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Multiplicative Infinitesimals

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Re: Multiplicative Infinitesimals

#11

Post by Ericson2314 »

philipov wrote: post #8
tzs wrote: post #6

> 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.

> Despite their formal rigor, Infinitesimals are liked in some settings, ... Even better without a split clause. "In some settings" was also repeated.

I did end up doing a bigger rearranging; I think it addresses your point also?

Re: Multiplicative Infinitesimals

#12

Post by gowld »

andrewla wrote: post #4

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…

> we'll typically just call this f_2(x) = 3x^2,

"we" who? You're projecting infinitesimals down onto reals, and then complaining that the infinitesimals are gone. That seems like a "you" problem. You can keep the ifinitesimals if you don't want to lose then.

Re: Multiplicative Infinitesimals

#14

Post by farrelle25 »

Qem wrote: post #9
farrelle25 wrote: post #2

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"…

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

Thanks for the tip... it seems to mention Robinson's 'hyperreals' too...!

Re: Multiplicative Infinitesimals

#16

Post by Smaug123 »

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.

Re: Multiplicative Infinitesimals

#17

Post by Ericson2314 »

xeonmc wrote: post #15

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

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!

Re: Multiplicative Infinitesimals

#18

Post by johnnyb_61820 »

farrelle25 wrote: post #2

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"…

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 )

Re: Multiplicative Infinitesimals

#20

Post by madhadron »

farrelle25 wrote: post #2

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"…

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.
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