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Questions tagged [infinitesimals]

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In constructive set theory, is it consistent for there to be a ring that models smooth infinitesimal analysis?

In a constructive set theory such as CZF, it is consistent to assume that every function $f : \mathbb R \to \mathbb R$ is continuous. However, it is not consistent to assume that every such function ...
Christopher King's user avatar
9 votes
2 answers
2k views

Are the models of infinitesimal analysis (philosophically) circular?

Infinitesimal analysis (by which I mean that originating from topos theory---not the nonstandard analysis of Robinson) seeks to recover the pre-limit notions of calculus (which are sufficiently useful ...
Duncan W's user avatar
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4 votes
1 answer
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Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$

Now we fix an ultrafilter of $\mathbb{N}$ that contains the cofinite filter, consider a hyperreal field ${}^{*}\mathbb{R}$. Let $\varepsilon$ be a positive infinitesimal. We doubt that a power series ...
M masa's user avatar
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6 votes
1 answer
708 views

What does "ultimately vanishing" mean? (Needham)

In the Prologue of the book Visual Differential Geometry and Forms by Needham the notion of two functions $A(\epsilon)$ and $B(\epsilon)$ being ultimately equal is defined: it means that $$ \lim_{\...
user489693's user avatar
-5 votes
1 answer
170 views

Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones? [closed]

Below, we interpret divergent integrals as germs of partial integrals at infinity: $$\int_0^\infty f(x) dx=\operatorname{bigpart} \int_0^\omega f(x) dx$$ where $\operatorname{bigpart}$ means taking ...
Anixx's user avatar
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1 vote
0 answers
74 views

infinitesimal generators for G/G/1 queue

I read the infinitesimal generator for the M/M/1 queue and thought to generalize to the G/G/1 queue. More specifically, though the queue length process is not Markovian anymore, we could consider an ...
Robert's user avatar
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38 votes
1 answer
1k views

Rigid non-archimedean real closed fields

Update. The question has been recently answered in the positive by David Marker and Charles Steinhorn (as in indicated in Marker's answer). Note that Remark 3 below is now expanded by reference to a ...
Ali Enayat's user avatar
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24 votes
1 answer
2k views

Has anything (other than what is in the obituary written by M. Noether) survived of Paul Gordan's defense of infinitesimals?

Question. Has anything other than what can be guessed from this obituary written by Max Noether survived of the 'defense' of infinitesimals that Paul Gordan gave in his doctoral disputation on March 1,...
Peter Heinig's user avatar
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1 vote
1 answer
361 views

Has anybody proposed such a generalization of integration?

Has anybody ever proposed a generalization of integration in which an integral of a function that has non-zero values only on a countable set produces non-zero (but maybe infinitesimal) result? For ...
Anixx's user avatar
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13 votes
2 answers
2k views

Fermat's opponents

It is well known among historians of Fermat that, while his technique of adequality prepared the ground for the general framework later developed by Leibniz and Newton, Fermat himself gave very little ...
Mikhail Katz's user avatar
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12 votes
1 answer
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Multiplicative infinitesimals in q-analogs?

Risking to be downvoted, here is a very lightweight question. In various fields - say, algebraic geometry, nonstandard analysis, synthetic differential geometry - infinitely small quantities, i. e. ...
მამუკა ჯიბლაძე's user avatar
1 vote
0 answers
125 views

Obtaining graphics of functions in non-standard analysis [closed]

In the context of $R(\varepsilon)$ or more broad fields, Levi-Civita field or $No(\omega_1)$, how can we obtain the graphics of functions on the infinitesimal range? For instance, it is alleged that ...
Anixx's user avatar
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3 votes
0 answers
169 views

Analysing functions on zero-length intervals and super-small values

Suppose a function that has a pole in $x=0$: Here we see the graphic of the real part of the Gamma function. As we can see on it, there is a vertical line at $x=0$ that comes from $-\infty$ to $\...
Anixx's user avatar
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21 votes
3 answers
2k views

Felix Klein on mean value theorem and infinitesimals

This is a reference request prompted by some intriguing comments made by Felix Klein. In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...
Mikhail Katz's user avatar
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7 votes
2 answers
514 views

Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book Robinson, A.; Laurmann, J. A. Wing theory....
Mikhail Katz's user avatar
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3 votes
1 answer
217 views

Can elements of Weil algebras be detected by maps into truncated symmetric algebras?

Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R. Such algebras form the basis of the Weil approach to differential geometry, pioneered ...
Dmitri Pavlov's user avatar
11 votes
3 answers
2k views

Nontrivial trivial integrals

I posted this question to stackexchange and after 24 hours it's got five votes and no answers, so let's see if mathoverflow can say more than that. Consider two propositions in geometry: ...
Michael Hardy's user avatar