# Questions tagged [nonstandard-analysis]

Non-standard ordered fields are fields which have infinitesimals, that is, positive numbers which are smaller than any positive *real* number. Non-standard analysis is analysis done over such fields (e.g. hyperreal fields).

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### Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
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### Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
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### About 0.999… = 1

I've just happened to read this question on MO (that of course has been closed) and some of the answers to a similar question on MSE. I know almost nothing of nonstandard analysis and was asking ...
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### Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
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### What is infinity divided by infinity?

This should be a simple question but I just want to make sure. I know $\infty/\infty$ is undefined. However, if we have 2 equal infinities divided by each other, would it be 1? And if we have an ...
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### A few questions on nonstandard analysis

I know that nonstandard analysis is analysis plus the existence of infinitesimal numbers. Does it mean that nonstandard analysis is the same theory as $ZF+\exists$infinitesimal numbers? From what I ...
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### Has anybody ever considered “full derivative”?

When differentiating we usually take a limit and drop the infinitesimal terms. But what if not to drop anything? First, we extend the real numbers with an infinitesimal element $\varepsilon$ which ...
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### What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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### Why continuum hypothesis implies the unique hyperreal system, ${}^{\ast}{\Bbb R}$?

On page 33, Robert Goldblatt, Lectures on Hyperreals(1998): Now it has been shown under certain set-theoretic assumption called continuum hypothesis the choice of $\mathcal F$ is irrelevant: All ...
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### Is mathematical history written by the victors?

The question is the title of a 2013 publication in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics ...
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### Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an ...
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### Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
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### Is it to the students' advantage to learn the language of infinitesimals? [closed]

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...
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### What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
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### What is the point of making dx an infinitesimal hyperreal?

It seems fairly common to describe $\mathrm{d}x$ in nonstandard analysis as an infinitesimal. But after thinking it through (and then skimming Keisler's text), I can't see the point and actually think ...
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### Basic Geometric intuition, context is undergraduate mathematics

At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. For me the first instance that comes to memory was in 7th ...
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### A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
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### Euler and infinity

What do people mean when they say that Euler treated infinity differently? I read in various books that, today, mathematicians would not approve of Euler's methods and his proofs lacked rigor. Can ...
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### l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ (...
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### Why adjoining non-Archimedean element doesn't work as calculus foundation?

Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used ...
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### What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \begin{equation} \forall a,b,c \in \Bbb ...
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### Is there an algebraic-geometric solution to the problem of the Leibnizian formalism?

The precise question appears at the end of this entry. With all the recent advances in understanding infinitesimals, we still don't fully understand why Leibniz's definition of $\frac{dy}{dx}$ as ...
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### Differential form vs Hyperreal vs Dual number

Differential forms, hyperreals, and dual numbers all seem to sort of do something similar: formalize the notion of the infinitesimal. How are they related to each other, and in what ways are they ...
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### Is there a universal property for the ultraproduct?

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i$ by the identification \$x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in ...
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### Trouble understanding how the Transfer Principle is applied for the Extreme Value theorem.

I am reading Keisler's Elementary Calculus (which can be downloaded here). I am having trouble understanding his proof sketch of Extreme Value Theorem and how he is applying the Transfer Principle. ...
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### Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...