[HN Gopher] Multiplicative Infinitesimals
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Multiplicative Infinitesimals
Author : Ericson2314
Score : 41 points
Date : 2025-01-08 19:58 UTC (3 hours ago)
(HTM) web link (github.com)
(TXT) w3m dump (github.com)
| farrelle25 wrote:
| 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."_
| Ericson2314 wrote:
| My recollection from real analysis was that I liked sequential
| continuity a lot (https://en.wikipedia.org/wiki/Continuous_func
| tion#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/ 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 wrote:
| 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
| farrelle25 wrote:
| Thanks for the tip... it seems to mention Robinson's
| 'hyperreals' too...!
| andrewla wrote:
| 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 wrote:
| > 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 wrote:
| You might be more interested in the linked page on proper
| multiplicative calculus then,
| 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. :)
| gowld wrote:
| > 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.
| tzs wrote:
| > 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 wrote:
| Thanks, fixed (check the commit log :))
| philipov wrote:
| > _Despite their formal rigor, Infinitesimals are liked in some
| settings, ..._
|
| Even better without a split clause. "In some settings" was also
| repeated.
| Ericson2314 wrote:
| I did end up doing a bigger rearranging; I think it addresses
| your point also?
| gowld wrote:
| The parent page is funny:
|
| https://github.com/Ericson2314/baccumulation/tree/main
|
| The Gen Z OP, after overexposure to blogs, has invented a new
| term for a classic "web site"
| xeonmc wrote:
| isn't this essentially Grossman's bi-geometric calculus[0]?
|
| [0] https://sites.google.com/site/nonnewtoniancalculus/brief-
| his...
| Smaug123 wrote:
| 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 Petry's
| "Analyse Infinitesimale: une presentation non standard" for an
| undergraduate first course in analysis expressed through
| nonstandard analysis.
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