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Multiplicative Infinitesimals

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Re: Multiplicative Infinitesimals

#31

Post by credit_guy »

farrelle25 wrote: post #30
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?

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 i…

> somehow infinitesimals allowed the giants like Newton & Lebnitz to work their way to some amazing results

I'm not that familiar with Leibnitz's work, but Newton understood calculus from many different angles. I heard this (probably apocryphal) saying by Feynman that you truly understand something if you understand it in three different ways.

Newton was like that with calculus, and he probably understood it in more than three ways. In particular he presented the world the theory of gravity using only geometry. Just take a look at [1], and see if you find anything that looks like limits, derivatives or integrals. You only see geometrical figures.

Newton was great at manipulating polynomials. He introduced what we call nowadays the "Newton-Raphson" method via an example of finding a root of a cubic polynomial. He never mentioned derivatives or tangents or slopes, or anything that we would now associate with calculus.

Of course, we know that Newton knew the binomial formula, some people wrongly think he invented it. What he did was that he generalized it to non-integer powers, so he could calculate the infinite series of things like sqrt(1+x) or sqrt(1-x^2). From here it doesn't take that long to derive the series for sine and cosine, especially if your name is Newton, and he did the arcsine and arctan for good measure too. (And from here he calculated many more digits of pi than anyone before him, by a good margin).

And Newton was intimately familiar with interpolation. Even today we have the concept of Newton interpolating polynomial [2]. Interpolation was indispensable in those times, even Briggs used it in his logarithmic tables which he published in 1617. Here's a quote from [3]: "Briggs’ quinquisection is actually a special case of Newton’s formula seen from a different vantage point". But "Newton was apparently unaware of Briggs’ work on finite differences and subtabulation".

[1] https://www.gutenberg.org/cache/epub/28233/pg28233-images.ht...

[2] https://en.wikipedia.org/wiki/Newton_polynomial

[3] https://inria.hal.science/inria-00543939/PDF/briggs1624doc.p...

Re: Multiplicative Infinitesimals

#32

Post by [deleted] »

Ericson2314 wrote: post #24
Ericson2314 wrote: post #19

Earlier quoted context omitted.

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.…

[deleted]

Re: Multiplicative Infinitesimals

#33

Post by numpy-thagoras »

credit_guy wrote: post #29
SyzygyRhythm wrote: post #28

Earlier quoted context omitted.

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?

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.

Re: Multiplicative Infinitesimals

#34

Post by farrelle25 »

credit_guy wrote: post #31
farrelle25 wrote: post #30

Earlier quoted context omitted.

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 i…

> somehow infinitesimals allowed the giants like Newton & Lebnitz to work their way to some amazing results I'm not that familiar with Leibnitz's work, but Newton understood calculus from many different angles. I heard this (probably apocryphal) saying by Feynman that you truly understand something if you understand it in three different ways. Newton was like that with calculus, and he probably understood it in more…

Totally agree about Newton and his 3 ways. I remember reading in Burton's History of Mathematics:

"Newton developed 3 different versions of his calculus, apparently searching for the best approach; or maybe each version served a different purpose.

- 'Infinitesimals': largely a geometric approach, - 'Fluxions': a kinematic approach, - 'Prime and ultimate ratios': his most rigorous, "algebraic" approach.

The 3 methods weren't always kept apart when solving problems. See: DT Whiteside, Mathematical Papers Isaac Newton."

You might enjoy Tristan Needham's book on Visual Differential Geometry where he really dives into Newton's geometric approach.

Thanks for the other links... must go through them. Lots of gold there.

Re: Multiplicative Infinitesimals

#35

Post by Buttons840 »

johnnyb_61820 wrote: post #18
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 enti…

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-Bartle...

Re: Multiplicative Infinitesimals

#36

Post by LegionMammal978 »

johnnyb_61820 wrote: post #18
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 enti…

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.

Re: Multiplicative Infinitesimals

#37

Post by akalin »

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....

Re: Multiplicative Infinitesimals

#38

Post by singularity2001 »

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"…

  > 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).

Re: Multiplicative Infinitesimals

#39

Post by marxisttemp »

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.

Stodgy people will insist you shouldn’t begin a sentence with a conjunction, but I agree that it’s a better sentence.

Re: Multiplicative Infinitesimals

#40

Post by johnnyb_61820 »

Buttons840 wrote: post #35
johnnyb_61820 wrote: post #18

Earlier quoted context omitted.

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 enti…

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…

Indeed! I'm glad you enjoyed the book! I hope you wrote it a nice Amazon review :)
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