What is the Euler's stand on infinitesimals?

[Vinay080]

Euler was the master in analysisng anything. This can be seen in his words in the preface of his book "Mmathematica" (translated by Ian Bruce), where he speaks on the text of Hermann "Phoronomiam":

"..Besides, and what distracts the reader the most, is the fact that everything is carried out synthetically, with the demonstrations presented in the manner of the old geometry, and the analysis hidden, and recoginiton of which is given only at the end of work..."

Euler has given many insightful words on analysisng things in his preface of many other books, another piece of statements of interest would be of his statements in the preface of "Introduction to Analysis of the infinite" (English translated verstion of Blanton).

All the above indicates his personlaity of not leaving anything unresolved, consider the following passage from his book "Foundations of Differential Calculus" (English vesion of Blanton translation):

"...ratios might be more easily gathered together and represented in calculations, the vanishing increments themselves, although they are really nothing, are still usually represented by certain symbols. Along with these symbols, there is no reason not to give them a certain name. They are called differentials, and since they are without quantity, they are also said to be infinitely small. Hence, by their nature they are to be so interpreted as absolutely nothing, or they are considered to be equal to nothing..."

As I am beginner in understanding there differentials, I don't understand him. At one place, he says differentials to be "vanishing" increments or "nothing" quantities, and in the other place he says it to be "infinitely small".

Is there any other fragment of statements where Euler has analysed the concept of "differentials" properly, or is there any different meaning that can be given to the above statements to understand them?

I am still reading this book, so I don't know whether he has analysed later on, if it is the case, I will report, but until then, I want to know whether anyone has idea on his stand on this concept; papers or books on this matter by Euler or any other person would be really helpful.

I think differentials to be closely related to infinitesimals, so I want know Eulers analysis on both of them (if they are different).

 

[Vinay080]

Great, Euler is really the master of analysis, for his reasoning style, chaining everything, this is really a happiest moment for me to have got words from Euler on what I wanted, here is it, enjoy it and think about it, and add anything you want to add ahead in the thread, like books, etc.. Before that wait, I am quoting a large piece from the book, for I think it is worth quoting here, as it is of Euler's ofcourse, and it is an important piece, and I don't think text is acessible to everyone, so here is it, enjoy!

..To many who have discussed the rules of differential calculus, it has seemed that there is a distinction between absolutely nothing and a special order of quantities infinitely small, which do not quite vanish completely but retain a certain quantity that is indeed less than any assignable quantity. Concerning these, it is correctly objected that geometric rigor has been neglected. Because these infinitely small quantities have been neglected, the conclusions that have been drawn are rightly suspected. Although these infinitely small quantities are conceived to be few in number, when even a few, or many, or even an innumerable number of these are neglected, an enormous error may result. There is an attempt wrongfully to refute this objection with examples of this kind, whereby conclusions are drawn from differential calculus in the same way as from elementary geometry. Indeed, if these infinitely small quantities, which are neglected in calculus, are not quite nothing, then necessarily an error must result that will be the greater the more these quantities are heaped up. If it should happen that the error is less, this must be attributed to a fault in the calculation whereby certain errors are compensated by other errors, rather than freeingcalculation from suspicion of error. In order that there be no compensating one error by a new one, let me fix firmly the point I want to make with clear examples. Those quantities that shall be neglected must surely be held to be absolutely nothing. Nor can the infinitely small that is discussed in differential calculus differen in any way from nothing. Even less should this business be ended when the infinitely small is described by some with the example wherein the tiniest mote of dust is compare to a huge mountain or even to the whole terrestrial globe. If someone undertakes to calculate the magnitude of the whole terrestrial globe, it is the custom easily to grant him an error not only of a single grain of dust, but of even many thousands of these. However, geometric rigor shrinks from even so small an error, and this objection would be simply too great were any force granted to it. Then it is difficult to say what possible advantage might be hoped for in distinguishing the infinitely small from absolutely nothing. Perhaps they fear that if they vanish completely, then will be taken away their ratio, to which they feel this whole business leads. It is avowed that it is impossible to conceive how two absolutely nothings can be compared; They think that some magnitude must be left for them that can be compared. They are forced to admit that this magnitude is so small that it is seen as if it were nothing and can be neglected in calculations without error. Neither do they dare to assign any certain and definite magnitude, even though incomprehensibly small. Even if they were assumed to be two or three times smaller, the comparisions are always made in the same way. From this it is clear that this magnitude gives nothing necessary for undertaking a comparision, and so the comparision is not taken away even though that magnitude vanishes completely...

Still I am understanding difficult parts of his reasoning; if you guys have got any important reflections, please do add below.

 

[lavinia]

Great, Euler is really the master of analysis, for his reasoning style, chaining everything, this is really a happiest moment for me to have got words from Euler on what I wanted, here is it, enjoy it and think about it, and add anything you want to add ahead in the thread, like books, etc.. Before that wait, I am quoting a large piece from the book, for I think it is worth quoting here, as it is of Euler's ofcourse, and it is an important piece, and I don't think text is acessible to everyone, so here is it, enjoy!
Still I am understanding difficult parts of his reasoning; if you guys have got any important reflections, please do add below.

Euler is struggling with the idea of a differential which seems not to have been clarified in his time. His description of infinitesimals or differentials as "nothing at all" , if I understand what he means, is wrong.

Modern mathematics defines differentials rigorously.

Euler is questioning the idea of an infinitesimal change because it is 'smaller than any quantity' and therefore for him it is "nothing at all".

 

[Vinay080]

Euler is struggling with the idea of a differential which seems not to have been clarified in his time. His description of infinitesimals or differentials as "nothing at all" , if I understand what he means, is wrong.

Oh, no..why?

In modern mathematics defines differentials are defined rigorously.

I will be happy if you can say me some of the pages or modern math topics which defines them, or it would be great if you can say the definition itself.

Euler is questioning the idea of an infinitesimal change because it is 'smaller than any quantity' and therefore for him it is "nothing at all".

From my analysis, he is saying differential to be nothing because of people being "sloppy" in neglecting it in calculations, he wants to restore the rigour of math. This is what I understood in my further study of that book.

 

[WWGD]

Oh, no..why?I will be happy if you can say me some of the pages or modern math topics which defines them, or it would be great if you can say the definition itself.
<Snip>

.

See non-standard analysis for its development. https://www.google.com/webhp?sourceid=chrome-instant&rlz=1C1AVNE_enUS633US654&ion=1&espv=2&ie=UTF-8#q=nonstandard analysis

If you want to go deeper into it, look into model theory, compactness and lowenheim-skolem for models of different cardinalities. The non-standard Reals are a model for the axioms of the Real numbers that have uncountable cardinality. Compactness and LS guarantee that if a model exists , then models of any/every infinite cardinality also exist. All models are elementary-equivalent to each other, but not isomorphic (as models), meaning all models satisfy the 1st- order , but not 2nd order properties. The NS reals do not, unlike the standard ones, satisfy, e.g., the Archimedean principle.

 

[WWGD]

<Snip>
As I am beginner in understanding there differentials, I don't understand him. At one place, he says differentials to be "vanishing" increments or "nothing" quantities, and in the other place he says it to be "infinitely small".

Is there any other fragment of statements where Euler has analysed the concept of "differentials" properly, or is there any different meaning that can be given to the above statements to understand them?

I am still reading this book, so I don't know whether he has analysed later on, if it is the case, I will report, but until then, I want to know whether anyone has idea on his stand on this concept; papers or books on this matter by Euler or any other person would be really helpful.

I think differentials to be closely related to infinitesimals, so I want know Eulers analysis on both of them (if they are different).

Not really. A differential is a function ( a linear approximation) while an infinitesimal is a certain type of number (please see my other post of today)

 

[zinq]

The question of infinitesimals in the real numbers and elsewhere is utterly fascinating, touching as it does on both mathematics and philosophy.

Sound logical foundations for calculus were certainly not established by Newton or Leibniz. It was apparently not until the early 1800s that the rigorous definition of the limit of a function was discovered. I can't tell from quick googling whether it was Bolzano (1817), Cauchy (1823), or Weierstrass (ca. 1860?); various online sources seem to disagree.

But I love Bishop George Berkeley's 1734 zinger about the shaky foundations of calculus at that time. He said about infinitesimals:

May we not call them the Ghosts of departed Quantities?​

I am perfectly happy with the standard ε-δ definition of the limit, including its use to define differentiability and derivatives. But that sidesteps infinitesimals entirely.

I'm not entirely happy with the "nonstandard reals". Although the nonstandard reals puts infinitesimals on a rigorous foundation, it seems a contrivance fabricated to show that the concept of a infinitesimal need not involve any contradictions, by scotch-taping an infinitesimal onto the reals. That's all well and good, but how are the nonstandard reals a natural outgrowth of the ordinary reals, for which calculus was originally defined? Not clear.

Actually, I think there is a perfectly good meaning that can be given to infinitesimals in the realm of probability. An infinitesimal ξ is some kind of number for which |ξ| is supposed to be smaller than any positive real number, but greater than 0. If you pick a random real number from the unit interval, say [0,1), what is the probability of picking some preassigned number, say 1/3 ? Although I don't know that this has appeared in any research paper yet, such a probability has just the right properties to be an inifinitesimal: It's infinitely unlikely that 1/3 will be picked, but it's not impossible.

On the third hand, John H. Conway discovered / invented the number system now known as the Surreal Numbers (or just the Surreals), which includes reals and infinitesimals in an entirely natural way: https://en.wikipedia.org/wiki/Surreal_number. (It's still not clear how this relates to calculus, but the connection can be made.)

 

[lavinia]

Oh, no..why?I will be happy if you can say me some of the pages or modern math topics which defines them, or it would be great if you can say the definition itself.
From my analysis, he is saying differential to be nothing because of people being "sloppy" in neglecting it in calculations, he wants to restore the rigour of math. This is what I understood in my further study of that book.

There are several ways to define differentials as others have already described in this post. Perhaps the non-standard analysis way is the most direct or at least the most literal.

In practice, the definition that I have always seen used is the idea of the exterior derivative of a function. Any book on calculus on manifolds will go through this.

If one considers tangent vectors to be infinitesimals e.g. instantaneous velocities then functions defined on tangent vectors are also infinitesimal quantities. The exterior derivative of a function,df, also called its "differential" is a linear map on tangent spaces to a manifold. It is defined as df(v) = v.f.

They key insight is that instantaneous quantities exist on different spaces, separate spaces of their own, that are not observed directly but are still there and can be imputed from physical and geometric measurements. For instantaneous velocities, the ambient space is called that tangent bundle of the underlying space(manifold) and for differentials of functions the space is called the cotangent bundle.

For the real line one can take ∂/∂t as the basis vector for the tangent spaces. df then maps ∂/∂t into the real numbers and thus can be represented as a number i.e. the number df(∂/∂t) or in classical notation as df/dt. For the function t, dt(∂/∂t) = 1. So at any point one has the equation

df = (df(∂/∂t)) dt where this is an equation between differentials. This equation has meaning on the domain of tangent vectors to the real line, not on the real line itself.

Note that the usual approximation equation applies. That is Δf = (df(∂/∂t)Δt + terms that go to zero faster than Δt.
In sciences e.g. Physics,this equation is often written as df= (df/dt)dt when the other terms are too small to measure.

I wonder if Euler would have also thought of the idea of instantaneous velocity as nothing at all since how can one have motion without the passage of time? However, an instantaneous velocity is well defined as an element of the tangent space of a manifold and can be computed by taking limits. A function of instantaneous velocities is also instantaneous. The differential is one example but more generally one has the idea of 1 forms with are linear maps on tangent spaces but may not be exterior derivatives of functions. All of these are infinitestimals since they are functions of tangent spaces.

 

[Vinay080]

@lavinia, @zinq ,@WWGD : Thank you very much for the support. I haven't still read non-standard analysis; I will go through it and then come back with reflections. I must say that Euler has made wonderful analysis in that book, what I have posted is a small piece, he has analysed each and every "thinking root"; "you should all see it", if you are a curious hunter of knowledge.

 

[lavinia]

@lavinia, @zinq ,@WWGD : Thank you very much for the support. I haven't still read non-standard analysis; I will go through it and then come back with reflections. I must say that Euler has made wonderful analysis in that book, what I have posted is a small piece, he has analysed each and every "thinking root"; "you should all see it", if you are a curious hunter of knowledge.

Thank you for suggesting reading Euler's works, something that I have not done. I also suggest the works of Gauss and Riemann. They are jointly often considered to be the fathers of modern mathematics.

One last point about tangent spaces. The tangent bundle and its relatives like the cotangent bundle are examples of vector bundles. These are spaces which assign a vector space to each point of the manifold. In general there are many different vector bundles on a manifold but the tangent bundle plays a special role. In some sense almost all of the intrinsic topological information about smooth manifolds in general - not specific manifolds - is contained in the tangent bundle. For instance the Euler characteristic,first discovered by Euler himself but whose significance was not completely understood until the 20'th century may be computed from the tangent bundle. This early idea of infinitesimals which now is framed as the tangent bundle of a manifold has profound implications for geometry, topology, and also Physics.