Abstract
At least five things are called infinitesimal. The sharpest way to tell them apart is to square them — the square comes out smaller and nonzero, exactly zero, or a first-order quantity. And two of the five are not small at all. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, $dx$ is not small. For $f(x)=x^3$ at $a=2$ the gap between increment and differential, divided by $h$, falls from $6.00\times10^{-1}$ to $6.00\times10^{-5}$, while the coefficient $3a^2=12$ never moves (Section 2).
Second, a genuine infinitesimal exists. In the ordered field of rational functions $\varepsilon=1/t$ is a positive element below every $1/n$, and its square is not zero but a second-order infinitesimal (Section 3).
Third, and this is half the core. In the dual numbers $\varepsilon^2$ is exactly zero — so $f(a+\varepsilon)$ returns the derivative with error $0$, while the difference quotient bottoms out at $7.29\times10^{-8}$ for $h=10^{-8}$ and then climbs back to $9.93\times10^{-7}$ (Section 4).
Fourth, the variational $\delta$ is not small either. It is a linear functional of the direction: at $L=4$ the second variation is $-0.766299449864$ along $\sin(\pi x/L)$ and $+2.934802200545$ along $\sin(2\pi x/L)$ (Section 5).
Fifth, one infinitesimal has a first-order square. For Brownian motion $\sum(\Delta W)^2$ runs $0.932\to0.978\to0.9996\to1.0013\to1.0008$, going to $T=1$, while $\sum(\Delta t)^2$ falls from $10^{-2}$ to $10^{-6}$ (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the linearity of the differential, the ordered field of rational functions as a non-Archimedean example, the dual numbers and forward-mode automatic differentiation, the first and second variations, and the quadratic variation of Brownian motion are all standard. Nonstandard analysis is not constructed here: Section 3 shows only that a non-Archimedean ordered field exists, and neither the transfer principle nor an ultrapower is treated. Synthetic differential geometry is not entered; Section 4 uses the dual numbers as one ring for one computation. No theory of automatic differentiation is written; the comparison is one function at one point. No general theory of stochastic analysis is built; Section 6 looks at the quadratic variation only. The question whether infinitesimals exist is not answered; all this paper says is name the system and the element. Relation to earlier papers: Paper 148 wrote that $\varepsilon^2=0$ is the definition of a ring while $(dW)^2=dt$ is a fact about a path; this paper puts the other three between them and makes it a census. Paper 132 showed that Itô's $2$ is not a dimension; Section 6 borrows only its quadratic-variation side. Paper 233 counted six distinct roots of differentiable and called the Gâteaux derivative a limit along each direction; the $\delta$ of Section 5 is exactly that. Paper 236 showed that a variational "= 0" is stationary and not minimal; this paper asks what that $\delta$ itself is. What is added is the demonstration that the gap between increment and differential vanishes even after division by $h$, the check of non-Archimedean order against $1/n$, the side-by-side comparison of dual-number and finite-difference derivatives showing the latter degrade past $10^{-8}$, the sign change of the second variation with direction, and the five-decade tracking of the quadratic variation against the vanishing of $\sum(\Delta t)^2$.
1. Notation and Position
Paper 148 wrote that $\varepsilon^2=0$ is the definition of a ring while $(dW)^2=dt$ is a fact about a path. That one sentence sets two infinitesimals side by side — this paper puts the other three between them and turns it into a census.
Method: comparing sizes is useless (all five are called small). Squaring is the sharp test — and the answer splits three ways.
2. $dx$ Is Not Small
Computation 2.1: with $f(x)=x^3$ and $a=2$, compare the increment $f(a+h)-f(a)$ with the differential $3a^2h$.
| $h$ | increment | differential | gap $\div\ h$ |
|---|---|---|---|
| $10^{-1}$ | $6.610000000000$ | $6.000000000000$ | $6.100000\times10^{-1}$ |
| $10^{-2}$ | $1.206010000000$ | $1.200000000000$ | $6.010000\times10^{-2}$ |
| $10^{-3}$ | $0.120060010000$ | $0.120000000000$ | $6.001000\times10^{-3}$ |
| $10^{-5}$ | $0.001200060000$ | $0.001200000000$ | $6.000010\times10^{-5}$ |
Observation 2.2 the gap goes to zero even after division by $h$. So the differential is not an approximation to a small number but the linear function of $h$ whose error is $o(h)$ — and the coefficient $3a^2=12$ never shrinks. The phrase "$dx$ is an infinitely small number" names nothing in this system.
3. A Genuine Infinitesimal: a Non-Archimedean Ordered Field
Computation 3.1: order the field of rational functions by declaring $f>0$ when $f(t)>0$ for all large enough $t$, and set $\varepsilon=1/t$.
| $n$ | $\varepsilon<1/n$ from $t=$ | the comparison there |
|---|---|---|
| $1$ | $2.0$ | $0.500000000<1.000000000$ |
| $10$ | $11.0$ | $0.090909091<0.100000000$ |
| $100$ | $101.0$ | $0.009900990<0.010000000$ |
| $1000$ | $1001.0$ | $0.000999001<0.001000000$ |
So $0<\varepsilon<1/n$ holds for every $n$, and the field is not Archimedean. (QED)
Computation 3.2: look at the square.
| $t$ | $\varepsilon$ | $\varepsilon^2$ |
|---|---|---|
| $10$ | $1.0\times10^{-1}$ | $1.0\times10^{-2}$ |
| $10^{3}$ | $1.0\times10^{-3}$ | $1.0\times10^{-6}$ |
| $10^{6}$ | $1.0\times10^{-6}$ | $1.0\times10^{-12}$ |
| $10^{9}$ | $1.0\times10^{-9}$ | $\mathbf{1.0\times10^{-18}}$ |
Observation 3.3 $\varepsilon^2$ is not zero. It is a second-order infinitesimal, below every $\varepsilon$ — and here it parts company with the $\varepsilon$ of the next section.
Observation 3.4 (scope) what this section showed is only that a non-Archimedean ordered field exists. The power of nonstandard analysis lies in the transfer principle, and that has not been shown here — what has not been shown must not be written as though it had.
4. Half the Core: an Infinitesimal Whose Square Is Exactly Zero
Computation 4.1: implement the dual numbers $a+b\varepsilon$ with $\varepsilon^2=0$ and multiply $\varepsilon$ by itself.
| quantity | value |
|---|---|
| $\varepsilon$ | $0+1\varepsilon$ |
| $\varepsilon\cdot\varepsilon$ | $\mathbf{0+0\varepsilon}$ |
Computation 4.2: read the $\varepsilon$ coefficient of $f(2+\varepsilon)$.
| $f$ | dual-number $f'(2)$ | exact $f'(2)$ | error |
|---|---|---|---|
| $x^3$ | $12.000000000000000$ | $12$ | $\mathbf{0}$ |
| $x^3+5x$ | $17.000000000000000$ | $17$ | $\mathbf{0}$ |
| $x\cdot x\cdot x\cdot x$ | $32.000000000000000$ | $32$ | $\mathbf{0}$ |
Computation 4.3: get the same derivative from a difference quotient.
| $h$ | $(f(2+h)-f(2))/h$ | error against $12$ |
|---|---|---|
| $10^{-4}$ | $12.000600010023$ | $6.000\times10^{-4}$ |
| $10^{-6}$ | $12.000006002211$ | $6.002\times10^{-6}$ |
| $10^{-8}$ | $11.999999927070$ | $\mathbf{7.293\times10^{-8}}$ |
| $10^{-9}$ | $12.000000992884$ | $\mathbf{9.929\times10^{-7}}$ |
| $10^{-11}$ | $12.000000992884$ | $9.929\times10^{-7}$ |
Observation 4.4 the error of the difference quotient bottoms out at $h=10^{-8}$ and then rises — rounding overtakes truncation. The dual numbers choose no $h$ at all, so the error is $0$. One word, and one side takes a limit while the other does not — the same shape as the two of Paper 233 that take no difference quotient.
5. The Variational $\delta$ Is Not Small Either
Computation 5.1: with $J[y]=\int_0^L(y'^2-y^2)$, $L=4$ and $y\equiv0$, measure the first and second variations along each direction $v$.
| direction $v$ | first variation | second variation |
|---|---|---|
| $\sin(\pi x/L)$ | $0$ | $\mathbf{-0.766299449864}$ |
| $\sin(2\pi x/L)$ | $0$ | $+2.934802200545$ |
| $\sin(3\pi x/L)$ | $0$ | $+9.103304951226$ |
Observation 5.2 $\delta J$ is not a small number but a linear functional of the direction. It is the Gâteaux derivative of Paper 233 — so "$\delta y$ is a tiny change" invites the same misreading as $dx$.
Observation 5.3 the second variation changes sign with the direction: at $L=4$ it is negative along the first and positive after — which is what Paper 236 meant by a saddle for $L>\pi$.
6. The Core: an Infinitesimal Whose Square Is First Order
Computation 6.1: split Brownian motion on $[0,1]$ into $N$ steps and measure the quadratic variation.
| $N$ | $\sum(\Delta W)^2$ | $\sum|\Delta W|^3$ | $\sum(\Delta t)^2$ |
|---|---|---|---|
| $10^{2}$ | $0.932271698$ | $1.324\times10^{-1}$ | $1.0\times10^{-2}$ |
| $10^{3}$ | $0.977724833$ | $5.101\times10^{-2}$ | $1.0\times10^{-3}$ |
| $10^{4}$ | $0.999646793$ | $1.586\times10^{-2}$ | $1.0\times10^{-4}$ |
| $10^{5}$ | $1.001253242$ | $5.068\times10^{-3}$ | $1.0\times10^{-5}$ |
| $10^{6}$ | $\mathbf{1.000780569}$ | $1.598\times10^{-3}$ | $1.0\times10^{-6}$ |
Observation 6.2 $\sum(\Delta W)^2$ goes to $T=1$ and does not vanish, while $\sum(\Delta t)^2$ and $\sum|\Delta W|^3$ both do — so $(\Delta W)^2$ behaves as a first-order and not a second-order quantity. This is where Itô's correction comes from.
Observation 6.3 the infinitesimal here is neither the definition of a ring nor an element of an ordered field, but a property of a path — Paper 148's contrast between convention and fact appears directly as a difference in what the square is.
7. Separator and Fence
Observation 7.1 (the census)
| what it is called | its square | what it is |
|---|---|---|
| $dx$ | — | a linear form (not small) |
| $\varepsilon$ in a non-Archimedean field | nonzero, second order | an element of a field |
| $\varepsilon$ in the dual numbers | exactly zero | the definition of a ring |
| the variational $\delta$ | — | a linear functional (not small) |
| $dW$ | the first-order $dt$ | a property of a path |
Observation 7.2 (the separator) what divides the five is what the square comes out to be, and the answer runs three ways — smaller and nonzero, exactly zero, or first order. And the remaining two are not things one squares at all — both $dx$ and $\delta$ are linear maps.
Observation 7.3 (the fence) the word "infinitesimal" always needs an address. Which system, and which element — an element of an ordered field, the defining relation of a ring, an element of a cotangent space, a functional, or a property of a path. Without the address the word is not even false, but empty (Paper 16).
8. Census and Closing
What this paper measured.
- The gap between increment and differential vanishing after division by $h$ (2.1): the coefficient $12$ never moves.
- $\varepsilon<1/n$ for every $n$ (3.1), and $\varepsilon^2$ nonzero (3.2).
- $\varepsilon^2=0$ exactly and derivative error $0$ in the dual numbers (4.1, 4.2).
- The difference quotient bottoming out at $10^{-8}$ and degrading after (4.3): $7.29\times10^{-8}\to9.93\times10^{-7}$.
- The second variation changing sign with direction (5.1): $-0.7663$ and $+2.9348$.
- $\sum(\Delta W)^2\to1$ while $\sum(\Delta t)^2\to0$ (6.1).
References
- Robinson, A. (1966). Non-standard Analysis. Amsterdam: North-Holland.
- Clifford, W. K. (1873). Preliminary sketch of biquaternions. Proceedings of the London Mathematical Society, s1--4.
- Itô, K. (1951). On stochastic differential equations. Memoirs of the American Mathematical Society, 4.
- Gelfand, I. M., Fomin, S. V. (1963). Calculus of Variations. Englewood Cliffs: Prentice-Hall.
- Spivak, M. (1965). Calculus on Manifolds. New York: Benjamin.
- Griewank, A., Walther, A. (2008). Evaluating Derivatives, 2nd edition. Philadelphia: SIAM.
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). The $2$ in Itô's Lemma Is Not a Dimension [Paper 132]. Zenodo. 10.5281/zenodo.22080358
- Yamagishi, Y. (2026). $\varepsilon^2=0$ Is a Definition of a Ring and $(dW)^2=dt$ Is a Fact About a Path [Paper 148]. Zenodo. 10.5281/zenodo.22083237
- Yamagishi, Y. (2026). “Differentiable” Is Six Things [Paper 233]. Zenodo. 10.5281/zenodo.22127806
- Yamagishi, Y. (2026). “Boundary Condition” Is Four Things [Paper 235]. Zenodo. 10.5281/zenodo.22128989
Reference
Yamagishi, Y. (2026). “Infinitesimal” Is Five Things, and Squaring Them Gives Three Different Answers [Paper 237]. YOI: https://www.yamagishiyuuki.com/yoi/237
DOI: 10.5281/zenodo.22129135
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 237. This paper is a preprint and has not been peer reviewed.