# 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 learnt some non-standard analysis and generalisations of limits introduced via (ultra-)filters. The way I see things is strongly inspired by Tao's short article on these issues. The rough and basic idea (to make precise which piece of mathematics I am talking about) seems to be that almost all structure on $\mathbb{R}$ can be "imported" to the sequences in $\mathbb{R}^\mathbb{N}$ by the following construction:

1. Take an ultrafilter $p$ (an subset of $\mathcal{P}(\mathbb{N})$ that can be thought as the family of all "large" subsets of $\mathbb{N}$). Use this to get a well-behaved notion of $p$-almost all $n$.

2. Declare a sentence about real numbers $\phi(x,y,\dots)$ to be $p$-true about sequences $(x_n)_{n=1}^\infty,(y_n)_{n=1}^\infty,\dots$ if and only if $\phi(x_n,y_n,\dots)$ holds for almost all $n$. Notice that $p$-truth same rules as "ordinary" truth.

3. Transfer as much structure as you need in accordance to $p$-truth. For instance $x + y = z$ should mean that $x_n + y_n = z_n$ for $p$-almost all $n$. If needs be, divide out the equivalence relation $x \sim x' \Leftrightarrow x_n = x_n' \text{ for$p$-almost all$n$}$. Call $\mathbb{R}^\mathbb{N}$ hyperreals with this structure, and denote it by $*\mathbb{R}$

The key facts about these hyperreals seem to be that:

1. There is the transfer principle, stating very roughly that same statements are true for reals and hyperreals, as long as you keep them intrinsic (i.e. quantification is only on the underlying set ($\mathbb{R}$ or $*\mathbb{R}$ rather than some external set like $\mathbb{N}$) and not too complex (no quantification over sets, etc.)

2. There is the countable saturation stating very roughly that if any finite number of sentences $\phi_1, \phi_2, \dots, \phi_n$ can be simultaneously realised, then all these sentences can be simultaneously realised.

3. There are infinitesimals (like the one given by $x_n = \frac{1}{n}$), infinities (like the one given by $x_n = n$), and ideas such as (uniform) continuity/differentiability etc. admit more elegant definitions and proofs.

4. A notion of a fully generalised limit emerges naturally by taking $p\text{-}\!\lim_n x_n = l$ where $l$ is the unique real that is infinitesimally close to $x$ (provided $x$ is bounded).

Being a student of mathematics, I have a rather technical interest in those issues. I have read a number of proofs using ultrafilters (mostly in ergodic theory/number theory, so it did not involve much analysis as such), and I might try to use these concepts working on mathematical problems.

I am going to present a short talk on the concepts discusses above, and I have learnt that the audience will include many non-mathamaticians: there will be physicists, economists, even biologists. Hence the question:

The question: How can one motivate non-standard analysis to someone with just a basic mathematical background? What are the reasons why a beginner mathematician would find it interesting? Motivation known to me includes mainly:

• Existence of actual infinitesimals/infinities and fully generalised limits.
• Simplified $\varepsilon$-management. (but: only interesting if you actually get your hands dirty with $\varepsilon/\delta$-type calculus)
• Applications of ultrafilters in combinatorics, like Ramsey's theorem for graphs or Arrow's theorem for voting.

Is there more? Are the above valid and convincing? Are there some additional conceptual key properties of $*\mathbb{R}$ that I did not mention?

Additional request: I would also highly appreciate any corrections to the thoughts presented above. They are rather vague by design, but where my thinking is flawed or suboptimal, I will be grateful to anyone who sets it right.

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Since your audience will have people who have struggled through compulsory calculus classes as part of their scientific education, emphasizing how easy certain manipulations become in a nonstandard context might be a good way to grab their attention. People who have had trouble proving, say, $(d/dx) x^2 = 2x$ using $\epsilon$-$\delta$ methods might be amused to see a NS proof. On the other hand if they've never done $\epsilon$-$\delta$ stuff it may be wasted on them. –  mt_ Sep 25 '12 at 12:26
@mt_: Thanks for the comment! My main concern with this kind of justification ($\varepsilon/\delta$-arguments simplified, as in explicit computation of derivatives, chain rule, the fundamental theorem of arithmetic, etc.) is that many among the audience might never have seen the actual proofs behind the classical analysis. In these cases, this justification amounts to showing a not-too-nice piece of computation and saying that with the standard approach things are uglier still. That being said, the argument is very much valid for all those who had to do the proofs. –  Feanor Sep 25 '12 at 17:20
you're welcome. You might be interested in the notes from a nonstandard talk I once gave: sites.google.com/site/matthewtowers/home/nsa.pdf It was to mathematicians, but aimed at people who did not know any nsa. –  mt_ Sep 25 '12 at 17:26

I think you're downplaying "simplified $\epsilon$-management": that's one of the main reasons why you want infinitesimals in the first place!

e.g. the formula for the limit

$$\lim_{x \to a} f(x) = \operatorname{st} f(\epsilon)$$

whenever the limit exists and $f$ and $a$ are standard and $\epsilon$ is a nonzero infinitesimal. $\text{st}$ is the "standard part": i.e. rounding a limited number to the nearest standard number. (also, given the conditions on $a$ and $f$, this limit exists if and only if the right hand side has the same value for all nonzero infinitesimals $\epsilon$)

Another example is that there's a really neat argument that any continuous standard function on $[a,b]$ has a maximum. The sketch is:

• "Enumerate" the interval $[a,b]$ by splitting it into (hyper)finitely many intervals
• The set of left endpoints is (hyper)finite, and so among them, $f$ has a maximum at, say, $c$.
• We must have $f(\operatorname{st} c) = \operatorname{st} f(c)$, and it is the maximum of $f(x)$ at standard points.

(and then, transfer the theorem to get the corresponding fact for all continuous functions and closed intervals)

Another example of why having infinitesimals is useful off the top of my head is the definition of the differential. Given a variable $x$ and infinitesimal variable $\Delta x$ if $y$ depends on $x$ via $y = f(x)$ (with $f$ differentiable), we introduce a $dy$ dependent on $x$ and $\Delta x$ by $dy = f'(x) \Delta x$. $dy/dx$ is literally ordinary division, without any thing subtle going on.

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Thank you! What you wrote is most enlightening. When you express limit like that it indeed becomes strikingly simple. I think it is also similar with (uniform) continuity. –  Feanor Sep 23 '12 at 11:40
A reason why I might be downplaying the role of $\varepsilon$-management is that (essential as it is), you only make use of it inside proofs (and definitions). Thus, it is useful only if you are ready to get your hands dirty and carry out (or follow) a proof, and usually you can get to the same place through classical $\varepsilon/\delta$-manipulation (though not that fast). As I see it, through non-standard analysis you can make an argument shorter and more intuitive, which is of course very important, but not so convincing in my particular situation. –  Feanor Sep 23 '12 at 11:47
@Feanor: One of the overarching patterns of calculus is to look at a problem, zoom in and work out the infinitesimal picture, and weave them back together to get the answer. Having honest infinitesimals makes it much easier to make intuition correspond directly to the math. Implicit in your comment is the observation that in the standard approach, intuition doesn't correspond to the math, and there is a much wider gulf between the "vague intuitive ideas I use to understand the problem" and "the actual mathematics involved in carrying out these ideas". –  Hurkyl Sep 23 '12 at 12:09
Integrals are another good example, because in practice (even without learning non-standard ideas), people form an idea of an integral adding up infinitesimal areas. I really like the example of the proof of attaining a maximum because it vividly demonstrates this. Compactness is an important idea, because once you've used it a lot and understand it, you get an idea that it's some sort of analog to finiteness. But in the non-standard approach you can use things that are actually (hyper)finite. –  Hurkyl Sep 23 '12 at 12:12
For a mixed audience, a bit of history might be useful, the doubts about the "infinitesimal calculus" of Leibniz (and Newton), Bishop Berkeley's quip about the "ghosts of departed quantities." Continued use of infinitesimals into the $19$-th century, and by applied people much longer. Rigorous treatment (Weierstrass et al) banishing infinitesimals. Abraham Robinson showing infinitesimals could be used in a precise way. –  André Nicolas Sep 23 '12 at 15:54