Absence of limits in Sylvanus P. Thompsons Book

[dkotschessaa]

I'm currently re-learning on my pre-calculus and calculus and have been reading Sylvanus P Thompson's _Calculus Made Easy_.

I am trying to figure out the place of this book within the framework of mathematics in general. He tells you he is going to teach you "those beautiful methods which are generally called by the terrifying names of the Differential Calculus and the Integral Calculus."

He does indeed make this look very easy, and I'm able to follow along in the book by simply reading along. But it occurred to me at some point that he has left out a few things that I remember from calculus (about 13 years ago), and am made to understand that this is a different approach.

According to http://en.wikipedia.org/wiki/Calculus_Made_Easy" (the completely accurate source of information for everything in the world) "Calculus Made Easy ignores the use of limits with its epsilon-delta definition, replacing it with a method of approximation directly to the correct answer in the spirit of Leibniz, now formally justified in modern nonstandard analysis."

Could somebody first explain to me in simpler terms what this means?

Later it is explained on the http://en.wikipedia.org/wiki/Leibniz#Calculus" that "Leibniz is credited, along with Sir Isaac Newton, with the inventing of infinitesimal calculus (that comprises differential and integral calculus)."

On the page on http://en.wikipedia.org/wiki/Infinitesimal_calculus" it is then remarked that "In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Bishop Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 1734."

Am I to assume that this is referring to the fact that Mr. Thompson has us discard those parts of the equation which he regards as "infinitesimally minute?" I did find this troubling when I first read the book - though of course one winds up with a perfectly reasonable answer - and can someone explain this in regards to the use of limits as another approach?

Thanks,

Dave KA

 

[snipez90]

According to http://en.wikipedia.org/wiki/Calculus_Made_Easy" (the completely accurate source of information for everything in the world) "Calculus Made Easy ignores the use of limits with its epsilon-delta definition, replacing it with a method of approximation directly to the correct answer in the spirit of Leibniz, now formally justified in modern nonstandard analysis."

Could somebody first explain to me in simpler terms what this means?

It most likely means that the text uses http://en.wikipedia.org/wiki/Infinitesimal" until relatively recently. Leibniz, who is credited as one of the founders of calculus, probably used what would now be considered rather handwavy and informal methods that nevertheless were based on correct intuitions and thus gave correct answers (to say physical problems).

The epsilon-delta definition is the standard axiomatic definition of the limit of a function used today. It is the basis for all of the mathematical analysis you're likely to encounter as a beginning math major. Thus, if you think you might major in math or physics in the future, this definition of the limit is crucial.

Bottom line is, limits probably aren't so much absent in Thompson's text, but it's highly unlikely that the methods he uses are mathematically rigorous. However, for developing good intuition about calculus concepts (and not worrying so much about the theoretical underpinnings), this is probably less harmful than some mathematicians might suggest.

 

[dkotschessaa]

It most likely means that the text uses http://en.wikipedia.org/wiki/Infinitesimal" until relatively recently. Leibniz, who is credited as one of the founders of calculus, probably used what would now be considered rather handwavy and informal methods that nevertheless were based on correct intuitions and thus gave correct answers (to say physical problems).

The epsilon-delta definition is the standard axiomatic definition of the limit of a function used today. It is the basis for all of the mathematical analysis you're likely to encounter as a beginning math major. Thus, if you think you might major in math or physics in the future, this definition of the limit is crucial.

Thank you. This is very helpful.

Bottom line is, limits probably aren't so much absent in Thompson's text, but it's highly unlikely that the methods he uses are mathematically rigorous. However, for developing good intuition about calculus concepts (and not worrying so much about the theoretical underpinnings), this is probably less harmful than some mathematicians might suggest.

Yes - I feel like I get a decent intuitive understanding of calculus through Sylvanus' book. Though since I'll be starting the traditional calculus courses in due time I will have the benefit of both approaches, I think.

-Dave KA

 

[mathwonk]

As I recall, he simply states that in an expression of form x + x^2 + x^3 +...+x^n, all terms with exponent higher than 1, "vanish to a higher order" than the term x. Hence x is the best linear approximation (derivative) near x=0. He does not define this concept precisely, hence does not prove anything about it however.

To do so, one could use limits by saying o(x) vanishes to higher order than one if

the limit o(x)/x --> 0 as x-->0.
then it is easy to show that x^2 + x^3 is of form o(x) near x=0, provided one defines limits first.
To most people this would not make it more convincing that x is the best linear approximation to x+x^2+x^3 near x=0.

Descartes similarly threw away all terms of order higher than one to do calculus of polynomials successfully. Si as goofy as it seems, after a while I became convinced everything silvanus thompson said was actually correct and even more insightful than what I was learning in limit calculus. I.e. I understood it only after having advanced, linear approximation, calculus of several variables.

 

[Hells]

Both are exactly the same methods. Non-standard calculus just formalize it.

 

[ShinyRedCar]

I'm currently re-learning on my pre-calculus and calculus and have been reading Sylvanus P Thompson's _Calculus Made Easy_...Am I to assume that this is referring to the fact that Mr. Thompson has us discard those parts of the equation which he regards as "infinitesimally minute?" I did find this troubling when I first read the book - though of course one winds up with a perfectly reasonable answer - and can someone explain this in regards to the use of limits as another approach?

Thanks,

Dave KA

I found this to be quite reasonable after reading chapter 2 "On Different Degrees of Smallness". What did you think of the chapter?

I believe what Thompson was after was putting the beauty of calculus back into calculus. The technical details can sometimes rob students of any enthusiasm for the subject. Those going on to be math majors would have a definite interest in the limit definitions however most physics majors and engineering majors probably do not.


Best of luck to you in your studies!