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Browsing through Calculus Made Easy, I found it curious that the text begins with “On Different Degrees of Smallness” and the limit is mentioned qualitatively b

Browsing through Calculus Made Easy, I found it curious that the text begins with “On Different Degrees of Smallness” and the limit is mentioned qualitatively but not used formally. It reminds me of the preface to Elementary Calculus [1] [2] an excellent text that I discovered years after learning the subject using the limit: > The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely small number. But for the past one hundred years infinitesimals have been banished from the calculus course for reasons of mathematical rigor. Students have had to learn the subject without the original intuition. This calculus book is based on the work of Abraham Robinson, who in 1960 found a way to make infinitesimals rigorous. While the traditional course begins with the difficult limit concept, this course begins with the more easily understood infinitesimals. [1]: https://www.math.wisc.edu/~keisler/calc.html https://www.math.wisc.edu/~keisler/calc.html [2]: https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infinitesimal_Approach https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...

cuddlybacon 6y ago
While I learned calculus, both in highschool and in university, both introductions belabored the concept of an infinitesimal. I remember a good portion of both classes finding the concept confusing. EDIT: It seems like the top-level comment by user woah also finds it confusing.
- apricot 6y ago
  When I took my first calculus class, in the 1980s, we spent the first few weeks of the course just computing limits. No reason was given, and no context, just techniques. I remember having to compute the limit of 2x+1 as x goes to 3 by writing it as the limit of 2x + the limit of 1, and then saying the first limit is 2 times the limit of x, and so on. Utterly boring, especially when the teacher couldn't even answer us when we asked "if we already know we'll get 7 by setting x to 3, why do we have to write all this?" The whole thing almost put me off math. There's a time and place to introduce properties of limits, but the first month of calculus 1 isn't it.
  - canjobear 6y ago
    My first month of calculus was identical, in the 2000s.
bmitc 6y ago
You might be interested in the book A Primer of Infinitesimal Analysis by John L. Bell, which is another approach to infinitesimals that uses intuitionistic logic. https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bell/dp/0521887186 https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Be... Any interesting property of this logic and model is that all functions are infinitely differentiable. Other introductions are An Invitation to Smooth Infinitesimal Analysis by John L. Bell and Synthetic Differential Geometry by Michael Shulman. http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.pdf http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...
omginternets 6y ago
>Browsing through Calculus Made Easy, I found it curious that the text begins with “On Different Degrees of Smallness” and the limit is mentioned qualitatively but not used formally. You might want to read the chapter of Calculus Made Easy entitled Epilogue and Apologue ;) Excerpt: >Thirdly, among the dreadful things they will say about “So Easy” is this: that there is an utter failure on the part of the author to demonstrate with rigid and satisfactory completeness the validity of sundry methods which he has presented in simple fashion, and has even dared to use in solving problems! But why should he not? You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus. You're not wrong, of course, but neither is Silvanus P. Thompson.

cuddlybacon 6y ago

While I learned calculus, both in highschool and in university, both introductions belabored the concept of an infinitesimal. I remember a good portion of both classes finding the concept confusing. EDIT: It seems like the top-level comment by user woah also finds it confusing.

apricot 6y ago
When I took my first calculus class, in the 1980s, we spent the first few weeks of the course just computing limits. No reason was given, and no context, just techniques. I remember having to compute the limit of 2x+1 as x goes to 3 by writing it as the limit of 2x + the limit of 1, and then saying the first limit is 2 times the limit of x, and so on. Utterly boring, especially when the teacher couldn't even answer us when we asked "if we already know we'll get 7 by setting x to 3, why do we have to write all this?" The whole thing almost put me off math. There's a time and place to introduce properties of limits, but the first month of calculus 1 isn't it.
- canjobear 6y ago
  My first month of calculus was identical, in the 2000s.

bmitc 6y ago

You might be interested in the book A Primer of Infinitesimal Analysis by John L. Bell, which is another approach to infinitesimals that uses intuitionistic logic. https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bell/dp/0521887186 https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Be... Any interesting property of this logic and model is that all functions are infinitely differentiable. Other introductions are An Invitation to Smooth Infinitesimal Analysis by John L. Bell and Synthetic Differential Geometry by Michael Shulman. http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.pdf http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...

omginternets 6y ago

>Browsing through Calculus Made Easy, I found it curious that the text begins with “On Different Degrees of Smallness” and the limit is mentioned qualitatively but not used formally. You might want to read the chapter of Calculus Made Easy entitled Epilogue and Apologue ;) Excerpt: >Thirdly, among the dreadful things they will say about “So Easy” is this: that there is an utter failure on the part of the author to demonstrate with rigid and satisfactory completeness the validity of sundry methods which he has presented in simple fashion, and has even dared to use in solving problems! But why should he not? You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus. You're not wrong, of course, but neither is Silvanus P. Thompson.