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Views on non standard analysis?

  1. Jul 8, 2012 #1
    Hey guys!

    I'm debating learning a little non standard analysis but I have two problems;
    1. I've seen a lot of critisism directed towards non standard analysis and I don't want to spend time learning about non standard analysis only to find out that it is logically flawed in some way
    2. I've had a hard time finding any textbooks on the subject and the ones I have seen are mostly out of print.

    So the questions I ask are; If you know about non standard analysis, what are your views on it and what good textbooks would you reccomend for someone looking to get started on it?

    Thanks in advance :biggrin:
     
  2. jcsd
  3. Jul 8, 2012 #2

    mathman

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    I suggest that you postpone looking into non-standard analysis until you have a good mathematics background, particularly in (standard) analysis.
     
  4. Jul 8, 2012 #3
    I've already got an okay background, I know at least the contents of rudins principles of mathematical analysis, apostols analysis book and some other bits and bobs from other places.
    If you mean anything on top of that though, I'd be happy to accept reccomendations at that level.
     
  5. Jul 8, 2012 #4
    Do you want constructive analysis?
     
  6. Jul 8, 2012 #5
    According to Wikipedia, the issue isn't with the content being logically sound or rigorous enough. The big issue is whether or not it's even needed, or whether it brings anything new to the table, something that can't already be obtained through other, existing methods.

    Here's the article: http://en.wikipedia.org/wiki/Criticism_of_non-standard_analysis

    I've never studied it, so I don't know about any textbooks. But a quick amazon search returned this, and the reviewer says that it is a reprint of a classic book in the subject: https://www.amazon.com/Non-standard...1784456&sr=1-1&keywords=Non-standard+analysis

    But also mentions that if you don't have a strong background or need an intro book, that this is better: https://www.amazon.com/Infinitesima...thematics/dp/0486428869/ref=cm_cr_dp_asin_lnk
     
  7. Jul 8, 2012 #6

    lavinia

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    I am sure that non-standard analysis is interesting since it makes sense out of the idea of an infinitesimal - something that Leibniz talked about when he invented calculus.

    But I do not think that mathematicians use it - or use it much - and would suggest learning more standard analysis.
     
  8. Jul 8, 2012 #7

    Bacle2

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    If you want to do some research of your own,

    1) Non-Standard analysis is a model for Analysis of a cardinality different from that of the

    standard model. All this follows from the compactness theorem and Lowenheim-Skolem:

    if there is a model of a certain infinite cardinality, then there are models of all cardinalities.

    2)Transfer Principle /Elementary Equivalence:

    Every first-oder statement in a given model is also true in a model of higher cardinality,

    since all of above models are what you call elementary equivalent. This is weaker than

    isomorphism between models. The usual --to avoid using the word 'standard'--

    example of a 2nd order statement that holds in non-standard but not in standard, you

    have the Archimedean Principle. Second order has to see with the objects over which you

    quantify in your statements in a given structure.

    From it, you can conclude in Standard that if x<1/n for all

    naturall n, then x is 0. In Non-Standard, x can be < 1/n and be a non-zero infinitesimal.


    Then every thing that holds in standard also holds in non-standard, but not viceversa.
     
  9. Jul 8, 2012 #8

    Bacle2

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    It may be a good idea to read arguments both for- and against- the non-standards.


    An interesting question, I think, is , given that Mathematics helps model the

    real world, which of the models is the best model?


    Moreover, there are models of the reals other than the standard and non-standard

    ones.
     
  10. Jul 8, 2012 #9

    HallsofIvy

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    Non-standard analysis is perfectly valid and gives simple "intuitive" results. For example, in non-standard analysis "dy" and "dx" have precise meanings and the derivative, dy/dx, really is a fraction so that all of the properties of the derivative are almost trivial.

    However, you pay a high price for that. In order to give a precise, logically correct, definition of "differential", and then "dx" and "dy", you have to appeal to the "compactness" property of logic- "if every finite subset of a set of axioms has a model then the entire set of axioms has a model." And students of Calculus do not have the deep background in Logic required for that.
     
  11. Jul 8, 2012 #10

    HallsofIvy

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    There simply is no "best" model because we cannot "linearly order" models. All we know of the real world comes to us through imperfect senses so that, when select a model, the best we can do is select one with axioms that approximately matches the real world.
     
  12. Jul 8, 2012 #11
    There's nothing wrong with NSA and perhaps it can give better (or at least different) intuitive ways of getting our minds around the problem of formalizing the vague notion of "arbitrarily close."

    The only issue with it is that it's truly nonstandard. Nobody uses it.

    In the 70's, Keisler's calculus text came out and it might have seemed at the time that in the future, NSA would win the pedagogy war. But it's now 40 years later and it hasn't happened.

    It's not that the subject isn't interesting or useful; it's just far enough out of the mainstream that you'd be swimming alone.
     
  13. Jul 9, 2012 #12

    Bacle2

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    I think all the models you get using compactness thm- Lowenheim Skolem are ordered

    by elementary-equivalence , aka the Transfer Principle, but I am not 100%. There are

    also models that are non-equivalent , I think, by using non-decidable statements.
     
  14. Jul 9, 2012 #13

    HallsofIvy

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    I was responding to the post about models of the real world. The "models" defined in logic are a different meaning of the word.
     
  15. Jul 10, 2012 #14

    lurflurf

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    1. Non standard analysis is logically sound and is in no way controversial (at least regards its foundation). If you care enough about calculus it is worthwhile to consider it from different directions. Some people use the fact that many theorems have shorter proofs to justify nonstandard analysis. Surely if some theorems proof were 10^4 pages using standard analysis and 10^10 pages by nonstandard analysis that would be a quite practical reduction (no such theorems have been found).

    2. An in print (~11$ by Dover) book for doing some light reading is Infinitesimal Calculus by James M. Henle. Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler is out of print, but has been made freely available by the author. Neither of these is a technical reference, they are at a level similar to a usual calculus book.
     
  16. Jul 10, 2012 #15
    This discussion reminds of a thing that's been bugging me. In differential equations you can take a something like f(x)g(y)(dy/dx) = G(x) and turn it into g(y)dy = (G(x)/f(x))dx and then integrate each side and get a solution, but why is it okay to split up dy/dx? dy/dx was first presented to me as an intuitive wave of denoting the derivative that made the chain rule easy to apply but then the numerator and denominator are split without the dx or dy ever being defined and the book shoves it under the rug saying, "this formal method is justified". Can non-standard analysis make this natural?
     
  17. Jul 10, 2012 #16

    lurflurf

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    ^Yes nonstandard analysis can make that natural at the previously mentioned cost of introducing the additional structures. This is an example of a useful idea common in mathematics, it is often easier to check an answer than it is to find it. Often given some questionable method instead making it rigorous and enumerating the cases it applies to, it is better to check the results.
     
  18. Jul 11, 2012 #17

    pwsnafu

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    NSA does make this natural. In standard analysis should be done as follows:
    ##f(x) \, g(y) \, \frac{dy}{dx} = G(x)##
    ##g(y) \, \frac{dy}{dx} = \frac{G(x)}{f(x)}##
    ##\int g(y) \, \frac{dy}{dx} \, dx= \int \frac{G(x)}{f(x)} \, dx##
    ##\int g(y) \, dy = \int \frac{G(x)}{f(x)} \,dx##
     
  19. Jul 11, 2012 #18
    Oh wow, my internet went out and I had no idea I'd get a whole page of replies :p

    To Bacle2 and HallsofIvy
    My background in logic isn't too great - it's pretty much just the material in Bourbakis set theory book

    To lurflurf
    I'll see if I can find a copy of one of those somewhere and give it a little read then!

    To Super Kirei
    This has also bothered me from time to time when I go from math mode to physics mode
     
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