What are the applications of Nonstandard Analysis in mathematical physics?

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In summary, the conversation is discussing the use and applications of NSA (non-standard analysis) in mathematical physics, specifically in the areas of differential equations, analysis of singularities, resolution methods, and bifurcation. The conversation also mentions two different perspectives on NSA, the Robinson point of view and the internal set theory of Nelson.
  • #1
MathematicalPhysicist
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can someone please provide a link to an ebook about nsa?

p.s
what are the application of this theory in mathematical physics?

thanks in advance.
 
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  • #2
To find books about NSA is very difficult. Now, you want the Robinson point of view or the internal set theory of Nelson?
In math. physics they are used for differential equations and the analysis of singularirties, as well as resolution mmethods and bifurcation.
 
  • #3
Originally posted by rutwig
Now, you want the Robinson point of view or the internal set theory of Nelson?
could you please ellaborate about them both?
 
  • #4
Originally posted by loop quantum gravity
could you please ellaborate about them both?

The work of Robinson is the analysis made with infinitesimals, while the Nelson IST is an extension of the ZF-axioms (plus choice axiom).
 

1. What is nonstandard analysis?

Nonstandard analysis is a branch of mathematics that extends the traditional real number system to include infinitesimal and infinite numbers. It was first developed by Abraham Robinson in the 1960s as an alternative to the more classical approach to real analysis.

2. What are the applications of nonstandard analysis?

Nonstandard analysis has applications in various fields of mathematics, including calculus, differential equations, and functional analysis. It has also been used in physics and economics to model continuous processes and systems.

3. How does nonstandard analysis differ from traditional analysis?

Traditional analysis is based on the concept of limits, where infinitesimal and infinite quantities are not allowed. Nonstandard analysis, on the other hand, allows for the use of infinitesimals and infinities as part of the number system.

4. Is nonstandard analysis widely accepted in mathematics?

Nonstandard analysis has gained acceptance in the mathematical community, although it is still considered a relatively new and developing field. It has been used to prove results that were previously unattainable using traditional analysis, and has sparked new research and developments in various areas of mathematics.

5. Are there any drawbacks to using nonstandard analysis?

Some mathematicians argue that nonstandard analysis is not rigorous enough and relies too heavily on intuition. It also requires a thorough understanding of the theory and can be difficult to apply in certain situations. However, many believe that the benefits of using nonstandard analysis outweigh these drawbacks.

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