I was wondering why an infinitesimal step in different coordinates are not the same size unless specifically stated. For example, dx and dy are both infinitesimal steps in spatial coordinates, but dx/dy is not equal to 1. Is this related to how there are different 'size' infinities?
Infinitesimal step in different coordinates
calculusinfinitesimals
Related Solutions
I was asked to write down my comments on the insights of the thread https://mathoverflow.net/questions/226277/what-is-a-grossone, which is about the framework of these articles and my thoughts. This is not really an answer as statements on this "new" infinity calculus are contradictory, so the very existence of such entities "numerical (or numeral) infinitesimal" is in doubt.
non-standard infinitely large integers
One very old introductory justification for non-standard calculus (from Reeb?) is the following: The Peano axioms define the set of natural numbers as the minimal set that contains $1,2,3$ and any other number obtained by counting up. Now there is an abstract decision to make: Do these "constructable" numbers fill up $\Bbb N$ or is there some additional unavoidable fluff that is larger than any of these "standard" numbers. With the first choice, nothing interesting happens. Formalizing the second variant gives a non-standard arithmetic in the flavour of the internal set theory of Nelson, or the ultra-filter construction of the $\Omega$-calculus of Laugwitz.
One could get the impression from http://www.theinfinitycomputer.com/The_second_paper_to_read_(Lagrange_Lecture).pdf that Sergeyevs "grossone" is obtained by taking one of these "i-large" natural numbers $N$ and setting $①=N!$. Then he restricts "his" natural numbers to the segment $[1,2,...,①]$, so that this restricted set indeed has $①$ elements, and calls all numbers $①+1,...,2①,...$ "extended natural numbers". This seems to be some kind of Humpty-Dumpty logic.
“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.” (Lewis Carroll: Through the looking glass)
non-standard "algebraic" extensions of number sets
Another, more classical, way to introduce infinite and infinitesimal "numbers" is to add to the number system an external symbol, say $X$, and the rules $1<X$, $2<X$, $3<X$, ... The arithmetic closure of this extension is the field of rational functions in $X$ with the order relation based on the definition that a rational expression is positive if the leading coefficients in numerator and denominator have the same sign. The algebraic closure of this field is the field of Puiseux series $\sum_{k\in \Bbb Z}c_kX^{k/n}$. The arithmetic of Taylor series is part of this.
The "numeral system" Sergeyev introduces is more oriented on this algebraic approach to infinities and thus also infinitesimals. It consists of finite sums $\sum_{k\in I}c_k\,①^{p_k}$ where the pairs $(c_k,p_k)$ are represented by standard floating point numbers. Arithmetic is done following usual arithmetic rules and and function evaluation seems to follow Taylor series arithmetic using not really apparent truncation rules (see the screen shot in https://mathoverflow.net/a/227536).
Despite the apparent symbolic nature of $①$ in this calculus, Sergeyev insists that this is not symbolic calculus. Despite the fact that automatic differentiation contains truncated Taylor series arithmetic as one of its tools, he insists that his techniques are different and unrelated to automatic differentiation. Operation counts are done relative to the full numerals, independent of truncation depth.
"Numerical infinitesimals" appear as elements $c\,①^p$ with $p<0$ and their combinations, most often $①^{-1}$ as initial infinitesimal. The religiously repeated claim is then that the evaluation of $f(x_0+①^{-1})$ is in no way related to Taylor series or automatic differentiation, without going into details.
If one allows all these pre-existing techniques, the patented methods as in
could be replicated by standard CAS software like Magma, using G for $①$, E for the infinitesimal $①^{-1}$
P<E>:=PuiseuxSeriesRing(RealField(8),400);
ratio:=func<p|Truncate(p*100)/100>;
G:=E^(-1);
num:=G^14-2.8+3*G^(-ratio(0.3));
den:=5*G^(ratio(7.2))+2.4*G^(-ratio(3.1));
num/den;
with output
0.20000000*E^(-34/5) - 0.096000000*E^(7/2) - 0.56000000*E^(36/5) +
0.59999999*E^(15/2) + 0.046080000*E^(69/5) + 0.26880000*E^(35/2) -
0.28800000*E^(89/5) - 0.022118400*E^(241/10) - 0.12902400*E^(139/5) +
0.13824000*E^(281/10) + O(E^(166/5))
where one would only have to replace E with G^(-1), which is string manipulation, no further calculation.
Best Answer
In the context of a modern theory of infinitesimals such as nonstandard analysis, $\frac{dy}{dx}$ can indeed be defined as a fraction of infinitesimals. Just as standard numbers can be of different sizes, infinitesimal numbers can also be of different sizes, entailing no contradiction at all.