22
votes
Accepted
Are the models of infinitesimal analysis (philosophically) circular?
It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of ...
- 236k
18
votes
Accepted
Rigid non-archimedean real closed fields
Charles Steinhorn and I have answered this question positively by constructing a rigid non-archimedean real closed field of transcendence degree 2. Our preprint is now posted on arxiv.
https://arxiv....
- 3,530
12
votes
Accepted
Has anything (other than what is in the obituary written by M. Noether) survived of Paul Gordan's defense of infinitesimals?
Paul Gordan's theses were published in De linea geodetica and digitised by Google, from which I reproduce the relevant page:
Translation:
I. The method of functional division, proposed by the ...
- 188k
8
votes
In hyperreal field, can ln(ε) and ln(ω) be expressed as infinite sums?
To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the ...
- 236k
7
votes
Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$
Let $A$ denote the convex subring of hyperreal numbers $y$ for which there exists an $n \in \mathbb{N}$ with $-\varepsilon^{-n}<y<\varepsilon^{-n}$. This has the set $\mathfrak{m}$ of numbers $z$...
- 2,519
5
votes
Has anybody proposed such a generalization of integration?
In some sense, that's what the theory of tempered distributions provides.
If $f$ is integrable, we have
$$\hat{f}(0) = \int_{\bf R} f(x) \, dx$$
If $f$ is a tempered distribution, the expression $...
- 18.7k
5
votes
What does "ultimately vanishing" mean? (Needham)
Since on page 275 Needham refers to Newton for the notion of an "ultimately vanishing" quantity, I would interpret that in the sense of Newton, where an ultimately vanishing quantity is an ...
- 188k
4
votes
Are the models of infinitesimal analysis (philosophically) circular?
Yes, there is some degree of philosophical circularity, if you take the view that the only "non-circular" way to build up a subject is to start with conceptually simple primitives, and work ...
- 82.6k
3
votes
In hyperreal field, can ln(ε) and ln(ω) be expressed as infinite sums?
For positive $\epsilon$, the expression $\ln \epsilon$ will be equal to its power series at $x=1$ (in the $\delta, N$ sense).
To help avoid any misunderstanding that may arise for readers of this ...
- 16.6k
2
votes
Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones?
This isn't an answer, just a long comment.
If this is from an established field, and I'd guess it is, that needs to be part of the question. Not knowing one, I will blindly sally forth because I am ...
- 30.1k
2
votes
Felix Klein on mean value theorem and infinitesimals
In our 2018 publication in Journal of Humanistic Mathematics we analyze the criterion of effectiveness as formulated by Klein and by Fraenkel, briefly summarize the controversy (over a proof of the ...
- 16.6k
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