Talk:Infinitesimal

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Geometry guy 21:13, 9 July 2007 (UTC)

1 Suggestion to stop the discussion for some time
2 What do you hope to accomplish?
3 What is an infinitesimal?
4 answer to remark above
5 Example?
6 The Law of Excluded Middle
7 Unit Vector Differentiation
8 Multiplication Of Two Infinitesimal Numbers
9 Merger?
10 Criticism of this article
11 Definition

[edit] Suggestion to stop the discussion for some time

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)

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)

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)

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)

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)

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)

"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)

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)

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)

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)

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)

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)

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)

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)

*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)

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)

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)

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)

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)

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)

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)

[edit] What do you hope to accomplish?

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)

[edit] What is an infinitesimal?

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)

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)

[edit] answer to remark above

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)

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)

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)

[edit] Example?

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)

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)

[edit] The Law of Excluded Middle

"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)

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)

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)

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)

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)

[edit] Unit Vector Differentiation

In dynamics, when a reference frame rotates we need to find the derivative of unit vectors $\vec u_r, \vec u_\theta$ .

In finding the derivative we arrive at $\lim_{\Delta \theta \to 0}\Delta \vec u_r = 2\lim_{\Delta \theta \to 0}[sin(\theta/2) (\vec u_\theta + \Delta \vec u_\theta)]$ .

For small $Δθ$ we can write $d\vec u_r = d\theta (\vec u_\theta + d\vec u_\theta)]$ .

As it is mentioned in the article, it is not rigorous to write $\vec u_\theta + d\vec u_\theta = \vec u_\theta$ .

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

If we expand, we can write $d\vec u_r = \vec u_\theta d\theta + d\theta d\vec u_\theta$ .

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

[edit] Multiplication Of Two Infinitesimal Numbers

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 $0$ and $d x$
Obviously $0 < d x < 1 = > d x 2 < d x$
If dx was a Real Number we could write $0 < d x 2 < d x$ but here we can not. If there is a number $d x 2$ such that $d x 2 > 0$ 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)

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)

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)

[edit] Merger?

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

[edit] Criticism of this article

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)

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).

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)

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

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

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)

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)

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)

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)

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

$\sum_{n=-k}^\infty x_n \epsilon^{\frac n m}$ ,

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

[edit] Definition

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)

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)

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)

"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)

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)

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)

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)

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)

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)

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)

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)

(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)