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So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta arguments (both these points are debatable).

However, although many theorems have been proven by non-standard analysis and transferred via the transfer principle, as far as I know all of these results were already known to be true. So, my question is:

Is there an example of a result that was first proved using non-standard analysis? To wit, is non-standard analysis actually useful for proving new theorems?

Edit: Due to overwhelming support of Francois' comment, I've changed the title of the question accordingly.

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Note that your question is really about how helpful non-standard analysis is. If you wanted to know how unhelpful it is, you would ask for theorems that cannot be proved using non-standard methods. – François G. Dorais Feb 24 '10 at 22:57
I believe that Ben Green, Terry Tao, and Tamar Ziegler are writing there forthcoming paper on the Inverse Conjecture for the Gowers norm (which combined with earlier work of Green and Tao will resolve many cases of the Hardy--Littlewood conjectures on linear equations in primes, including precise asymptotics for primes in arithmetic progressions) in the language of non-standard analysis. That seems pretty helpful! (By the way, I strongly recommend Terry Tao's blog for several discussions of the applicability of non-standard analysis to "everyday" mathematics.) – Emerton Feb 25 '10 at 2:53
I added the nonstandard-analysis tag. – Joel David Hamkins Feb 25 '10 at 14:23

16 Answers 16

up vote 23 down vote accepted

From the Wikipedia article:

the list of new applications in mathematics is still very small. One of these results is the theorem proven by Abraham Robinson and Allen Bernstein that every polynomially compact linear operator on a Hilbert space has an invariant subspace. Upon reading a preprint of the Bernstein-Robinson paper, Paul Halmos reinterpreted their proof using standard techniques. Both papers appeared back-to-back in the same issue of the Pacific Journal of Mathematics. Some of the ideas used in Halmos' proof reappeared many years later in Halmos' own work on quasi-triangular operators.

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Great, this is what I was looking for. Thanks for the links. This certainly does answer my question, as Halmos was actually one of the original posers of that problem. It is quite interesting that the papers appear back-to-back in the same journal. – Tony Huynh Feb 24 '10 at 22:53
As Greg Lawler put it in his article in the recent volume dedicated to Nelson: "There are some theorems that were first published with nonstandard proofs but, at least in all the cases where I understand the result, they could have been done standardly." In a footnote he adds, "Of course, it is harder to answer the question: would the proofs have been found without nonstandard analysis? In fact, there are probably some proofs that have been done originally using nonstandard analysis but the author chose to write a standard proof instead." – Steve Huntsman Feb 24 '10 at 22:54
The reason for choosing standard proofs over nonstandard ones is obvious, and Lawler himself brings it up in the same article. Proving something with NSA hurts when trying to communicate results to a wide audience. In this sense NSA and experimentation are in the same boat--they can help, but generally behind the scenes. – Steve Huntsman Feb 24 '10 at 23:01
The story repeats itself many times. For example, Kamae's proof of the ergodic theorem using nonstandard analysis was rewritten by Weiss, and the two articles appeared back-to-back. I interpret this as antipathy to Robinson himself: "Look, your methods aren't so innovative as you tell everybody!" Perhaps if Robinson had been more likable, we could have a nicer analysis already. – Kevin O'Bryant Feb 26 '11 at 2:15
@katz: Wikipedia evolves. If you look at the version which Steve Huntsman quoted from (considering that this post has not been edited since it was first posted, that'd be the version from Feb 22, 2010), the quote is accurate.… – Willie Wong Apr 8 '13 at 11:00

The other answers are excellent, but let me add a few points.

First, with a historical perspective, all the early fundamental theorems of calculus were first proved via methods using infinitesimals, rather than by methods using epsilon-delta arguments, since those methods did not appear until the nineteenth century. Calculus proceeded for centuries on the infinitesimal foundation, and the early arguments---whatever their level of rigor---are closer to their modern analogues in nonstandard analysis than to their modern analogues in epsilon-delta methods. In this sense, one could reasonably answer your question by pointing to any of these early fundamental theorems.

To be sure, the epsilon-delta methods arose in part because mathematicians became unsure of the foundational validity of infinitesimals. But since nonstandard analysis exactly provides the missing legitimacy, the original motivation for adopting epsilon-delta arguments appears to fall away.

Second, while it is true that almost any application of nonstandard analysis in analysis can be carried out using standard methods, the converse is also true. That is, epsilon-delta arguments can often also be translated into nonstandard analysis. Furthermore, someone raised with nonstandard analysis in their mathematical childhood would likely prefer things this way. In this sense, the preference between the two methods may be a cultural matter of upbringing.

For example, H. Jerome Keisler wrote an introductory calculus textbook called Elementary Calculus: an infinitesimal approach, and this text was used for many years as the main calculus textbook at the University of Wisconsin, Madison. I encourage you to take a look at this interesting text, which looks at first like an ordinary calculus textbook, except that in the inside cover, next to the various formulas for derivatives and integrals, there are also listed the various rules for manipulating infinitesimals, which fill the text. Kiesler writes:

This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960. Robinson's modern infinitesimal approach puts the intuitive ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners to understand than the more common approach via limits.

Finally, third, some may take your question to presume that a central purpose of nonstandard analysis is to provide applications in analysis. But this is not correct. The concept of nonstandard models of arithmetic, of analysis and of set theory arose in mathematical logic and has grown into an entire field, with hundreds of articles and many books, with its own problems and questions and methods, quite divorced from any application of the methods in other parts of mathematics. For example, the subject of Models of Arithmetic is focused on understanding the nonstandard models of the first order Peano Axioms, and it makes little sense to analyze these models using only standard methods.

To mention just a few fascinating classical theorems: every countable nonstandard model of arithmetic is isomorphic to a proper initial segment of itself (H. Friedman). Under the Continuum Hypothesis, every Scott set (a family of sets of natural numbers closed under Boolean operations, Turing reducibility and satisfying Konig's lemma) is the collection of definable sets of natural numbers of some nonstandard model of arithmetic (D. Scott and others). There is no nonstandard model of arithmetic for which either addition or multiplication is computable (S. Tennenbaum). Nonstandard models of arithmetic were also used to prove several fascinating independence results over PA, such as the results on Goodstein sequences, as well as the Paris-Harrington theorem on the independence over PA of a strong Ramsey theorem. Another interesting result shows that various forms of the pigeon hole principle are not equivalent over weak base theories; for example, the weak pigeon-hole principle that there is no bijection of n to 2n is not provable over the base theory from the weaker principle that there is no bijection of n with n2. These proofs all make fundamental use of nonstandard methods, which it would seem difficult or impossible to omit or to translate to standard methods.

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Regarding NSA vs epsilons and deltas, isn't 'not using the Axiom of Choice unnecessarily' a good reason to use the latter? – HJRW Feb 25 '10 at 21:01
Well, many ordinary uses of epsilon-delta also use choice. For example, to know that the epsilon-delta definition of continuity for a function on the reals is equivalent to the convergent sequence characterization relies on AC, since you need to pick the points inside those delta-balls. – Joel David Hamkins Feb 25 '10 at 21:19
Joel, I enjoyed your answer and learned from it, but I wondered whether "nonstandard analysis exactly provides the missing legitimacy" [of early calculus] was overstating it. I'm more familiar with the "other" way of putting infinitesimals on a firm footing, that of Synthetic Diff Geom (as e.g. in Bell's text A Primer of Infinitesimal Analysis). As I understand it, a crucial difference is that the infinitesimals of SDG can, for instance, have square equal to 0, but the infinitesimals of NSA can't. I'd guess that to be an important part of providing that "missing legitimacy". Any thoughts? – Tom Leinster Feb 27 '10 at 2:21
Tom, I didn't mean to suggest that NSA is the only possible justification of infinitesimals. It is true, however, that many early calculus writers described infinitesimals as having all the properties of real numbers, except infinitely smaller (smaller than 1/n for every natural number n). That idea sounds like a primordial Trasfer Principle, which holds for NSA but not SDG. – Joel David Hamkins Feb 27 '10 at 12:56
... I concur with Joel David's opinion that this is in fact "a primordial Transfer Principle". The connection between Leibniz's Law of continuity and the Los-Robinson Transfer Principle was explored in a paper by David Sherry and myself in the Notices of the AMS, see – Mikhail Katz Apr 11 '13 at 8:50

Nonstandard hulls of spaces are used all the time in Banach space theory, so much so that books devote sections to the construction of ultraproducts of Banach spaces (e.g. Absolutely summing operators by Diestel, Jarchow, and Tonge). There are cases where NSA is used to prove the existence of an estimate, yet no one knows how directly to compute an estimate. For example, the unconditional constant of any basis for the span of the first n unit basis vectors in the James' space of sequences of bounded quadratic variation must go to infinity, but the only known proof involves NSA.

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In 1986 C. Ward Henson and H. J. Keisler published “On the Strength of Nonstandard Analysis” (The Journal of Symbolic Logic, Vol. 51, No. 2 (Jun., 1986), pp. 377-386), which is a seminal contribution to the meta-mathematics of nonstandard analysis. Since their result bears directly on the issue in this thread which has been reopened after laying dormant for some time now, and since no reference to their work is referred to in the original thread, I am taking the liberty of quoting the introduction to Henson and Keisler’s important paper (which I believe is as current today as when it was published).

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis. The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes. This suggests that mathematical practice usually takes place in a conservative extension of some system of second order arithmetic, and that it is difficult to use the higher levels of sets. In this paper we shall consider systems of nonstandard analysis consisting of second order nonstandard arithmetic with saturation principles (which are frequently used in practice in nonstandard arguments). We shall prove that nonstandard analysis (i.e. second order nonstandard arithmetic) with the $\omega_{1}$-saturation axiom scheme has the same strength as third order arithmetic. This shows that in principle there are theorems which can be proved with nonstandard analysis but cannot be proved by the usual standard methods. The problem of finding a specific and mathematically natural example of such a theorem remains open. However, there are several results, particularly in probability theory, whose only known proofs are nonstandard arguments which depend on saturation principles; see, for example, the monograph [Ke]. Experience suggests that it is easier to work with nonstandard objects at a lower level than with sets at a higher level. This underlies the success of nonstandard methods in discovering new results. To sum up, nonstandard analysis still takes place within ZFC, but in practice it uses a larger portion of full ZFC than is used in standard mathematical proofs.

[F] S. FEFERMAN. Theories of finite type related to mathematical practice, Handbook of mathematical logic (J. Barwise, editor), North-Holland, Amsterdam, .1977, pp. 913-971.

[Ke] H. J. KEISLER, An infinitesimal approach to stochastic analysis, Memoirs of the American Mathematical Society, No. 297 (1984).

[S] S. SIMPSON, Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations? JSL, vol. 49 (1984), pp. 783-802.

It is perhaps worth adding that Keisler (making use of work of Avigad) subsequently published a sequel to his paper with Henson in which he introduces what might be regarded as a system of Reverse Mathematics for nonstandard analysis with the hope of being able to establish the strength of particular theorems proved using nonstandard analysis. (See “The Strength of Nonstandard Analysis” by H.J. Keisler in The Strength of Nonstandard Analysis ed. By imme van den berg and vitor nerves, Springer, 2007).

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I first understood what the Thurston-type-compactification of the space of properly strictly convex real projective structures on a closed surface was using non-standard methods. What had been murky and confusing was suddenly clear. I have struggled with the question of whether or not to use NSA in the written proof. It is so much easier to use NSA I think we will.

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The asymptotic cone of a metric space (and hence of a finitely generated group endowed with the word metric) is constructed using non-standard analysis, and has been used to prove many nice theorems. To take just one example, asymptotic cones are an important tool in the proof that mapping class groups are quasi-isometrically rigid.

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Freiman conjectured a classification of finite sets $A$ of integers that have

$$|A+A| = 3|A|-3+b$$

for some $0\leq b \leq |A|/3-2$. Renling Jin recently resolved this using nonstandard analysis. He has quite a few other nice results that appeared first with nonstandard analysis.

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Indeed, I have heard Renling Jin say that although one could translate (many of) his arguments to standard methods, it is better not to do so; they are more illuminating in their nonstandard form. – Joel David Hamkins Feb 28 '10 at 2:07
The first link sent me to a Not Found page. – Todd Trimble Apr 8 '13 at 13:25

This reminded me of a talk by Mircea Mustata in which he mentioned that non-standard analysis type arguments were used to prove some things related to algebraic geometry. I can't remember what the talk was about, but I found the paper that it was based on:

The paper mentions that later Kollár found proofs avoiding these techniques (but they are similar in spirit).

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Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first proved via non-standard methods:

1) $T_n$ is closed in $\mathbb R$ for all $n$.

2) The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emerged later, but the first one used non-standard technique.

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Ah, Sam and me gave the same answer within 55 seconds of each other (:. Sorry I did not see Sam's answer. – Hailong Dao Feb 24 '10 at 23:01

I think the only known solution to the local version of the Hilbert's fifth problem heavily uses nonstandard analysis. To be more precise the result is: every locally euclidean local group is locally isomorphic to a Lie group. You can find details in Isaac Goldbring's paper.

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In mathematical economics, one often faces the following problem: One wants to formalize the idea of a large, relatively anonymous group of people (an atomless measure space of agents) that all face some risk that is iid of these people. Since there are lots of people, this risk should cancel out in the aggregate by some law of large numbers. The expost empirical distribution should be the ex ante distribution of the risk. If one uses something like the unit interval endowed with Lebesgue measure, this does not work. Most sample realizations are not measurable in that case.

Yeneng Sun has shown that there are exact laws of large numbers with a continuum of random variables for certain types of measure spaces. The only known examples were obtained using the Loeb measure construction that relies heavily on NSA. Later, Konrad Podczeck has shown how to construct appropriate measure spaces using conventional methods.

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Gromov was writing in one of his books (among other things) about some new mathematics coming from nonstandard analysis. Another example is proving that some statistical field theories (and lattice QFTs) are well-defined by Sergio Albeverio et al. (look at their book about that kind of applications to physics). Kiesler has been emphasising that some functional spaces are much richer in nonstandard analysis and that this power is one of the main arguments for the theory. Analysts say that one should look for applications where one has several degrees of infinitesimals or asymptotics, to somewhat reduce fitting complicated estimates to satisfy all.

There are some other approaches to infinitesimals which are not nonstandard analysis (no general transfer principle), but are similar in spirit, namely the synthetic differential geometry.

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I am not aware of any comments by Gromov directly on nonstandard analysis. In his book "metric structures for riemannian and non-riemannian spaces" , page 97, he does comment on the construction of asymptotic cones using nonprincipal ultrafilters, and cites the paper by van den Dries and Wilkie from 1984. In a recent interview, he praised their work more explicitly, but still without mentioning nonstandard analysis. – Mikhail Katz Apr 8 '13 at 12:28

Here is one paper with some results I have only seen being done in non-standard analysis so far, perhaps it is helpful to you:

A mathematical proof of the existence of trends in financial time series by Michel Fliess & C´edric Join

From the abstract: "We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis [...] Those trends, which might coexist with some altered random walk paradigm and efficient market hypothesis, seem nevertheless difficult to reconcile with the celebrated Black-Scholes model. They are estimated via recent techniques stemming from control and signal theory. Several quite convincing computer simulations on the forecast of various financial quantities are depicted. We conclude by discussing the role of probability theory."

See also this question/answers on Mathoverflow

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I just came across a 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals", see where probability and stochastic analysis are done without having to develop the complexities of measure and integration theory first. Ever since E.Nelson, such an approach is called "radically elementary" and it really is. What this proves is the new result that stochastic calculus can be done without measure theory.

To give a historical parallel, recall that Leibniz's mentor in mathematics was Huygens. When Huygens first learned of Leibniz's invention of infinitesimal calculus, Huygens was sceptical, and wrote to Leibniz that he is merely doing what Fermat and others have done before him in a different language. What Huygens failed to recognize immediately (but did recognize later) was the generality of the methods and the lucidity of the presentation of Leibniz's new approach. The Nelson-Herzberg approach to stochastic calculus is in a way more significant than merely a new "result", since it provides a new methodology.

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Steve Huntsman's claim attributed to wikipedia that "the list of new applications in mathematics is still very small" is patently false. In fact, I was unable to find such a claim there. To mention just the most famous results, there is the recent work by T. Tao et al, by I. Goldbring on the local version of Hilbert's 5, Albeverio (several applications in math physics), Arkeryd (see his piece in the American Mathematical Monthly at in hydrodynamics, the works on "canards" in perturbation theory, Jin's work in additive number theory, as well as numerous applications in statistics and economics. Robinson's work also occasioned a critical re-evaluation of whig history dominated by a reductive epsilontist agenda.

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@katz: Wikipedia evolves. See the version from which Steve Huntsman quoted when he wrote his answer three years ago:… – Willie Wong Apr 8 '13 at 11:01
(BTW, several of the results you mention are already discussed in the various other answers to this question below, and it would be great if you can add links to the ones which aren't [for example, a link or actual citation reference to the relevant papers of Arkeryd would be wonderful!]) – Willie Wong Apr 8 '13 at 11:09
The version you cite dates from 2010. The claim was as false in 2010 as it is in 2013. It is not appropriate to hide behind anynomous claims posted in the public domain if such claims are incorrect. – Mikhail Katz Apr 8 '13 at 11:12
I make no comment on the verity of the sentiments expressed by that quote. I take issue with your statement "In fact, I was unable to find such a claim there", which for better or for worse sounds like you are accusing Steve Huntsman of fabricating the quote out of thin air. – Willie Wong Apr 8 '13 at 11:51
You are putting words in my mouth. Most people know that wiki is a work in progress. If this claim was deleted, there must have been good reasons for this. Since posting material on wiki involves little personal responsibility, it is inappropriate to rely on negative claims made there. My objection to Huntsman's presentation of his comment stands. – Mikhail Katz Apr 8 '13 at 12:15

Edward Nelson was working on a book on NSA mentioned here:

His existing book "Radically elementary probability theory" (linked from that page) uses some NSA. I've been wanting to read it but don't understand much of it.

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That's related (and interesting), but doesn't directly address the question, viz. what results have been proven first using NSA. – Robert Haraway Apr 8 '13 at 21:02

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