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Multiplicative Infinitesimals

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Re: Multiplicative Infinitesimals

#21

Post by Ericson2314 »

Ericson2314 wrote: post #17
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!

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 :).)

Re: Multiplicative Infinitesimals

#22

Post by farrelle25 »

madhadron wrote: post #20
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 infinitesima…

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...

Re: Multiplicative Infinitesimals

#23

Post by Ericson2314 »

farrelle25 wrote: post #22
madhadron wrote: post #20

Earlier quoted context omitted.

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 infinitesima…

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…

https://ncatlab.org/nlab/show/real+numbers+object you can definitely have real numbers without the infinitesimals in constructive math, however.

Re: Multiplicative Infinitesimals

#24

Post by Ericson2314 »

Ericson2314 wrote: post #19
gowld wrote: post #13

[flagged]

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 :).

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.

Re: Multiplicative Infinitesimals

#25

Post by kkylin »

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"…

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

Re: Multiplicative Infinitesimals

#26

Post by credit_guy »

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"…

> 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?

Re: Multiplicative Infinitesimals

#27

Post by Ericson2314 »

Smaug123 wrote: post #16

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.

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...) and then fixed it. I think it should be good now.

Re: Multiplicative Infinitesimals

#28

Post by SyzygyRhythm »

credit_guy wrote: post #26
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"…

> 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?

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.

Re: Multiplicative Infinitesimals

#29

Post by credit_guy »

SyzygyRhythm wrote: post #28
credit_guy wrote: post #26

Earlier quoted context omitted.

> 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?

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.

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?

Re: Multiplicative Infinitesimals

#30

Post by farrelle25 »

credit_guy wrote: post #26
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"…

> 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?

I think that's why I'm redoing my old Physics problem sets - but using the infinitesimal approach this time. To see is it more useful. So far the gains are modest but I find it easier to 'reason' about some of the calculations.

The author Seth Braver has two nice examples of reasoning with infinitesimals in the book intro - the first few chapters are available for free: https://www.bravernewmath.com/

Time will tell if the study will pay off. In later years of the Physics degree I ended up doing lots of algebraic manipulation without much understanding. Maybe because I had no intuitive 'feel' for the Calculus and it all felt like symbol manipulation ... As another commenter said, somehow infinitesimals allowed the giants like Newton & Lebnitz to work their way to some amazing results (especially about motion...)

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