6 ms·

Multiplicative Infinitesimals

83 points by Ericson2314 2y ago

tzs 2y ago
> Infinitesimals are liked, despite their formal rigor (in most settings), are liked in some settings, like informally solving differential equations, and other applied tasks. There seems to be one too many "are liked" in that sentence. Deleting either one of them makes the sentence read a lot better. I think deleting the second one reads better than deleting the first one.
- Ericson2314 2y ago
  Thanks, fixed (check the commit log :))
- philipov 2y ago
  > Despite their formal rigor, Infinitesimals are liked in some settings, ... Even better without a split clause. "In some settings" was also repeated.
  - Ericson2314 2y ago
    I did end up doing a bigger rearranging; I think it addresses your point also?
  - marxisttemp 2y ago
    Stodgy people will insist you shouldn’t begin a sentence with a conjunction, but I agree that it’s a better sentence.
gowld 2y ago
[flagged]
- Ericson2314 2y ago
  OK, it is a fair that a "classic 'web site'" is a 1-person wiki is this. But, I must set the record straight that I am a Gen Y citing a Gen X for coming up with the idea. No Gen Zs were involved :).
  - Ericson2314 2y ago
    Thinking some more, I suppose I am excited about never-ending editing and curation (1) and hypertext (2), and the 1990s were excited too, but a bit distracted by these revolutionary concepts by flashy hypermedia. What is being conveyed, edited, linked (text vs media) is not so interesting to me --- in the same way that the cool thing about container data structures is that they are parameterized over their contents. And I have a bias for text because it is "sober" and less likely to rot my only-so-non-fragile mind than other flashier things.
    - deleted 2y ago
      [deleted]
xeonmc 2y ago
isn't this essentially Grossman's bi-geometric calculus[0]? [0] https://sites.google.com/site/nonnewtoniancalculus/brief-history https://sites.google.com/site/nonnewtoniancalculus/brief-his...
- Ericson2314 2y ago
  I cite (on the adjacent page linked at the top) doi:10.1016/j.jmaa.2007.03.081 which sites them. But, as far as I know, all that stuff is doing regular formal limits-based calculus. I haven't yet come-across the notion of these non-Newtonian infinitesimals in particular. Thanks for the link to that page though, it is a good starting point for seeing what other things may be going on!
  - Ericson2314 2y ago
    Thanks again, that link is really good! I think the bi in bigeometic refers to how the domain and range are both made greometric? That corresponds to elasticity, but not the multiplicative derivation and integral in my examples. That (which is exactly the concepts from the paper I cited above) would be called by them the geometric calculus (mono, no bi) I think. (I since did a larger edit which makes good use of that :).)
Smaug123 2y ago
By the way, the computation involving nonstandard reals is correct (to the best of my ten-year-old memory of studying this stuff). As usual, I will recommend Goldblatt's _Lectures on the Hyperreals_ for an intro to how it all works, and Pétry's "Analyse Infinitésimale: une présentation non standard" for an undergraduate first course in analysis expressed through nonstandard analysis.
- Ericson2314 2y ago
  Thanks. I actually now think it was a bit incomplete or even wrong 3 hours ago when you wrote that :), but then I thought a bit harder, read a bit more (other Wikipedia and https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Davis.pdf https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers...) and then fixed it. I think it should be good now.
akalin 2y ago
I'm partial to Caratheodory's definition of the derivative, which avoid limits like the infinitesimal approach, but doesn't pull in all the extra baggage that come with infinitesimals (if you do it rigorously). djb (yes, that one) has a pretty good primer on it: https://cr.yp.to/papers/calculus-19970403-retypeset20220326.pdf https://cr.yp.to/papers/calculus-19970403-retypeset20220326....
- Ericson2314 2y ago
  Oh yes that's a good write-up! At the very end > I follow the Kurzweil-Henstock approach to integration. ... I learned about this from advertisements by Robert G. Bartle in the Bulletin and the Monthly. I had just read https://math.vanderbilt.edu/schectex/ccc/gauge/letter/ https://math.vanderbilt.edu/schectex/ccc/gauge/letter/ so I got that reference :)
farrelle25 2y ago
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...
andrewla 2y ago
I'm generally a little skeptical about approaches that treat infinitesimals in a symbolic computation. The approach typically is to solve traditional analysis problems but throw in some infinitesimals and show that you can get the same answers. But if you approach infinitesimals from first principles I feel like you run into a lot of problems. For example it is almost always the case that you can remove higher order infinitesimals, like (dx)^2, when computing things like derivatives. But this always necessitates a step where you translate from infinitesimals to "standard" reals, and then continue on your merry way. We happily round away the higher-order terms when we compute something like ((x + dx)^3 - x^3) / dx, but if we're continuing to do infinitesimal math they may become relevant again. Call that function, f_1(x) = 3x^2 + 3xdx + dx^2; we'll typically just call this f_2(x) = 3x^2, but these functions are not equivalent if we then proceed to compute (f(x) - 3x^2) / dx, which, presumably, we can just do because we've admitted this horror of a syntax into our formal language. I'm very skeptical of this being useful outside of being able to reason about trivial limits for this reason.
- scotty79 2y ago
  > you can remove higher order infinitesimals, like (dx)^2, when computing things like derivatives I always found that iffy and a bit of a (completely legal) hack. It's a nice point that what enables this hack is promptly leaving the world of infinitesimals and retreating back to reals.
- Ericson2314 2y ago
  You might be more interested in the linked page on proper multiplicative calculus then, https://github.com/Ericson2314/baccumulation/blob/main/math/multiplicative-calculus.md https://github.com/Ericson2314/baccumulation/blob/main/math/... . That, in turn, is mostly just a retelling of doi:10.1016/j.jmaa.2007.03.081 I submit two claims basically, in response to what you are saying: - The "proper" multiplicative calculus with limits is no more broken than its additive counterpart - These multiplicative infinitesimals are no more broken than their additive counterparts It seems like you were trying to hold these multiplicative infinitesimals to the standard of calculus with limits, and rejecting them on those grounds. To that rejection, I just say that these infinitesimals were never meant to meet that standard. :)

83 points by Ericson2314 2y ago

tzs 2y ago

> Infinitesimals are liked, despite their formal rigor (in most settings), are liked in some settings, like informally solving differential equations, and other applied tasks. There seems to be one too many "are liked" in that sentence. Deleting either one of them makes the sentence read a lot better. I think deleting the second one reads better than deleting the first one.

Ericson2314 2y ago
Thanks, fixed (check the commit log :))
philipov 2y ago
> Despite their formal rigor, Infinitesimals are liked in some settings, ... Even better without a split clause. "In some settings" was also repeated.
- Ericson2314 2y ago
  I did end up doing a bigger rearranging; I think it addresses your point also?
- marxisttemp 2y ago
  Stodgy people will insist you shouldn’t begin a sentence with a conjunction, but I agree that it’s a better sentence.

gowld 2y ago

[flagged]

Ericson2314 2y ago
OK, it is a fair that a "classic 'web site'" is a 1-person wiki is this. But, I must set the record straight that I am a Gen Y citing a Gen X for coming up with the idea. No Gen Zs were involved :).
- Ericson2314 2y ago
  Thinking some more, I suppose I am excited about never-ending editing and curation (1) and hypertext (2), and the 1990s were excited too, but a bit distracted by these revolutionary concepts by flashy hypermedia. What is being conveyed, edited, linked (text vs media) is not so interesting to me --- in the same way that the cool thing about container data structures is that they are parameterized over their contents. And I have a bias for text because it is "sober" and less likely to rot my only-so-non-fragile mind than other flashier things.
  - deleted 2y ago
    [deleted]

xeonmc 2y ago

isn't this essentially Grossman's bi-geometric calculus[0]? [0] https://sites.google.com/site/nonnewtoniancalculus/brief-history https://sites.google.com/site/nonnewtoniancalculus/brief-his...

Ericson2314 2y ago
I cite (on the adjacent page linked at the top) doi:10.1016/j.jmaa.2007.03.081 which sites them. But, as far as I know, all that stuff is doing regular formal limits-based calculus. I haven't yet come-across the notion of these non-Newtonian infinitesimals in particular. Thanks for the link to that page though, it is a good starting point for seeing what other things may be going on!
- Ericson2314 2y ago
  Thanks again, that link is really good! I think the bi in bigeometic refers to how the domain and range are both made greometric? That corresponds to elasticity, but not the multiplicative derivation and integral in my examples. That (which is exactly the concepts from the paper I cited above) would be called by them the geometric calculus (mono, no bi) I think. (I since did a larger edit which makes good use of that :).)

Smaug123 2y ago

By the way, the computation involving nonstandard reals is correct (to the best of my ten-year-old memory of studying this stuff). As usual, I will recommend Goldblatt's _Lectures on the Hyperreals_ for an intro to how it all works, and Pétry's "Analyse Infinitésimale: une présentation non standard" for an undergraduate first course in analysis expressed through nonstandard analysis.

Ericson2314 2y ago
Thanks. I actually now think it was a bit incomplete or even wrong 3 hours ago when you wrote that :), but then I thought a bit harder, read a bit more (other Wikipedia and https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Davis.pdf https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers...) and then fixed it. I think it should be good now.

akalin 2y ago

I'm partial to Caratheodory's definition of the derivative, which avoid limits like the infinitesimal approach, but doesn't pull in all the extra baggage that come with infinitesimals (if you do it rigorously). djb (yes, that one) has a pretty good primer on it: https://cr.yp.to/papers/calculus-19970403-retypeset20220326.pdf https://cr.yp.to/papers/calculus-19970403-retypeset20220326....

Ericson2314 2y ago
Oh yes that's a good write-up! At the very end > I follow the Kurzweil-Henstock approach to integration. ... I learned about this from advertisements by Robert G. Bartle in the Bulletin and the Monthly. I had just read https://math.vanderbilt.edu/schectex/ccc/gauge/letter/ https://math.vanderbilt.edu/schectex/ccc/gauge/letter/ so I got that reference :)

farrelle25 2y ago

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...

andrewla 2y ago

I'm generally a little skeptical about approaches that treat infinitesimals in a symbolic computation. The approach typically is to solve traditional analysis problems but throw in some infinitesimals and show that you can get the same answers. But if you approach infinitesimals from first principles I feel like you run into a lot of problems. For example it is almost always the case that you can remove higher order infinitesimals, like (dx)^2, when computing things like derivatives. But this always necessitates a step where you translate from infinitesimals to "standard" reals, and then continue on your merry way. We happily round away the higher-order terms when we compute something like ((x + dx)^3 - x^3) / dx, but if we're continuing to do infinitesimal math they may become relevant again. Call that function, f_1(x) = 3x^2 + 3xdx + dx^2; we'll typically just call this f_2(x) = 3x^2, but these functions are not equivalent if we then proceed to compute (f(x) - 3x^2) / dx, which, presumably, we can just do because we've admitted this horror of a syntax into our formal language. I'm very skeptical of this being useful outside of being able to reason about trivial limits for this reason.

scotty79 2y ago
> you can remove higher order infinitesimals, like (dx)^2, when computing things like derivatives I always found that iffy and a bit of a (completely legal) hack. It's a nice point that what enables this hack is promptly leaving the world of infinitesimals and retreating back to reals.
Ericson2314 2y ago
You might be more interested in the linked page on proper multiplicative calculus then, https://github.com/Ericson2314/baccumulation/blob/main/math/multiplicative-calculus.md https://github.com/Ericson2314/baccumulation/blob/main/math/... . That, in turn, is mostly just a retelling of doi:10.1016/j.jmaa.2007.03.081 I submit two claims basically, in response to what you are saying: - The "proper" multiplicative calculus with limits is no more broken than its additive counterpart - These multiplicative infinitesimals are no more broken than their additive counterparts It seems like you were trying to hold these multiplicative infinitesimals to the standard of calculus with limits, and rejecting them on those grounds. To that rejection, I just say that these infinitesimals were never meant to meet that standard. :)