensus among historians and mathematicians alike that Archimedes used indivisibles. Tkuvho (talk) 17:48, 21 December 2011 (UTC)

New comment by IP
The IP recently commented as follows: ''ZFC is far more sound than infinitesimal theory and even though I don't think it's important, it is factual for the most part. So I won't debate it. The equivalence class of 1, 1/2, 1/3, ... is *zero*, not an infinitesimal.'' It should be pointed out that the infinitesimals discussed in this page are indeed constructed in a ZFC framework. Therefore a claim to the effect that "ZFC is far more sound" is erroneous. As far as the equivalence class of the null sequence you mentioned, the usual equivalence relation on Cauchy sequences can be relaxed in such a way that the (refined) equivalence class of the sequence is a nonzero infinitesimal. As I mentioned, this material does not belong on the page. Tkuvho (talk) 13:05, 18 December 2011 (UTC)


 * If you want to debate how sound Cauchy's equivalence classes are, then this is a different topic. Which material are you referring to?


 * I am referring to a recent construction (in ZFC of an infinitesimal-enriched continuum, obtained by refining Cantor's equivalence relation on Cauchy sequences. Infinitesimals are at least as "real" as real numbers in this sense.  Tkuvho (talk) 17:22, 19 December 2011 (UTC)


 * All I am stating is that the article contains non-factual statements:

1. Archimedes used the method of exhaustion (nothing to do with infinitesimals or indivisibles). There are more false statements: "Archimedes, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals."

This is in direct contradiction to other articles claiming that Robinson was the first to rigorize the theory of infinitesimals.
 * Archimedes (or more precisely Eudoxus) proposed a coherent definition of infinitesimals. Robinson was able to construct them in ZFC, thereby implementing Eudoxus' definition.  Tkuvho (talk) 17:21, 19 December 2011 (UTC)


 * You claim Robinson constructed them in ZFC but cannot give even one example. Nor can you show how Archimedes "used" them. 12.176.152.194 (talk) 17:31, 21 December 2011 (UTC)


 * I gave you an example already. The equivalence class of the sequence sequence 1, 1/2, 1/3, ... modulo a suitable equivalence relation defines an infinitesimal.  Tkuvho (talk) 17:50, 21 December 2011 (UTC)

There are several more false statements/contradictions:

a) His Archimedean property defines a number x as infinite ...

Nonsense. Archimedes rejected infinite numbers.
 * He certainly used arguments using indivisibles/infinitesimals informally, and in most of his publications preferred to replace them by arguments by exhaustion. The Method is one exception  to this practice.  Tkuvho (talk) 17:24, 19 December 2011 (UTC)


 * Not true. Archimedes' arguments used only rational numbers. If he used any concept informally, this would be the concept of an incommensurable magnitude (known as an irrational or real number today). This decidedly has nothing to do with an infinitesimal and is only indirectly related to an indivisible in the sense that Archimedes' rational approximation would be more complete if it were possible to measure the position of an indivisible on the real number line. 12.176.152.194 (talk) 17:31, 21 December 2011 (UTC)


 * In fact Archimedes used only relations among natural numbers, but he used indivisibles (in the sense of Cavalieri) all the same.  Tkuvho (talk) 17:51, 21 December 2011 (UTC)

b) In the ancient Greek system of mathematics, 1 represents the length of some line segment which has arbitrarily been picked as the unit of measurement.

Not entirely true. 1 represents the comparison of a magnitude to itself.

c) When Newton and Leibniz invented the calculus, they made use of infinitesimals.

False. Newton and Leibniz (aside from NOT inventing calculus) knew very little of the theory which you call infinitesimal theory today. Their ideas were wrong.


 * I suggest you consult the page Law of Continuity. Leibniz definitely had some right ideas.  Tkuvho (talk) 17:25, 19 December 2011 (UTC)


 * They both had some ideas but their definitions are both faulty. I am not discrediting them, just stating they were wrong about certain important fundamental facts and concepts. 12.176.152.194 (talk) 17:31, 21 December 2011 (UTC)

2. The article which is supposed to be about "infinitesimals" tries to support its validity by making false statements about Archimedes. 3. Infinitesimal is not a well-defined concept. Ask yourself how logical is it for a subset of (0,1) to be the infinitesimals? What is the LUB of this set where magnitudes cease to be infinitesimal? Can you demonstrate two infinitesimals and do useful arithmetic with the same?


 * That's exactly what Abraham Robinson proved. Take Abraham Fraenkel's word for it!  Tkuvho (talk) 17:26, 19 December 2011 (UTC)


 * I do not take anyone's word for it. I am not inferior to anyone in terms of intelligence. Fraenkel was a "non-mathematician" in my opinion. I have little respect for Zermelo and no respect for Fraenkel.


 * An attitude of disdain toward Ernst Zermelo and Abraham Fraenkel is totally unacceptable. Tkuvho (talk) 17:53, 21 December 2011 (UTC)


 * A mathematician is like an artist: the objects arising from concepts in a mathematician's mind, are only as appealing as they are well-defined. 12.176.152.194 (talk) 00:51, 23 December 2011 (UTC)

Once again, Robinson proved nothing. Please show me an infinitesimal and do some arithmetic with it. 12.176.152.194 (talk) 17:31, 21 December 2011 (UTC)



What I am telling you (most confidently and as a mathematician) is that infinitesimals have no place in calculus or mathematics in any respect.

I am an educator who cannot look any of my students in the eye and tell them the rubbish written in this article is true.


 * As far as educational issues are concerned, I would be interested in your reactions to Robert Ely's recent education study concerning infinitesimals, which tends to go counter to your conclusions. Tkuvho (talk) 17:28, 19 December 2011 (UTC)


 * Do you have a specific link that I can read? I am not convinced but I am curious. 12.176.152.194 (talk) 17:31, 21 December 2011 (UTC)


 * The article by Ely is referenced at 0.999.... Tkuvho (talk) 17:54, 21 December 2011 (UTC)

Does Wikipedia actually care about article content at all? 12.176.152.194 (talk) 17:01, 18 December 2011 (UTC)
 * Yes. The criticism in item (b) above is entirely correct.  I haven't noticed that passage before.  It's gone now.  Tkuvho (talk) 11:59, 19 December 2011 (UTC)

Rubin attempts to discredit my new calculus
Both f and g have derivatives of 0 in your system (as in the common system), but f+g is 2 x3, which you claim does not have a derivative at 0. — Arthur Rubin (talk) 16:11, 22 December 2011 (UTC)
 * This is correct because in the New Calculus, a function is differentiable at a given point if and only if a finite tangent line can be constructed. Therefore the rule $$(f+g)' = f' + g'$$ would not generally apply as you noticed. There are also some rules in the standard calculus which would not apply generally in the New Calculus. As an example, consider f(x)=|x|, in the New calculus, it makes no sense to talk about a derivative at x=0 because the function is not smooth at that point. This is due to the fact that concepts in the New Calculus are well-defined, for example the derivative. In standard calculus, if f(x)=1/x and g(x)=-1/x, the general rule you stated, fails. Moral of the story is that there are always exceptions - even to the general rules. Mr. Rubin, if you ask a question and I answer it, this does not mean I care to discuss my theory on Wikipedia where you think it is inappropriate. 12.176.152.194 (talk) 16:50, 22 December 2011 (UTC)
 * There is nothing wrong with the "general rule" in the case that you give. If f is differentiable and g is differentiable then f+g is always differentiable.  The only difficulty with 1/x and -1/x is that neither function is defined at 0, there sum is not the zero function but the sum is also undefined at zero.  So you've given examples where f, g and f+g all fail to be differentiable at 0, which doesn't quite make your point.  Thenub314 (talk) 22:02, 22 December 2011 (UTC)


 * I disagree. Their sum is the zero function, but f and g are not defined at 0. So whilst (f+g) is differentiable at 0, f and g are not but the general rule does not fail, because 1/x^2 -1/x^2 = 0. It all depends on where one decides to stop, that is, at which step do things fall apart. There are always exceptions to the general rule and this is the point I was trying to make. This example shows how general rules can often be misleading because of the order in which operations are done. And while on the subject of infinitesimals which you claim exist but I know do not exist, then I can argue that the sum of two infinitesimals 1/x and -1/x where x approaches infinity is 0. This would support Rubin's stance. Yet we know that f and g are not even defined at x=0. The reason for this confusion is that Cauchy's derivative is ill-defined.

None of this (whether correct or not) alters the fact that a tangent line cannot be constructed at x=0 for the function x^3. Finally, if there are any results from standard calculus that don't work the same way in the New Calculus, one of two things are possible. a) The concept in standard calculus is ill-defined/flawed or b) there is a new approach that is no longer compatible with the wrong ideas of standard calculus. 12.176.152.194 (talk) 23:51, 22 December 2011 (UTC)
 * You may find it interesting to notice that others may disagree. For example if you take a look at "Range, R. Michael (May 2011), "Where Are Limits Needed in Calculus?", American Mathematical Monthly 118 (5): 404-417" you'll see a development of differential calculus that doesn't use limits or infinitesimals.
 * It is based on a method of Descartes which is effectively geometric/algebraic in nature. But it does construct tangent lines to x^3 at 0. Thenub314 (talk) 15:33, 23 December 2011 (UTC)


 * It is impossible to construct any finite tangent line to the function x^3 at x=0. In order to convince me otherwise, you would have to find any a and b, such that f'(0)= [f(b)-f(a)]/(b-a) = 0. I put it to you that you can't. Descartes used tangent circles to find the gradients of tangent lines to points on given curves. Even with Descartes's method, you cannot find a tangent to x^3 at x=0. Do you have a link to the article by Range, R. Michael? I do not believe there is any other source besides my new calculus which is limit or infinitesimal free. BTW: Although Descartes's method is interesting, you will find that it is almost impossible to use except for very simple problems. 12.176.152.194 (talk) 00:48, 24 December 2011 (UTC)


 * Did you mean this link? http://www.jstor.org/pss/10.4169/amer.math.monthly.118.05.404 No, I have not read it and it is not free. As I have studied Descartes' method, I don't believe there is anything else Range can teach me. 12.176.152.194 (talk) 01:02, 24 December 2011 (UTC)


 * He can teach you that the x-axis is tangent to the cubic y=x^3 at the origin. Tkuvho (talk) 12:40, 28 December 2011 (UTC)


 * And Rubin calls you a mathematician? Wow. More like a shotgun mathematician... A tangent meets a curve or surface in a single point if a sufficiently small interval is considered (Webster). And gee, let me see, your own Wikipedia entry says:  "More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f." This is implied directly by the mean value theorem. The reason your understanding is faulty is due to your education - Cauchy's ill-defined derivative.  In very simple language and as the Greeks invented it, a tangent is a finite straight line that meets another curve in one point and crosses it nowhere. Now take a deep breath and tell me if the x-axis crosses the cubic. Allow me to educate you a little bit, the tangent is the movable part of a trapezium (the non-parallel side) which is a tangent object in planar geometry. The Greeks were trying to use tangents to determine if curves were smooth given the same curves are continuous. This was the main reason they came up with the idea of tangent. The Greeks knew only intuitively that the conical curves they knew were smooth. Much later, curvature was measured using tangent line gradients. For more, you'll have to wait for the publication of the most important mathematics book ever written - What you had to know in mathematics but your educators could not tell you. To learn more about single variable calculus, you can read the file called NewCalculusAbstract-Part1 at http://india-men.ning.com/forum/topics/meaning-of-the-differential-quotient?page=1&commentId=2238831%3AComment%3A46087&x=1#2238831Comment46087 12.176.152.194 (talk) 22:35, 28 December 2011 (UTC)
 * I am not sure which Webster if any you are quoting, but the definition "A tangent meets a curve or surface in a single point if a sufficiently small interval is considered" is erroneous. Thus, the function x^2 cos(1/x) has x-axis for a tangent at the origin, but, contrary to you alleged "webster" definition, it does not meet the graph in a single point no matter how small the interval considered, but rather at infinitely many points.  Tkuvho (talk) 08:33, 29 December 2011 (UTC)


 * The definition is not erroneous. This is what it means for a line to be a tangent line. The function x^2 cos(1/x) is not defined at 0, so how can it have a tangent line there? You are incorrect about this function not meeting the x-axis - it intersects the x-axis infinitely many times except at the origin. You must be the only one who has such an absurd understanding of what it means to be a tangent because no one else I know would agree with you. 12.176.152.194 (talk) 14:01, 29 December 2011 (UTC)
 * I omitted to mention that the function is defined to be zero at the origin and x^2 cos(1/x) everywhere else. One can construct similar functions without mentioning such two cases, as well, with the property that the tangent will meet the graph at infinitely many points.  Tkuvho (talk) 14:06, 29 December 2011 (UTC)
 * See earlier note about removing discontinuity at x=0. The methods of calculus apply to continuous and smooth functions. As slippery as the concept of continuity is, it can be defined simply as follows: A function is continuous over an interval if there are no disjoint paths (geometric definition). A function is smooth (and therefore differentiable) if at each point of the function, exactly one finite tangent line can be constructed. Your assertion that the tangent to the function x^2cos(1/x) will meet at infinitely many points (even if it were defined at the origin) is false. The curve of this function is sinusoidal from both sides of the origin where it is undefined, hence it is impossible for it to be a straight line ever. 12.176.152.194 (talk) 14:43, 30 December 2011 (UTC)

A gentle suggestion for the IP
If you would like to make a constructive contribution to wiki, I suggest that you should try to come to terms with the following two items at the very least: (A) the x-axis is the tangent line to the graph of the cubic y=x^3; (B) Ernst Zermelo and Abraham Fraenkel are mathematical giants who are fully deserving of our respect. Otherwise you should refrain from contributing to wiki. Tkuvho (talk) 18:29, 25 December 2011 (UTC)

(A) The x-axis is not a tangent line to the graph of x^3. I challenged you to prove it but you could not. It does not matter how many times you say something unless you can prove it - do you understand this? Now I have given you an assignment - In order to convince me otherwise, you would have to find any a and b, such that f'(0)= [f(b)-f(a)]/(b-a) = 0. If you can do this, I'll concede a tangent exists. Good luck! Just to let you know, I can prove no such a and b exist very ingeniously. Or maybe I can't... What do you think? (B) I asked you several times to well-define an infinitesimal - you could not. You still have not answered my questions regarding what is the LUB of the infinitesimal set? Where does it end and the real numbers begin? (C) Ernst Zermelo and Abraham Fraenkel are fools who are not worthy of my respect. In fact if I could have my way, I would list their names in a Mathematics Book of Infamy. However, you are naturally free to worship whomsoever you wish. Perhaps you should worship your idols while you can, because the time is coming when more mathematicians will realize they have been duped.12.176.152.194 (talk) 20:02, 25 December 2011 (UTC)


 * If you could name one mathematician who doesn't accept the additivity of the derivative, it might help your cause. Regardless, of WP:TRUTH, your original definition of derivative and your original conclusion that infinitesimals do not exist have no place on Wikipedia unless they are at least commented on in a reliable source.  That's not going to happen.  (In my professional opinion.)  I think even intuitionists accept the infinitesimals in R(((&epsilon;))).  — Arthur Rubin  (talk) 22:18, 25 December 2011 (UTC)


 * I for one accept the rule in "standard calculus" given Cauchy's ill-defined derivative which is the source of numerous other ill-defined concepts, besides the one discussed here, that is, infinitesimals. This discussion is about infinitesimals, not my calculus or anything else. So, I suggested you quote reliable sources to back up your false claims in this article or you continue to lose credibility as a "reliable" source of information. Out of curiosity, reliable probably means what - if it can pass Rubin-Hardy and sub-ordinates' scrutiny? So, once again: 1) Before you claim Archimedes used infinitesimals anywhere, please show how he used them. Do not refer me to this article (or the one on Mechanical Theorems), because neither has anything about how Archimedes used infinitesimals. The only numbers Archimedes used were rational numbers. 2) If the infinitesimal set is well-defined, please tell me what is its LUB? Where do infinitesimals end and real numbers begin? 3) Other than telling me k is an infinitesimal member of R(((e))), show me k and how Archimedes may have used it. If you can't do this, then you should consider removing the false claims regarding Archimedes. Whatever your conception or notion about an infinitesimal, it is not possible Archimedes could have had the same ideas. To deny this, is to deceive yourself and those foolish enough to think your article has any worth. 12.176.152.194 (talk) 02:11, 26 December 2011 (UTC)


 * 1) I'm not convinced that Archimedes used infinitesimals; however, he certainly considered them, or the concept of an Archimedean field would not have arisen.  I don't presently have access to the references used in indivisibles to support the statement, so I can't confirm that Archimedes did use them.
 * 2) In non-standard set theory (or "internal set theory"), the set of infinitesimals is not an "standard" or "internal" set, so the LUB property of R doesn't transfer to R* with respect to that set. R(((&epsilon;))) has no trace of the LUB property, so there is no need to assert there's a problem there.
 * 3) "k" (wherever you got that from) could be &epsilon; itself.
 * You can certainly request verification of the citation for the claim that Archimedes used infinitesimals. You may not say that the theory of infinitesimals is not mathematically consistent, unless you are prepared to prove it using reliable sources.
 * And, although the reasoning is somewhat circular, a reliable source is one that is recognized as reliable (according to Wikipedia standards) by other reliable sources. We're probably not going to get to your "new calculus" by any chain starting with a reliable source, but I could be surprised.
 * As for your letters above:
 * A) If you want to redefine tangent line to a curve at a point to mean a line parallel to arbitrarily short secants of arcs containing that point, I can't stop you, but it doesn't have any place on Wikipedia unless used in a reliable source.
 * B) I can define an infinitesimal in any field of characteristic 0. Whether or not they exist, and whether or not they are useful if they do exist, depends on the field.  You may deny the reality of ultrafilters, the axiom of choice, or set theory in general, but you cannot deny that most mathematicians use them, and that they have not been proved inconsistent.
 * C) Is your opinion.
 * — Arthur Rubin (talk) 06:41, 26 December 2011 (UTC)


 * "I'm not convinced that Archimedes used infinitesimals; however, he certainly considered them, or the concept.." -  Now this is acceptable. Why don't you change the article to reflect this fact? Instead of "Archimedes used indivisibles in The Method of Mechanical Theorems to find areas of regions and volumes of solids.", you could say "Archimedes certainly considered using infinitesimals in The Method of Mechanical Theorems to find areas of regions and volumes of solids, but no evidence exists to support this notion." 12.176.152.194 (talk) 16:48, 26 December 2011 (UTC)
 * Whether or not Archimedes used infinitesimals pales in comparison with the fact that the x-axis is tangent to the cubic y=x^3 at the origin. Please come to grips with this fact.  Tkuvho (talk) 12:39, 28 December 2011 (UTC)
 * I do not agree with 12.176.152.194's opinions about calculus more then anyone else, but reading this it seems as if something has gone wrong. The comments above are focused on improving the article, so we should at least not dismiss all of his comments because he has a unusual notion of what it means for a line to be tangent.  Since the comment about Archimedes is disupted, then we should tag it as such or find a reference to support it. Thenub314 (talk) 15:43, 28 December 2011 (UTC)
 * There is a reference given; it's just that I haven't read it and the IP claims that it doesn't support the statement. Perhaps vs is a better tag than disputed-inline.  It would be better if someone who is familiar with the given reference could comment.  — Arthur Rubin  (talk) 15:55, 28 December 2011 (UTC)
 * The only reference I saw was the one given to Archimedes actual work itself, which would clearly be in ancient greek, and as with all historical documents would probably require an expert in how the language was used at the time to make sense of what was being said. But perhaps there is a good translation etc.  Even better would be a secondary source though. Thenub314 (talk) 16:18, 28 December 2011 (UTC)


 * Good. I read, write and speak Greek. Nowhere in the Greek is anything said about indivisible (*). Now to claim that he used what came to be known as The Method of Indivisibles is still not true. The only mathematical objects Archimedes knew about were the rational numbers and incommensurable magnitudes (what you like to think of as irrational numbers). Nothing else. Note that indivisible is neither a "magnitude", nor is it a "number". So whilst this last change is an improvement, it still misleads the reader. You might say that Archimedes knew of the Cavalieri principle but did not use it. Tkuvho - please come to grips with the fact that the x-axis is NOT a tangent to x^3. Unless you can find a and b on either side such that the mean value theorem is true, then the cubic is not differentiable at x=0. If you are interested in a proof, you may contact me privately and I will show you. You cannot claim the cubic is differentiable on a given interval and in the same breath claim that the mean value theorem [f'(c)=[f(b)-f(a)]/(b-a)=0] does not apply when c=0. For if it does not apply, then the cubic cannot be differentiable at x=0. The usual statement of the mean value theorem is what Cauchy used to derive his derivative definition. And will you stop calling me "The IP" please. My name is Gabriel. (*) It is obvious to me that none of you have studied The Works of Archimedes. In his translation Thomas Little Heath does not use the word indivisible even once! There's a good reason for this - it is not in the original Greek. Not in Heath's manuscripts, nor in the later palimpsest discoveries.12.176.152.194 (talk) 18:39, 28 December 2011 (UTC)
 * See my comment in the previous section. Tkuvho (talk) 08:35, 29 December 2011 (UTC)

Tkuvho Responses
You cannot create piecewise or conditional fuctions, and draw general conclusions from these functions. Even if the function x^2cos(1/x) is defined to be 0 at the origin, the x-axis cannot be its tangent - ever. You have a fundamentally incorrect understanding of what it means for a line to be tangent. Webster's online was the source I was referring to. In any event, unless you can respond to the questions I put to you, you are wasting your time. So far, you have not been able to respond to any of the questions or challenges. I suggest you stay with sound logical arguments rather than your opinions which are entirely wrong. — Preceding unsigned comment added by 12.176.152.194 (talk) 14:15, 29 December 2011 (UTC)
 * OK, let's stick with logical arguments: note that the derivative of x^2cos(1/x) at the origin exists even according to your definition of derivative. Namely, the function is even, so that its values at h and -h are equal.  Therefore the quotient you propose as the definition of derivative will be zero.  Therefore the derivative exists and equals 0.  Since the graph passes through the origin, the tangent line is the x-axis.  Thus your "webster" definition of a tangent line in terms of a unique point of intersection does not work in all cases.  At any rate, to calculate more general derivatives, one will have to drop the infinitesimal remainder at the end if one wants a usable theory.  Tkuvho (talk) 14:20, 29 December 2011 (UTC)


 * Wrong. The derivative does not exist at x=0 using any definition. Even the derivative is undefined at 0. And once again, NO, we do not need to drop any 'remainder'. Standard calculus is in error and that's what I prove conclusively in the file I referred you to (Cauchy's Kludge). 12.176.152.194 (talk) 14:25, 29 December 2011 (UTC)
 * If I follow your definition in Cauchy's Kludge correctly, the derivative does exist and equals zero at the origin. Tkuvho (talk) 14:33, 29 December 2011 (UTC)


 * You are not following it correctly, hence your incorrect conclusion. There is no related distance pair that supports a tangent to the cubic at the origin. 12.176.152.194 (talk) 14:41, 29 December 2011 (UTC)


 * Where is my error? The function is even, hence f(h)=f(-h), and therefore the ratio f(h)-f(-h)/h vanishes, precisely as you state.  Hence the derivative is zero, and the x-axis is the tangent line.  Tkuvho (talk) 14:42, 29 December 2011 (UTC)


 * There certainly are such pairs, for instance when h is the inverse of (&pi;/2 + 2&pi;k) for integer k. These are the points of intersection with the x-axis you yourself mentioned in an earlier post, noting that there are infinitely many of them.  Tkuvho (talk) 14:45, 29 December 2011 (UTC)


 * Such pairs must be contained in the same segment of the arc. A tangent does not cross the curve - ever, for otherwise it cannot be a tangent. You are in denial of this fact but it is a simple fact. 12.176.152.194 (talk) 15:03, 29 December 2011 (UTC)
 * This is not the same definition of a tangent line we were working with before. It used to be based on the slope at the point, without reference to intersections with the curve.  Tkuvho (talk) 16:03, 29 December 2011 (UTC)
 * I have always used the same definition. A tangent line is not based on slope. Rather slope is based on the tangent line. You cannot define a tangent line without reference to intersections with the curve. This is the key property of tangents. Otherwise they are just other curves intersecting other curves. So, the fact that an intersecting curve does not cross another curve is the identifying feature of a tangent. In all Ancient Greek texts (they invented the concept btw), this is exactly what a tangent line is. 12.176.152.194 (talk) 01:42, 30 December 2011 (UTC)

(Separated comments)
(I don't know how these comments got separated from that of the IP, or which comments it was in response to.)
 * Fair enough. Your article is incomprehensible to a mathematician — at least to the editors who have commented on it here on Wikipedia, most of whom are mathematicians.
 * However, there isn't a single definition. The second sentence of the "history" section provides a usable definition, however.  Is that early enough for you?  — Arthur Rubin  (talk) 16:23, 16 December 2011 (UTC)

Archimedes
We should reference our claims about Archimedes. This section can serve as a centralized place for discussion. Thenub314 (talk) 21:06, 28 December 2011 (UTC)
 * Looking at "" has a nice section on Archimedes work.  He does use the term indivisibles but, as a indication that Archimedes himself did not use the term he places it in quotes the first time he uses it in this section.  Thenub314 (talk) 21:43, 28 December 2011 (UTC)


 * The most reliable non-Greek language text is The Works of Archimedes (Thomas Heath). There is no other reliable source that claims Archimedes used indivisibles or the ill-defined concept of infinitesimal. Heath in my opinion was the greatest mathematics scholar whose command of the Greek language surpassed any other non-Greek. Heath translated not only The Works of Archimedes, but also the Elements and the geometry masterpiece of Apollonius. In fact Heath clearly writes about how the Greeks rejected the ill-defined notion of anything infinitely small or infinitely large. In fact, they rejected infinity because it is an ill-defined concept. One needn't look too far to see what nonsensical results have arisen out of this concept in the form of limits, infinitesimals, set theory, real analysis, etc. 12.176.152.194 (talk) 22:28, 28 December 2011 (UTC)
 * Correct, they were suspicious of infinity. That's why they typically replaced their arguments a la Cavalieri (as in The Method) by arguments by exhaustion in "official" publications.  The Method was a private letter where Archimedes did use indivisibles.  Tkuvho (talk) 08:37, 29 December 2011 (UTC)
 * Once again, aside from your first sentence, every other sentence is false, so I can see how your opinions are entirely wrong. Now, you cannot base an article without reliable sources on your "opinion". 12.176.152.194 (talk) 14:05, 29 December 2011 (UTC)
 * Please see the article by Netz et al. Tkuvho (talk) 14:07, 29 December 2011 (UTC)

Tkuvho Responses 2
You cannot create piecewise or conditional fuctions, and draw general conclusions from these functions. Even if the function x^2cos(1/x) is defined to be 0 at the origin, the x-axis cannot be its tangent - ever. You have a fundamentally incorrect understanding of what it means for a line to be tangent. Webster's online was the source I was referring to. In any event, unless you can respond to the questions I put to you, you are wasting your time. So far, you have not been able to respond to any of the questions or challenges. I suggest you stay with sound logical arguments rather than your opinions which are entirely wrong. As for Netz, he has not written anything noteworthy so I don't understand why you keep mentioning his name. 12.176.152.194 (talk) 14:18, 29 December 2011 (UTC)


 * As far as the function is concerned, see my reply above. The article by Netz, Saito, and Tchernetska received a very favorable review in MathSciNet. Do you believe in mathscinet? Tkuvho (talk) 14:23, 29 December 2011 (UTC)


 * Rather than mentioning other sources (which are invariably incorrect), I suggest you try to understand these facts. 12.176.152.194 (talk) —Preceding undated comment added 14:28, 29 December 2011 (UTC).


 * Can wikipedia readers be expected to accept your "facts" rather than mathscinet's? Tkuvho (talk) 14:31, 29 December 2011 (UTC)


 * I don't know which 'facts' you are referring to. Thus far, you have been discussing only your opinions. mathscinet does not trump the original sources which clearly indicate no presence of infinitesimals or indivisibles. You may not like this very much but that's just the way it is. 12.176.152.194 (talk) 14:39, 29 December 2011 (UTC)


 * Netz et al state in their article that Archimedes used indivisibles, and mathscinet reviewer also states this explicitly. Do you think it is possible that wiki readers may be more interested in Netz's and Mathscinet's opinion than in yours?  Tkuvho (talk) 14:41, 29 December 2011 (UTC)


 * Perhaps or perhaps not. It does not change the fact that those views are only the opinion of Netz et al. Opinion is NOT reliable source. By the way, please place all your responses here. I cannot search through the text anymore.12.176.152.194 (talk) 14:51, 29 December 2011 (UTC)

I thought you were referring to the cubic. Sorry. Your reasoning would be correct if your interpretation of tangent were correct but it's not. Once again, there is no related distance pair even for x^2cos(1/x) that produces a tangent at the origin. Remember, a tangent by definition meets a curve at one point only and crosses it nowhere. You have to be careful when using graphical software to study curves. The cubic which you erroneously think has a tangent at the origin has no zero ordinates on either side of the origin. However, the software makes it appear there are 'infinitesimal' ordinates on either side. 12.176.152.194 (talk) 14:51, 29 December 2011 (UTC)


 * If you wish to continue this discussion, I would prefer email. You can reach me at john underscore gabriel at yahoo dot com. Sorry, my eyesight is not so good and I feel great discomfort searching for your responses. You are welcome to share the conversations with others here if you wish. 12.176.152.194 (talk) 14:55, 29 December 2011 (UTC)


 * Unfortunately, your response here contradicts what you wrote in your Cauchy text. You can't change your definitions when you run into logical difficulties. By wiki rules, Netz's opinions are reliable since they are published in a reputable venue.  You should inform your interlocutors at the indian site that your theory contains errors.  Many, many people have tried to develop calculus without resorting to discarding the infinitesimal remainder at the end (or to an equivalent method in terms of epsilontics), but they were not successful.  Unfortunately, your theory does not seem to be any different.  Tkuvho (talk) 14:57, 29 December 2011 (UTC)
 * Please be kind enough to tell in what way there is contradiction for I see none. Rather than make false accusations about me changing my definitions, you would gain more support if you used facts only. I have developed the first rigorous calculus without the use of limits - this is an indisputable fact. It's not debatable. Rather than write silly comments here, I would suggest you study the new calculus. If you have any questions, I will be glad to help you. BTW: There is no infinitesimal remainder at the end as you claim. This is part of the problem with Cauchy's flawed definition and the reason it is jury-rigged. This error by Cauchy gave birth to his incorrect theory regarding infinitesimals. We can discuss the new calculus but this is not the place for it. I prefer private email. Let's stay with the topic which is about fixing the many incorrect claims in this article. 12.176.152.194 (talk) 15:08, 29 December 2011 (UTC)
 * You defined the derivative in your pdf in terms of "f(x+k)-f(x-h)", where one can set k=h in the case of the parabola. Applying this definition to x^2 cos(1/x), we get zero slope at the origin.  You can't change your definition in midsteam and claim that you have a different definition of the tangent line.  Tkuvho (talk) 15:45, 29 December 2011 (UTC)
 * You cannot simply assume that h=k in every instance. In fact, this is not the case here. Now if you had studied the links I referred you to, you would have seen that the relationship between the distances varies for each dissimilar function. That is, you have to find it unless you are interested only in the general derivative, in which case you can use the (0,0) pair provided the function is differentiable at a given point. So, there is no changing of definitions only a problem with your interpretation and understanding. One more thing - calculus applies only to smooth continuous functions. The minute you introduce conditional functions (piece-wise), all bets are off. Newton had no clue about conditional functions. These came much later and in the hands of amateurs we now have theory that confuses the likes of you. If I defined a function as follows, f(x)=1/x for all x except 0 and f(x)=0 when x=0, then any conclusions I try to draw from calculus will be suspect because f(x) is not defined at 0. You can't "just" remove the discontinuity as you feel like it. As yet another example, consider the absolute value function. It is irrelevant to talk about differentiability at the origin because the conditional function is not smooth there. It is also non-remarkable that one can construct infinitely many finite tangents to the function at x=0. One does not require calculating the limit from the left and the right to see this. It's a no-brainer. Yet I have seen incompetent professors ask this irrelevant question in assessments once too often. 12.176.152.194 (talk) 18:38, 29 December 2011 (UTC)