Mathematics / Mathematik / Matemática
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Winfried Weber Group moderatorThe company name is only visible to registered members.Nonstandard Analysis
Hello,
I start with a discussion about "Nonstandard Analysis", introducted by Abraham Robinson in the 60's.
Before I explain the details I would like to know, who has experience in this interesting field of mathematics.
Best Regards,
Winfried Weber
- 21 Jul 2006, 10:33 am
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Dr. Andreas EckerThe company name is only visible to registered members.Re: Nonstandard Analysis
Hello Winfried,
I have some experience with Nonstandard Analaysis, but not too much.
Also it is closely related to Model Theory, the area in which I worked.
Cheers,
Andreas
- 26 Jul 2006, 10:00 pm
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Winfried Weber Group moderatorThe company name is only visible to registered members.
Re^2: Nonstandard Analysis
Dr. Andreas Geißler wrote:
Hallo Herr Weber,
ich habe ... eine schöne Abhandlung der Uni Tübingen gefunden.
Vielleicht hilft Ihnen das weiter ?
Hallo Andreas,
danke für die Abhandlung. Ist sehr interessant, ich muss mir die 72 Seiten erstmal genauer ansehen.
[Bemerkung am Rande: Statt Definition steht durchweg "Deffnition". Jaja, der Computer...]
Hast Du Dir das Skript auch durchgelesen?
Viele Grüsse,
Winfried
- 08 Aug 2006, 09:15 am
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Marc GoossensThe company name is only visible to registered members.Infinitesimals (Re: Nonstandard Analysis)
Hi Winfried,
Non-standard analysis is not specifically my line of business, but indeed an intriguing subject nonetheless.
I assume the references already suggested above should provide some more information. As an outsider, it is my impression that in actual calculus applications, non-standard infinitesimal have not quite lived up to their original promise (or hype?). Any other opinions on this?
One nice illustration recasting an infinitesimal argument of C.F. Gauss in the form of non-standard infinitesimals is given in the "Handbook of Mathematical Logic" edited by Barwise. But it seems the non-standard reals are mostly used as an illustration of model theory nowadays.
The "real life" realization of non-standard models using germs / jets of functions is noteworthy also, and of course these notions are useful in modern treatments of differential equations.
There is an argument by the great contemporary mathematician Alain Connes somewhere, as to why these (by now traditional) infinitesimals are not so useful after all. Of course, Connes, the father of non-commutative geometry, has introduced a powerful, rigorous notion of infinitesimals himself, based on notions from quantum mechanics (compact operators on Hilbert space). I should be able to dig up the Connes reference if anyone is interested.
Now I'm vaguely wondering about Yet Another Flavour of infinitesimals somewhere. Can't remember now. To do with quantum groups (Hopf algebra's)? Perhaps someone around here knows more about this?
Cheers,
Marc
This post was modified on 29 Aug 2006 at 03:20 pm.- 29 Aug 2006, 3:17 pm
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Marc GoossensThe company name is only visible to registered members.
more on infinitesimals & noncommutative geometry
Rob:
Now there's a challenge! As a mere physicist, I'm truly not an expert in any of these fields. Did a bit of soul-, mind-, file- and internet-searching. Took me better part of the day, and results not very spectacular; still - if you care to read on: I did my best.
You may find more on non-standard analysis and infinitesimals – ok if I call them infs? – on (English) Wikipedia.
Here’s some of what I collected:
• Some references? Perhaps Joel Tropp’s “Infinitesimals: History and Application” (
http://www-personal.umich.edu/~jtropp/papers/Tro99-Infinites...) is among the most accessible accounts.
And of course there is Jerome Keisler’s classic “Elementary Calculus: an Approach Using Infinitesimals”. The book is only a thousand pages or so, which is fine for a weekend; read it, and you will definitely know a lot more about infs then I do; you will find the links athttp://www.math.wisc.edu/~keisler/calc.html).
• As for simple explanations... At some point, the story always gets complicated. A “construction” of the “hyperreals” (= real numbers with infs) as extensions of the real numbers starts with sequences of (ordinary) real numbers, like (1,2,3,4,5,...), which “diverges” and is a candidate for an “infinitely large” number, or (1, 1/2, 1/3, 1/4, ...), which “becomes infinitesimally small”.
In the next step, you proceed according to a well-known scheme, by declaring certain such sequences to be “equivalent”. In this case, we want to regard sequences that “shrink equally fast” as being the same. But rigourously defining a suitable “equivalence of sequences of real numbers” involves the subtle notion of “ultrafilters” and the like.
• Essentially, the infinitesimals complete the “<” ordering of the real numbers, by adding elements “smaller than every ordinary real number, however small”, yet non-zero, and appending elements “larger then every other real number”. The result is more subtle then the classical completions with “infinity”. But similar to the classical case, there is a price to be paid: the extension destroys most of the nice algebraic (“group”) structure of the real numbers (such as every real number r having an inverse 1/r for multiplication)...
• I thought infs could also be realised in the form of jets (germs of) of real functions, but I was unable to find any reference.
• I think it’s fair to say that the appeal of infs seems mostly a nostalgic (or maybe exotic) one. Infintesimalistic (?) arguments and proofs were given by great mathematicians such as Euler, Leibniz and Gauss. It is probably true that the now current epsilon-delta approach to analysis is mundaine and a tad mechanical. But so what? Tropp points out that inf notions are helpful in topology and the theory of distributions (including Dirac “delta functions”).
• Apart from that, the hyperreals provide the paradigmatic examples of “non-standard models” in model theory (as part of formal logic). Which is nice. Yet even Wikipedia concludes that non-standard analysis has not lead to many new, original applications or discoveries.
• Another “modern” way of going infinitesimal is by way of categories, by the way, but here binary logic (“tertium non datur”) has to be abandoned.
• Now watch out: here comes the link to geometry. Indeed, the Grand Master of noncommutative geometry, Alain Connes (http://www.alainconnes.org/), has criticised “robinsonian” infs, as non-constructible entities. It appears that an existence proof of such non-standard sets requires one to adopt the axiom of choice. His own approach is based on compact operators, inspired on quantum mechanics and applied in noncommutative geometry.
• As for noncommutative geometry: you will find a lot on the web, but all of it pretty advanced stuff. Personally, I rather like the “motivation” given in Wikipedia; let me summarise it as follows:
“There is a close relationship between a space X, which is geometric in nature, and the set C(X) of (complex) numerical functions on this space. In general, such functions will form a commutative ring (meaning that f x g = g x f). Often (e.g., if X is a compact Hausdorff space), we can recover X from C(X), and therefore it makes some sense to say that X has commutative geometry. From other fields (like physics or functional analysis) we know of noncommutative rings. Reversing our procedure and starting from these, we may “reconstruct” a space with a non-commutative notion of geometry.”
Of course, this is far from saying what noncommutative geometry is really all about. That’s another story... Anyone here who can tell us more?
At any rate, and this in contrast with Robinson's infinitesimals, non-commutative geometry certainly has opened up an extremely rich avenue of original research, both in pure math and in theoretical physics!
Oh, here’s one noteworthy property of infs, which I discovered myself. Type the word “infinitesimals” ten times, and it will come out differently each time.
Whew.
Marc
This post was modified on 22 Oct 2006 at 09:26 pm.- 22 Oct 2006, 5:02 pm
