"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."
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! :)
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 )
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...
The advantage to the revised notation is that you can describe things that are difficult or impossible to describe in the other notation. For example, you can legitimately look at d^2y/d^2x (note the placement of the 2 on the denominator to see how this is different). This is a valid ratio under my system but invalid under the standard system (though I actually consider my system to be the standard system just with prior mistakes corrected).
In practice, I always end up needing to work in cotangents, but deriving them is always roundabout in terms of the limit definition of pushforwards. Never found a nice way to swap which is primary and which is secondary, but it feels like there should be a clean view of it that way somewhere.
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...
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?
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.
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...)
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...
"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.
> infinitesimals more intuitive than the formal 'limits-based' approach.
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.
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.
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. :)
"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.
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.
Even better without a split clause. "In some settings" was also repeated.
[0] https://sites.google.com/site/nonnewtoniancalculus/brief-his...
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!
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 :).)
djb (yes, that one) has a pretty good primer on it: https://cr.yp.to/papers/calculus-19970403-retypeset20220326....