Talk:Infinitesimal

Rubin tries to define infinitesimal

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)

In the Levi-Civita field, ε is an element of the field, corresponding to the function "a" which maps 1 to 1 and all other rationals to 0. That it is infinitesimal follows from the definition of "<" in the field (which isn't specified in the article, but is specified in the definition of the field.) I suppose you're going to tell me that the definition of the field (the collection of all functions a from the rationals to the reals such that the set of indices of the nonvanishing coefficients $\{q\in \mathbb {Q} :a_{q}\neq 0\}$ is be a left-finite set) is not a definition, either.

If I recall correctly, the definition of addition, multiplication, and less that, where a and b are in the field, are as follows:

$(a+b)_{q}=a_{q}+b_{q}$

$(a\cdot b)_{q}=\sum _{r+s=q}(a_{r}\cdot b_{s})$

$a<b{\text{ if the least rational }}q{\text{ such that }}a_{q}\neq b_{q}{\text{ satisfies }}a_{q}<b_{q}$

where the sum and the existence of "the least rational" follow from the left-finite properties. — Arthur Rubin (talk) 16:53, 2 January 2012 (UTC)

"My" field R(((ε))) can be defined the same way, except that the support $\{q\in \mathbb {Q} :a_{q}\neq 0\}$ is required to be bounded below and have a common denominator, making the verification of closure easier. — Arthur Rubin (talk) 16:59, 2 January 2012 (UTC)

For what it's worth, for the purpose of this argument, I define mathematically trained as anyone who can understand the definition of the Levi-Civita field, as defined in our article. — Arthur Rubin (talk) 17:03, 2 January 2012 (UTC)

There are so many problems with what you have written, that it is difficult to know where I can begin to show how wrong you are.

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.

If Robinson is an idiot I'm happily a moron in your view. It sounds like a great offer. iNic (talk) 10:29, 3 January 2012 (UTC)

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)

I see you don't understand the concept of abstract algebra. The function f, as you call it, considered as an element of the field, is an infinitesimal. And the set of infinitesimals never has an LUB, in the Levi-Civita field, because there is no attempt to claim the field is complete, and in the hyperreals or ultraproduct analysis, because the set is not "internal" or "standard". I wasn't going to say that you are a moron, but anyone who doesn't understand the article Levi-Civita field, whether or not you agree with it, is not a mathematician. It's clear that you don't understand it. — Arthur Rubin (talk) 20:37, 2 January 2012 (UTC)

I think I understand very well. What I am saying is you are wrong and these are two different things. You can call the function f anything you like, but it does not remotely resemble anything close to the idea associated with the infinitesimal concept. In fact you can call your subset (which is ill-defined because it has no LUB) of infinitesimals "rubins" if you like, but it does not change the fact that it's all feces. There is absolutely ZERO relation between your abstract little set and infinitesimals in the traditional meaning thereof. The Levi-Cevita field is ill-defined. In plain words, it's a load of rubbish. And you are no mathematician. Also, no honest mathematician who has completed studies in Abstract Algebra (as I have) will agree with what you claim. The difference between us is this: I am a real mathematician without a PhD. You have a PhD but are a fake mathematician. 12.176.152.194 (talk) 22:07, 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)

Inic: I know this is all too overwhelming for you. I suggest you follow your advice. Perhaps you can contribute in a more positive way. What do you say? 12.176.152.194 (talk) 02:53, 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)

Another non-factual and misleading claim in the article

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) }}

List of names that agree with Rubin's/Abraham Robinson's views

Inappropriate personal attacks that don't contribute to article creation. TenOfAllTrades(talk) 14:55, 3 January 2012 (UTC)
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. So how will the history books of the future refer to this great man that is superior to Newton, Leibniz and Cauchy? As an IP address? You are a big joke. iNic (talk) 10:19, 3 January 2012 (UTC) Calling me a joke is not a personal attack? Hypocrite! 12.176.152.194 (talk) 11:35, 3 January 2012 (UTC) No, it's an euphemism. iNic (talk) 12:13, 3 January 2012 (UTC) 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. 12.176.152.194 (talk) 22:24, 2 January 2012 (UTC) That's 1 to 0, so far. (You haven't signed with your real name.) — Arthur Rubin (talk) 06:04, 3 January 2012 (UTC) There is a good reason my name is not on the list. I claim that this knowledge is flawed and therefore incomprehensible. There is no amount of understanding that can help the situation. 12.176.152.194 (talk) 11:43, 3 January 2012 (UTC) ... and, in case someone is taking the IP seriously, it should be pointed out that I said that one who does not understand the article Levi-Civita field is not a mathematician; Well, you have been taking me seriously. Now what does that say for you eh? I think you know that you are BSing. 12.176.152.194 (talk) 11:43, 3 January 2012 (UTC) In other words, if x read the Wikipedia article Levi-Civita field, and x does not understand it (and, implied, x speaks English), then x is not a mathematician. There is a difference between "understanding" and "agreeing". What gives you the right to call anyone a non-mathematician? You were the first to imply this of me. My suggestion to you is don't dish it out if you can't take it. To call someone a non-mathematician when it is false, is an attack on one's character. I returned your favour. Now, if what you claim is true, there were no real mathematicians until Levi-Civita came along. This smacks to me of Jewish superiority complex. 12.176.152.194 (talk) 11:43, 3 January 2012 (UTC) The conclusion is obvious. — Arthur Rubin (talk) 06:11, 3 January 2012 (UTC) The conclusion is obvious - when you don't like what someone says because it's true, do what the Nazis did and silence that person. How Jewish of you! 12.176.152.194 (talk) 11:47, 3 January 2012 (UTC)

lede

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

n extensions of R

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

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

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

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

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

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

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

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

Define constant symbols $c_{x}$ for each x in X, constant symbols $d_{\beta }$ for each β < α, and constant symbol ε with axioms:

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

$c_{x}<c_{y}$ for x < y in X

$0<\epsilon$

$\epsilon <c_{x}$ for x > 0 in X

$d_{\beta }\neq d_{\gamma }$ for β < γ < α

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

Subject to opinion

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)

An infinitesimal number by itself is useless

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)

Do you have a source for your "uselessness" claim? Tkuvho (talk) 16:35, 25 December 2012 (UTC)

No, I don't. I was about to comment on this myself and ask what the person who wrote the original comment meant by "useless" (exactly like you also did), only to notice that it was myself who had written it, twelve years ago. :P I think the way I think about numbers has changed a bit over time; I guess that studying abstract mathematics tends to have that effect.

I think that what I meant when I wrote that in order to give an infinitesimal number a meaning it usually has to be compared to another infinitesimal object in the same context, was that it (for example) is impossible to say whether a single infinitesimal number is large or small without putting it into a context and comparing it with another (non-zero) infinitesimal number, just like it is impossible to say whether a physical quantity is large or small without comparing it with another quantity with the same dimensions. To say whether something that is unitless is large or small without being given any further context is in principle possible since you always can compare it with unity, but if it is dimensionful, you can't do that, and an infinitesimal number can be considered a dimensionful quantity where ε is the unit (unless you want to say that all infinitesimal numbers always are small, but that's not especially helpful). —Kri (talk) 09:46, 9 April 2024 (UTC)

Lede edits

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)

I am not sure I understand your question "Since when does 1 - .999... = 1/x?" How does "1/x" come into the picture? The point is that students naturally relate to the string "0.999..." as being infinitely close to 1, so that 1-"0.999..." is a kind of a "naturally occurring" infinitesimal. As has been explained in a number of recent articles including the ones cited in the lede, student intuitions of an infinite string of 9s falling just short of 1 can be rigorously justified. As a "naturally occurring" infinitesimal, 1-"0.999..." is helpful in setting the stage in this article, because it is an immediate way of giving the reader an idea of what we are talking about, that he or she can perfectly well relate to. It therefore fits nicely in the lede. Tkuvho (talk) 10:05, 28 April 2013 (UTC)

The point of my question is that, according to all sources, that for the same reason there is absolutely no infinitesimal of any kind between .333... and 1/3, there is also absolutely no infinitesimal between .999... and (1/3)*3. That a single student journal points out a couple of fringe worthy guys that dispute these facts and references a math wizard's hyperreal expansion (without even actually discussing it) and then having something perhaps vaguely like it placed prominently here? No. Sure, its vital that students and teachers can and should talk about the infinitesimal difference between .999...;...999 and 1, since .999...;...999 < .999...;...999... = 1. Of course, they need to be astute and explicit with their math when going about this too and to not be so vague or inaccurate that they mistakenly promulgate misconceptions in the process. -Modocc (talk) 13:54, 28 April 2013 (UTC)

The references in question are: (1) as well as (2) . These are by no means "student journals". Thus, Journal for Research in Mathematics Education is a leading (some would say, the leading) education journal. Meanwhile, The Mathematics Educator is published by students, but it is a refereed journal that publishes established education scholars, as reference (2) clearly illustrates. As far as your claim of "promulgating misconceptions" is concerned, I believe wiki policy is dictated by standards of verifiability rather than personal opinions of individual editors. Tkuvho (talk) 14:33, 28 April 2013 (UTC)

The relevant text with regard to your specific claim is: "Nonstandard analysis provides a sound basis for treating infinitesimals like real numbers and for rejecting the equality of 0.999... and 1(Katz & Katz, 2010)." Thus verifiable, but it is from a student journal and its not clear to me that this is the accepted position. Per wp:due, if you can provide other statements like this one by others, I'd like to see them (I don't have access to all the text in question). -Modocc (talk) 15:09, 28 April 2013 (UTC)

I am not sure why you choose to focus on the article in The Mathematics Educator and to emphasize the fact that it is student edited (again, it is not a "student journal" as you call it). The article in Journal for Research in Mathematics Education is more notable, makes the point about usefulness of an "0.999..." entity falling short of 1 as far as student learning is concerned (based on a field study), and is cited by no fewer than 24 articles by a wide variety of authors, see here Given the hostile tone of your earlier messages concerning this issue, it is hard to understand your claims to the effect that "more references are needed". Does one need 48 references to get this past this particular editor? 96? Tkuvho (talk) 12:08, 29 April 2013 (UTC)

To begin with, do any of the articles actually refer to the "difference" between 0.999... and 1 as "infinitesimal"? If not, the discussion do not belong here. If so, I would need to see the context to see if it is appropriate, and need to do research to see if it's "fringe". — Arthur Rubin (talk) 12:39, 29 April 2013 (UTC)

Certainly, Ely's article specifically discusses nonstandard student intuitions of an infinitesimal number 0.000... ...1 and its usefulness in learning calculus. To the extent that there is a crystal clear formalisation of this in the context of the hyperreals, it seems to escape the "fringe" label. The point, of course, is not that the students should be taught about ultrafilters, but that their intuitions are "nonstandard" rather than "erroneous". User:Modocc obviously adheres to the latter view, and is moreover perfectly within his rights to view such intuitions as "erroneous". As soon as he publishes his research, we will be able to cite it. Tkuvho (talk) 12:45, 29 April 2013 (UTC)

I focused on the authors' math statement because that is precisely what you want inserted into the article (I refer to the editors involved because they do bear responsibility to correct mistakes). Since 0.999... = 1, you need statements that clearly and unambiguously state otherwise (you need specific statements from math articles like 0.999... <> 1 to make that claim!). The alternative (and not to be placed in the lede please) is to write about how the hyperreal conception of ".999...;X" falls short and makes rigorous the students' misconception of ".999...". --Modocc (talk) 13:01, 29 April 2013 (UTC)

It's unsuitable for the lead, because the students' intuition is erroneous. Reliable sources making statements contrary to fact may be disregarded. It might still be helpful in student learning, although I dislike the concept of teaching using erroneous concepts. The statement "The argument that 0.999… only approximates 1 has grounding in formal mathematics.", in Norton & Baldwin, is erroneous. I'm sure we can find reliable sources, if necessary. Neither Norton nor Baldwin appears to have any published papers in non-standard analysis, nor in any field related to it, nor is non-standard analysis within the primary field of study of either journal. I could argue that my statement here that the intuition is erroneous might quailify as a WP:RS, but I won't; my field being mathematical logic and set theory, not specifically including non-standard analysis. — Arthur Rubin (talk) 13:45, 29 April 2013 (UTC)

Arthur, I don't think the leading math education journal would have published a paper with such a glaring mathematical error, nor would such a paper have gotten widely cited. Obviously, the author is referring to student intuitions of "0.999..." being a nonstandard number, rather than a real number. After all, before the students have been taught anything about the real numbers, they can legitimately hold opinions about a variety of possible numbers. We have a large subsection in 0.999... explaining this. Your point that the citation is unsuitable for the lede may be well-taken, but we should agree about the mathematics before we go on to discuss notability! Tkuvho (talk) 14:18, 29 April 2013 (UTC)

Thanks for the new reference that I can access. Inclusion of this proposed reassignment of 0.999... is fine in the body of this article (not the lede) iff its shown with the explicit context that was included in the 0.999... article. Actually, since this article only briefly summarizes the hyperreals, it should only be noted that such a reassignment is proposed by these authors. --Modocc (talk) 16:16, 29 April 2013 (UTC)

We're not discussing the proposed addition. I have little objection to the proposed text appearing in some article, but I don't think students could have an useful intuitive understanding unless they could also see "0.999..." < 1 < 2 − "0.999...", which I see no reference to in the papers I've been able to read. I think it should more likely be in a hyperreal article than in infinitesimal — Arthur Rubin (talk) 16:58, 29 April 2013 (UTC)

The proposed addition, wherever it lives, does not show that "0.999..." (Katz & Katz) is the reassignment of "00.999..." to a number different than 1, nor does it make it clear that it was recently proposed. I can see the motivation for this though, because if the rank goes to infinity then the limit of the reciprocal term equals zero, and one ends up with the standard real 0.999... = 1. Like I said at the beginning, context matters and this is missing from the proposed addition. My apologizes if I seemed arrogant, indifferent, or "in the way" of the contribution I removed. My goal now is to improve the text so I and others have a better understanding on a first read without trudging though discussions such as this. In addition, the reason I wrote 1/x is because I didn't type out the markup for 1/infinity when 1/x works, because the reciprocal of any quantity, of any size, is always nonzero and the reason infinitesimals (and limits) exist. -Modocc (talk) 17:19, 29 April 2013 (UTC)

@Arthur: I am not sure I understand your query with regard to the inequality [1 < 2 − "0.999..."]. By very simple algebra this reduces to ["0.999..." < 1] so a student who can relate to the one, can also relate to the other. What does the other inequality add? The comment about a "naturally occurring" infinitesimal is accessible to a large audience which is certainly broader than the readership of hyperreal number. Tkuvho (talk) 12:35, 30 April 2013 (UTC)

@User:Modocc: I re-read your comment on 1/x several times, but I am still not sure what you are getting at. If you are asking what the reciprocal of such a non-standard [1 - "0.999..."] is, the answer is precisely 10^H where H is the (infinite) rank where the last "9" occurs. Tkuvho (talk) 14:29, 30 April 2013 (UTC)

I'm not asking what the reciprocal of the non-standard [1 - "0.999..."] is, but I am implying that the math of that is incorrect for an infinite rank, but I'm not a mathematician and I have not studied hyperreals: thus I might be gulping down someone's porridge if its not to hot, or not to cold, thus bear with me please, :), and I'll do my best to explain my rationale: I'm claiming that for the set of hyperreals and its subset of reals, the formula from the 0.999... article (attributed to Karin Katz and Mikhail Katz) does not consistently hold for an infinite rank H. To show this (with a rigor that is only sufficient for me at the moment) consider that a standard number is 3/10 + 3/100 + ... = 0.333... = 1/3 (this should be true whether we are working with reals or hyperreals, since hyperreals include all reals). Therefore: 0.999... = 9/10 + 9/100 + ... = 3*(3/10 + 3/100 + ...) = 3*(1/3) = 1 = 1 - 0 <> 1 - |1/x| for all x because the reciprocal function f(x)=1/x<>0 can not be zero for any and all x including all possible infinitely small numbers given by the infinitesimals such as the specific one given by Katz&Katz: 1/10^H. QED. In short, the standard summation of "9/10 + 9/100 + ..." does not equate with the summation of 1 and an infinitesimal. Now like I said, I am not a mathematician, but if the maths I learned for the reals somehow do not apply to the hyperreals, and I need to study that article too I'm possibly in the wrong house and ... -Modocc (talk) 16:59, 30 April 2013 (UTC)

@User:Modocc: You apparently feel that there is an error in these articles published in refereed journals. This is certainly possible. However, there is a detailed discussion of this calculation in the "infinitesimal" section of 0.999.... A number of editors have gone over that section without detecting errors. My impression is that you are taking the expression "infinite sum" too literally. You could raise this issue at Talk:0.999... if you still feel there is an error. Tkuvho (talk) 19:32, 30 April 2013 (UTC)

I've no intention of trying to contravene policy (see my comments below). --Modocc (talk) 20:43, 30 April 2013 (UTC)

(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)

The student R. Ely interviewed described this ε as 0.000... ...1 so she would certainly not have any problem describing 1+ε as 1.000... ...1 --what exactly are you getting at? Tkuvho (talk) 19:09, 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)

Please see Talk:0.999...#Consistency_across-the-board_for_all_the_reals where I responded in more detail. Tkuvho (talk) 08:38, 1 May 2013 (UTC)

Partially resolved

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)

Norton and Baldwin

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)

Thanks for your interest in this page. I don't think the Norton-Baldwin paper is either "incorrect" or "fringe" but the citation issue makes it hard to insist on its inclusion. Let me take this opportunity to explain why the term "infinite sum" cannot be taken too literally. Consider the infinite sum 1 + 1/2 + 1/4 + 1/8 + etc. The partial sums are clearly less than 2, whereas the series yields 2. Now consider the following "thought experiment": suppose you are working inside a system containing infinitesimals. Let e be a positive infinitesimal. The number 2-e is infinitely close to 2. Now notice that 2-e is in fact an upper bound for each of the partial sums! Yes the "infinite sum" is bigger than 2-e, namely 2. You get the same "paradox" if you concatenate intervals of lengths 1, 1/2, 1/4, 1/8, etc. At each stage the resulting interval has length less than 2-e, yet the "infinite concatenation" (if you think of the series in such terms) somehow overcomes 2-e. There is of course no paradox here if one interprets "infinite sum" as it should be, namely as a series defined via the concept of limit. There is another way of seeing this by decomposing the "taking the limit" procedure into two steps. Tkuvho (talk) 12:38, 6 May 2013 (UTC

If students apply nonstandard analysis to the problem, they still must accept the fact that the standard non-terminating number 0.999...;...999... is equal to 1 regardless of the partial sums they might consider instead (a rank of 2 if they like or a terminating infinite sequence) when learning about real numbers and limits. These facts are straight forward enough, thus 1)my reasons for not keeping the Norton and Baldwin reference are unchanged, 2)we seem to have better references regarding these partial sums than it, and 3)we agree that its lack of citations is a problem for it. -Modocc (talk) 13:49, 12 May 2013 (UTC)

Thuvho: I think you may have overlooked something which may mitigate the need to think in terms of limits. All the finite partial sums are less than 2-e but if you are viewing this from inside the system which contains the infinitesimal, you need to carry the sum out "all the way" to get 2, and that will include infinite terms which bring the partial sums above 2-e. In particular, if N is an infinite hyperinteger larger than 1/e, then the partial sum 1 + 1/2 + 1/4 + ... + 1/N will be larger than 2 - e. Note further that, if you're doing this from inside the nonstandard system, you have no way of taking the sum over the reals without including the nonstandard terms, since, from inside the system, you can't tell the difference between a standard and a nonstandard term. Similar objections apply to the idea that 0.999... could be considered less than 1, of course -- inside the system, you have partial sums, you have the whole series, but you have no way to specify that you will include only standard terms. Salaw (talk) 16:01, 11 February 2014 (UTC)

I believe the Norton-Baldwin reference was removed in the end, so I am not sure what we are discussing exactly. I fully agree that 0.999...=1 :-) Tkuvho (talk) 16:07, 11 February 2014 (UTC)

First-order properties

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)

Actually, the Archimedean property is in $L_{\omega _{1}\omega }$, as the quantification over N is in the metalanguage. (see Infinitary logic#Definition of Hilbert-type infinitary logics for notation.) You'll note that hyperreals satisfy the displayed definition of the Archimedean property, with N being the nonnegative hyperintegers. — Arthur Rubin (talk) 23:05, 12 June 2013 (UTC)

I spent some time digging through the history of this section last night. In case anyone's interested in what went wrong with this section, here's the deal. It started out as a completely conventional exposition of first-order infinitesimal theory, just like I learned it in school. In particular, it asserted that not all properties of the hyperreals are identical to the reals (after all, they'd be kinda useless if they were totally identical to the reals, eh?), and it asserted that, since the reals are the only complete ordered field, we can't expect the hyperreals to be complete. An IP address then came in and added some rather angry (my description) and ungrammatical (objective judgement) edits to the section to assert that one can extend the reals in a way that preserves all properties, and also added the last sentence of paragraph 2 that begins "This is also wrong..." which contradicts the sentence just before it.
  Some long-suffering editor came in afterwards and cleaned up the grammar and made things read a little better, but the last sentence in the paragraph is still standing there as a direct (and clearly intentional) contradiction of the earlier assertions.
  I'd try to fix it up but unfortunately my knowledge of this subject is too soft for me to be sure what's correct here. I know that simple expositions of the hyperreals carry over only first order properties. I also know that Robinson spent what seemed to me to be a large (and difficult) amount of space in his book working to carry forward the second order properties of the reals as well (at least, I think that's what he was doing), which would seem to imply he was striving for a version of the hyperreals which were complete. That agrees with some things I've read elsewhere on the web. Presumably, the catch is that from inside the hyperreals, you can't tell the difference between a real and a hyperreal, so you can't form, say, the set of all values infinitesimally close to 1, which obviously doesn't have either a GLB or LUB. Salaw (talk) 15:41, 11 February 2014 (UTC)

Thanks for your interest. The IP is probably influenced by the internal set theory viewpoint, where they really are identical to the reals (in fact, they are the reals). I remember a few years ago there was an editor making mistakes of this sort that had to be corrected by other users. He apparently moonlighted as an IP without anybody noticing it. If the last sentence is incorrect the simplest thing would be to remove it. Tkuvho (talk) 15:56, 11 February 2014 (UTC)

If you check the French wiki page for non-standard analysis you will notice that it is dominated by the IST outlook. Checking through its history may help identify the editor but who cares? Tkuvho (talk) 15:57, 11 February 2014 (UTC)

By the way, the IST viewpoint is fabulous, but obviously it should not be confused with the NSA viewpoint. Tkuvho (talk) 16:01, 11 February 2014 (UTC)

Infinitesimals in teaching

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)

Merely an oversight. Please go ahead and add this! Tkuvho (talk) 13:16, 11 February 2014 (UTC)

OK -- I'll have to think a little more about what to say, tho, and find some time to say it. May be a couple more days.
BTW, I should say thanks for the reference to Keisler -- it looks good, and I may end up using it. We're doing homeschool calculus this semester and I've been finding that, when it comes down to actually using a book to teach the subject, I don't like any of the four or five calculus texts I've got lying around the house. And writing up my own notes, instead of working from a book, is not only proving (very) time consuming, but I'm afraid that, in the end, it's unlikely to produce something better than a published textbook (no matter what my overwhelming hubris may be telling me). Salaw (talk) 05:53, 13 February 2014 (UTC)

After explaining the basic concepts in terms of infinitesimals, Keisler also provides an explanation in terms of epsilon, deltas. This difficult approach is easier for the students to understand once they have a grasp of the notions already. If they will continue at the university at some point they should be acquainted with the epsilon, delta approach as well (unless the epsilon, delta approach is dropped by the universities in favor of the infinitesimal approach by the time they go to college). Tkuvho (talk) 12:22, 13 February 2014 (UTC)

It would be cool if that happened, but then, 30 years ago we were expecting it any day now. On the other hand, if Keisler's third edition can be taken as evidence, the double standard which made it so hard to use the infinitesimal approach may be fading away. Unless I've overlooked it in leafing through the book, he nowhere shows the construction of the hyperreals.
  Long ago, I asked a math teacher about using hyperreals to teach introductory calculus, and his response was that it was a great idea except that you had to spend the first three months of the course developing the hyperreals. That's what I mean by a "double standard" -- in algebra, in calculus, they teach the reals by introducing the axioms and taking it for granted that they're consistent and we can find a model for them. How many elementary calculus students even know what a "cut" is, or have heard of the Peano integers, or would know what it meant to form equivalence classes of all equal fractions? For that matter, what fraction of analysis students have actually read Landau's Foundations? Yet there seemed to be a feeling that to use the hyperreals, it was necessary to first show how to build a model for them -- upward lowenheim-skolem, ultrafilters, uncountable symbol sets, the works.
  Oh, well, rant rant rant. Time to go do something useful. Salaw (talk) 18:28, 13 February 2014 (UTC)

My comment about the universities was tongue-in-cheek. Meanwhile I do use Keisler in teaching. As far as the "double standard" is concerned, one of the reasons may be an insufficient awareness of just how much of an idealization the "ordinary reals" really are. But that's a vast subject... Tkuvho (talk) 08:40, 14 February 2014 (UTC)

Symbolic Representation 1/∞

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)

(See Draft:1/_∞)

Do have a very careful look at our policies regarding wp:reliable sources and wp:original research. Make sure that anything you create is backed by solid wp:secondary sources. I have put a welcome message on your talk page with some pointers. Good luck. - DVdm (talk) 06:41, 11 July 2014 (UTC)

This is an interesting project. I encourage you to pursue it. Tkuvho (talk) 12:50, 1 January 2015 (UTC)

The article was created at 1/∞. Tkuvho (talk) 08:35, 23 February 2015 (UTC)

A record of the discussion of the original draft is at User:Tkuvho/1/_∞. Tkuvho (talk) 09:05, 23 February 2015 (UTC)

Improvements and Article Quality Concerns

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)

equivalent infinitesimals

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 $\lim {\frac {\alpha _{1}}{\alpha _{2}}}=1$ 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)

equivalent infinitesimal are two variables whose limits are both zero at the same point and they satisfy $\lim {\frac {\alpha _{1}}{\alpha _{2}}}=1$. For example, $\lim _{x\to 0}{\frac {\sin x}{x}}=1$. Their main use is to determine the indeterminate form 0/0, which can be helpful when using L'Hopital's rule is complicated. In fact, I also edit Indeterminate form to reflect the change.Degenerate prodigy (talk) 15:27, 16 December 2014 (UTC)

Prodigy, all this is true, but this article does not use the term "infinitesimal" in the sense of a variable tending to zero. The viewpoint taken in this article is the viewpoint of Leibniz, Euler, and many others that an infinitesimal is a number that's smaller than every positive assignable (ordinary) number. Therefore there is a problem with using the term in a different sense. Tkuvho (talk) 08:32, 17 December 2014 (UTC)

Infinitesimals are not a basic building block of infinitesimal calculus

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)

Hi User:Chatul. There is a problem here with the link infinitesimal calculus. The page used to be an explanation that the term "infinitesimal calculus" meant something different historically from its common meaning today. The page has been redirected since to calculus. Perhaps the passage should be rephrased to emphasize that infinitesimals were a basic components in the procedures of the infinitesimal calculus as developed by Leibniz. Tkuvho (talk) 20:26, 20 December 2014 (UTC)

I see you have a joint publication with W. A. J. Luxemburg. Congratulations! Tkuvho (talk) 20:38, 20 December 2014 (UTC)

Proposed merge with 1/∞

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)

If this is merged it may be better to merge to John Wallis which page is more specific than infinitesimal. Tkuvho (talk) 11:07, 30 March 2015 (UTC)

Well, the fact that John Wallis invented the symbol is so compact (so to speak), that it can be probably repeated both in this article, in his biography, and even in the article about the lemniscate symbol, without causing much inconvenience. — Keφr 13:36, 30 March 2015 (UTC)

How is "transseries" defined?

The article gives an example but doesn't really define the term. Equinox (talk) 06:44, 19 January 2016 (UTC)

It's (supposedly) defined in the reference; as the reference may or may not be a reliable source, and I didn't check, I need to say "supposedly". The question of whether this is definition has any importance is unclear. — Arthur Rubin (talk) 11:04, 19 January 2016 (UTC)

Section 'Smooth infinitesimal analysis' needs just a bit more?

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)

Notation

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)

"Smallest number" listed at Redirects for discussion

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)

Lead Section

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)

Archimedean property on infinitesimals is ambiguous

"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)

Lede too long?

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)

Is zero not an infinitesimal?

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)