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Anyone who reads this article and believes it is truly naive. It contradicts itself numerous times. First it states:
"When we consider numbers, the naive definition is clearly flawed: an infinitesimal is a number whose modulus is less than any non zero positive number. Considering positive numbers, the only way for a number to be less than all numbers would be to be the least positive number. If h is such a number, then what is h/2? Or if h is undividable, is it still a number?"
Then in a section called 'A definition':
"An infinitesimal number is a nonstandard number whose modulus is less than any nonzero positive standard number".
It is true that such a definition is absolute nonsense. Not only this, but there is no evidence that an infinitesimal exists. To talk about the plural is ludicrous. Isaac Newton was groping in darkness when he coined this term. He was himself uncertain how to explain the calculation of a gradient or average 'at a point'. Furthermore the article states Archimedes used infinitesimals but till this day there is no coherent definition of an 'infinitesimal' and non-standard analysis is at most wishy. How could Archimedes have used infinitesimals if they do not exist and he had no idea what these are?
However, what surprises me is that Wikipedia allows this to be published. Another article on nonstandard numbers is also dreamy. 70.120.182.243 17:55, 30 April 2007 (UTC)
I think I understand Wikipedia's policy now. It does not matter what "facts" are true or false. As long as there is a publication of such facts, these qualify to be part of an article.
Is my understanding correct? If so, then I think my suggestions can be discarded. I think you should warn your readers that no information on your site can be trusted. It is also clear to me now why most academics warn their students to steer clear of Wikipedia. What I tell my students is to read everything and believe nothing unless it makes sense to them. I shall never suggest anything here again. 12.176.152.194 ( talk) 18:32, 18 December 2011 (UTC)
12.176.152.194 ( talk) 15:52, 22 December 2011 (UTC)
Perhaps you need to study English or mathematical reasoning more thoroughly. That states that for any secant there is a parallel tangent. You have asserted that for any tangent that there is a parallel secant. Not at all the same. — Arthur Rubin (talk) 17:02, 30 December 2011 (UTC)
The mean value states that if there is a tangent, there must be parallel secants. So for a given interval, it is exactly the same. You ought to address the previous paragraph to yourself. 12.176.152.194 ( talk) 00:40, 2 January 2012 (UTC)
Netz is not an authority by any stretch on Archimedes. His main contribution is/was in the restoration effort. I would go as far as saying that very few mathematicians after Heath understand Archimedes' Works. I am one of the few who has studied and understands his works well. 12.176.152.194 ( talk) 15:16, 29 December 2011 (UTC)
(ec) That no sense makes. And this still has nothing to do with anything which should be on Wikipedia. As I see it, the "new calculus" definition of the derivative is:
although, I can't come up with a definition which also requires to exist. I'm pretty sure that that definition implies that exists, and that, if you then define , then under the usual definition. — Arthur Rubin (talk) 17:33, 1 January 2012 (UTC)
I thought about your comment regarding circularity and it occurred to me that you are getting confused. So I will try to help you understand this. Gradient = rise/run. Rise = f(x+n)-f(x-m) Run = m+n So, gradient k = f(x+n)-f(x-m)/(m+n) => f(x+n)-f(x-m) = k(m+n). We don't know k but we know both rise and run so we can find k. To convince yourself there is no "infinitesimal remainder", divide both sides of f(x+n)-f(x-m) = k(m+n) by (m+n). On the left you have what you started out with and on the right you have k. For any difference quotient, you work with the left hand side so that after cancellation you will have one term without any m or n in it. This term denotes the gradient when m=n=0, that is, both distances on the side of the tangent at the tangent point are zero. Study diagrams in files to get a better understanding. Now, if you wish to find k for any one of the other secant lines that are parallel to the tangent, then you must know their (m,n) pairs. Finding a relationship between m and n helps. However, k will be the same for all the secant lines that are parallel to the tangent. BTW: The terms in m and n are not remainders! But their sum is always zero because the secants are parallel to the tangent. Each secant has its own (m,n) pair which makes all these terms zero. For example, consider f(x)=x^2. f'(x)=2x+(n-m). This is exactly the derivative. 2x+(n-m)=2x always. If x=1, then all the following are valid gradients: 2(1)+(0-0); 2(1)+(0.005-0.005); 2(1)+(3-3); 2(1)+(m-n) Note that m=n in the case of the parabola. This is not always true for every function. In fact, it's hardly ever true for most other functions. 12.176.152.194 ( talk) 17:05, 1 January 2012 (UTC)
An important step in learning the New Calculus is first realizing where the standard calculus is wrong. You cannot divide by h ever in the standard difference ratio. You can divide by (m+n) always in the New Calculus. Why? Every term of the numerator f(x+n)-f(x-m) contains a factor of (m+n). After cancellation (taking the quotient), exactly one term will be the gradient of the tangent line [distance pair (0;0)]. To find the gradients of all the parallel secants we use the terms in m and n if we want to be "devout". However, there is no need to do this because their gradients are all equal to the tangent gradient. Now we can find distance pairs in (m,n) for other reasons and there are many interesting reasons - especially in the theory of differential equations which I have researched using the new calculus. So, what you have to do is forget everything you learned and interpret what you read literally. It will take a while even if you are extremely smart. I have found that it's much easier to teach someone who has not learned standard calculus. 12.176.152.194 ( talk) 19:47, 1 January 2012 (UTC)
This discussion is not about my New Calculus. Now, although I do not mind whether you mention my New Calculus or not, I will mind if you mention it without proper attribution (my name and web page). I will win any argument in a court of law if it comes to this. Not threatening, just warning. Cauchy's kludge, secant method, distance pairs, etc are also my copyright phrases, not to be mentioned without correct attribution. What I have noticed about academics is that they are cynical until they understand and then they think it's no big deal. Well, it is a big deal because I was the first to think of it. It is also a big deal that I have corrected three great mathematicians: Newton, Leibniz and Cauchy. Although I can't stop you from quoting my work with the correct attribution, I would prefer that you do not quote my work at all. 12.176.152.194 ( talk) 19:05, 1 January 2012 (UTC)
Dear 12.176.152.194, Wikipedia should not be used for self-promotion. Neither in the articles nor in the talk pages. If you have your own version of the Calculus I urge you to get it published in a peer reviewed journal. In any case, Wikipedia is definitely the wrong place for publishing or discussing original research. Please respect that. iNic ( talk) 04:39, 2 January 2012 (UTC)
OK so what does the non factual statements have to do with your own OR? If there are non factual statements you should be able to point these out without referring to your own opinion about it or your own research. Have you done that? Please do not ever answer any questions about your own research on Wikipedia. Ever. Please just ignore all questions and comments about it here from now on. Those interested can contact you directly. If we stick to the Wikipedia rules we should all be cool. iNic ( talk) 13:18, 2 January 2012 (UTC)
If you have good arguments you should not have to resort to personal attacks like this. By the way, it's not allowed here and you can be banned from wikipedia if you continue like this. iNic ( talk) 15:34, 2 January 2012 (UTC)
How can he attack you if you stop talking about your own ideas? iNic ( talk) 16:09, 2 January 2012 (UTC)
Aha so you proved Einstein wrong too? Did you publish it? iNic ( talk) 15:34, 2 January 2012 (UTC)
This is very much off topic but please tell me when and in what context Einstein was proved wrong? iNic ( talk) 15:48, 2 January 2012 (UTC)
My work has been published online. That I have a website means it is copyrighted. Furthermore, it is dated so no one can say it's not original. Don't give me that nonsense regarding your knowledge of legal matters. One more thing - I did not bring up the topic, I have been asked several questions and referred those readers to the material. They did not have to read it or continue to ask me further questions. 12.176.152.194 ( talk) 05:03, 2 January 2012 (UTC)
So why don't you publish your own ideas proving Abraham Robinson to be wrong? Why wasting your time here while you have your important mission? Wikipedia can only take into account already published work and so far Robinson has published his ideas whereas you haven't. In the meantime please only talk about work that is not your own and that is published. "Published online" doesn't count I'm afraid, unless it's in a peer reviewed online journal. iNic ( talk) 15:43, 2 January 2012 (UTC)
Claiming that Robinson was an idiot is just stupid. Period. iNic ( talk) 10:23, 3 January 2012 (UTC)
As for infinitesimals, this article should be adequate to explain them to any mathematically trained person. My R(((ε))) is a subfield of the Levi-Civita field, which has most of the same properties, but is easier to calculate with. — Arthur Rubin (talk) 05:47, 2 January 2012 (UTC)
Nonsense. The Levi-Civita field definition assumes ε is an infinitesimal. It does not define an infinitesimal. The definition is circular. It is also a misnomer in my opinion because it follows from Cauchy's wrong ideas regarding infinitesimals. Like Cauchy, you appear to have missed this circularity in your reasoning (or lack thereof). 12.176.152.194 (talk) 15:36, 2 January 2012 (UTC)
The Levi-Civita field defines ε, and it can be shown it is an infinitesimal. — Arthur Rubin (talk) 15:45, 2 January 2012 (UTC)
That is false. Care to define ε? Care to define "mathematically trained" person? (*) To say that ε is an infinitesimal is not a definition. In order to say that something can be shown to be infinitesimal, you first have to define infinitesimal, that is, you have to know what you are talking about. Of course in your misguided thoughts this did not occur to you, did it? 12.176.152.194 (talk) 15:54, 2 January 2012 (UTC) — Preceding unsigned comment added by 12.176.152.194 ( talk)
(*) Mathematically trained according to you would be someone who believes the same rot as you do? Hmm, Archimedes and most great mathematicians did not possess degrees. So please do tell what this means? 12.176.152.194 ( talk) 16:35, 2 January 2012 (UTC)
By your definition:
f(x) = 1 if x=1 f(x) = 0 if x=a/b and a/b is rational
then x is infinitesimal. I don't think so.
"That it is infinitesimal follows from the definition of "<" in the field." - is the most ridiculous nonsense I have ever read.
I define one mathematically trained if one can see immediately that what you've written is absolute rot.
You are correct about your left-finite set definition - it is a non-definition. Aside from being completely irrelevant, it only makes your attempt to define an infinitesimal more complex. Furthermore, the fact that your imaginary set of infinitesimals has no LUB tells me immediately it is ill-defined even in terms of set theory. I don't care about the transfer principle because it is BS and there are mathematicians who agree with me on this.
Rubin, no well-trained mathematician will honestly believe in infinitesimals. What you have written is such nonsense that it's almost laughable. I'll go one step further: any mathematician who thinks infinitesimals are a sound concept is not a mathematician. More like a fool.
I suppose you are going to tell me this is just my opinion. Well, I'll tell you, anyone who claims infinitesimal theory borders on being a moron.
Rubin, I am sorry to say this (really) but you may be a bigger moron than I thought, if you sincerely believe in the garbage you've written.
One more thing: I can tell that you don't understand the theory very well. Most mathematicians will simply allow you to pull the wool over their dull eyes. Perhaps you should get your buddy Hardy to help you? But he is a statistician who claims that dy/dx is not a ratio. Tsk, tsk. 12.176.152.194 ( talk) 18:03, 2 January 2012 (UTC)
Dear 12.176.152.194, why do you write in the talk page section of an article if you don't understand the subject? There are many wikipedia articles and I'm sure you can contribute in a positive way to Wikipedia if you find a topic that you understand. Good luck! iNic ( talk) 00:29, 3 January 2012 (UTC)
Please take my advice and leave this page. You are just making a fool of yourself. iNic ( talk) 10:38, 3 January 2012 (UTC)
It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods. This is outright false. The reference is subject to opinion and debate. Khodr Shamseddine, "Analysis on the Levi-Civia Field: A Brief Overview," http://www.uwec.edu/surepam/media/RS-Overview.pdf 12.176.152.194 ( talk) 23:15, 2 January 2012 (UTC) }}
Inappropriate personal attacks that don't contribute to article creation. TenOfAllTrades( talk) 14:55, 3 January 2012 (UTC) |
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The following discussion has been closed. Please do not modify it. |
I think this section can serve as evidence. Anyone who thinks infinitesimal theory is sound and pertinent to the idea of infinitesimal in this article, is welcome to write his/her full name in this section followed by a link to his/her home web page. Only full real name and web page will qualify as entry. "Anyone who doesn't understand the article Levi-Civita field is not a mathematician" - gee, I guess that counts out all the great mathematicians starting from Archimedes and ending with those who came just before Abraham Robinson. It does not matter that Rubin has no idea whereof he is talking about. All he has to do is throw out the "correct" terminology and he can say whatever he likes because he has a PhD. Here I am, superior to Newton, Leibniz and Cauchy. Now Rubin who is a worm next to me, claims I do not understand. Ha, ha. This is too funny. Please add your name otherwise you are not a mathematician.
Seriously: There are no infinitesimals. I proved that Cauchy's derivative definition is a Kludge. Rubin could not understand it. It has every bit of relevance to this article and all the wrong theory of infinitesimals that has arisen from it. But of course it will be rejected because unlike Robinson's ideas, it has not been inked. My closing sentence is: The Emperor has no clothes. — Preceding unsigned comment added by 12.176.152.194 ( talk) 03:00, 3 January 2012 (UTC) 1. Arthur Rubin http://en.wikipedia.org/wiki/Arthur_Rubin 2.
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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)
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.
( 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
I removed the last sentence from this paragraph on Levi Field - "It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods." It is a matter of opinion. The stated reference (8) does not provide any evidence this is true. 166.249.134.226 ( talk) 17:51, 17 June 2012 (UTC)
I added the following statements to the second paragraph:
"An infinitesimal object by itself is often useless and not very well defined; in order to give it a meaning it usually has to be compared to another infinitesimal object in the same context (as in a derivative) or added together with an extremely large (an infinite) amount of other infinitesimal objects (as in an integral)."
I know that there maybe exists other ways to give infinitesimal numbers a meaning, but I didn't really know how to continue the lasts sentence. "Or in any other way give it a meaning" does just not sound right. Feel free to extend this statement to complete it. — Kri ( talk) 22:42, 17 June 2012 (UTC)
Since when does 1 - .999... = 1/x? As stated [1] its incorrect because 1/x <>0 and with limits it can be shown that .999... equals one, thus 1 - .999... = 0 (and not 1/x). In any case, even if you can show its properly sourced with some twisted interpetation of .999... (verifiable, not truth and all that), as I said in my edit summary, the first paragraph is supposed to define and summarize the article, not inadequately go into the minutia of how students are taught, per wp:lede. Thus, it needs to be removed. I just removed it again [2]. I removed it thinking the maths were sourced to a primary source, but I was mistaken on that. The authors referenced (Katz & Katz, 2010). I've not yet looked at that reference, but from another source I see that the expansion being referenced does not involve standard notation, since 0.999...;...999... is the hyperreal version of .999..., so the claim that .999... is different from 1 is either misleading or missing appropriate context, thus it does not belong in the lede. - Modocc ( talk) 21:55, 27 April 2013 (UTC)
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(ec; I think I agree with Modocc, but expressed it differently.) The claim being made in the papers is that students understand "0.999..." < 1 as an infinitesimal difference. If they don't understand 1 − ε < 1 < 1 + &epsilon (where ε = 1 − "0.999..."), then I would argue that what they "understand" doesn't act like an infinitesimal. Although mathematical education is not my field, if the papers don't comment on that one way or the other, then they seem questionable. — Arthur Rubin (talk) 17:02, 30 April 2013 (UTC)
Tkuvho, I am fine with the Wikipedia's verifiability not truth policy and I intend to keep my editor's hat on when it comes to deciding appropriate placement of content. Essentially, Katz & Katz is proposing to subtract terms in a way that with the hyperreal notation, defines some sets of numbers that according to their rank, gives numbers such that 1 - infinitesimal. But, as my demonstration above shows, I don't see how this allows for consistency across-the-board for all the reals, thus its a pedagogically nightmare for me to understand, and I wouldn't want to try to teach such a major conceptual revision without learning it for myself. But putting my opinion of this wiki's coverage of these papers (which I will read once I get a chance to visit the library) aside, there shouldn't be any need for me to discuss the content's veracity further. Anyway, this article is not about teaching students Katz & Katz's work on hyperreals! Since its nonstandard, its not a notable "introduction"! Further, without any tertiary sources (which cover well-seasoned research) actually discussing the K&K's recent proposal, it does not yet come close to meeting wp:DUE to warrant being placed in any article's lede. - Modocc ( talk) 20:43, 30 April 2013 (UTC)
I clarified the nonstandard reinterpretation of "0.999..." that is being taught, with this edit. [3]. And I started a new subsection regarding the Norton and Baldwin reference that led to this discussion. - Modocc ( talk) 21:12, 5 May 2013 (UTC)
What is the justification for including the Norton and Baldwin reference? Clicking on the one citation [4] in google scholar brings up an article that doesn't even appear to cite it. Which leaves no citations for it. Since it presents arguments that the standard number is somehow incorrect (its not), it seems prudent that this research paper be removed per wp:fringe since "exceptional claims require high-quality reliable sources". The emphasis in bold is mine. - Modocc ( talk) 22:20, 5 May 2013 (UTC)
I presume some legitimate points are being made in this section, but it is such a muddle.
Perhaps the subject head should be "elementary properties". The term "elementary" is introduced here, and it is true that the Archimedean property is a consequence of the completeness property of the real numbers, and stating the completeness property as the LUB property does involve making statements about sets of real numbers. So it evidently is not an elementary property of the real numbers.
The section also refers repeatedly to first-order logic (FOL), and that somehow non-elementary properties cannot be stated in FOL. The section states that logic with quantification restricted to elements and not sets "is referred to as first-order logic". But FOL is not only compatible with set theory, but widely used in combination with it throughout mathematics;
The second paragraph seems to contradict itself, and I do not have the energy to try to sort it out. It states that the Archimedean property can be expressed by quantification over sets. The LUB property is indeed expressed using sets, but the Archimedean property can be expressed as:
∀x, y ∈ R. x > 0 ∧ y > 0 ⇒ ∃ n ∈ N. n ⋅ x > y
and this does not involve quantification over sets.
The second paragraph then goes on to make further points, but I am unable to follow the argument. — Preceding unsigned comment added by Crisperdue ( talk • contribs) 20:09, 12 June 2013 (UTC)
Comments in this talk page section so far are by me, sorry I omitted the explicit signature originally. Crisperdue ( talk) 21:06, 12 June 2013 (UTC)
Perhaps this is not a good question to ask here, but given the very small set of published calculus books using Robinson's methods, I'm curious as to why Henle & Kleinberg's Infinitesimal Calculus isn't mentioned. I'm no doubt biased, since, as a college student some decades ago, I was totally entranced by Kleinberg's lectures on the subject. None the less, though perhaps not a complete calculus course, the text seems like a pretty nice exposition of the use of hyperreals in elementary calculus. Is the issue that its circulation was too small to make it notable, or that it's not widely enough used -- or was it just an oversight? Salaw ( talk) 05:58, 11 February 2014 (UTC)
I have submitted a new article for review specifically dedicated to the symbolic representation of the concept of infinitesimal given by 1/∞. All though that it is clear that a line has to be drawn as to which mathematical concepts such as operators etc. warrant a unique article (ie a unique article for the del operator etc)...my opinion is that this one definitely does because it represents the concept in the form of a relationship between two other well defined/accepted mathematical concepts and their associated symbols (whose origins preceded it). The article might make mention of other symbolic representations such as dx but only as supplementary discussion. Would appreciate a heads up if there are any objections and would surely appreciate recommendations and edits. YWA2014 ( talk) 04:12, 11 July 2014 (UTC)
Its a concern when an article has glaring issues right from the start. Such as the first sentence which gives one the impression that this a Physics article and not a math article. The use of the word "object" conveys the notion that infinitesimals are a concept that has some connection with something tangible (ie such that they would be something physical one would find discussed in a physics book)...which they are not. Likewise the discussion of their "measurability" as "objects" is rather strange as nowhere in history has a physics experiment ever been performed in order to test the hypothesis that one will find something that one cannot measure due to it not having a finite size. The proper word if one is trying to find some symmetry here between the math and the physics would be "singularity". Just saying this because I may want to make some changes here in the interest of confluence if the article above gets approved. YWA2014 ( talk) 04:12, 11 July 2014 (UTC)
If I can add that at least one reference doesn't appear to be checked. "Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.[7]" references a work of fiction, i.e. a "historical novel". 203.184.26.158 ( talk) 03:17, 21 August 2016 (UTC)
The new subsection created by User Prodigy uses the term "infinitesimal" in a different sense from the rest of this page, namely as a function that tends to zero. In particular the expression is meaningless. This could perhaps be interpreted in terms of the standard part function but the current version is confusing. Tkuvho ( talk) 09:58, 16 December 2014 (UTC)
While Isaac Newton and Gottfried Leibniz mostly thought in terms of infinitesimals, they never put the notion on a rigorous footing and in fact sometimes used what is recognizable as a precursor to Karl Weierstrass's more modern concept of the (ε, δ)-definition of limit (aka Epsilontics).
It was only when Abraham Robinson introduced Non-standard analysis that it became possible to build calculus rigorously from infinitesimals, and that approach is not dominant.
As a side note, the reference to Archimedes does not belong in the articles, since neither the method of exhaustion nor the method of indivisibles involves the notion of infinitesimals.
Shmuel (Seymour J.) Metz Username:Chatul ( talk) 22:17, 19 December 2014 (UTC)
While certainly interesting, this is just a single item of mathematical notation that might just as well be described in an article about the concept it denotes. — Keφr 14:27, 26 March 2015 (UTC)
I agree, however, the article is in dire need of proper citation. Bekamancer ( talk) 17:06, 28 March 2015 (UTC)
The article gives an example but doesn't really define the term. Equinox ( talk) 06:44, 19 January 2016 (UTC)
Last sentence reads: "Since the background logic is intuitionistic logic, it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first."
Might be helpful to less informed readers (like me :), to flesh out a little what "1,2, and 3" are in this context. Just a few words would probably suffice. (BTW, I did try to figure it out from the Smooth infinitesimal analysis main page as well as that of the Intuitionistic logic page to no avail. Granted, some deeper knowledge of such subjects is required of the reader, but it's relatively inexpensive to add just a bit more meat to such bare bones descriptions.) thx — Preceding unsigned comment added by 2602:306:CF8C:98D0:0:0:0:3E8 ( talk) 04:05, 27 August 2016 (UTC)
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I came to this page to figure out the correct mathematical symbol to represent an infinitesimal. I think it's small-capital-Greek-letter-Epsilon. But the page didn't actually answer my question! Would a short section titled "Notation" be helpful and appropriate? — Preceding unsigned comment added by 73.12.48.32 ( talk) 21:15, 19 December 2018 (UTC)
An editor has asked for a discussion to address the redirect Smallest number. Please participate in the redirect discussion if you wish to do so. Steel1943 ( talk) 22:06, 20 September 2019 (UTC)
I think most of the information in the lead section, while useful, might do better in the 'history' section; it's very long and there's information in there that doesn't show up elsewhere in the article. Perciv ( talk) 18:01, 9 April 2020 (UTC)
"His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers." Which similar set of conditions? It could be interpreted as |x|> 1, |x|> 1/(1+1), |x|> 1(1+1+1),..., or as |x|< 1, |x| < 1/(1+1), |x| < 1/(1+1+1),.... L1ucas ( talk) 00:26, 31 August 2021 (UTC)
In spite of a comment from 2012 about a "statutory" 4 paragraphs, I think the lede gets too far into details. The lede should be a summary. The third paragraph, for example, seems to be "detailed content" and not summary content. Maybe also the fourth paragraph. Why is (or was) there a desire to make the lede as long as possible? Thanks. David10244 ( talk) 13:42, 30 December 2022 (UTC)
According to the picture, it looks like the infinitesimals are a set of numbers that include zero, but the lead of the article mentions explicitly that an infinitesimal number "is not 0." So, is zero included in the infinitesimals or not? If it is not included, I think this should be indicated in some way in the image, because right now I would say that it is at best misleading. — Kri ( talk) 10:41, 9 April 2024 (UTC)
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Anyone who reads this article and believes it is truly naive. It contradicts itself numerous times. First it states:
"When we consider numbers, the naive definition is clearly flawed: an infinitesimal is a number whose modulus is less than any non zero positive number. Considering positive numbers, the only way for a number to be less than all numbers would be to be the least positive number. If h is such a number, then what is h/2? Or if h is undividable, is it still a number?"
Then in a section called 'A definition':
"An infinitesimal number is a nonstandard number whose modulus is less than any nonzero positive standard number".
It is true that such a definition is absolute nonsense. Not only this, but there is no evidence that an infinitesimal exists. To talk about the plural is ludicrous. Isaac Newton was groping in darkness when he coined this term. He was himself uncertain how to explain the calculation of a gradient or average 'at a point'. Furthermore the article states Archimedes used infinitesimals but till this day there is no coherent definition of an 'infinitesimal' and non-standard analysis is at most wishy. How could Archimedes have used infinitesimals if they do not exist and he had no idea what these are?
However, what surprises me is that Wikipedia allows this to be published. Another article on nonstandard numbers is also dreamy. 70.120.182.243 17:55, 30 April 2007 (UTC)
I think I understand Wikipedia's policy now. It does not matter what "facts" are true or false. As long as there is a publication of such facts, these qualify to be part of an article.
Is my understanding correct? If so, then I think my suggestions can be discarded. I think you should warn your readers that no information on your site can be trusted. It is also clear to me now why most academics warn their students to steer clear of Wikipedia. What I tell my students is to read everything and believe nothing unless it makes sense to them. I shall never suggest anything here again. 12.176.152.194 ( talk) 18:32, 18 December 2011 (UTC)
12.176.152.194 ( talk) 15:52, 22 December 2011 (UTC)
Perhaps you need to study English or mathematical reasoning more thoroughly. That states that for any secant there is a parallel tangent. You have asserted that for any tangent that there is a parallel secant. Not at all the same. — Arthur Rubin (talk) 17:02, 30 December 2011 (UTC)
The mean value states that if there is a tangent, there must be parallel secants. So for a given interval, it is exactly the same. You ought to address the previous paragraph to yourself. 12.176.152.194 ( talk) 00:40, 2 January 2012 (UTC)
Netz is not an authority by any stretch on Archimedes. His main contribution is/was in the restoration effort. I would go as far as saying that very few mathematicians after Heath understand Archimedes' Works. I am one of the few who has studied and understands his works well. 12.176.152.194 ( talk) 15:16, 29 December 2011 (UTC)
(ec) That no sense makes. And this still has nothing to do with anything which should be on Wikipedia. As I see it, the "new calculus" definition of the derivative is:
although, I can't come up with a definition which also requires to exist. I'm pretty sure that that definition implies that exists, and that, if you then define , then under the usual definition. — Arthur Rubin (talk) 17:33, 1 January 2012 (UTC)
I thought about your comment regarding circularity and it occurred to me that you are getting confused. So I will try to help you understand this. Gradient = rise/run. Rise = f(x+n)-f(x-m) Run = m+n So, gradient k = f(x+n)-f(x-m)/(m+n) => f(x+n)-f(x-m) = k(m+n). We don't know k but we know both rise and run so we can find k. To convince yourself there is no "infinitesimal remainder", divide both sides of f(x+n)-f(x-m) = k(m+n) by (m+n). On the left you have what you started out with and on the right you have k. For any difference quotient, you work with the left hand side so that after cancellation you will have one term without any m or n in it. This term denotes the gradient when m=n=0, that is, both distances on the side of the tangent at the tangent point are zero. Study diagrams in files to get a better understanding. Now, if you wish to find k for any one of the other secant lines that are parallel to the tangent, then you must know their (m,n) pairs. Finding a relationship between m and n helps. However, k will be the same for all the secant lines that are parallel to the tangent. BTW: The terms in m and n are not remainders! But their sum is always zero because the secants are parallel to the tangent. Each secant has its own (m,n) pair which makes all these terms zero. For example, consider f(x)=x^2. f'(x)=2x+(n-m). This is exactly the derivative. 2x+(n-m)=2x always. If x=1, then all the following are valid gradients: 2(1)+(0-0); 2(1)+(0.005-0.005); 2(1)+(3-3); 2(1)+(m-n) Note that m=n in the case of the parabola. This is not always true for every function. In fact, it's hardly ever true for most other functions. 12.176.152.194 ( talk) 17:05, 1 January 2012 (UTC)
An important step in learning the New Calculus is first realizing where the standard calculus is wrong. You cannot divide by h ever in the standard difference ratio. You can divide by (m+n) always in the New Calculus. Why? Every term of the numerator f(x+n)-f(x-m) contains a factor of (m+n). After cancellation (taking the quotient), exactly one term will be the gradient of the tangent line [distance pair (0;0)]. To find the gradients of all the parallel secants we use the terms in m and n if we want to be "devout". However, there is no need to do this because their gradients are all equal to the tangent gradient. Now we can find distance pairs in (m,n) for other reasons and there are many interesting reasons - especially in the theory of differential equations which I have researched using the new calculus. So, what you have to do is forget everything you learned and interpret what you read literally. It will take a while even if you are extremely smart. I have found that it's much easier to teach someone who has not learned standard calculus. 12.176.152.194 ( talk) 19:47, 1 January 2012 (UTC)
This discussion is not about my New Calculus. Now, although I do not mind whether you mention my New Calculus or not, I will mind if you mention it without proper attribution (my name and web page). I will win any argument in a court of law if it comes to this. Not threatening, just warning. Cauchy's kludge, secant method, distance pairs, etc are also my copyright phrases, not to be mentioned without correct attribution. What I have noticed about academics is that they are cynical until they understand and then they think it's no big deal. Well, it is a big deal because I was the first to think of it. It is also a big deal that I have corrected three great mathematicians: Newton, Leibniz and Cauchy. Although I can't stop you from quoting my work with the correct attribution, I would prefer that you do not quote my work at all. 12.176.152.194 ( talk) 19:05, 1 January 2012 (UTC)
Dear 12.176.152.194, Wikipedia should not be used for self-promotion. Neither in the articles nor in the talk pages. If you have your own version of the Calculus I urge you to get it published in a peer reviewed journal. In any case, Wikipedia is definitely the wrong place for publishing or discussing original research. Please respect that. iNic ( talk) 04:39, 2 January 2012 (UTC)
OK so what does the non factual statements have to do with your own OR? If there are non factual statements you should be able to point these out without referring to your own opinion about it or your own research. Have you done that? Please do not ever answer any questions about your own research on Wikipedia. Ever. Please just ignore all questions and comments about it here from now on. Those interested can contact you directly. If we stick to the Wikipedia rules we should all be cool. iNic ( talk) 13:18, 2 January 2012 (UTC)
If you have good arguments you should not have to resort to personal attacks like this. By the way, it's not allowed here and you can be banned from wikipedia if you continue like this. iNic ( talk) 15:34, 2 January 2012 (UTC)
How can he attack you if you stop talking about your own ideas? iNic ( talk) 16:09, 2 January 2012 (UTC)
Aha so you proved Einstein wrong too? Did you publish it? iNic ( talk) 15:34, 2 January 2012 (UTC)
This is very much off topic but please tell me when and in what context Einstein was proved wrong? iNic ( talk) 15:48, 2 January 2012 (UTC)
My work has been published online. That I have a website means it is copyrighted. Furthermore, it is dated so no one can say it's not original. Don't give me that nonsense regarding your knowledge of legal matters. One more thing - I did not bring up the topic, I have been asked several questions and referred those readers to the material. They did not have to read it or continue to ask me further questions. 12.176.152.194 ( talk) 05:03, 2 January 2012 (UTC)
So why don't you publish your own ideas proving Abraham Robinson to be wrong? Why wasting your time here while you have your important mission? Wikipedia can only take into account already published work and so far Robinson has published his ideas whereas you haven't. In the meantime please only talk about work that is not your own and that is published. "Published online" doesn't count I'm afraid, unless it's in a peer reviewed online journal. iNic ( talk) 15:43, 2 January 2012 (UTC)
Claiming that Robinson was an idiot is just stupid. Period. iNic ( talk) 10:23, 3 January 2012 (UTC)
As for infinitesimals, this article should be adequate to explain them to any mathematically trained person. My R(((ε))) is a subfield of the Levi-Civita field, which has most of the same properties, but is easier to calculate with. — Arthur Rubin (talk) 05:47, 2 January 2012 (UTC)
Nonsense. The Levi-Civita field definition assumes ε is an infinitesimal. It does not define an infinitesimal. The definition is circular. It is also a misnomer in my opinion because it follows from Cauchy's wrong ideas regarding infinitesimals. Like Cauchy, you appear to have missed this circularity in your reasoning (or lack thereof). 12.176.152.194 (talk) 15:36, 2 January 2012 (UTC)
The Levi-Civita field defines ε, and it can be shown it is an infinitesimal. — Arthur Rubin (talk) 15:45, 2 January 2012 (UTC)
That is false. Care to define ε? Care to define "mathematically trained" person? (*) To say that ε is an infinitesimal is not a definition. In order to say that something can be shown to be infinitesimal, you first have to define infinitesimal, that is, you have to know what you are talking about. Of course in your misguided thoughts this did not occur to you, did it? 12.176.152.194 (talk) 15:54, 2 January 2012 (UTC) — Preceding unsigned comment added by 12.176.152.194 ( talk)
(*) Mathematically trained according to you would be someone who believes the same rot as you do? Hmm, Archimedes and most great mathematicians did not possess degrees. So please do tell what this means? 12.176.152.194 ( talk) 16:35, 2 January 2012 (UTC)
By your definition:
f(x) = 1 if x=1 f(x) = 0 if x=a/b and a/b is rational
then x is infinitesimal. I don't think so.
"That it is infinitesimal follows from the definition of "<" in the field." - is the most ridiculous nonsense I have ever read.
I define one mathematically trained if one can see immediately that what you've written is absolute rot.
You are correct about your left-finite set definition - it is a non-definition. Aside from being completely irrelevant, it only makes your attempt to define an infinitesimal more complex. Furthermore, the fact that your imaginary set of infinitesimals has no LUB tells me immediately it is ill-defined even in terms of set theory. I don't care about the transfer principle because it is BS and there are mathematicians who agree with me on this.
Rubin, no well-trained mathematician will honestly believe in infinitesimals. What you have written is such nonsense that it's almost laughable. I'll go one step further: any mathematician who thinks infinitesimals are a sound concept is not a mathematician. More like a fool.
I suppose you are going to tell me this is just my opinion. Well, I'll tell you, anyone who claims infinitesimal theory borders on being a moron.
Rubin, I am sorry to say this (really) but you may be a bigger moron than I thought, if you sincerely believe in the garbage you've written.
One more thing: I can tell that you don't understand the theory very well. Most mathematicians will simply allow you to pull the wool over their dull eyes. Perhaps you should get your buddy Hardy to help you? But he is a statistician who claims that dy/dx is not a ratio. Tsk, tsk. 12.176.152.194 ( talk) 18:03, 2 January 2012 (UTC)
Dear 12.176.152.194, why do you write in the talk page section of an article if you don't understand the subject? There are many wikipedia articles and I'm sure you can contribute in a positive way to Wikipedia if you find a topic that you understand. Good luck! iNic ( talk) 00:29, 3 January 2012 (UTC)
Please take my advice and leave this page. You are just making a fool of yourself. iNic ( talk) 10:38, 3 January 2012 (UTC)
It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods. This is outright false. The reference is subject to opinion and debate. Khodr Shamseddine, "Analysis on the Levi-Civia Field: A Brief Overview," http://www.uwec.edu/surepam/media/RS-Overview.pdf 12.176.152.194 ( talk) 23:15, 2 January 2012 (UTC) }}
Inappropriate personal attacks that don't contribute to article creation. TenOfAllTrades( talk) 14:55, 3 January 2012 (UTC) |
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The following discussion has been closed. Please do not modify it. |
I think this section can serve as evidence. Anyone who thinks infinitesimal theory is sound and pertinent to the idea of infinitesimal in this article, is welcome to write his/her full name in this section followed by a link to his/her home web page. Only full real name and web page will qualify as entry. "Anyone who doesn't understand the article Levi-Civita field is not a mathematician" - gee, I guess that counts out all the great mathematicians starting from Archimedes and ending with those who came just before Abraham Robinson. It does not matter that Rubin has no idea whereof he is talking about. All he has to do is throw out the "correct" terminology and he can say whatever he likes because he has a PhD. Here I am, superior to Newton, Leibniz and Cauchy. Now Rubin who is a worm next to me, claims I do not understand. Ha, ha. This is too funny. Please add your name otherwise you are not a mathematician.
Seriously: There are no infinitesimals. I proved that Cauchy's derivative definition is a Kludge. Rubin could not understand it. It has every bit of relevance to this article and all the wrong theory of infinitesimals that has arisen from it. But of course it will be rejected because unlike Robinson's ideas, it has not been inked. My closing sentence is: The Emperor has no clothes. — Preceding unsigned comment added by 12.176.152.194 ( talk) 03:00, 3 January 2012 (UTC) 1. Arthur Rubin http://en.wikipedia.org/wiki/Arthur_Rubin 2.
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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)
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.
( 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
I removed the last sentence from this paragraph on Levi Field - "It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods." It is a matter of opinion. The stated reference (8) does not provide any evidence this is true. 166.249.134.226 ( talk) 17:51, 17 June 2012 (UTC)
I added the following statements to the second paragraph:
"An infinitesimal object by itself is often useless and not very well defined; in order to give it a meaning it usually has to be compared to another infinitesimal object in the same context (as in a derivative) or added together with an extremely large (an infinite) amount of other infinitesimal objects (as in an integral)."
I know that there maybe exists other ways to give infinitesimal numbers a meaning, but I didn't really know how to continue the lasts sentence. "Or in any other way give it a meaning" does just not sound right. Feel free to extend this statement to complete it. — Kri ( talk) 22:42, 17 June 2012 (UTC)
Since when does 1 - .999... = 1/x? As stated [1] its incorrect because 1/x <>0 and with limits it can be shown that .999... equals one, thus 1 - .999... = 0 (and not 1/x). In any case, even if you can show its properly sourced with some twisted interpetation of .999... (verifiable, not truth and all that), as I said in my edit summary, the first paragraph is supposed to define and summarize the article, not inadequately go into the minutia of how students are taught, per wp:lede. Thus, it needs to be removed. I just removed it again [2]. I removed it thinking the maths were sourced to a primary source, but I was mistaken on that. The authors referenced (Katz & Katz, 2010). I've not yet looked at that reference, but from another source I see that the expansion being referenced does not involve standard notation, since 0.999...;...999... is the hyperreal version of .999..., so the claim that .999... is different from 1 is either misleading or missing appropriate context, thus it does not belong in the lede. - Modocc ( talk) 21:55, 27 April 2013 (UTC)
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(ec; I think I agree with Modocc, but expressed it differently.) The claim being made in the papers is that students understand "0.999..." < 1 as an infinitesimal difference. If they don't understand 1 − ε < 1 < 1 + &epsilon (where ε = 1 − "0.999..."), then I would argue that what they "understand" doesn't act like an infinitesimal. Although mathematical education is not my field, if the papers don't comment on that one way or the other, then they seem questionable. — Arthur Rubin (talk) 17:02, 30 April 2013 (UTC)
Tkuvho, I am fine with the Wikipedia's verifiability not truth policy and I intend to keep my editor's hat on when it comes to deciding appropriate placement of content. Essentially, Katz & Katz is proposing to subtract terms in a way that with the hyperreal notation, defines some sets of numbers that according to their rank, gives numbers such that 1 - infinitesimal. But, as my demonstration above shows, I don't see how this allows for consistency across-the-board for all the reals, thus its a pedagogically nightmare for me to understand, and I wouldn't want to try to teach such a major conceptual revision without learning it for myself. But putting my opinion of this wiki's coverage of these papers (which I will read once I get a chance to visit the library) aside, there shouldn't be any need for me to discuss the content's veracity further. Anyway, this article is not about teaching students Katz & Katz's work on hyperreals! Since its nonstandard, its not a notable "introduction"! Further, without any tertiary sources (which cover well-seasoned research) actually discussing the K&K's recent proposal, it does not yet come close to meeting wp:DUE to warrant being placed in any article's lede. - Modocc ( talk) 20:43, 30 April 2013 (UTC)
I clarified the nonstandard reinterpretation of "0.999..." that is being taught, with this edit. [3]. And I started a new subsection regarding the Norton and Baldwin reference that led to this discussion. - Modocc ( talk) 21:12, 5 May 2013 (UTC)
What is the justification for including the Norton and Baldwin reference? Clicking on the one citation [4] in google scholar brings up an article that doesn't even appear to cite it. Which leaves no citations for it. Since it presents arguments that the standard number is somehow incorrect (its not), it seems prudent that this research paper be removed per wp:fringe since "exceptional claims require high-quality reliable sources". The emphasis in bold is mine. - Modocc ( talk) 22:20, 5 May 2013 (UTC)
I presume some legitimate points are being made in this section, but it is such a muddle.
Perhaps the subject head should be "elementary properties". The term "elementary" is introduced here, and it is true that the Archimedean property is a consequence of the completeness property of the real numbers, and stating the completeness property as the LUB property does involve making statements about sets of real numbers. So it evidently is not an elementary property of the real numbers.
The section also refers repeatedly to first-order logic (FOL), and that somehow non-elementary properties cannot be stated in FOL. The section states that logic with quantification restricted to elements and not sets "is referred to as first-order logic". But FOL is not only compatible with set theory, but widely used in combination with it throughout mathematics;
The second paragraph seems to contradict itself, and I do not have the energy to try to sort it out. It states that the Archimedean property can be expressed by quantification over sets. The LUB property is indeed expressed using sets, but the Archimedean property can be expressed as:
∀x, y ∈ R. x > 0 ∧ y > 0 ⇒ ∃ n ∈ N. n ⋅ x > y
and this does not involve quantification over sets.
The second paragraph then goes on to make further points, but I am unable to follow the argument. — Preceding unsigned comment added by Crisperdue ( talk • contribs) 20:09, 12 June 2013 (UTC)
Comments in this talk page section so far are by me, sorry I omitted the explicit signature originally. Crisperdue ( talk) 21:06, 12 June 2013 (UTC)
Perhaps this is not a good question to ask here, but given the very small set of published calculus books using Robinson's methods, I'm curious as to why Henle & Kleinberg's Infinitesimal Calculus isn't mentioned. I'm no doubt biased, since, as a college student some decades ago, I was totally entranced by Kleinberg's lectures on the subject. None the less, though perhaps not a complete calculus course, the text seems like a pretty nice exposition of the use of hyperreals in elementary calculus. Is the issue that its circulation was too small to make it notable, or that it's not widely enough used -- or was it just an oversight? Salaw ( talk) 05:58, 11 February 2014 (UTC)
I have submitted a new article for review specifically dedicated to the symbolic representation of the concept of infinitesimal given by 1/∞. All though that it is clear that a line has to be drawn as to which mathematical concepts such as operators etc. warrant a unique article (ie a unique article for the del operator etc)...my opinion is that this one definitely does because it represents the concept in the form of a relationship between two other well defined/accepted mathematical concepts and their associated symbols (whose origins preceded it). The article might make mention of other symbolic representations such as dx but only as supplementary discussion. Would appreciate a heads up if there are any objections and would surely appreciate recommendations and edits. YWA2014 ( talk) 04:12, 11 July 2014 (UTC)
Its a concern when an article has glaring issues right from the start. Such as the first sentence which gives one the impression that this a Physics article and not a math article. The use of the word "object" conveys the notion that infinitesimals are a concept that has some connection with something tangible (ie such that they would be something physical one would find discussed in a physics book)...which they are not. Likewise the discussion of their "measurability" as "objects" is rather strange as nowhere in history has a physics experiment ever been performed in order to test the hypothesis that one will find something that one cannot measure due to it not having a finite size. The proper word if one is trying to find some symmetry here between the math and the physics would be "singularity". Just saying this because I may want to make some changes here in the interest of confluence if the article above gets approved. YWA2014 ( talk) 04:12, 11 July 2014 (UTC)
If I can add that at least one reference doesn't appear to be checked. "Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.[7]" references a work of fiction, i.e. a "historical novel". 203.184.26.158 ( talk) 03:17, 21 August 2016 (UTC)
The new subsection created by User Prodigy uses the term "infinitesimal" in a different sense from the rest of this page, namely as a function that tends to zero. In particular the expression is meaningless. This could perhaps be interpreted in terms of the standard part function but the current version is confusing. Tkuvho ( talk) 09:58, 16 December 2014 (UTC)
While Isaac Newton and Gottfried Leibniz mostly thought in terms of infinitesimals, they never put the notion on a rigorous footing and in fact sometimes used what is recognizable as a precursor to Karl Weierstrass's more modern concept of the (ε, δ)-definition of limit (aka Epsilontics).
It was only when Abraham Robinson introduced Non-standard analysis that it became possible to build calculus rigorously from infinitesimals, and that approach is not dominant.
As a side note, the reference to Archimedes does not belong in the articles, since neither the method of exhaustion nor the method of indivisibles involves the notion of infinitesimals.
Shmuel (Seymour J.) Metz Username:Chatul ( talk) 22:17, 19 December 2014 (UTC)
While certainly interesting, this is just a single item of mathematical notation that might just as well be described in an article about the concept it denotes. — Keφr 14:27, 26 March 2015 (UTC)
I agree, however, the article is in dire need of proper citation. Bekamancer ( talk) 17:06, 28 March 2015 (UTC)
The article gives an example but doesn't really define the term. Equinox ( talk) 06:44, 19 January 2016 (UTC)
Last sentence reads: "Since the background logic is intuitionistic logic, it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first."
Might be helpful to less informed readers (like me :), to flesh out a little what "1,2, and 3" are in this context. Just a few words would probably suffice. (BTW, I did try to figure it out from the Smooth infinitesimal analysis main page as well as that of the Intuitionistic logic page to no avail. Granted, some deeper knowledge of such subjects is required of the reader, but it's relatively inexpensive to add just a bit more meat to such bare bones descriptions.) thx — Preceding unsigned comment added by 2602:306:CF8C:98D0:0:0:0:3E8 ( talk) 04:05, 27 August 2016 (UTC)
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I came to this page to figure out the correct mathematical symbol to represent an infinitesimal. I think it's small-capital-Greek-letter-Epsilon. But the page didn't actually answer my question! Would a short section titled "Notation" be helpful and appropriate? — Preceding unsigned comment added by 73.12.48.32 ( talk) 21:15, 19 December 2018 (UTC)
An editor has asked for a discussion to address the redirect Smallest number. Please participate in the redirect discussion if you wish to do so. Steel1943 ( talk) 22:06, 20 September 2019 (UTC)
I think most of the information in the lead section, while useful, might do better in the 'history' section; it's very long and there's information in there that doesn't show up elsewhere in the article. Perciv ( talk) 18:01, 9 April 2020 (UTC)
"His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers." Which similar set of conditions? It could be interpreted as |x|> 1, |x|> 1/(1+1), |x|> 1(1+1+1),..., or as |x|< 1, |x| < 1/(1+1), |x| < 1/(1+1+1),.... L1ucas ( talk) 00:26, 31 August 2021 (UTC)
In spite of a comment from 2012 about a "statutory" 4 paragraphs, I think the lede gets too far into details. The lede should be a summary. The third paragraph, for example, seems to be "detailed content" and not summary content. Maybe also the fourth paragraph. Why is (or was) there a desire to make the lede as long as possible? Thanks. David10244 ( talk) 13:42, 30 December 2022 (UTC)
According to the picture, it looks like the infinitesimals are a set of numbers that include zero, but the lead of the article mentions explicitly that an infinitesimal number "is not 0." So, is zero included in the infinitesimals or not? If it is not included, I think this should be indicated in some way in the image, because right now I would say that it is at best misleading. — Kri ( talk) 10:41, 9 April 2024 (UTC)