Talk:Infinitesimal

Mathematics Start‑class Mid‑priority

This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Start This article has been rated as Start-class on Wikipedia's content assessment scale. Mid This article has been rated as Mid-priority on the project's priority scale.

Apparently there are conflicting views. Repeating that it is right or that it is wrong will get us nowhere. I suggest that we stop writing back at each other. Then in, say, a month or two, we can meet again. And it would be good if those who write regularly signed their remarks. Odonovanr 13:49, 9 May 2007 (UTC)[reply]

A month has passed and this article has neither been deleted nor revised. Last week, out of a group of 12 people (9 of whom had 'respectable' PHds in this field) and a discussion on the use of Wikipedia by our students, the concensus was that students not refer to Wikipedia. Articles like

this one (and Wiki's non-standard analysis article) were mentioned as reasons. It is a shame because Wikipedia is free. But, if it is not reliable, then what use is it? I informed the group that in my opinion most of the articles are reliable, generally error free and of an exceptionally high quality. Over half of the group do not believe that non-standard analysis has been proved or is rigorous by any stretch. Almost all of the group believes that 'infinitesimal' is an ill-defined word.

I suggest that you change your policy of free editing and get rid of your current editors/sysops. You could establish the validity of an article by having a random panel of voters with exceptional knowledge in this field. I know this is difficult and might be a lengthy process. However, the article could state that its contents have not yet been approved or majority concensus not reached until such a process is complete. 70.120.182.243 14:39, 24 June 2007 (UTC)[reply]

Your suggestions for the article have been totally wrong, so I can see why you might feel that way. — Arthur Rubin | (talk) 15:04, 24 June 2007 (UTC)[reply]

I am surprised that anybody considering maths should think that "believing" has anything to do with the subject. Robinson "proved" his theorems which make nonstandard analysis a valid theory. If someone finds an error in the proof, them OK, it is wrong. Otherwise, until such error is shown, the theory has to be accepted. Goldblatt gives another proof which I find easier. I find no error, but then it is only me. But it is not fair that people assert that the thory is wrong or ill-defined without proving their claim. As for infinitesimal as an ill-defined word: well it is not the word that needs defining, but the concept the word represents. And the concept is given a definition in nonstandard analysis.

Is wikipedia reliable? Just as most thingson the net: it needs cross checking. This is why there is a list of references at the end of the article.Odonovanr 16:38, 24 June 2007 (UTC)[reply]

According to whom has Robinson's work been proved? Your argument is lame. You continue to ask me to disprove Robinson's work. I have responded logically by stating my objections:

- The concept of infinitesimal is ill-defined. On this point alone, I reject Robinson's work. Please do not point me to Robinson's book - it is an incomprehensible load of rubbish and rants by a Jewish (American?) fool who desperately tried to gain recognition in the 1960s. - Nothing is proved until it is proved.

Here is my proof that Robinson's ideas are nonsense: Robinson assumes an infinitesimal is well-defined and draws all his conclusions from this, thus all his subsequent assumptions are incorrect and so are any deductions based on the same. Please tell me why you think Goldblatt's proof is correct. What is his definition of an infinitesimal? 70.120.182.243 22:55, 24 June 2007 (UTC)[reply]

It's perfectly valid in mathematics to form theorems based on assumptions you can't falsify. I've seen theorems that start with "Suppose there exists a strongly innaccsessible cardinal" or "Suppose P=NP". If you can *prove* that Robinson's constructions are inconsistent, then publish. If not, Robinson has valid mathematics. It doesn't mean the conclusions are true (you can reject the assumptions), but you can't disagree with theorems. Endomorphic 05:39, 25 June 2007 (UTC)[reply]

"An infinitesimal number is a nonstandard number whose modulus is less than any nonzero positive standard number." This states that if |x| < a for any real positive a,

your mistake is here: you must say whether a is standard or not.Odonovanr 13:06, 25 June 2007 (UTC)[reply]

What difference does it make? I still don't know what a non-standard number is. You are implying a circular definition here that makes no sense. 70.120.182.243 00:18, 27 June 2007 (UTC)[reply]

then x is infinitesimal unless Wikipedia means something else by 'modulus'. It's not hard to show that the only quantity/number that satisfies this condition is zero. If the foundation is incorrect, then no matter how good Robinson's constructions are, they do not convince. If what you say is true, anyone can start off with a false premise and build theorems that no one else can disagree with. For example, Russell's paradox is not a paradox because Russell's logic is faulty from step 1. It all has to do with the definitions Russell uses. Definition provides meaning. Meaningless definitions lead to paradoxes and rubbish such as Robinson's work that you claim can't be disproved. What good is this? Have you looked at Robinson's non-standard analysis book at all? 70.120.182.243 12:58, 25 June 2007 (UTC)[reply]

Goldblatt (following Luxemburg) consider sequences of real numbers and equivalence classes of such sequences. Two sequences are equivalent if the set of indices where they are equal belongs to some non principle ultrafilter. This new set of equivalence classes is the set of hyperreals. This is similar to the Cauchy sequences and equivalence classes to construct the reals. And just as a rational is "included" in the new set by considering constant sequences of rationals as canonical representatives of a class, the reals are included in the hyperreals by considering a constant real sequence as canonical representative of a class. Then of course, the usual operations must be defined, which is done quite naturally. Then if it is possible to find a positive element of the hyperreals which is less in modulus than any real number (by the inclusion mentioned above) then this element is called infinitesimal. It is not assumed to exist. It is proven to exist, assuming that we can agree that a positive number less in modulus than any positive non zero real number deserves to be called infinitesimal. Even if you don't agree, the definition is there in a mathematically consistent manner. Similarly, the reciprocal of a positive infinitesimal is greater than any positive real so it is infinitely large.

So does this force you to accept infinitesimals? No. It all depends on what axioms your mathematics are based. The existence of a non principal ultrafilter depends (if I remember correctly) on the axiom of choice. If you choose not to accept this axiom, it is perfectly reasonable. But then you also accept the consequences of this, one being that the possibility to compare transfinite cardinalities disappears so you can't say that there are more reals than integers. But even if I talk with an intuitionist, he will not say that infinitesimals are ill defined. He will say that their existence depends on an axiom he does not have.

I am not certain, but I think there are some constructions which yield infinitesimals which do not use the axiom of choice.

Also, Nelson's extra axioms do, in a way, assume axiomatically the existence of infinitesimals. But he then shows that these axioms add no contradictions to the theory given by the former set of axioms. It all goes down to axioms and consistency proofs.

as for the remark on Robinson, I resent it as being racist. — Odonovanr 13:03, 25 June 2007 (UTC)[reply]

Apologies to 70.120, but modern mathematics is no way based on meaning. From Mathematics: "Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while most systems of axioms are derived from our perceptions and experiments, they are not dependent on them."

Secondly, at no point did I say that Robinson's work was beyond refutation. His structures are explicitly non-Archimedian, which might give you difficulties were you to try to make your argument rigorous. If you can disprove his infintesimals, why not publish? In a peer reviewed journal of course; Wikipedia accepts no original research.

For Odonovanr: Infintesimals can lead to constructions of non-Lebesgue measurable subsets of [0,1] and free ultrafilters on N, so even if you don't have AC you've still got all the nasty nonconstructive stuff that leads people to complain about AC in the first place. Endomorphic 22:52, 25 June 2007 (UTC)[reply]

How can you make a statement such as "..mathematics is no way based on meaning." ? Of course it is based on meaning. Mathematics would not exist were it not for definitions. Definitions are only definitions when these have meaning. You can't divorce the two. As for disproving Robinson's claims about non-standard numbers, it's easy: I start with his definitions. 70.120.182.243 00:18, 27 June 2007 (UTC)[reply]

How? Like this. From Mathematics: "Mathematical concepts and theorems need not correspond to anything in the physical world."

Look at the axioms of set theory (ZFC by default). They don't tell you what a set actually is, yet all mathematics is built upon these foundations. That's the point: you and I can disagree on what a set *means*, and yet share meaningful analysis since we both agree on which definitions to use. The drive for axiomatic rigor in foundational mathematics was *exactly* a division between definitions and meaning. That's why you have models seperate to your axioms and theorems.

A lot of this "infintesimals" content involves swapping in and out the difinitions you're using. If you're doing Smooth infinitesimal analysis, you don't have the Law of excluded middle. If you're playing with Robinson's stuff, then you don't get to use the Archimedean property. 70.120 previously said "It's not hard to show that the only quantity/number that satisfies this [infintesimal] condition is zero" - true, but only if you can use the Archimedean property or an equivalent. Thing is, you can't: Robinson's definitions are *explicitly* non-Archimedean.

Finally, x is a standard number if it satisfies x > 1/n for some natural number n, and non-standard if x < 1/n holds for all natural numbers n. It's equivalent to the previous definition, without circularity or ambiguity. You need the Archimedean property to prove that ' x is non-standard' implies ' x=0'. Endomorphic 03:15, 27 June 2007 (UTC)[reply]

The axioms of set theory did not exist when mathematics was created. Zermelo and Frankel were not born when the foundations of mathematics were laid. There are a lot of problems with set theory just as there are problems with real analysis and especially with so called non-standard analysis. For each step forward it seems a hundred steps backward are taken. Mathematics was built many centuries before the concept of set was conceived. To say that axiomatic rigour is a division between definitions and meanings is like saying a 'division between meanings and meanings'. It makes no sense. You claim "Finally, x is a standard number if it satisfies x > 1/n for some natural number n, and non-standard if x < 1/n holds for all natural numbers n." Well, this is new. The article does not state this. Besides, the fact that a number is non-archimedean does not necessarily make it non-standard. The previous statement defines only which numbers are archimedean. How do you define a non-standard number? You cannot simply negate the statement regarding the archimedean definition. 70.120.182.243 23:17, 27 June 2007 (UTC)[reply]

*sigh*

Your first three lines *highlight* the divide between mathematical definitions and reality. Axiomatic mathematics studies the things which Frankel invented, which aren't intrinsically connected to reality, as we've been counting oranges since before Frankel was born.

The x<1/n construction is indeed mentioned in the article.

From the *introduction* of Archimedean property: "Roughly speaking, [being Archimedean] is the property of having no infinitely large or infinitely small elements (i.e. no nontrivial infinitesimals)" and "An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is called Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is called non-Archimedean".

In summary: non-standard, infintesimal, and non-Archimedean all go hand in hand. Endomorphic 23:43, 1 July 2007 (UTC)[reply]

Sigh? I am not interested in what Frankel did or did not invent. 'Axiomatic mathematics' - what do you mean? The construction x < 1/n is

indeed mentioned in the article but not in the same context as you imply it is mentioned. It is used to define Archimedean numbers, not non-standard numbers. Saying that anything else is non-standard is meaningless. You wrote: "Roughly speaking, [being Archimedean] is the property of having no infinitely large or infinitely small elements." Well, again, what are infinitely large or infinitely small elements? You have not defined either of these. The Archimedean property does not define these as you think. It defines only real numbers. You write: "In summary: non-standard, infintesimal, and non-Archimedean all go hand in hand." Really? What is an infinitesimal and how is it different from non-Archimedean? Look, I don't mean to be rude but if you don't know what you are talking about, you should at least have the decency to be honest about it. I asked Michael hardy to give me a definition over a month ago - he has yet to respond. The reason is simple: He will make a bigger fool of himself than he already has. Finally, check your english before you post. 70.120.182.243 02:22, 2 July 2007 (UTC)[reply]

If you want the quoted statements explained, perhaps you could *read* the article they were directly copied from (Archimedean property). Also, "Archimedean" is a property which applies to fields or number systems as a whole; a single number can't be "Archimedean" by itself.

The use of x<1/n in the Infintesimals article is not taken out of context; it follows "This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them" and is the formulation used when showing that infintesimals can be formally justified.

I've already given you a definition which seperates non-standard numbers from all others without circularity or ambiguity, referring only to natural numbers. A better version is "A positive number x is non-standard (aka an infintesimal) if and only if x < 1/n for every natural number n" because -1 is certainly standard.

Finally, check your ad hominems before you post. Endomorphic 05:09, 2 July 2007 (UTC)[reply]

It is 'correct' to say that a number is Archimedean - this means it belongs to

an archimedean field. The article states: "In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them." Do you call this in context? Since when was the compactness theorem the authority for proving infinitesimal existence? This theorem is nonsense. And yes, it is taken out of context. So you have not provided a definition; only a regurgitation of the nonsense propagated by Michael Hardy and others on the Wiki team. "Ad hominem" is a favourite Wiki phrase used by sysops and administrators. You ought to look up its meaning before you use it. Look, I don't want to have an endless dialogue with you - either answer the questions with solid, appropriate definitions or proofs or let someone else try. 70.120.182.243 14:12, 2 July 2007 (UTC)[reply]

Okay, so lets get this straight. You claim it's out of context to take a definition of infintesimals from a theorem about the existence of infintesimals? Just because you don't like the theorem?

Look, I hate to be blunt, but there's no indication you're so much as reading the sentances surrounding those I quote, nor reading articles I link to, nor instilling any energy to comprehend technical statements made by myself (or Micheal Hardy). I'm not going to bother responding until you make some kind of effort; I'm not here to motivate you to learn set theory. Endomorphic 21:17, 2 July 2007 (UTC)[reply]

Just a quick reminder to all here that Wikipedia is not a discussion forum. Nor is it an online university. If one editor doesn't understand or accept some results in mathematics this is the wrong place to educate him or her. An encyclopedia article can never be written in a way that replaces a university course, let alone several years of education. I think we all should keep this in mind and simply put an end to the current discussion. iNic 23:26, 2 July 2007 (UTC)[reply]

agreed with iNic. Lets concentrate on how to make the article better, whether others understand it or not, whether they like it or not. But it could interesting if someone started an article about mathematical existence. Odonovanr 10:18, 3 July 2007 (UTC)[reply]

Endomorphic: Of course it is out of context - not because I don't like the theorem but because it has nothing to do with infinitesimals. The compactness theorem is about propositional logic. You should study it carefully. iNic: I am a mathematician and if any one needs education, it would probably be you. Who is saying an article should replace a university course? I am asking you to define an infinitesimal. So far, none of you have been able to provide a definition that makes any sense at all. Someone with a BS or PHd in mathematics will not understand this article. Why? It provides no sensible definition for an infinitesimal. What do you hope to accomplish? Infinitesimals do not exist. The concept is a contradiction in itself. Non-standard analysis is a bunch of baloney because it is based on an ill-defined definition, namely the infinitesimal. Newton would be having a good laugh at the lot of you. He knew more than you because he knew that he did not know what he was talking about when he used this term. 70.120.182.243 22:28, 3 July 2007 (UTC)[reply]

Here's a definition for you: "Let F be an ordered field. A (nonzero) element e of F is infinitesimal if, for every positive integer n, we have ${\displaystyle -n^{-1}<e<n^{-1}}$. If we are regarding F as a field extension of the rational numbers, we may call e an infinitesimal number." Note that this definition does not guarantee that any interesting field extensions of the rational numbers actually have infinitesimals: that is a theorem of Robinson. --Ian Maxwell (talk) 21:45, 9 October 2009 (UTC)[reply]

Infinitesimals were known to exist in some ordered fields before Robinson came along. But Robinson gave us the transfer principle, the concept of concurrence, and the concept of internal and external objects. Michael Hardy (talk) 23:22, 9 October 2009 (UTC)[reply]

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)[reply]

You are making an ancient mistake. The only numbers we "see" are the natural numbers...starting not at zero but at one. The Romans didn't believe in zero. Europeans of the Middle Ages had no clear idea of negative numbers, perhaps because loaning money for interest was forbidden and most people lived in a cashless agrarian society. The Romans and the mediaeval Europeans would have made your argument: that the number zero or negative numbers were unreal.

The question is, are you for real? You see, there probably was a time when primitive man had no concept of number whatsoever, perhaps in a primitive communist society. Then, there may have been a long time in which people only counted up to a small integer to keep track of sheep and wives.

We can imagine a primitive guy proposing bigger integers such as "one hundred sheep" to the other guys. We can imagine a guy like you saying to the primitive Einstein, "what, you stupid, or what"?

Math progresses when we use our imagination to extend our concepts; oftentimes, a use for the extension is only found after the extension: for example, complex numbers were introduced to physics after they were invented in math. Guys like you retard progress when you claim that the mathematician should not be allowed to ask what would happen if we extended a number system and harass real mathematicians. —Preceding unsigned comment added by 202.82.33.202 (talk) 05:55, 26 April 2008 (UTC)[reply]

What surprises me is that people talk about what they don't know. Nonstandard analysis is a rigorous part of modern mathematics. You may choose not to use it but it still is there. Read Robinson's papers, Keisler's articles or the books by Luxemburg or Goldblatt. There is "evidence" that infinitesimals exist because they have been formalised. I don't see the contradictions in the article: read carefully, they do NOT contradict themselves. You don't like it? fine. But you cannot dictate that it is not maths, that it should disappear. An encyclopedia is about knowledge. Nonstandard analysis is part of it. The philosophical aspects of infinitesimals and technical considerations are also part of what can be in an encyclopedia. Robinson proves (yes: a proof) that the real numbers can be extended to a set containing infinitesimals. Nelson proves (again: proof) that the set of axioms can be extended so as to allow for (nonstandard) infinitesimals in the set of real numbers.

This is mathematics, subject to proof not opinion.

But maybe the article is not sufficiently clear yet. Positive remarks would help.

Odonovanr 09:08, 8 May 2007 (UTC)[reply]

Robinson's proofs are not accepted by everyone. Many mathematicians today still reject his ideas. You got one thing right: an encyclopedia is about knowledge. This article is pure speculation and opinion, not proof or mathematics. 24.167.4.177 14:32, 8 May 2007 (UTC)[reply]

A proof (if it is a proof) is either right or wrong. It is possible to follow Robinson's proof step by step. Goldblatt gives a simpler proof which is not exactly the same but points to the same hyperreals. If someone does not accept a proof there can be only two reasons: the proof is wrong, then the error must be shown and it is no longer a proof until (and if) it is "mended". The other reason is that the proof is incomplete so again it is not a proof until (and if) it is completed. Apart from this, there is no other grounds to accept or reject a proof. If someone can show an error or a hole in Robinson's work, and Nelson's and Maltsev's, then let it be shown for all to see. Being a scientist, if there is an error I will accept it and stop writing about infinitesimals. (But many mathematicians don't like the ideas of Robinson so if they had found errors, no doubt they would have exposed them.)Odonovanr 13:30, 9 May 2007 (UTC)[reply]

Could the distance between the graph of y = 1/x and the x-axis (or y-axis, if you prefer) be used as an example of an infinitesimal quantity? This distance, if it is defined at all, must be smaller than any positive real number and yet cannot be zero since the graph does not intersect the axis. Just thinking that an example (apart from calculus) would be nice... - dcljr (talk) 7 July 2005 06:59 (UTC)

Or how about the distance between two line segments obtained by removing a single point from a given line segment? Is there a problem with using the term "distance" for either of these examples? - dcljr (talk) 05:49, 20 December 2005 (UTC)[reply]

Indeed, that first example can be considered an infinitesimal. It's a hyperreal number, to be specific. As for the second... well, that's essentially "the number after 0", which can probably be formalized somehow. Neither one is actually a real number--the reals have no infinitesimals, unless you consider 0 to be infinitesimal--but you can call them numbers nonetheless. --Ihope127 18:55, 11 March 2007 (UTC)[reply]

"This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of the excluded middle--i.e., NOT (a ≠ b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x ² = 0 is true, but x ≠ 0 can also be true at the same time."

If nilpotent infinitesimals deny the law of excluded middle, then so do negative numbers. -1 can be defined as "a number x where x ² = 1 is true, but x ≠ 1 can also be true at the same time." How are nilpotent infinitesimals any different? --Slobad 22:45, 21 December 2005 (UTC)[reply]

Good point here about negative numbers.

The comment about excluded middle is about another approach to infinitesimals. Maybe one thing that appears when studying different flavours of nonstandard analysis is that "infinity" is not a very well defined concept. This has led to different formalisations which are not all compatible. The nilpotent infinitesimals come from a theory by J.L. Bell. The more or less intuitive image given above to explain how one can understand that there is space between real points on the geometric line describes the Robinson hyperreals, so the debate about excluded middle, though interesting per se, is slightly out of place. 85.1.144.180 07:51, 23 April 2006 (UTC)[reply]

No, it is not a good point. If x ² = 0, but x is not zero, then, as a nonzero element of a field, it is invertible. Hence, multiplying by its inverse 1/x, we obtain x = (1/x) 0 = 0, a contradiction. This is why nilpotent infinitesimal real numbers are incompatible with classical logic. On the other hand applying a similar argument when x² = 1 yields only x = 1/x, for which 1 and -1 are both perfectly valid solutions. Geometry guy 22:26, 11 March 2007 (UTC)[reply]

My problem with smooth infinitesimal analysis is that negating the law of the excluded middle (LEM) precludes the powerful method of proof by contradiction (reductio ad absurdum). If the LEM is negated only in certain situations, then it needs to be made clear when it is safe to do so. Moreover, nilsquare infinitesimals are not needed when one can construct "working models" of infinitesimals using polynomial ratios or Robinson's hyperreals. The existence of a model of infinitesimals within the real number system demonstrates that they are as consistent as the real number system itself.Alan R. Fisher 00:35, 25 February 2007 (UTC)[reply]

Some people (e.g. constructivists) argue that this is a strength, in that the method of proof by contradiction is confusing and should only be used when it is really needed. For example, I recently had to prove that some vectors were linearly independent. For my first attempt, I assumed that there was a nontrivial linear dependence relation between the vectors, and derived a contradiction. Then I rewrote it: I assumed instead that there was a linear dependence relation and showed that it must be trivial (all coefficients zero). The second proof was much clearer (and would also be valid in the presence of nilpotent infinitesimals). Geometry guy 22:26, 11 March 2007 (UTC)[reply]

In dynamics, when a reference frame rotates we need to find the derivative of unit vectors ${\displaystyle {\vec {u}}_{r},{\vec {u}}_{\theta }}$.

In finding the derivative we arrive at ${\displaystyle \lim _{\Delta \theta \to 0}\Delta {\vec {u}}_{r}=2\lim _{\Delta \theta \to 0}[sin(\theta /2)({\vec {u}}_{\theta }+\Delta {\vec {u}}_{\theta })]}$.

For small ${\displaystyle \Delta \theta }$ we can write ${\displaystyle d{\vec {u}}_{r}=d\theta ({\vec {u}}_{\theta }+d{\vec {u}}_{\theta })]}$.

As it is mentioned in the article, it is not rigorous to write ${\displaystyle {\vec {u}}_{\theta }+d{\vec {u}}_{\theta }={\vec {u}}_{\theta }}$.

The articles about limit, derivative and differential do not provide insight.

If we expand, we can write ${\displaystyle d{\vec {u}}_{r}={\vec {u}}_{\theta }d\theta +d\theta d{\vec {u}}_{\theta }}$.

It is a common practice (and I think an erroneous one) among engineers to set ${\displaystyle d\theta d{\vec {u}}_{\theta }=0}$ since the multiplication of two infinitesimal numbers is assumed to be zero. Any ideas? Skorkmaz 09:15, 30 October 2006 (UTC)[reply]

The rigorous way of doing this in nonstandard analysis is to define derivatives in terms of the standard part of the quotient dy/dx. The article now has an example of this. The dx^2 in the numerator gets divided by dx, and then when we take the standard part of the result, the dx term goes away.--76.167.77.165 (talk) 01:36, 8 March 2009 (UTC)[reply]

Is it possible to proove that multiplication of two infinitesimal numbers is exactly zero by the proof given below?

• Let's take an infinitesimal number dx
• Assumption: Assume that dx is an infinitesimal number such that there is no any other infinitesimal number between ${\displaystyle 0}$ and ${\displaystyle dx}$
• Obviously ${\displaystyle 0<dx<1=>dx^{2}<dx}$
• If dx was a Real Number we could write ${\displaystyle 0<dx^{2}<dx}$ but here we can not. If there is a number ${\displaystyle dx^{2}}$ such that ${\displaystyle dx^{2}>0}$ this is in conflict with our assumption.
• So if there is a number smaller than dx that number must be exactly zero. This is the end of the proof.

Fgeridonmez 15:24, 30 October 2006 (UTC)[reply]

Your second assumption is impossible with hyperreal numbers, but I can't rule it out if we're talking about other sorts of systems. Michael Hardy 18:04, 30 October 2006 (UTC)[reply]

In Nelson's IST it is also impossible but Bell uses a category definition in which there are nilsquares, nonzero numbers such that their square is zero (as mentioned above).

Sure, but Bell also has a bunch of numbers that you can't prove aren't zero. And a bunch of other *different* numbers that you can't tell apart from zero. Bell also discards the law of excluded middle, so the 4th point doesn't work in Bell's context. Lastly, as long as people are using infantesimals called dx, I'll point out that dydx (and higher order terms) see a lot of use, and that dydx = dy * dx wouldn't be all that usefull if it were always zero. Endomorphic 23:04, 18 March 2007 (UTC)[reply]

I've rewritten the article originally known as Differential (calculus) into an article Differential (infinitesimal). There is some overlap with the article here, although the purpose of the Differential (infinitesimal) article is to give an overview of the historical meaning of differentials such as dx, as well as ways to the notion rigorous. Comments would be welcome. Also if anyone wants to propose some sort of merger or any other way to pool ideas related to infinitesimals, please discuss below. Geometry guy 22:32, 11 March 2007 (UTC)[reply]

I think that's a bad idea. Infintesimals and differentials are confused for each other enough already. Infintesimals are numbers smaller than any positive real number; they are elements that might live within a field structure extending the reals. If t is an infintesimal, you can always say t < 3 and be sure of the truth. Differentials in integration and differential equations are very different creatures; within the context of dy/dx = 3x, the equation dx < 3 is not sensible at all. Endomorphic 23:28, 18 March 2007 (UTC)[reply]

Why is it not sensible? One can have dy/dx = 3x and of course dy and dx are infinitesimals, and so of course |dy| and |dx| < 1. Michael Hardy 23:32, 18 March 2007 (UTC)[reply]

It's not sensible because dx and dy are either measures, differential 1-forms, linear mappings, or whatnot. They're not something you can intuitively extend the reals to include. You can't take the wedge product of 2 and 3, for instance. There are contexts for numbers like 3, such as being an equivalence class of certain Cauchy sequences. There are contexts for differential forms and measures, such as dy = F dx where F = dy/dx. Infintesimals (in the sense of number smaller than all other strictly positive reals) work in the first context, but not in the second. Endomorphic 00:12, 19 March 2007 (UTC)[reply]

In some contexts they may by differential 1-forms or the like. But Leibniz thought of dy and dx as being infinitely small incrememnts of y and x, so that dy/dx is the ratio of increments. For integrals, if you think of dx as an infinitely small increment of x, then, for example, if f(x) is measured in meters per second, and x (and so also dx) in seconds, then f(x) dx is in meters, and is an infinitely small distance, and the integral is the sum of infinitely many such infinitely small distances. So we're talking about things that you can "intuitively extend the reals to include". Of course there are other ways of looking at things, in which dx is for example a measure; no one denies that. Michael Hardy 00:53, 19 March 2007 (UTC)[reply]

The infintesimals used by Leibnitz and Newton to justify calculus were found to be lacking rigor, hence the developments of forms and measures and suchlike. Calculus from infintesimals is now only of historical interest and as an educational aid prior to limits, but that's all. The idea of derivatives being ratios of infintesimals is like goldfish heaven; a comforting story you tell your kids, not something considered fact. You can differentiate with limits of reals, with linear operators on function spaces, or with connections on tangent bundles, but there aren't any infintesimals anywhere. Endomorphic 01:59, 19 March 2007 (UTC)[reply]

Let's see... first you use the word "intuitively", but then you want logical rigor. I disagree that intuitive unrigorous ideas should be used ONLY until rigorous methods are learned. I can write an argument using infinitesimals in an intuitive way, knowing that later if I prepare something for publication, I may need to write out a rigorous rather than intuitive version of the argument, but in the early stage I'm more interested in where the argument will ultimately lead than in what it looks like after I've dotted the last "i". Rigor has its place, and it's an important place, and rigor is important. But rigor isn't everything. Michael Hardy 02:16, 19 March 2007 (UTC)[reply]

I didn't want to open with a rigorous discussion of why dy and dx aren't infintesimals and a bunch differential geometry links :)

Your points are good, but the article gives a different impression. It doesn't do nearly enough to point out that modern mathematics uses rather different methods. No mention of measures. No forms. No little-o notation. One link to limits. No mention that epsilon and dx often denote things which are *not* infintesimals, but are easily mistaken for them. Nothing to say that infintesimals are now really only used for brief informal back-of-the-envolope type calculations. The present article makes it sound like infintesimals had a few hiccups with Berkely, but are otherwise cool. They're not. Endomorphic 04:14, 19 March 2007 (UTC)[reply]

Oh dear, I seem to have opened a can of worms here. For the record, I agree with Endomorphic that the articles should not be merged, but I also agree with Michael Hardy that differentials and infinitesimals are closely related: the unrigorous arguments are useful intuitively, and there are many ways to make them rigorous. However, I hope you won't be offended if (as a relative novice here), I issue a reminder that this page is for discussing improvements to the article, rather than the pros and cons of infinitesimals. Though, if you continue the discussion on your user pages, let me know, and I'll add them to my watch list, since I find the discussion rather interesting. Geometry guy 19:45, 19 March 2007 (UTC)[reply]

I really think there is a lot of confusion here. If this is supposed to be an article about infinitesimals, then we define what they are, where they historically come from and maybe also we explain why at some time in the history of mathematics is was found safe to avoid them. Still, since Robinson in the sixties, it has been shown that infinitesimals can be made rigourous, so anybody saying that they are not should return to his/her books. Whether we choose to use them or not is another question (just as you can choose to use complex numbers or not). Then Nelson extended the syntax and showed that it is possible to rigourously define infinitesimals within the real numbers. There are standard real numbers and nonstandard real numbers. Infinitesimals are nonstandard real numbers less in modulus than any positive standard real number. You can like it or not: they are rigourously defined. Recent work by Hrbacek adapts Nelson's approach and makes it easier to use.

In all approaches, dy/dx is a quotient. The differentials are defined and all the intuitive ideas get rigourous definitions.Odonovanr 09:08, 8 May 2007 (UTC)[reply]

The infinitesimal is not rigourously defined - it is not even well-defined. IN fact, it is an ill-defined concept that makes no sense whatsoever. You need to go back to your books but more than this, you may need to start thinking for yourself. 70.120.182.243 19:24, 30 April 2007 (UTC)[reply]

Sorry, wrong and completely wrong. I do think for myself and books, yes, read them too. (who are you and what are you trying to prove? Do you read mathematics? are you profficient?) Odonovanr 09:08, 8 May 2007 (UTC)[reply]

Why don't you learn to spell first? (Proficient has one eff) I am sorry you are taking my comments so personally. I am not attacking you or anyone. I am commenting on the article. So please keep your opinion to yourself also. Thanks. 24.167.4.177 14:38, 8 May 2007 (UTC)[reply]

History of Infinitesimal: 1) Archimedes did not use infinitesimals because: a) He did not have an idea what these might be b) The concept of 'infinitesimal' had not been born till Newton.

2) Newton and Leibniz never used infinitesimals because: a) An infinitesimal had never been defined b) The typical argument you provide as an example is in fact the calculation of an average between two points. The problem of the day in Newton's time was to find a general method for calculating the average at a point. Archimedes and the ancients knew about finding an average at a point. For your example f(x) = x^2 , the average function becomes: A(x) = 2x + h. One can use this to calculate the average (or rate or slope) between any two points (x,f(x)) and (x+h, f(x+h)). c) There is no use of limits or real analysis in the example. d) The above result is both appealing and mathematically rigorous. Newton's foolish mistake was to coin the term infinitesimal. e) Karl Weierstrass complicated matters by introducing the notion of limit. His claim to fame is: |f(x)-L| < epsilon <=> 0<|x-c|<delta This says that as values of x approach c, the values of f(x) approach some limiting value L. Moreover, it confirms that epsilon and delta approach zero. Finally if f(x) is defined at x=c, then the function is also continuous at c. Poor students waste countless hours learning how to find a formula relating epsilon and delta and this is subsequently treated as proof that the limit exists. However, the methods involved are axiomatic – they require no proof.

Modern uses of infinitesimals: a) There are no modern uses of infinitesimals. What is an infinitesimal? b) “Infinitesimal is a relative concept.” - You got this one right! The last sentence of this section is completely nonsense: The crucial point? Either a number is an infinitesimal or it is not.

The path to formalization: This section is such a bowel movement, it is hardly worth commenting on. It is a good example of analysis gone horribly wrong. The logic in this paragraph is riddled with errors. It states the axioms can be extended. This is equivalent to stating that something which is self-evident can be extended. But the kind of reasoning applied here requires solid foundational concepts upon which an extension can be made. How can you attach an ill-defined concept to the axioms governing the real number system? The reals do not have holes. The reals are complete. You would of necessity require a completely new set of axioms – this is the crucial point!! To say that an infinitesimal is a nonstandard real number which is less, in absolute value than any positive standard real number is to say that it is zero. You cannot just define a standard and nonstandard number (that do not exist or even begin to make sense) into the real number system. In fact by defining a nonstandard number, you have already created a concept where an object is defined in terms of itself!!

Nonstandard number = real part + nonstandard part (or infinitesimal)

If |x| < a for any real positive number a, then x = 0. According to your logic:

   Small nonstandard number = 0 + infinitesimal 

You are trying to build the small nonstandard numbers from a subset of the very small real numbers and the large nonstandard numbers from infinity which is not a number. When one talks about very small numbers, one must remember that just because they cannot be represented finitely, it does not mean these are not finite quantities. They are finite! Supposing you do create nonstandard numbers, where or when do these become (using your jargon) “infinitesimal with respect to real numbers”?

The article is sheer speculation and nonsense – much like most of the other articles on Wikipedia. Unless the idiots that run this site are not fired, you will continue to be ridiculed for your gross stupidity. 70.120.182.243 13:12, 9 May 2007 (UTC)[reply]

This comment is anti-mathematical nonsense. The concept of infinitesimals in non-standard analysis or the hyperreals is mathematically well-defined. It's only when you mix the "common" (Leibniz) and the formally defined version that you may get into trouble with non-(real)-analytic functions. —The preceding unsigned comment was added by Arthur Rubin (talk • contribs) 16:55, 9 May 2007 (UTC).[reply]

The above comment from a PHd who could not explain what an infinitesimal because it is apparently too simple a concept for him. Look, although I don't agree with Odonovanr, I think his idea to wait some time until this article is completely revised is a good one. I will be watching out for the revisions. Remember, the article will remain worthless unless you can logically define the infinitesimal. A definition is only logical if it is also axiomatic. This means you cannot just make up nonsense definitions as you please - a definition must be self-evident. 70.120.182.243 17:16, 9 May 2007 (UTC)[reply]

Infinitesimals are defined differently in the contexts of non-standard analysis and in the hyperreals. The simplest definition that I've used is that if X ⊂ Y are real-closed fields, then an element a of Y is infinitesimal over X if

${\displaystyle \left(\forall b\in \mathbf {X} \right)\left(0<b\implies \left|a\right|<b\right)}$

X is normally taken to be the reals, but the definition makes sense over any real-closed field.

I'm afraid I don't have any of my non-standard analysis references with me, so I can't confirm that reference. — Arthur Rubin | (talk) 17:55, 9 May 2007 (UTC)[reply]

For every b that is an element of X, then the following holds: if b>0 then |k| < b where k is an infinitesimal. This definition is nonsense - it says nothing different to what the article already states.Sorry to be blunt, but your response is simply unacceptable. What is an infinitesimal? Your definition tells me nothing except that for every element of X > 0, there is an element K from an undefined infinitesimal set Y. Your statement claims that Y is a set of infinitesimals and assumes that all the elements of Y are smaller than any element of X which is greater than 0. In other words, the definition itself is cranky and not even worthy of consideration. You will have to do a lot better than this to convince me. 70.120.182.243 22:10, 9 May 2007 (UTC)[reply]

You said there wasn't a definition. I provided one, even though the precise use depends on which formulation of non-standard analysis one works with. The proof that such may exist requires use of advanced logical theorems (compactness theorem) or of advanced set-theoretical techniques (ultraproducts), which are probably beyond the scope of the article. — Arthur Rubin | (talk) 22:38, 9 May 2007 (UTC)[reply]

These advanced logical theorems you refer to are a load of BS. There are mathematicians who do not accept the axiom of choice and also reject Godel's completeness theorem which is used to prove the compactness theorem - with good reason. 70.120.182.243 23:43, 9 May 2007 (UTC)[reply]

Well — there's the filtered product (not ultraproduct) formalism, and the formal power series field R(((ε))), if you want specific examples of real-closed fields in which there are infinitesimals over R. (The third level of parenthesis indicates that the exponents are rational with a common denominator, rather than necessarily being integers. As the theory of real-closed fields is complete and decidable, it doesn't matter which infinitesimal you use.

A "typical" element of R(((ε))) is

${\displaystyle \sum _{n=-k}^{\infty }x_{n}\epsilon ^{\frac {n}{m}}}$,

where k is an arbitrary integer, m is an arbitrary positive integer, and x_n are arbitrary reals.

— Arthur Rubin | (talk) 23:59, 9 May 2007 (UTC)[reply]

You provided an example that assumes an infinitesimal already exists without actually defining it. The typical element you provided as an example is real, not infinitesimal. 70.120.182.243 14:26, 10 May 2007 (UTC)[reply]

False. R(((ε))) is a real-closed field which has infinitesimals in it, as the formal symbol ε is infinitesimal. The filtered product construction is a relative consistency proof, which is probably beyond you, even if it doesn't require the axiom of choice. — Arthur Rubin | (talk) 18:23, 10 May 2007 (UTC)[reply]

I was not planning to make any further posts. However, an unrebuked fool remains in his folly. I am going to reveal several problems with what you have written so that all out there can judge for themselves. "R(((e))) is a real-closed field which has infinitesimals in it, as the formal symbol e is infinitesimal."

1). To state that R(((e))) is a real-closed field confirms that you are referring to a construction of the small nonstandard numbers from a subset of the very small real numbers. See my earlier post to understand why such a filter is nonsense in criticism of article section. 2). You assume that R(((e))) contains infinitesimals - once again you do not define these. You simple assume they exist. 3). The definition for a filter product in set theory applies to reals which exist, as opposed to infinitesimals for which you cannot provide any proof or even one example. Anyone with a just a little sense will after reading your last paragraph realize you are very confused. This is my final response to you. 70.120.182.243 16:16, 11 May 2007 (UTC)[reply]

I was not planning any further response either but I find the tone of our contradictor utterly unacceptable: he or she is rude and insulting, denying everything but giving no reference nor evidence about his or her claims. One aspect which could be of interest, but for a separate article I think, is what does "exist" mean in mathematics.

Stil, I plan to collaborate to extend the article. Odonovanr 16:21, 12 May 2007 (UTC)[reply]

Existence is shown by the construction of a mathematical object. Reals exist because there is a construction for them. A definition is insufficient for a logical construction. One must be able to derive a usable object from such a defintion. Perhaps you can wax philosophical at this point but the problem is evident: you cannot simply create a definition that is nonsense. Regarding the accusations in the previous paragraph, I suggest that whoever posted these take time to reflect upon himself. If I am contradicting myself, I wonder what it says about all the other contributors who don't have a clue what they are talking about? 70.120.182.243 17:57, 13 May 2007 (UTC)[reply]

R(((ε))) is an explicit formal construction, not requiring any advanced mathematics. It is clearly a real-closed field, and ε is clearly infinitesimal. I don't know what the anon wants. — Arthur Rubin | (talk) 22:46, 13 May 2007 (UTC)[reply]

I think it would be nice if we had a separate article at Wikipedia devoted to the tricky concept of mathematical existence. From ancient times, when mathematicians almost had a physical interpretation of mathematical existence, via the unexpected difficulties encountered when defining the existence of a function, to the current very abstract ideas based on model theory, set theory and logic. This page could also mention the still ongoing controversies among mathematicians about this (intuitionism vs. classical mathematics). Non-mathematicians quite naturally have a very concrete interpretation of mathematicla existence, as they are unaware of the long development of these ideas. I suspect that a difference in views of what we mean with mathematical existence are responsible for the current discussion. iNic 02:58, 15 May 2007 (UTC)[reply]

The difference of views is not only due to interpretation of mathematical existence; this article makes statements that are incorrect. The 'facts' are not true and the logic used leaves much to be desired. Archimedes knew nothing about 'infinitesimals' - how could he have? No one today has any idea what these are because they do not exist. To speak of an infinitesimal is stupid enough but to talk of the plural is retarded. Suppose that the numbers on either side of zero are infinitesimal, what other numbers are infinitesimal? Where does an infinitesimal end and a real number begin? The numbers on either side of of zero are real, no matter how small these are. The fact that we cannot represent these numbers using the decimal or any other system does not make them infinitesimal. We still cannot represent most real numbers... 70.120.182.243 14:07, 15 May 2007 (UTC)[reply]

It's clear that in the real real numbers, there are no infinitesimals. So, all of these extended fields and infinitesimals are constructed, in some sense. However, the construction I mentioned above doesn't require advanced mathematics to verify. — Arthur Rubin | (talk) 21:15, 15 May 2007 (UTC)[reply]

I'm not exactly an expert in the field but can it not simply be defined as the reciprocal of infinity? Prophile 09:06, 24 May 2007 (UTC)[reply]

I don't think so. In context (perhaps ordered field theory or the theory of real closed fields, rather than nonstandard analysis) it may be the reciprocal of an infinite number, but "infinity" is not quite the same. — Arthur Rubin | (talk) 18:15, 24 May 2007 (UTC)[reply]

Prophile: Infinity is 'not a number' so you cannot form its reciprocal in context or otherwise. Robinson's book and his ideas on nonstandard analysis is not accepted by many mathematics professors. You can read its contents online on the Google website. 70.120.182.243 19:12, 24 May 2007 (UTC)[reply]

"Robinson's book and his ideas on nonstandard analysis is not accepted by many mathematics professors. " is probably false. In any case, it's accepted (by transfer theorems) as a method of generating proofs of "standard" results in real analysis by all published mathematicians with whom I'm familiar. If the anon is willing to name a mathematician who doesn't believe in Robinson's methods, I'm willing to attempt to determine if he/she is sufficiently notable to be worthy of comment in this article or in nonstandard analysis. — Arthur Rubin | (talk) 21:01, 24 May 2007 (UTC)[reply]

Prophile: No one who is notable will bother responding to comments that are mathematically absurd such as Rubin's. In fact, if you keep an eye on these discussions, you will notice that they are strictly the views of Wiki sysops/administrators such as Michael hardy (An ex-MIT stats professor) and his cohorts (such as Rubin - a PHd whose dissertation isn't worth the paper it was written on). Hardy makes a comparison of infinitesimals to differentials and is bold enough to state that 'mathematical rigor isn't everything' (sic). Anyone who takes Robinson seriously is a fool. Unfortunately there are a lot of these around. 70.120.182.243 13:54, 25 May 2007 (UTC)[reply]

70.120.182.243, if you want the article to represent your views, you need to cite some verifiable sources. You're simply misinformed. The logical rigor of NSA is not at all controversial.--76.167.77.165 (talk) 02:52, 1 May 2009 (UTC)[reply]

Oh, anon is still around. And still hasn't given even the slightest hint as to why he/she considers infinitesimals to be ill-defined... Odonovanr 11:10, 29 May 2007 (UTC)[reply]

And you call yourself a scientist? It seems to me that in addition to not being able to spell properly, you also have a problem with reading. Try reading my comments. Perhaps the 'hints' will then be evident. 70.120.182.243 21:53, 30 May 2007 (UTC)[reply]

Anon, first off please read WP:NPA. Secondly, there are a lot of good books covering the interesting history of ideas in this area of mathematics you can read to catch up on the current situation. For example the nice collection of articles in the book Real numbers, generalizations of the reals, and theories of continua edited by Philip Ehrlich. I hope you will discover that it is fun to know how these ideas evolved. Best wishes iNic 11:07, 26 June 2007 (UTC)[reply]

I can't realy verify this, but I'm sure I've seen 'infinitessimal' defined as the multiplicative inverse of 'infinity' in some formulation of the extended real number line. Essentially, there is exactly one positive and one negative infinity on this number line, so one can define exactly one positive and one negative infinitessimal as well. Like I said, though, I can't really verify this--I hope somebody else can give you a better clue. Eebster the Great (talk) 02:38, 13 May 2008 (UTC)[reply]

On another note, I wonder what Anon would think of aleph, beth, omega and epsilon numbers, or other transfinite numbers. Surely he must think those are the most ridiculous of all! Eebster the Great (talk) 02:39, 13 May 2008 (UTC)[reply]

In defence of "Anon", mathematicians in the past have doubted the existence of zero, negative numbers, irrationals, imaginary numbers (etc.) so it is not surprising that the existence of infinitesimals is questioned here. Personally, I find the article fascinating, challenging to my imagination, and much less incredible than physicists' multiverse theories about the origin of the universe. I'm off to read Robinson's theories with an open but sceptical mind. Dbfirs 19:02, 28 May 2008 (UTC)[reply]

(later)I'll have to brush up on my set theory before I fully understand the formalities, but *R satisfies my intuitive feel for numbers better than R (e.g. 0.9 recurring being equal to 1.0), and I realise with amazement that I have been happily using these numbers for more than 40 years without realizing exactly what they were! Thanks for the article! Dbfirs 07:30, 29 May 2008 (UTC)[reply]

Can you say that "Archimedes, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals". This statement might give one the impression that Archimedes actually provided a logical rigorous definition of infinitesimals, whereas, as is correctly stated in the rest of Wikipedia, no logically consistent theory of infinitesimals existed until Robinson. 75.61.103.121 (talk) 21:33, 5 September 2009 (UTC)[reply]

Why does the article say "In 1936 Maltsev proved the compactness theorem"??? I thought Gödel proved it in 1930, and that it had been somewhat anticipated by Skolem in the 1920's. I also don't see the connection with infinitesimals. The section is both unclear and appears historically wrong (about the compactness theorem). Can someone fix it? 66.127.52.47 (talk) 01:21, 16 March 2010 (UTC)[reply]

One would need to look into the dates, but the connection is simple: the theorem is used in one of the constructions of the hyperreals. Namely, one enlarges the usual "list" of axioms by adding a countable list of inequalities ε<1/n, and then invokes the compactness theorem to conclude that a model exists. Tkuvho (talk) 10:43, 16 March 2010 (UTC)[reply]

In mathematical logic, Charles Sanders Peirce had interesting writings about infinitesimals. Secondary literature includes the mathematics historians Joseph W. Dauben, Carolyn Eisele, and John L. Bell: I quote from Bell's article (which was published in the Mathematical Intelligencer):

It is of interest to note in this connection Peirce’s awareness, even before Brouwer, of the fact that a faithful account of the truly continuous would involve abandoning the unrestricted applicability of the law of excluded middle. In a note written in 1903, he says:

Now if we are to accept the common idea of continuity...we must either say that a continuous line contains no points...or that the law of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual...but places being mere possibilities without actual existence are not individuals.

The prescience shown by Peirce here is all the more remarkable since in SIA the law of excluded middle does, in a certain sense, apply to individuals.

C.f. the wider discussions of Peirce's mathematics (and mathematical logic) by the mathematical logicians by Hilary Putnam (e.g. in Peirce's "Reasoning and the Logic of Things") and Jaakko Hintikka (e.g. in "Rule of Reason"). See also:

• Peirce on Infinitesimals |Author(s): P. T. Sagal | Source: Transactions of the Charles S. Peirce Society, Vol. 14, No. 2 (Spring, 1978), pp. 132-135 | Published by: Indiana University Press
• The Genesis of the Peircean Continuum |Moore, Matthew E. Transactions of the Charles S. Peirce Society: A Quarterly Journal in American Philosophy, Volume 43, Number 3, Summer 2007, pp. 425-469 (Article) | DOI: 10.1353/csp.2007.0037

(This duplicates a posting at talk page of the WP article "infinitesimal calculus".) Thanks, Kiefer.Wolfowitz (talk) 15:21, 4 July 2010 (UTC)[reply]

Very interesting. Could you elaborate on some of the details of what Sagal said? I only have access to the first page of his article from here. Tkuvho (talk) 15:47, 4 July 2010 (UTC)[reply]

The section lists the following two properties:

1. An ordered field obeys all the usual axioms of the real number system that can be stated in first-order logic, but does not necessarily obey the axiom of completeness. For example, the commutativity axiom x + y = y + x holds.
2. A real closed field has all the first-order properties of the real number system (regardless of whether they are usually taken as axiomatic) for statements involving the basic ordered-field relations +, ×, and ≤. This is a stronger condition than obeying the ordered-field axioms, since some additional first-order properties may be proved using the completeness axiom. For example, every number must have a cube root.

However, the references to completeness seem to be confusing. This should apparently be replaced by the real form of being algebraically closed, namely existence of zero of odd-degree polynomials, namely the defining property of a real closed field. Tkuvho (talk) 18:12, 12 September 2010 (UTC)[reply]

I should like to add a subsection 3.6 on irreal infinitesimals, which were introduced in my 2005 paper 'To Continue with Continuity,' Metaphysica 6, pp. 91-109. They are arguably what most mathematicians were actually talking about in the early calculus (or so I argue in this blogpost, which contains a link to that paper). They are primarily geometrical, since they would exist in actual continua if there were such things (e.g. space, maybe) and if continuity is as I describe it in that paper, i.e. if there are # points in any extension, where # is basically 1/0 (in the more precise sense that 1 is one of the values of 0.#). I am, however, thinking that my view is biased towards irreal infinitesimals, and that it would be much better if someone else wanted to write a small outline of what they are, for a new subsection 3.6. Does anyone want to? Does anyone else object? Username12321 (talk) 11:58, 8 October 2010 (UTC)[reply]

Is Metaphysica a reliable source. And, even so, what does it say? I would lean against inclusion, even if Metaphysica were a reliable source on the history of mathematical concepts, as the summary on your blog does not reflect what any mathematician since the 18th century was talking about. — Arthur Rubin (talk) 15:53, 8 October 2010 (UTC)[reply]

Three points: (1) Colyvan's article you mention is indeed very interesting. Perhaps you could try to summarize it in one of our pages. (2) your remark concerning the representation of numbers by infinite decimals is inaccurate. This is indeed a widely recognized way of defining the reals, going back to Simon Stevin. (3) Today one no longer needs to develop number systems that allow for "x + l = x". The logical paradox you are referring to was resolved by Abraham Robinson in the 1960s by means of the standard part function, see Ghosts of departed quantities. Thanks for your interesting contribution. Tkuvho (talk) 22:18, 9 October 2010 (UTC)[reply]

And thank you... Regarding your third point, Robinson's treatment did not answer the logical problem, which concerned reference (see below); but more importantly, it did not answer the mathematical question. Irreal infinitesimals are different to non-standard infinitesimals. The world may be such that instantiated continua contain the former but not the latter. If so, then our science would be more realistic if it included the former. Regarding your second point, I too take the reals to be infinite decimals... Regarding the preceding comment, please accept my apologies for not already including the following link to Metaphysica, which publishes work by academic philosophers. (It may mean something that after 5 years the mathematics in my paper has not been shown to be incorrect.) And note that the reference question concerns what the mathematicians using the early calculus were talking about. (The view that Euler was referring with his "2" to the modern 2 of mainstream mathematics, whose foundation is ZF set theory, is challenged by me in next month's issue of The Reasoner, incidentally).Username12321 (talk) 09:12, 12 October 2010 (UTC)[reply]

I don't understand what you mean by the "reference question". As for Colyvan, he seems to be arguing that systems containing apparent contradictions should not be rejected outright, and proposes a system where it is possible to have a "local contradiction" without invalidating the entire theory. My reading of Colyvan is that he is not arguing in favor of reconstructing historical theories that were once thought contradictory, but a modern resolution has been found. Your project seems to involve such a reconstruction. How many people are interested in it? Tkuvho (talk) 20:05, 16 October 2010 (UTC)[reply]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I do not agree that 1-0.999... is well-defined. Wikipedia has an article that states 0.999... = 1, therefore in this case the difference is 0, which is not an infinitesimal but a well-defined concept. If you are trying to express the magnitude that succeeds zero, it is still not well-defined. But even if you suppose that 1-0.999... is well-defined, then what would be the next infinitesimal? More precisely, where do infinitesimals begin and most importantly, where do these end? 12.176.152.194 (talk) 03:40, 13 December 2011 (UTC)[reply]

The article you mentioned specifically contains a section 0.999#Infinitesimals explaining how to define 1-"0.999..." in such a way as to make it a well-defined concept. I am interested in your contention that infinitesimals are ill-defined but it has to be backed up by sources before it can be implemented at wiki. Specifically, what kind of framework do you accept as well-defined? Is it ZFC? Tkuvho (talk) 11:57, 13 December 2011 (UTC)[reply]

Well, one must be able to derive an object - either theoretically or physically from any concept in order for it to be well-defined. I am a mathematician and do not accept that an infinitesimal/s is/are well-defined. If I cannot understand the nonsense in this article, I have no doubt that any lay person or beginner or even a graduate of mathematics will be able to understand it. What I am saying is, this article and all other articles that mention infinitesimals are not factual. I put it to you that anyone who reads this does not understand because it makes no sense. Now if you say 1-0.999... is infinitesimal, call it phi. So show me, what can you do with phi? 12.176.152.194 (talk) 17:00, 13 December 2011 (UTC)[reply]

See Special:Contributions/70.120.182.243 for his/her similar comments in 2007. The ones in this talk page are presently in the first 4 sections of the talk page. I agree that his/her contentions there make little sense. I tend to agree that the comments here about 1/inf and 1 - 0.999999... are not particularly helpful, either, but the anon's contentions were refuted in 2007. — Arthur Rubin (talk) 17:36, 13 December 2011 (UTC)[reply]

That my objections "were refuted" is false.

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

I have no intention of having the same discussion with you again. In fact, I think you should draw attention to your biographical article on Wikipedia which is not notable and has survived so many attempts at deletion. Mr. Rubin, you are a poor excuse for a mathematician. 12.176.152.194 (talk) 05:52, 14 December 2011 (UTC)[reply]

You do not recall correctly, and you (if you are the author of the web page you refer to below) have no credibility as a mathematician. I have over a dozen published papers in peer-reviewed journals, and (if you're interested in speculation), my analysis of 3-dimensional unitary algebras over the reals was taken from sci.math and published in a real journal. — Arthur Rubin (talk) 07:14, 14 December 2011 (UTC)[reply]

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

I would like to follow up on the IP's suggestion to denote by phi an infinitesimal of the form 1-"0.999...". One needs one further assumption, namely that phi is an element of an ordered field extending the real numbers. If we are in such a situation, then we can usefully apply phi to develop the calculus. Thus, we can define the deritive of a function f by forming the quotient f(x+phi)-f(x)/phi. This quotient in general will not be a real number, but it will be infinitely close to a real number; that real number is the derivative of f at x. If Leibniz thought that's a good approach to the "differential quotient", perhaps some students may think so, as well. To pursue this a little bit further: you did not respond to my request to clarify the nature of mathematics that you do find "well-defined and factual". Does ZFC meet this criterion, in your opinion? Tkuvho (talk) 18:01, 13 December 2011 (UTC)[reply]

Well I would like to agree with Arthur Rubin, infinitesimals most certainly have a place in wikipedia. As far as 1-0.999..., I would have to agree that this is not a particularly helpful discussion as the example is quite misleading. The whole point as I understand it is that is that there is some ambiguity as to what an ellipses means in this setting, and so the quantity might refer to an infinite family of infinitesimals, but it is not a well defined set of symbols. This is complicated by the fact that in the real numbers ${\displaystyle 1-0.{\bar {9}}=0}$ which is of course a necessarily true in the non-standard setting as well, written in this way there is no confusion with the ellipses. Thenub314 (talk) 19:36, 13 December 2011 (UTC)[reply]

I am familiar with your opinion on this matter. I was hoping to find out the IP's. Tkuvho (talk) 20:10, 13 December 2011 (UTC)[reply]

I had forgotten we had ever discussed this particular issue. Thenub314 (talk) 20:25, 13 December 2011 (UTC)[reply]

Sorry if I misunderstood your position. What is it? Tkuvho (talk) 12:42, 14 December 2011 (UTC)[reply]

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

http://thenewcalculus.weebly.com

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

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

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

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

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

Which cubic are you referring to? Tkuvho (talk) 16:50, 14 December 2011 (UTC)[reply]

You were talking about 3x^2 which is the derivative of x^3. Did I misunderstand what you were saying? Please explain clearly if I did. There are no errors in my New Calculus - I am certain of this. What do you mean by "the function which vanishes for negative x" ?12.176.152.194 (talk) 16:53, 14 December 2011 (UTC)[reply]

The example you treat in your Cauchy text is 3x^2, whose derivative is 6x. What I am proposing is to consider the function which is equal to zero for negative x, and is equal to 3x^2 for nonnegative x. If you draw its graph, you will notice that it is smooth at the origin, in the sense that the tangent line exists there. Therefore one would expect the derivative to exist, as well. Are you implying that this function is not differentiable in your approach? That would be a major drawback. Tkuvho (talk) 17:06, 14 December 2011 (UTC)[reply]

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

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

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

I feel I would comment here that, even within constructive mathematics, the specific function you mention is fine. Though not all piecewise functions are defined, that one is. So your correct that even within constructive mathematics. Mostly thought Gandalf61 is correct. After the books above are results above regarding new calculus are published in a reliable source they then might be reasonable to include. But removing references to infinitesimals is not possible. Even if you disagree with their existence, they are certainly something that is discussed frequently in the wider world and so are suitable for inclusion. Thenub314 (talk) 18:43, 15 December 2011 (UTC)[reply]

You are in error. For the function to be defined on all of R, we would need the law of trichotomy. This law is often not assumed in constructive mathematics. For example, van Dalen's book contains counterexamples based on the violation of the law of trichotomy. At any rate, as far as the IP's theory is concerned, it turns out that the cubic x^3 is also not differentiable in his sense! So constructive mathematics is a bit of an overkill. Tkuvho (talk) 20:31, 15 December 2011 (UTC)[reply]

I agree with you fully about the law of trichotomy, but that is more a problem with your description of the function then the function itself. For example ${\displaystyle |x|}$ is constructively defined (Techniques of Constructive Analysis Bridges and Simona page 30) as is ${\displaystyle y(x)=(x+|x|)/2}$. Finally, your function is simply ${\displaystyle 3y(x)^{2}}$. Sums and compositions are fine within the constructive framework, so your function would be fine also. Thenub314 (talk) 23:10, 15 December 2011 (UTC)[reply]

Good point, thanks. Tkuvho (talk) 09:05, 16 December 2011 (UTC)[reply]

The function |x| is a set of conditional functions, that is, f(x) = x when x>0; f(x)=-x when x<0 and f(x)=0 when x=0. By the way, how can you define it any other way except "constructively"? 64.134.230.145 (talk) 23:40, 15 December 2011 (UTC)[reply]

Perhaps you can point to a source that shows Archimedes knew what an infinitesimal is? I do not believe there is any such source. 12.176.152.194 (talk) 16:57, 14 December 2011 (UTC)[reply]

According to our article on The Method, he used infinitesimals, whether or not he knew what they were. — Arthur Rubin (talk) 17:05, 14 December 2011 (UTC)[reply]

The article on the Method does not show how Archimedes used infinitesimals; rather it shows how he used the method of exhaustion.

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

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

So, write down an infinitesimal that Archimedes used?

You cannot say he used them, if you cannot even provide one example. 12.176.152.194 (talk) 19:10, 14 December 2011 (UTC)[reply]

You got me on this one, thanks. Archimedes used indivisibles, not infinitesimals. I corrected the page yesterday as soon as I read your message. It is true that some of our pages may report Archimedes as using infinitesimals, but this is not exactly true, unless of course you extend the meaning of "infinitesimal" to include "indivisible" (in Cavalieri's sense), of course. Tkuvho (talk) 13:51, 15 December 2011 (UTC)[reply]

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

At the following link:

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

You will find three articles that might interest you:

   The Positional Derivative.pdf
   The definition and proof of the Mean Value Theorem.pdf
   John Gabriel's Average Sum Theorem.pdf

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

It is interesting that you are willing to talk about "an average of infinitely many values", but not about "infinitesimals". I think they are actually equivalent. Tkuvho (talk) 10:14, 16 December 2011 (UTC)[reply]

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

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

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

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

64.134.230.145 (talk) 23:25, 15 December 2011 (UTC)[reply]

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

ZFC is far more sound than infinitesimal theory and even though I don't think it's important, it is factual for the most part. So I won't debate it. The equivalence class of 1, 1/2, 1/3, ... is *zero*, not an infinitesimal.

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

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

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

${\displaystyle f(x)={\begin{cases}0&{\text{if }}n<0\\3x^{2}&{\text{if }}n\geq 0\end{cases}}}$

is not a function because it involves two rules. But we can rewrite this in a single rule:

${\displaystyle g(x)={\frac {3x(x+{\sqrt {x^{2}}})}{2}}}$

By your definition g is a function but f is not - and yet f and g take the same value for all real values of x. So your definition creates a contradiction. Gandalf61 (talk) 10:59, 16 December 2011 (UTC)[reply]

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

        ${\displaystyle g(x)=1.5x(x+|x|)}$

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

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

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

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

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

12.176.152.194 (talk) 16:01, 16 December 2011 (UTC)[reply]

In my opinion, anyone who thinks that what you've written is a constructive^{[note 1]} is not a mathematician. — Arthur Rubin (talk) 16:45, 16 December 2011 (UTC)[reply]

In my opinion, I know that you are not a mathematician. As I have already stated Mr. Rubin, having a PhD does not make you a mathematician. Neither do 1000 publications which have no real significance and are understood only by you. — Preceding unsigned comment added by 12.176.152.194 (talk • contribs)

Yes, it is correct to say that you have provided no evidence that any mathematicians support your analysis. If you can provide evidence that someone using mathematical or logical reasoning believes that the "new Calculus" has any mathematical or pedagogical value, you might try supplying such on your web site, or pointing to a reliable source to that effect on Wikipedia. — Arthur Rubin (talk) 17:52, 27 December 2011 (UTC)[reply]

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

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

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

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

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

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

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

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

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

This is in direct contradiction to other articles claiming that Robinson was the first to rigorize the theory of infinitesimals.

Archimedes (or more precisely Eudoxus) proposed a coherent definition of infinitesimals. Robinson was able to construct them in ZFC, thereby implementing Eudoxus' definition. Tkuvho (talk) 17:21, 19 December 2011 (UTC)[reply]

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

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

There are several more false statements/contradictions:

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

Nonsense. Archimedes rejected infinite numbers.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Once again, Robinson proved nothing. Please show me an infinitesimal and do some arithmetic with it.

12.176.152.194 (talk) 17:31, 21 December 2011 (UTC)[reply]

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

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

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

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

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

Does Wikipedia actually care about article content at all? 12.176.152.194 (talk) 17:01, 18 December 2011 (UTC)[reply]

Yes. The criticism in item (b) above is entirely correct. I haven't noticed that passage before. It's gone now. Tkuvho (talk) 11:59, 19 December 2011 (UTC)[reply]

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)[reply]

Your first sentence is exactly the point. Spot on. Thenub314 (talk) 22:13, 18 December 2011 (UTC)[reply]

Also if you look at the bottom of the page there should be a "Disclaimers" link. I think the words printed largely on the page linked to address part of your concern. Thenub314 (talk) 22:14, 18 December 2011 (UTC)[reply]

Perhaps the "Disclaimers" link should appear in large bold font at the top of each page? I for one, never noticed it. 12.176.152.194 (talk) 17:17, 21 December 2011 (UTC)[reply]

There is no particular reason to emphasize the disclaimer on this page, as the information here is mostly correct. Tkuvho (talk) 17:22, 21 December 2011 (UTC)[reply]

See, it is this kind of attitude that is problematic. No knowledge is ever beyond investigation is all I will say and add that it is your opinion that is it "mostly correct". BTW: I read Robert Ely's research - I am not impressed. He really does not say anything that supports what you claim about students and the concept of infinitesimal. No use debating this also because I am not convinced that all real numbers can be represented in a given radix system. In fact, I am certain that real numbers are ill-defined. 12.176.152.194 (talk) 17:36, 21 December 2011 (UTC)[reply]

Interesting. The real numbers are not well-defined. But apparently a notion of derivative where x^3 is not differentiable at 0 is well-defined. Tkuvho (talk) 17:56, 21 December 2011 (UTC)[reply]

About real numbers and magnitudes: http://thenewcalculus.weebly.com/uploads/5/6/7/4/5674177/magnitude_and_number.pdf If you use Cauchy's definition which is flawed, then a derivative exists at 0. However, using my sound New Calculus, a derivative does not exist at 0 because it is impossible to construct a finite tangent line to x^3 at 0. 12.176.152.194 (talk) 03:44, 22 December 2011 (UTC)[reply]

Since we happen to be at the "infinitesimal" talkpage, I will mention that the reason that the tangent line to y=x^3 at the origin does exist is because you can pick two values of x infinitely close to the origin, and draw the line through the corresponding points on the graph. That line is infinitely close to the x-axis, and that's enough to declare the x-axis to be the tangent line to the graph. Tkuvho (talk) 12:07, 22 December 2011 (UTC)[reply]

I don't think so. Your line would cross x^3 so it cannot be a tangent line. Before I exit this discussion, I just want to say that I have seen the original Greek and nowhere is there any mention of infinitesimal or indivisible. Now, I know your English publication states this, but it is incorrect. The modern Greek word is composed of two words "infinite" and "minimum". There was no Ancient Greek word for infinitesimal. The word for indivisible literally means that which cannot be divided into smaller parts. Archimedes most definitely did not use infinitesimals because these don't exist. Did he use indivisibles? Only in the sense of lines, areas and volumes(ala Cavalieri). The method of exhaustion relies on the approach of finding the area (e.g parabola example) of a shape which represents the area (or volume) one wishes to find. The more one can make the shape similar to the desired area, the better the approximation. You could say Archimedes anticipated "limits" [in the sense that a given approximation approaches some incommensurable magnitude (pi) or some rational number (e.g. 1/3). Not at all like the modern definition of limit], but had no idea about indivisibles or infinitesimals. The only objects he knew about were the rational numbers and the incommensurable magnitudes. In the Works of Archimedes, one finds evidence of this in many places.

And out of curiosity, please tell me which two infinitely small values (*) you can pick close to zero? See, "infinitely small" is not well-defined, therefore it is nonsense. Many questions arise... if these infinitely close values are subtracted, then the difference is 0, thus your denominator of the finite difference quotient is also 0 which is undefined. Furthermore, the ordinates corresponding to these abscissas (*) you mention also have a difference close to 0, so that your difference quotient looks like 0/0. Hmm, so what can this mean? Sorry, you should never accept any concept that is ill-defined.

12.176.152.194 (talk) 15:52, 22 December 2011 (UTC)[reply]

Archimedes did not use the word "indivisible" (this word was introduced in the middle ages), but he used indivisibles in the sense of Cavalieri all the same. Congratulations upon reading Archimedes in the original, but you might want to brush up on some Archimedes scholarship, as well, for instance the work of Reviel Netz. Tkuvho (talk) 18:50, 22 December 2011 (UTC)[reply]

Archimedes could not have used indivisibles because the method of exhaustion does not use any concept of indivisibility. Also note that an infinitesimal (even according to your understanding) is not necessarily the same as an indivisible. For example, indivisible applies to line, area and volume whereas infinitesimal applies to number. Both have entirely different meanings: infinitesimal (vaguely some magnitude close to zero) and indivisible (a line or width of disc). Reviel Netz has not revealed anything new to my knowledge except that he worked on the restoration of the palimpsest? I have not read the entire palimpsest in Greek (only parts of it). I have studied the Works of Archimedes (Thomas Heath) from cover to cover. I can tell you that there are many things in Archimedes' Works that are to this day not well-understood. 12.176.152.194 (talk) 20:41, 22 December 2011 (UTC)[reply]

There are two different methods that Archimedes used: (1) method of indivisibles, and (2) method of exhaustion. I certainly agree that there are many things that Archimedes did that you apparently don't understand, particularly his application of the method of exhaustion. Tkuvho (talk) 08:25, 23 December 2011 (UTC)[reply]

You are clueless regarding the meaning of indivisible and even more confused regarding infinitesimals but I expect this from having read your comments which are obviously wrong. This article is a joke because the method of exhaustion uses neither of these concepts, yet your article claims it does. There are none so blind as those who will not see. 12.176.152.194 (talk) 16:12, 24 December 2011 (UTC)[reply]

The point about seeing is well put. Now draw the cubic y=x^3 and the x-axis, take a deep breath, and let us know what you see. Tkuvho (talk) 11:59, 27 December 2011 (UTC)[reply]

I see the x-axis intersecting and crossing the cubic at x=0. Common sense tells me it can't be a tangent. If you can show me a secant with defined gradient that is parallel to the x-axis, then a tangent must exist for the cubic at x=0. There is no such secant - this violates the mean value theorem. Conclusion is that cubic is not differentiable at x=0. The algebra proof is somewhat longer and more complicated. 12.176.152.194 (talk) 02:33, 29 December 2011 (UTC)[reply]

If you insist that the tangent line must be a secant line through a pair of points, then you will never find any differentiable functions other than the linear ones. Even for the parabola, you have to discard a remainder term to pass from a secant line to a tangent line. A theory that denies the differentiability of the parabola may be logically consistent but it will not be very useful. Dropping the infinitesimal remainder is not something you can avoid in doing calculus. If you can't beat them, join them :) Tkuvho (talk) 08:29, 29 December 2011 (UTC)[reply]

I don't insist anything - the mean value theorem requires that there be a parallel secant for any tangent line. That you will never find any "differentiable functions other than linear ones" is outright false. I have shown that every function that is differentiable in standard calculus is also differentiable in the new calculus. Your statement about discarding a remainder is also false. In fact, every one of your sentences in the previous paragraph is false. I have beaten "them" and before long they will be joining me! 12.176.152.194 (talk) 13:54, 29 December 2011 (UTC)[reply]

You have misunderstood the mean value theorem. The mean value theorem does not assert that for every tangent line there is a parallel secant line. Rather, it asserts that for every secant line, there is a parallel tangent line. In fact, the mean value theorem clearly illustrates the problem with your approach: if y=x^3 is not differentiable at the origin as you seem to claim, then one cannot apply the mean value theorem to this polynomial. This is a major inconvenience compared to the usual approach involving dropping the infinitesimal remainder at the end of the calculation of the slope. Tkuvho (talk) 14:03, 29 December 2011 (UTC)[reply]

Seems to me the misunderstanding is on your part, not mine. The mean value theorem asserts both statements, that is, for every secant line there is a parallel tangent line and vice-versa. And of course you cannot apply the mean value theorem to x^3 at the origin because it (cubic) is not differentiable at the origin. There is no inconvenience of any sort whatsoever. If you don't like this, then it's your problem but that does not prevent it from being fact. 12.176.152.194 (talk) 14:35, 29 December 2011 (UTC)[reply]

Do you have a source for your version of the mean value theorem? All the textbooks I have seen use the version I stated above. Tkuvho (talk) 14:36, 29 December 2011 (UTC)[reply]

I don't need a source. I learned the theorem when I was 14 years old. However, your own Wikipedia entry on the mean value theorem says: For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. 12.176.152.194 (talk) 05:04, 30 December 2011 (UTC)[reply]

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)[reply]

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)[reply]

Cite? (Other than your files?) — Arthur Rubin (talk) 14:59, 2 January 2012 (UTC)[reply]

Both f and g have derivatives of 0 in your system (as in the common system), but f+g is 2 x³, which you claim does not have a derivative at 0. — Arthur Rubin (talk) 16:11, 22 December 2011 (UTC)[reply]

This is correct because in the New Calculus, a function is differentiable at a given point if and only if a finite tangent line can be constructed. Therefore the rule ${\displaystyle (f+g)'=f'+g'}$ would not generally apply as you noticed. There are also some rules in the standard calculus which would not apply generally in the New Calculus. As an example, consider f(x)=|x|, in the New calculus, it makes no sense to talk about a derivative at x=0 because the function is not smooth at that point. This is due to the fact that concepts in the New Calculus are well-defined, for example the derivative. In standard calculus, if f(x)=1/x and g(x)=-1/x, the general rule you stated, fails. Moral of the story is that there are always exceptions - even to the general rules. Mr. Rubin, if you ask a question and I answer it, this does not mean I care to discuss my theory on Wikipedia where you think it is inappropriate. 12.176.152.194 (talk) 16:50, 22 December 2011 (UTC)[reply]

There is nothing wrong with the "general rule" in the case that you give. If f is differentiable and g is differentiable then f+g is always differentiable. The only difficulty with 1/x and -1/x is that neither function is defined at 0, there sum is not the zero function but the sum is also undefined at zero. So you've given examples where f, g and f+g all fail to be differentiable at 0, which doesn't quite make your point. Thenub314 (talk) 22:02, 22 December 2011 (UTC)[reply]

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

None of this (whether correct or not) alters the fact that a tangent line cannot be constructed at x=0 for the function x^3. Finally, if there are any results from standard calculus that don't work the same way in the New Calculus, one of two things are possible. a) The concept in standard calculus is ill-defined/flawed or b) there is a new approach that is no longer compatible with the wrong ideas of standard calculus. 12.176.152.194 (talk) 23:51, 22 December 2011 (UTC)[reply]

You may find it interesting to notice that others may disagree. For example if you take a look at "Range, R. Michael (May 2011), "Where Are Limits Needed in Calculus?", American Mathematical Monthly 118 (5): 404-417" you'll see a development of differential calculus that doesn't use limits or infinitesimals.

It is based on a method of Descartes which is effectively geometric/algebraic in nature. But it does construct tangent lines to x^3 at 0. Thenub314 (talk) 15:33, 23 December 2011 (UTC)[reply]

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

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

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

And Rubin calls you a mathematician? Wow. More like a shotgun mathematician... A tangent meets a curve or surface in a single point if a sufficiently small interval is considered (Webster). And gee, let me see, your own Wikipedia entry says: "More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f." This is implied directly by the mean value theorem. The reason your understanding is faulty is due to your education - Cauchy's ill-defined derivative. In very simple language and as the Greeks invented it, a tangent is a finite straight line that meets another curve in one point and crosses it nowhere. Now take a deep breath and tell me if the x-axis crosses the cubic. Allow me to educate you a little bit, the tangent is the movable part of a trapezium (the non-parallel side) which is a tangent object in planar geometry. The Greeks were trying to use tangents to determine if curves were smooth given the same curves are continuous. This was the main reason they came up with the idea of tangent. The Greeks knew only intuitively that the conical curves they knew were smooth. Much later, curvature was measured using tangent line gradients. For more, you'll have to wait for the publication of the most important mathematics book ever written - What you had to know in mathematics but your educators could not tell you. To learn more about single variable calculus, you can read the file called NewCalculusAbstract-Part1 at http://india-men.ning.com/forum/topics/meaning-of-the-differential-quotient?page=1&commentId=2238831%3AComment%3A46087&x=1#2238831Comment46087 12.176.152.194 (talk) 22:35, 28 December 2011 (UTC)[reply]

I am not sure which Webster if any you are quoting, but the definition "A tangent meets a curve or surface in a single point if a sufficiently small interval is considered" is erroneous. Thus, the function x^2 cos(1/x) has x-axis for a tangent at the origin, but, contrary to you alleged "webster" definition, it does not meet the graph in a single point no matter how small the interval considered, but rather at infinitely many points. Tkuvho (talk) 08:33, 29 December 2011 (UTC)[reply]

The definition is not erroneous. This is what it means for a line to be a tangent line. The function x^2 cos(1/x) is not defined at 0, so how can it have a tangent line there? You are incorrect about this function not meeting the x-axis - it intersects the x-axis infinitely many times except at the origin. You must be the only one who has such an absurd understanding of what it means to be a tangent because no one else I know would agree with you. 12.176.152.194 (talk) 14:01, 29 December 2011 (UTC)[reply]

I omitted to mention that the function is defined to be zero at the origin and x^2 cos(1/x) everywhere else. One can construct similar functions without mentioning such two cases, as well, with the property that the tangent will meet the graph at infinitely many points. Tkuvho (talk) 14:06, 29 December 2011 (UTC)[reply]

See earlier note about removing discontinuity at x=0. The methods of calculus apply to continuous and smooth functions. As slippery as the concept of continuity is, it can be defined simply as follows: A function is continuous over an interval if there are no disjoint paths (geometric definition). A function is smooth (and therefore differentiable) if at each point of the function, exactly one finite tangent line can be constructed. Your assertion that the tangent to the function x^2cos(1/x) will meet at infinitely many points (even if it were defined at the origin) is false. The curve of this function is sinusoidal from both sides of the origin where it is undefined, hence it is impossible for it to be a straight line ever. 12.176.152.194 (talk) 14:43, 30 December 2011 (UTC)[reply]

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

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

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

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

1) I'm not convinced that Archimedes used infinitesimals; however, he certainly considered them, or the concept of an Archimedean field would not have arisen. I don't presently have access to the references used in indivisibles to support the statement, so I can't confirm that Archimedes did use them.

2) In non-standard set theory (or "internal set theory"), the set of infinitesimals is not an "standard" or "internal" set, so the LUB property of R doesn't transfer to R^* with respect to that set. R(((ε))) has no trace of the LUB property, so there is no need to assert there's a problem there.

3) "k" (wherever you got that from) could be ε itself.

You can certainly request verification of the citation for the claim that Archimedes used infinitesimals. You may not say that the theory of infinitesimals is not mathematically consistent, unless you are prepared to prove it using reliable sources.

And, although the reasoning is somewhat circular, a reliable source is one that is recognized as reliable (according to Wikipedia standards) by other reliable sources. We're probably not going to get to your "new calculus" by any chain starting with a reliable source, but I could be surprised.

As for your letters above:

A) If you want to redefine tangent line to a curve at a point to mean a line parallel to arbitrarily short secants of arcs containing that point, I can't stop you, but it doesn't have any place on Wikipedia unless used in a reliable source.

B) I can define an infinitesimal in any field of characteristic 0. Whether or not they exist, and whether or not they are useful if they do exist, depends on the field. You may deny the reality of ultrafilters, the axiom of choice, or set theory in general, but you cannot deny that most mathematicians use them, and that they have not been proved inconsistent.

C) Is your opinion.

— Arthur Rubin (talk) 06:41, 26 December 2011 (UTC)[reply]

"I'm not convinced that Archimedes used infinitesimals; however, he certainly considered them, or the concept.." - Now this is acceptable. Why don't you change the article to reflect this fact? Instead of "Archimedes used indivisibles in The Method of Mechanical Theorems to find areas of regions and volumes of solids.", you could say "Archimedes certainly considered using infinitesimals in The Method of Mechanical Theorems to find areas of regions and volumes of solids, but no evidence exists to support this notion." 12.176.152.194 (talk) 16:48, 26 December 2011 (UTC)[reply]

Whether or not Archimedes used infinitesimals pales in comparison with the fact that the x-axis is tangent to the cubic y=x^3 at the origin. Please come to grips with this fact. Tkuvho (talk) 12:39, 28 December 2011 (UTC)[reply]

I do not agree with 12.176.152.194's opinions about calculus more then anyone else, but reading this it seems as if something has gone wrong. The comments above are focused on improving the article, so we should at least not dismiss all of his comments because he has a unusual notion of what it means for a line to be tangent. Since the comment about Archimedes is disupted, then we should tag it as such or find a reference to support it. Thenub314 (talk) 15:43, 28 December 2011 (UTC)[reply]

There is a reference given; it's just that I haven't read it and the IP claims that it doesn't support the statement. Perhaps {{vs}} is a better tag than {{disputed-inline}}. It would be better if someone who is familiar with the given reference could comment. — Arthur Rubin (talk) 15:55, 28 December 2011 (UTC)[reply]

The only reference I saw was the one given to Archimedes actual work itself, which would clearly be in ancient greek, and as with all historical documents would probably require an expert in how the language was used at the time to make sense of what was being said. But perhaps there is a good translation etc. Even better would be a secondary source though. Thenub314 (talk) 16:18, 28 December 2011 (UTC)[reply]

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

See my comment in the previous section. Tkuvho (talk) 08:35, 29 December 2011 (UTC)[reply]

You cannot create piecewise or conditional fuctions, and draw general conclusions from these functions. Even if the function x^2cos(1/x) is defined to be 0 at the origin, the x-axis cannot be its tangent - ever. You have a fundamentally incorrect understanding of what it means for a line to be tangent. Webster's online was the source I was referring to. In any event, unless you can respond to the questions I put to you, you are wasting your time. So far, you have not been able to respond to any of the questions or challenges. I suggest you stay with sound logical arguments rather than your opinions which are entirely wrong. — Preceding unsigned comment added by 12.176.152.194 (talk) 14:15, 29 December 2011 (UTC)[reply]

OK, let's stick with logical arguments: note that the derivative of x^2cos(1/x) at the origin exists even according to your definition of derivative. Namely, the function is even, so that its values at h and -h are equal. Therefore the quotient you propose as the definition of derivative will be zero. Therefore the derivative exists and equals 0. Since the graph passes through the origin, the tangent line is the x-axis. Thus your "webster" definition of a tangent line in terms of a unique point of intersection does not work in all cases. At any rate, to calculate more general derivatives, one will have to drop the infinitesimal remainder at the end if one wants a usable theory. Tkuvho (talk) 14:20, 29 December 2011 (UTC)[reply]

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

If I follow your definition in Cauchy's Kludge correctly, the derivative does exist and equals zero at the origin. Tkuvho (talk) 14:33, 29 December 2011 (UTC)[reply]

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

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

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

Such pairs must be contained in the same segment of the arc. A tangent does not cross the curve - ever, for otherwise it cannot be a tangent. You are in denial of this fact but it is a simple fact. 12.176.152.194 (talk) 15:03, 29 December 2011 (UTC)[reply]

This is not the same definition of a tangent line we were working with before. It used to be based on the slope at the point, without reference to intersections with the curve. Tkuvho (talk) 16:03, 29 December 2011 (UTC)[reply]

I have always used the same definition. A tangent line is not based on slope. Rather slope is based on the tangent line. You cannot define a tangent line without reference to intersections with the curve. This is the key property of tangents. Otherwise they are just other curves intersecting other curves. So, the fact that an intersecting curve does not cross another curve is the identifying feature of a tangent. In all Ancient Greek texts (they invented the concept btw), this is exactly what a tangent line is. 12.176.152.194 (talk) 01:42, 30 December 2011 (UTC)[reply]

(I don't know how these comments got separated from that of the IP, or which comments it was in response to.)

Fair enough. Your article is incomprehensible to a mathematician — at least to the editors who have commented on it here on Wikipedia, most of whom are mathematicians.

However, there isn't a single definition. The second sentence of the "history" section provides a usable definition, however. Is that early enough for you? — Arthur Rubin (talk) 16:23, 16 December 2011 (UTC)[reply]

We should reference our claims about Archimedes. This section can serve as a centralized place for discussion. Thenub314 (talk) 21:06, 28 December 2011 (UTC)[reply]

Looking at "Katz, V. (2008), A History of Mathematics:An Introduction, Addison Wesley" has a nice section on Archimedes work. He does use the term indivisibles but, as a indication that Archimedes himself did not use the term he places it in quotes the first time he uses it in this section. Thenub314 (talk) 21:43, 28 December 2011 (UTC)[reply]

The most reliable non-Greek language text is The Works of Archimedes (Thomas Heath). There is no other reliable source that claims Archimedes used indivisibles or the ill-defined concept of infinitesimal. Heath in my opinion was the greatest mathematics scholar whose command of the Greek language surpassed any other non-Greek. Heath translated not only The Works of Archimedes, but also the Elements and the geometry masterpiece of Apollonius. In fact Heath clearly writes about how the Greeks rejected the ill-defined notion of anything infinitely small or infinitely large. In fact, they rejected infinity because it is an ill-defined concept. One needn't look too far to see what nonsensical results have arisen out of this concept in the form of limits, infinitesimals, set theory, real analysis, etc. 12.176.152.194 (talk) 22:28, 28 December 2011 (UTC)[reply]

Correct, they were suspicious of infinity. That's why they typically replaced their arguments a la Cavalieri (as in The Method) by arguments by exhaustion in "official" publications. The Method was a private letter where Archimedes did use indivisibles. Tkuvho (talk) 08:37, 29 December 2011 (UTC)[reply]

Once again, aside from your first sentence, every other sentence is false, so I can see how your opinions are entirely wrong. Now, you cannot base an article without reliable sources on your "opinion". 12.176.152.194 (talk) 14:05, 29 December 2011 (UTC)[reply]

Please see the article by Netz et al. Tkuvho (talk) 14:07, 29 December 2011 (UTC)[reply]

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

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

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

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

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

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

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

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

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

Unfortunately, your response here contradicts what you wrote in your Cauchy text. You can't change your definitions when you run into logical difficulties. By wiki rules, Netz's opinions are reliable since they are published in a reputable venue. You should inform your interlocutors at the indian site that your theory contains errors. Many, many people have tried to develop calculus without resorting to discarding the infinitesimal remainder at the end (or to an equivalent method in terms of epsilontics), but they were not successful. Unfortunately, your theory does not seem to be any different. Tkuvho (talk) 14:57, 29 December 2011 (UTC)[reply]

Please be kind enough to tell in what way there is contradiction for I see none. Rather than make false accusations about me changing my definitions, you would gain more support if you used facts only. I have developed the first rigorous calculus without the use of limits - this is an indisputable fact. It's not debatable. Rather than write silly comments here, I would suggest you study the new calculus. If you have any questions, I will be glad to help you. BTW: There is no infinitesimal remainder at the end as you claim. This is part of the problem with Cauchy's flawed definition and the reason it is jury-rigged. This error by Cauchy gave birth to his incorrect theory regarding infinitesimals. We can discuss the new calculus but this is not the place for it. I prefer private email. Let's stay with the topic which is about fixing the many incorrect claims in this article. 12.176.152.194 (talk) 15:08, 29 December 2011 (UTC)[reply]

You defined the derivative in your pdf in terms of "f(x+k)-f(x-h)", where one can set k=h in the case of the parabola. Applying this definition to x^2 cos(1/x), we get zero slope at the origin. You can't change your definition in midsteam and claim that you have a different definition of the tangent line. Tkuvho (talk) 15:45, 29 December 2011 (UTC)[reply]

You cannot simply assume that h=k in every instance. In fact, this is not the case here. Now if you had studied the links I referred you to, you would have seen that the relationship between the distances varies for each dissimilar function. That is, you have to find it unless you are interested only in the general derivative, in which case you can use the (0,0) pair provided the function is differentiable at a given point. So, there is no changing of definitions only a problem with your interpretation and understanding. One more thing - calculus applies only to smooth continuous functions. The minute you introduce conditional functions (piece-wise), all bets are off. Newton had no clue about conditional functions. These came much later and in the hands of amateurs we now have theory that confuses the likes of you. If I defined a function as follows, f(x)=1/x for all x except 0 and f(x)=0 when x=0, then any conclusions I try to draw from calculus will be suspect because f(x) is not defined at 0. You can't "just" remove the discontinuity as you feel like it. As yet another example, consider the absolute value function. It is irrelevant to talk about differentiability at the origin because the conditional function is not smooth there. It is also non-remarkable that one can construct infinitely many finite tangents to the function at x=0. One does not require calculating the limit from the left and the right to see this. It's a no-brainer. Yet I have seen incompetent professors ask this irrelevant question in assessments once too often. 12.176.152.194 (talk) 18:38, 29 December 2011 (UTC)[reply]

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)[reply]

I would suggest you try again to go over the proof of Proposition 14 of The Method, where Archimedes uses Cavalieri's technique to compute the volume of the solid. Tkuvho (talk) 15:32, 29 December 2011 (UTC)[reply]

If you need help understanding Archimedes's works, I can provide some guidance for you. Nowhere in Proposition 14 does Archimedes use Cavalieri's principle. Do you enjoy misrepresenting facts always? I can see you have zero understanding. Reread the proposition and you will notice that it says "divide Qq into any number of equal parts". This is part of Archimedes's integration techniques that rely on natural averages. There are no infinitesimals or indivisibles mentioned anywhere. Let me give you a simple example. Suppose you wish to find the area between any curve and the x-axis. Furthermore, suppose there is no primitive function so you cannot use the fundamental theorem of calculus. So what do you do? Well, you will fall back on one of the numeric integration techniques that were taught to you. However, what you are actually doing in every case, is finding the average of the length of the ordinates in the interval and taking the product of this average with the interval width. This produces the area. You might call it rectangulation. I call it an average sum (my average sum theorem always uses equal parts or partitions whether area or volume). I define area as the product of two averages (average of ordinate lengths and average of horizontal lines which is equal to interval width). Volume as the product of 3 averages, etc. So, Archimedes used an averaging technique much as I do today. Since the area in such cases is *always* an approximation, it makes no sense to think about limits, infinitesimals or the like, because the more equal parts we divide Qq into, the better our approximation. So,once again, you need to study the Works of Archimedes in this light. Archimedes, Newton and all the great mathematicians before Cauchy would agree with me if they were able to know of this discussion. Cauchy really messed up with his definitions. Besides Archimedes, almost every other mathematician has ill-defined one or more concepts. Newton's definition of the derivative is ill-defined. Newton knew this and it was the main reason he did not publish his ideas sooner. He would have loved to know of my New Calculus. One of the characteristics of a great mathematician is the ability to well-define concepts. A mathematician is like an artist. The objects arising from concepts in a mathematician's mind are only as appealing as they are well-defined. In case you missed it earlier... 12.176.152.194 (talk) 18:54, 29 December 2011 (UTC)[reply]

I recommend the recent article by Pippinger here. Tkuvho (talk) 10:23, 30 December 2011 (UTC)[reply]

Sorry, I can't read the article without paying for it. I am not sure it's worth much, but as I haven't read it, I reserve my judgment. Because something is published in any journal, does not mean it automatically has great worth. It is only as good as he who reviewed it. Many false ideas have been published (Robinson's non-standard infinitesimals is a prime example) and are continually being published. Now, it's quite audacious for anyone to be saying Archimedes used Cavalieri's principle because Cavalieri was not even born till hundreds of years later. Thus, if anyone used anyone else's method, it would be Cavalieri who copied Archimedes. Archimedes's method of exhaustion (although it looks different to the integral) uses exactly the same methods as modern numeric integration with the exception that Archimedes's numeric integration is natural integration. The integral as Riemann defined it, is ill-defined since it uses the concept of infinity. It is fundamentally equivalent to Archimedes' method but adorned with different words (limit, infinity) and a disguised approach that is equivalent. The problem with Riemann's integral is that one is not summing anything infinite but the limit concept obscures this and other facts. It is very easy to show that the Riemann definite integral is in fact equivalent to the product of two averages, that is, the interval width (average of infinitely many horizontal lines all of same width as interval and average value of the vertical lines which are the ordinates). The indivisibles associated with Cavalieri, but used in finding the area between a curve and axis, are in fact the abscissas for corresponding ordinates in a given interval (abscissas are scaling factors or in common parlance, "integration with respect to x") where the area is being computed. Nothing magical or mysterious. If volume is being computed, the indivisibles are the areas of each disc (if you are using the disc method). Therefore, different terminology and appearance, but fundamentally still Archimedean in every respect. These and other misconceptions are explained in my unpublished book. Anyway, although this article is far from perfect, it is somewhat better than it was. I don't agree with Wikipedia that it's okay to state facts as long as these are published. I also don't agree with Rubin-Hardy (your math gods) ideas regarding reliability. The process you have in place is tedious, ineffective and time consuming. Look at the length of this page to get an idea of how difficult it was to change one sentence and it is still not completely correct - "Archimedes used what eventually came to be known as the Method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids." It should read: "Archimedes used the Method of Exhaustion (a key feature that involves 'equal partitioning' in determination of approximate averages that are used to compute area and volume) in his work, which eventually came to be known as the Method of Indivisibles." 12.176.152.194 (talk) 15:27, 30 December 2011 (UTC)[reply]

The concept of reliability that Wikipedia uses does lead to some anomalies, but using WP:FRINGE material, such as yours, would be even worse. There would be no means of distinguishing your — findings — from that of Archimedes Plutonium, other than (alleged, in both cases) expert analysis. As Wikipedia editors are not expected to be experts, neither can be used until published commentary is written. — Arthur Rubin (talk) 18:39, 30 December 2011 (UTC)[reply]

I agree. But, I am not advocating or suggesting my material be published, quite the contrary in fact. The article can be written without the author's opinion injecting his personal understanding (or lack thereof) and bias. "Archimedes used the Method of Exhaustion (a key feature that involves 'equal partitioning' in determination of approximate averages (*) that are used to compute area and volume) in his work, which eventually came to be known as the Method of Indivisibles." The previous sentence is 100% factual. If you have studied Archimedes' works, you will see that he uses the same methods over and over again as described in this sentence. Numerous proposition proofs are started with the same approach - "Let Qq be divided into equal parts,..." This is not my idea or new knowledge by any means. (*) I was the first to notice calculus is about averages, but if one does not like to mention averages, you can replace the word by areas or volumes. 12.176.152.194 (talk) 19:52, 30 December 2011 (UTC)[reply]

The method of exhaustion and the method of indivisibles are generally considered by scholars to be two distinct methods. As far as your ideas about calculus, do you also have a singularity at 0 when you integrate 3x^2? If integration goes smoothly even at the origin to produce x^3, then your calculus does not satisfy the fundamental theorem of calculus, which is a steep price to pay for a ban on dropping an infinitesimal remainder at the end of the calculation. Tkuvho (talk) 20:26, 31 December 2011 (UTC)[reply]

They may be generally considered distinct, but in fact they are the same thing. Regarding singularity - good question. Answer: The singularity you refer to only affects the derivative at the origin. There is no singularity when integrating 3x^2 and my calculus very much satisfies the fundamental theorem for any smooth and continuous function. For the cubic, although it is not differentiable at the origin, it is smooth (*) and continuous, but a point of inflection exists there. Now even in standard calculus there is provision for this property, that is, inflexion. Integration is possible because...when you integrate, you are calculating the 'infinite' average of the lengths of ordinates in a given interval and taking the product of this average with the interval length. There is never an infinitesimal remainder at the end of cancellation for a finite difference ratio, and this is exactly what I have debunked in the publication called Cauchy's Kludge. Look, k(m+n)/(m+n) is always equal to k. There is no remainder of any kind- ever. I am not a good writer, but if you can read my entire web page, I am certain you will have all your questions answered there. If you still have further queries or spot something you think is an error, I welcome your opinions and views. Once again, I prefer email because this page is not about my new calculus and I don't think others care to read our dialogue. (*) Generally a function is smooth when exactly one finite tangent line can be constructed at any given point. The only exception to this rule is a point of inflexion. Bear in mind that standard calculus has been around for almost 350 years since Newton. I actually address many of the standard 'singularities' such as points of inflection, saddle points (multi-variate), etc, in my book. If I ever get to publish it, these details will be explained. Although you can extend the New Calculus single variable formulation to multi-variable calculus as you can in standard calculus, there is also a different approach (far more intuitive and efficient) explained in dealing with tangent objects, which is in fact brand new mathematics (no limits or real analysis). I am certain any mathphile will find it tantalizing. Finally, I have shared some details of my New Calculus but I do hope to at some future time earn some money from the publication, therefore I have not revealed much of it for this reason. You can read my NewCalculusAbstract-Part1.pdf which was rejected by the AMS because "I regret that, due to the large volume of papers we receive, we are unable to accept your paper for publication."(sic) - whatever this means... 12.176.152.194 (talk) 23:36, 31 December 2011 (UTC)[reply]

Perhaps what bothered the AMS is the idea that x^n is not always differentiable. It is puzzling that you blame Cauchy for this. The derivative of x^n, at all points including 0, was already known before Newton and Leibniz. I have to check whether it was Hudde, Wallis, or Barlow, but at any rate this was two centuries before Cauchy. Tkuvho (talk) 08:03, 1 January 2012 (UTC)[reply]

I don't think so. No derivative exists at an inflection point. Anyway, I don't believe they even got round to reading it. x^n always has a general derivative but even so, one cannot be certain unless one checks the derivative at a point. For example, the upper half of a circle always has a general derivative but there is no derivative when x=r(or x>r) or x=-r (or x<-r). And of course we have seen some examples with Rubin where an actual numeric derivative does not exist, e.g. f(x)=1/x at the origin. The general derivative of f(x) is g(x)=-1/(x^2) but neither f(0) nor g(0) exist; both are undefined. I 'blame' Cauchy for the definition f'(x) = (lim as h approaches 0) {f(x+h)-f(x)} / h. It is from this flawed definition that all sorts of nonsense and misconceptions have arisen, to wit, the limit and the infinitesimal - neither of which are required and have no place in differential calculus or any other mathematics. The limit is somewhat sound but the infinitesimal is absolute rubbish. I think Newton knew that he didn't know. This is what kept him from publishing anything sooner than he did. Leibniz was trying to be more precise, but he also failed. I realized their ideas are wrong when I was 14 years old. Cauchy was trying to add rigor, but he made things more complicated than they actually were. Suppose the gradient of a parallel secant is given by [f(x+n)-f(x-m)] / (m+n) = k. Now k is some ratio. It follows that f(x+n)-f(x-m) is always equal to k(m+n). Now, when we are calculating a derivative, we form the numerator of the difference ration, that is, f(x+n)-f(x-m) and its denominator is (m+n). Therefore it follows k must be given by k(m+n)/(m+n). It's really that simple. Newton, Leibniz and Cauchy missed this. In fact you and every other academic missed it, until I came along. My divisibility identities although further confirmation of these facts are interesting, but not actually required. Related distance pairs are interesting and very useful but for a general derivative all one cares is about the distance pair (0;0). There is no need to study limits for 6 months or take a course in real analysis. One finds a quotient from the difference ratio and uses (0;0) to find the general derivative at a given point. This is sound and rigorous mathematics. From there on, everything else that follows is crystal clear. 12.176.152.194 (talk) 12:15, 1 January 2012 (UTC)[reply]

The problem with your approach is that it creates difficulties in the way of applications in physics. Suppose a car is traveling a distance s(t) as a function of time t according to the law s(t)=t^3. Is there any reason to suppose that its speedometer will not show the value 0 when t=0? Do you know of any physicist that would accept this? Tkuvho (talk) 12:21, 1 January 2012 (UTC)[reply]

Well, it's not always possible to match the model to a physical situation. There are frequently questions that need to be asked. If the car was at rest at t=0, then this is a special case and v(t)=3t^2 is a new function valid at every point t>0 or t=0. When any physicist analyzes the gradient or area characteristics for given data in planar representation, you seldom have a one to one correspondence between the model and physical events. The example you have provided is simple. Consider, an optimization problem using differential equations. It's almost never a case of exact correspondence, there may be existence problems, singularity problems, etc. (*) Once again, the only reason physicists may have some difficulty accepting this, is that they are used to flawed mathematics. Old habits are hard to break. A technician who has used an old tool all his life is generally reluctant to use a better tool. Going back to the example, it is illogical to consider speed at s(0) because s(t) itself is not defined for t<0 and makes no sense. In other words, there is no data for t<0 which implies a tangent gradient (v(t)) is not possible in the first place, therefore no gradient. (*) I have found the New Calculus to be more effective in these problems than standard calculus. There is much more that can be done. You have to ask yourself whether it's more efficient to learn a flawed calculus in much longer periods of time without ever fully understanding it, or learning a robust calculus in a couple of weeks which you can understand completely. 12.176.152.194 (talk) 16:05, 1 January 2012 (UTC)[reply]

Is your calculus robust? It may suffer from a logical circularity: in order to calculate the derivative, you insist on forming a finite ratio, but you must know in advance what the value of the ratio is supposed to be! And in order to calculate that value, you have to do the usual calculation involving discarding the infinitesimal remainder at the end. Tkuvho (talk) 16:09, 1 January 2012 (UTC)[reply]

It's the only robust calculus. What makes you think you have to know anything in advance? I think you are confusing yourself. If you use the (0;0) distance pair, you don't know anything in advance. 12.176.152.194 (talk) 16:22, 1 January 2012 (UTC)[reply]

What's a (0,0) distance pair? Tkuvho (talk) 16:25, 1 January 2012 (UTC)[reply]

Read the abstract. It explains what you need to know. http://india-men.ning.com/forum/topics/meaning-of-the-differential-quotient?page=1 12.176.152.194 (talk) 16:28, 1 January 2012 (UTC)[reply]

(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:

${\displaystyle f'(x){\text{ exists and is equal to }}L{\text{ if }}\bigcap _{\epsilon >0}\left\{{\frac {f(x+m)-f(x-n)}{m+n}}|0<m,n<\epsilon \right\}=\left\{L\right\},}$

although, I can't come up with a definition which also requires ${\displaystyle f(x)}$ to exist. I'm pretty sure that that definition implies that ${\displaystyle \lim _{h\to 0}f(x+h)}$ exists, and that, if you then define ${\displaystyle f(x)=\lim _{h\to 0}f(x+h)}$, then ${\displaystyle f'(x)=L}$ under the usual definition. — Arthur Rubin (talk) 17:33, 1 January 2012 (UTC)[reply]

You are missing the point. No limits are required. No epsilonics. It's actually very simple: you find the tangent gradient by finding the gradient of a parallel secant in the same interval. A derivative f'(x) exists in a given interval if f(x) exists; the distance pair (0;0) satisfies the tangent gradient and infinitely many other distance pairs (m;n) exist in the same interval that satisfy the parallel secant gradients. f'(x) = [f(x+n)-f(x-n)]/(m+n) To find a general derivative one uses only the distance pair (0;0). To show that no derivative exists, one must prove that there are no distance pairs other than (0;0) if indeed (0;0) is valid.12.176.152.194 (talk) 19:00, 1 January 2012 (UTC)[reply]

In the meantime, John seems to have proposed a different definition which sounds like what Fermat did: take f(x+e)-f(x), expand in powers of e, cancel constant terms, divide by e, and look for the coefficient of e^1. In certain situations, this can be done without any infinitary processes. However, he writes this down with two variables in place of one, which seems an unnecessary complication. It has to be acknowledged that for polynomials the derivative can be calculated by a finite process. Tkuvho (talk) 17:43, 1 January 2012 (UTC)[reply]

Not even remotely the same. Fermat was trying different optimization approaches. There is no similarity between any prior mathematician's work and mine. I'll say this: if Newton had known about Euler's function notation, there may have been a chance he could have discovered my approach sooner. The problem at hand was not finding a derivative at a point - Newton showed this was easy using his approximate difference ratio of non-parallel secants. The challenge was finding a method to compute the general derivative of a given function without having to explain away division by zero. I am the first to succeed in this regard with a rigorous calculus that excludes limits. BTW: The derivative can be calculated by a finite process for any differentiable function, not just a polynomial. Every time you stumble on this, remind yourself of the (0;0) distance pair.12.176.152.194 (talk) 19:00, 1 January 2012 (UTC)[reply]

That (your reply to my formula) is nonsense. First, I was trying explain to Tkuvko that if your derivative exists, it is equal to the standard derivative (after adjusting for removable singularities, as your definition does not require that the value of the function exist at the relevant point). Second, your definition does require epsilontics; it has almost the same quantifier-complexity as the standard derivative. However, your underlying mathematics is even more restrictive than constructive mathematics; not that I think your method has any value, but you need to define your underlying mathematics and mathematical logic before anyone can determine whether it has value; it's significantly different than anything in the literature, and your "calculus" makes no sense (except in the formulation I gave it above) without modifying the underlying mathematical logic. — Arthur Rubin (talk) 02:08, 2 January 2012 (UTC)[reply]

Rubin, do tell what's nonsense about it or shut up I say. Every one of your statements in the previous paragraph are false. You do not understand my new calculus. Please don't pretend that you do. Your previous response is a whole lot of illogical rambling. Then again this is what I expect from someone who claims infinitesimals are too simple a concept for him to explain, yet is unable to provide any evidence of this. "it's significantly different than anything in the literature, and your "calculus" makes no sense" is your opinion. In fact you could do us all a favour and just hold your opinions, okay? 12.176.152.194 (talk) 04:51, 2 January 2012 (UTC)[reply]

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)[reply]

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)[reply]

The Levi-Civita field defines ε, and it can be shown it is an infinitesimal. — Arthur Rubin (talk) 15:45, 2 January 2012 (UTC)[reply]

That is false. 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)[reply]

For your definition of derivative, it would be helpful if you wrote it out symbolically, as there does seem to be some confusion as to what you mean.

For your definition of function, you will have to explain why the absolute value function isn't well-defined, as even intuitionists seem to accept it. Also, whether the function ${\displaystyle x^{2}|x|}$ has a derivative at 0 in your system. — Arthur Rubin (talk) 06:07, 2 January 2012 (UTC)[reply]

The NewCalculusAbstract-Part1.pdf contains a perfect definition in it that uses symbols. Your previous definition is wrong. m and n can only take on the value of (0;0) pair (in addition to infinitely many other pairs before and after reduction) after the difference quotient is reduced in my calculus, but this is illegal in standard calculus even though it works. Cauchy's Kludge explains this. I am not answering any more questions regarding my Calculus on this web page. 12.176.152.194 (talk) 14:13, 2 January 2012 (UTC)[reply]

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)[reply]

You refer to "terms". I assume therefore that you are working with polynomials. The technique you outlined is interesting, but it seems to be what Pierre de Fermat did a few decades before Newton and Leibniz, in developing his method of adequality. If you figured this out on your own that's certainly brilliant. But, believe me, calculus has come a long way since then. In particular, dealing with "terms" can only be done in the context of polynomials. Alternatively, you need power series which would allow you to account for analytic functions. Already in power series you would need to delete the infinitesimal remainder at the end of the calculations. Furthermore, to apply this to functions that are not analytic, you would need the usual differential quotient and the standard part function. Tkuvho (talk) 17:14, 1 January 2012 (UTC)[reply]

Not at all the same. It does not matter what you are working with, polynomials or any other function. If the function is smooth and continuous, it will work. No, Fermat had no idea about this. In fact no mathematician before me knew any of it. And although I am trying to encourage you to study it, there is some learning to be done. If your neurons are firing connections with any previous mathematics, you are not getting it. There are no remainders - infinitesimal or otherwise. I hope you will somehow understand this. It seems to me that you have a big stumbling block where this is concerned. There are no infinitesimals. Not in theory and not in reality. 12.176.152.194 (talk) 17:17, 1 January 2012 (UTC)[reply]

A curve ball for you - the only objects we know about are the rational numbers and incommensurable magnitudes. Nothing has changed since Archimedes. Most of what you learned in real analysis is either unsound or just plain wrong. Usually the latter is true. 12.176.152.194 (talk) 17:26, 1 January 2012 (UTC)[reply]

You will find some here and very well explained, too. Tkuvho (talk) 17:26, 1 January 2012 (UTC)[reply]

Has lots of misconceptions and errors. Not much different from anything else like it. Are you Keisler?12.176.152.194 (talk) 17:29, 1 January 2012 (UTC)[reply]

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)[reply]

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)[reply]

I do believe that is a legal threat. You can stop us quoting your work if it's under copyright, except for "fair use", which our discussion trying to find out whether it has any possible validity seems to fall under. If you don't want it discussed, you shouldn't have brought it up. — Arthur Rubin (talk) 02:11, 2 January 2012 (UTC)[reply]

I can stop you quoting it if you do not use correct attribution and if you do quote it without correct attribution, I will stop you. Look Rubin, you annoy me intensely. I have repeatedly informed you that the original discussion was regarding infinitesimals. You kept coming back to my New Calculus. Thukvo continued to ask me about it and the dialogue is primarily with Thukvo, not you. I do not agree with most of your views because they are wrong. If you think this is a threat, that's your problem. 12.176.152.194 (talk) 04:57, 2 January 2012 (UTC)[reply]

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)[reply]

Self-promotion? Have you read any of my comments? I have been trying to contribute to this article. It is full of non-factual statements. Although Thukvo asks me questions regarding my calculus, I keep returning to the main topic. I am prepared to stop right here if Thukvo ceases to ask me questions. I even recommended he contact me via private email if he wishes to continue the discussion. Rubin is an annoying trouble maker with a lot of time on his hands. As for self-promotion, what do you call Rubin's page on Wikipedia? It has nothing notable or remarkable.12.176.152.194 (talk) 05:00, 2 January 2012 (UTC)[reply]

The only non-factual statements made are by you, and possibly those attempting to interpret your <redacted> "New Calculus". — Arthur Rubin (talk) 05:38, 2 January 2012 (UTC)[reply]

You are the main reason I would prefer no references are made to my work. Your previous <redacted> definition is your wrong interpretation of my definition. 12.176.152.194 (talk) 14:00, 2 January 2012 (UTC)[reply]

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)[reply]

Agreed. The relevant OR is Cauchy's Kludge. I will not respond to any more questions on my New Calculus or any mathematics not related to this topic. 12.176.152.194 (talk) 14:00, 2 January 2012 (UTC)[reply]

"Cauchy's Kludge" (the name, that it is a "kludge", and the alleged error in Cauchy's work) is also your original research. — Arthur Rubin (talk) 15:03, 2 January 2012 (UTC)[reply]

And so? I am not disputing this fact. iNic asked me what OR and I responded. What's your problem Rubin? Perhaps a new pair of reading glasses is in order? Still living with your mother? 12.176.152.194 (talk) 15:20, 2 January 2012 (UTC)[reply]

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) [reply]

Good arguments I have. Patience for Rubin I do not. Rubin and I go back a long way. There is no love lost between us. I can assure you he probably dislikes me more than I dislike him. What do you say about the tone of Rubin's comments? Do you think he is not attacking me? He is very disdainful and continues to accuse me falsely. A normal person experiences what's called annoyance. 12.176.152.194 (talk) 15:44, 2 January 2012 (UTC)[reply]

How can he attack you if you stop talking about your own ideas? iNic (talk) 16:09, 2 January 2012 (UTC)[reply]

Yes, it is a kludge and it feels good to see you squirming because just about everything you think is knowledge is based on this Kludge. I almost feel sorry for you Rubin. Let's see - Einstein proved to be wrong. Next to follow will be your fake hero Abraham Robinson? 12.176.152.194 (talk) 15:23, 2 January 2012 (UTC)[reply]

Aha so you proved Einstein wrong too? Did you publish it? iNic (talk) 15:34, 2 January 2012 (UTC)[reply]

Don't tell me you suffer from reading problems also? Sorry, I meant to say he has been proved wrong. 12.176.152.194 (talk) 15:41, 2 January 2012 (UTC)[reply]

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)[reply]

Have you been following the news lately? I think we must discuss only the topic here - this article. Practice what you preach! 12.176.152.194 (talk) 18:14, 2 January 2012 (UTC)[reply]

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)[reply]

You did bring it up, because it is the only source you have given for the assertion that infinitesimals are even problematic. I still don't know what you have against R(((ε)))). — Arthur Rubin (talk) 05:36, 2 January 2012 (UTC)[reply]

Rubbish. I did not bring it up. I was asserting that the modern theory of infinitesimals started with Cauchy's Kludge. Therefore in this respect, the file I referred to (Cauchy's Kludge on my web site) is indeed not only relevant but central to the discussion. There are other sources, many of which are published online, not necessarily also published in the form of a physical book. I have already explained that R(((ε)))) is a figment of your troubled imagination. We had this discussion years ago and neither you nor Michael Hardy could not see the light then. What makes you think you will see it now? The entire theory is absolute rot. But I can't argue against this because Abraham Robinson, bless his dead little Jewish heart, actually got to print his wrong ideas. 12.176.152.194 (talk) 14:07, 2 January 2012 (UTC)[reply]

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)[reply]

I have spent sufficient time satisfying myself that Robinson was an idiot. As for wasting time, it's not really a waste of time pointing out that your article contains many non-factual statements regarding Archimedes and infinitesimals. These have been changed but are still not correct. I don't really care one way or the other so no need to respond to me again. If you want to have the last word, be my guest. 12.176.152.194 (talk) 15:48, 2 January 2012 (UTC)[reply]

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)[reply]

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 ${\displaystyle \{q\in \mathbb {Q} :a_{q}\neq 0\}}$ 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:

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

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

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

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)[reply]

"My" field R(((ε))) can be defined the same way, except that the support ${\displaystyle \{q\in \mathbb {Q} :a_{q}\neq 0\}}$ 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)[reply]

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)[reply]

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.

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)[reply]

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)[reply]

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 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.12.176.152.194 (talk) 22:07, 2 January 2012 (UTC)[reply]

I think this section can serve as evidence. Anyone who thinks infinitesimal theory is sound and pertinent is welcome to write his/her full name in this section followed by a link to his/her home web page.

1. Arthur Rubin

12.176.152.194 (talk) 22:24, 2 January 2012 (UTC)[reply]