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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 ...
Joel David Hamkins's user avatar
16 votes
Accepted

Fermat's opponents

Maybe the following article http://arxiv.org/abs/1306.5973 (Is mathematical history written by the victors?) and references therein will be useful :-). Fermat's life and work is carefully ...
Zurab Silagadze's user avatar
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 ...
Carlo Beenakker's user avatar
11 votes

Fermat's opponents

The point (5) is at most doubtful. It was made by Mahoney in his book on Fermat, but Mahoney's interpretation of what Digby writes to Wallis does not correspond to what Digby actually writes, as ...
Joël's user avatar
  • 25.8k
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 $z$...
nombre's user avatar
  • 2,417
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 $...
coudy's user avatar
  • 18.6k
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 ...
Timothy Chow's user avatar
  • 79.3k
4 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 ...
Carlo Beenakker's user avatar
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 ...
Aaron Meyerowitz's user avatar
2 votes
Accepted

Multiplicative infinitesimals in q-analogs?

There is more than one question that is being asked here so I will leave aside the one about $q$-analogues for the simple reason that one can take any question, say $X$, in mathematics, and ask for ...
Mikhail Katz's user avatar
  • 15.6k
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 ...
Mikhail Katz's user avatar
  • 15.6k

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