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I find infinitesimals more intuitive than the formal 'limits-based' approach. I'm currently studying my old degree material but using a fairly interesting book:

I find infinitesimals more intuitive than the formal 'limits-based' approach. I'm currently studying my old degree material but using a fairly interesting book: "Full Frontal Calculus: An Infinitesimal Approach" by Seth Braver. I like his readable style. His poetic intro finally gave me an intuition why infinitesimals might be useful, compared to the good old reals: "Yet, by developing a "calculus of infinitesimals" (as it was known for two centuries), mathematicians got great insight into `real` functions, breaking through the static algebraic ice shelf to reach a flowing world of motion below, changing and evolving in time."

Qem 2y ago
Nice, I didn't know about this book. Did you try the Keisler book too, "Elementary Calculus: An Infinitesimal Approach"? See https://people.math.wisc.edu/~hkeisler/calc.html https://people.math.wisc.edu/~hkeisler/calc.html
- farrelle25 2y ago
  Thanks for the tip... it seems to mention Robinson's 'hyperreals' too...!
johnnyb_61820 2y ago
If you are going to use infinitesimals, though, it requires some additional doing for notation. The standard notation for higher-order derivatives (and partial derivatives) needs to be modified in order for them to work (but they do work great once you do this). Instead of the second derivative being "d^2y/dx^2" it is "(d^2y/dx^2) - (dy/dx)(d^2x/dx^2)" and the differentials can be manipulated just like any other entity. Additionally, you can infer this notation by simply applying the quotient rule to the first derivative (which is a quotient of infinitesimals). See more: "Extending the Algebraic Manipulability of Differentials" ( 10.48550/arXiv.1801.09553 ) "Total and Partial Differentials as Algebraically Manipulable Entities" ( 10.48550/arXiv.2210.07958 )
- Buttons840 2y ago
  You're the author of this paper? Johnathan Bartlett? If so, I used your calculus textbook to pass calculus at WGU. I had passed calculus in high school and university a long time ago, but when I finally decided to finish my degree I had to take it again, and got to choose my own text book; I liked your textbook best, I can see it sitting on my bookshelf right now. https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartlett/dp/1944918027 https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...
  - johnnyb_61820 2y ago
    Indeed! I'm glad you enjoyed the book! I hope you wrote it a nice Amazon review :)
    - Buttons840 2y ago
      I did. Glad to know you saw it.
- LegionMammal978 2y ago
  How I like to think about it is that given an expression with a derivative dy/dx, we can always insert an arbitrary variable s that varies with both x and y, so that we can obtain an ordinary quotient (dy/ds)/(dx/ds) by the chain rule, and manipulate it normally with no qualms about what it means. As you say, second (and higher) derivatives can be calculated with the quotient rule.
  - johnnyb_61820 2y ago
    What I did in my book to keep everything algebraic but not introduce weird notation is just set the derivative equal to a variable. So, say m = dy/dx. Then, the second derivative is just dm/dx. The advantage to the revised notation is that you can describe things that are difficult or impossible to describe in the other notation. For example, you can legitimately look at d^2y/d^2x (note the placement of the 2 on the denominator to see how this is different). This is a valid ratio under my system but invalid under the standard system (though I actually consider my system to be the standard system just with prior mistakes corrected).
madhadron 2y ago
It's a tradeoff. You can have excluded middle in your logic or infinitestimals in your extended reals. For mathematicians dealing with all the wild stuff coming out of studying infinities in the calculus, getting rid of excluded middle was a non-starter, so the system based on limits was created. If non-constructible proofs via contradiction aren't useful to you, as in physics, then you can certainly use infinitesimals.
- farrelle25 2y ago
  That's interesting - I read something similar in Bell's 'A primer of infinitesimal analysis' where he said the price for 'Smooth World' infinitestimals is giving up the Law of Excluded Middle (LEM). Don't really understand why (he said something about unconstrained use of LEM allows discontinuous functions...) Is there any link to Brouwer's Intuitionism where LEM is rejected too (?!) Ah it's all an interesting can of worms...
  - Ericson2314 2y ago
    https://ncatlab.org/nlab/show/real+numbers+object https://ncatlab.org/nlab/show/real+numbers+object you can definitely have real numbers without the infinitesimals in constructive math, however.
kkylin 2y ago
Side comment: anyone interested in calculus via infinitesimals may also be interested in taking a look at Radically Elementary Probability Theory by Ed Nelson: https://web.math.princeton.edu/~nelson/books/rept.pdf https://web.math.princeton.edu/~nelson/books/rept.pdf
credit_guy 2y ago
> why infinitesimals might be useful Ok, I'll ask. Why might they be useful? Is there any situation that you know of where infinitesimals can better attack a problem than the old-fashioned Calculus?
- SyzygyRhythm 2y ago
  Infinitesimal calculus is the old-fashioned calculus! It was what Newton and Leibniz invented. Limits only came into play later when mathematicians wanted a more robust foundation. But then Robinson proved that infinitesimals were perfectly rigorous. IMO, non-standard analysis is more intuitive than limit-based calculus.
  - credit_guy 2y ago
    Ok, it might be more intuitive. But in terms of applications, is there any example where there's any advantage of using infinitesimal calculus or non-standard analysis?
    - numpy-thagoras 2y ago
      Yes, any time you have to reduce something to a point for analysis in any geometric problem. You can also vary infinitesimals and utilize them not just in nonstandard analysis, but in fractional calculus, such as for inferring stock market motions. They have helpful applications in physics, especially field theory. * I can imagine, a long time from now, many elegant mathematical constructs simplified by the use of, e.g. infinitesimals, Clifford algebras, category theory, etc. There's a lot of complicated ideas that are nicely simplified, and are even more intuitive, easy to teach the fundamentals of, rather than the standard approach. I think it's important to understand that the canonical calculus approach came from rather mechanical questions in analysis and proofs, and the math is layered with that, as well as the notational conveniences of forms of calculus commonly used for electromagnetism, classical mechanics, etc. There's a lot of legacy syntax there, and we just live with it, but it's not optimal. Infinitesimals are a way to go back to applications and to better syntax.
- 2y ago
singularity2001 2y ago
> infinitesimals more intuitive than the formal 'limits-based' approach. I predict that infinitesimal/hyperreals will become mainstream in math one day the same way the 'complex' number i is now taught in school. Having probability ε instead of 0 just makes more sense (e.g. for hitting a number on an interval).
Ericson2314 2y ago
My recollection from real analysis was that I liked sequential continuity a lot (https://en.wikipedia.org/wiki/Continuous_function#Sequences_and_nets https://en.wikipedia.org/wiki/Continuous_function#Sequences_...). Sequences form a nice beginner-friendly monad (`bind` is the diagonal nth from nth), and lifting a real function over a real sequence is just `fmap`! (This is the same notion of sequence that https://clash-lang.org/ https://clash-lang.org/ uses for sequential circuits, but it skips the monad because circuits are first order.) Convergent sequences are like ordered binary tree sets, they do also form a monad, but one in a sub-category: sequentially continuous functions are precisely those that are in the domain of the underlying functor! :)

Qem 2y ago

Nice, I didn't know about this book. Did you try the Keisler book too, "Elementary Calculus: An Infinitesimal Approach"? See https://people.math.wisc.edu/~hkeisler/calc.html https://people.math.wisc.edu/~hkeisler/calc.html

farrelle25 2y ago
Thanks for the tip... it seems to mention Robinson's 'hyperreals' too...!

johnnyb_61820 2y ago

If you are going to use infinitesimals, though, it requires some additional doing for notation. The standard notation for higher-order derivatives (and partial derivatives) needs to be modified in order for them to work (but they do work great once you do this). Instead of the second derivative being "d^2y/dx^2" it is "(d^2y/dx^2) - (dy/dx)(d^2x/dx^2)" and the differentials can be manipulated just like any other entity. Additionally, you can infer this notation by simply applying the quotient rule to the first derivative (which is a quotient of infinitesimals). See more: "Extending the Algebraic Manipulability of Differentials" ( 10.48550/arXiv.1801.09553 ) "Total and Partial Differentials as Algebraically Manipulable Entities" ( 10.48550/arXiv.2210.07958 )

Buttons840 2y ago
You're the author of this paper? Johnathan Bartlett? If so, I used your calculus textbook to pass calculus at WGU. I had passed calculus in high school and university a long time ago, but when I finally decided to finish my degree I had to take it again, and got to choose my own text book; I liked your textbook best, I can see it sitting on my bookshelf right now. https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartlett/dp/1944918027 https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...
- johnnyb_61820 2y ago
  Indeed! I'm glad you enjoyed the book! I hope you wrote it a nice Amazon review :)
  - Buttons840 2y ago
    I did. Glad to know you saw it.
LegionMammal978 2y ago
How I like to think about it is that given an expression with a derivative dy/dx, we can always insert an arbitrary variable s that varies with both x and y, so that we can obtain an ordinary quotient (dy/ds)/(dx/ds) by the chain rule, and manipulate it normally with no qualms about what it means. As you say, second (and higher) derivatives can be calculated with the quotient rule.
- johnnyb_61820 2y ago
  What I did in my book to keep everything algebraic but not introduce weird notation is just set the derivative equal to a variable. So, say m = dy/dx. Then, the second derivative is just dm/dx. The advantage to the revised notation is that you can describe things that are difficult or impossible to describe in the other notation. For example, you can legitimately look at d^2y/d^2x (note the placement of the 2 on the denominator to see how this is different). This is a valid ratio under my system but invalid under the standard system (though I actually consider my system to be the standard system just with prior mistakes corrected).

madhadron 2y ago

It's a tradeoff. You can have excluded middle in your logic or infinitestimals in your extended reals. For mathematicians dealing with all the wild stuff coming out of studying infinities in the calculus, getting rid of excluded middle was a non-starter, so the system based on limits was created. If non-constructible proofs via contradiction aren't useful to you, as in physics, then you can certainly use infinitesimals.

farrelle25 2y ago
That's interesting - I read something similar in Bell's 'A primer of infinitesimal analysis' where he said the price for 'Smooth World' infinitestimals is giving up the Law of Excluded Middle (LEM). Don't really understand why (he said something about unconstrained use of LEM allows discontinuous functions...) Is there any link to Brouwer's Intuitionism where LEM is rejected too (?!) Ah it's all an interesting can of worms...
- Ericson2314 2y ago
  https://ncatlab.org/nlab/show/real+numbers+object https://ncatlab.org/nlab/show/real+numbers+object you can definitely have real numbers without the infinitesimals in constructive math, however.

kkylin 2y ago

Side comment: anyone interested in calculus via infinitesimals may also be interested in taking a look at Radically Elementary Probability Theory by Ed Nelson: https://web.math.princeton.edu/~nelson/books/rept.pdf https://web.math.princeton.edu/~nelson/books/rept.pdf

credit_guy 2y ago

> why infinitesimals might be useful Ok, I'll ask. Why might they be useful? Is there any situation that you know of where infinitesimals can better attack a problem than the old-fashioned Calculus?

SyzygyRhythm 2y ago
Infinitesimal calculus is the old-fashioned calculus! It was what Newton and Leibniz invented. Limits only came into play later when mathematicians wanted a more robust foundation. But then Robinson proved that infinitesimals were perfectly rigorous. IMO, non-standard analysis is more intuitive than limit-based calculus.
- credit_guy 2y ago
  Ok, it might be more intuitive. But in terms of applications, is there any example where there's any advantage of using infinitesimal calculus or non-standard analysis?
  - numpy-thagoras 2y ago
    Yes, any time you have to reduce something to a point for analysis in any geometric problem. You can also vary infinitesimals and utilize them not just in nonstandard analysis, but in fractional calculus, such as for inferring stock market motions. They have helpful applications in physics, especially field theory. * I can imagine, a long time from now, many elegant mathematical constructs simplified by the use of, e.g. infinitesimals, Clifford algebras, category theory, etc. There's a lot of complicated ideas that are nicely simplified, and are even more intuitive, easy to teach the fundamentals of, rather than the standard approach. I think it's important to understand that the canonical calculus approach came from rather mechanical questions in analysis and proofs, and the math is layered with that, as well as the notational conveniences of forms of calculus commonly used for electromagnetism, classical mechanics, etc. There's a lot of legacy syntax there, and we just live with it, but it's not optimal. Infinitesimals are a way to go back to applications and to better syntax.
2y ago

singularity2001 2y ago

> infinitesimals more intuitive than the formal 'limits-based' approach. I predict that infinitesimal/hyperreals will become mainstream in math one day the same way the 'complex' number i is now taught in school. Having probability ε instead of 0 just makes more sense (e.g. for hitting a number on an interval).

Ericson2314 2y ago

My recollection from real analysis was that I liked sequential continuity a lot (https://en.wikipedia.org/wiki/Continuous_function#Sequences_and_nets https://en.wikipedia.org/wiki/Continuous_function#Sequences_...). Sequences form a nice beginner-friendly monad (`bind` is the diagonal nth from nth), and lifting a real function over a real sequence is just `fmap`! (This is the same notion of sequence that https://clash-lang.org/ https://clash-lang.org/ uses for sequential circuits, but it skips the monad because circuits are first order.) Convergent sequences are like ordered binary tree sets, they do also form a monad, but one in a sub-category: sequentially continuous functions are precisely those that are in the domain of the underlying functor! :)