Several Questions About Smooth Infinitesimal Analysis

In summary, the conversation discusses smooth infinitesimal analysis and its applications to logic and set theory. The questions focus on the meaning of certain notations and the use of the Kock-Lawvere axiom in proving the law of excluded middle in SIA. The expert also raises concerns about the implications of working in a domain where traditional mathematical statements may not hold.
  • #1
Mike_bb
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Hello.
I read about smooth infinitesimal analysis and I have several questions:

1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6)

2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2) (https://www.fuw.edu.pl/~kostecki/sdg.pdf , page 21)

Thanks!
 

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  • #2
Mike_bb said:
I read about smooth infinitesimal analysis and I have several questions:
It is not possible to understand SIA without solid understanding of logic, set theory and real analysis; do you have these?

Mike_bb said:
1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6)
This is standard notation (although perhaps not typeset very well). ## \varepsilon \cdot 1 ## means ε multiplied by 1. Photo 1 is part of a proof that the law of the excluded middle (LEM) does not hold in SIA.

Mike_bb said:
2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2) (https://www.fuw.edu.pl/~kostecki/sdg.pdf , page 21)
I don't understand what you are asking. In photo 2 the author uses the Kock-Lawvere Axiom and LEM to derive a statement that is a contradiction, again proving that LEM does not hold in SIA.

What is your interest in SIA? Are you really comfortable working in a domain where ## x^2 = 0 ## does not mean ## x = 0 ##? Or where ## x \ne 0 ## being not true does not mean ## x = 0 ##?
 
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1. What is Smooth Infinitesimal Analysis (SIA)?

Smooth Infinitesimal Analysis (SIA) is an alternative approach to traditional calculus. It is based on the concept of infinitesimals, which are numbers that are infinitely small but not equal to zero. SIA provides a rigorous foundation for calculus by using infinitesimals to define the derivative and integral, rather than limits.

2. How is SIA different from traditional calculus?

SIA differs from traditional calculus in its use of infinitesimals. In traditional calculus, limits are used to define the derivative and integral, while in SIA, infinitesimals are used instead. This allows for a more intuitive and direct approach to calculating derivatives and integrals, and also allows for the use of non-standard analysis.

3. What are the advantages of using SIA?

There are several advantages to using SIA. One of the main advantages is that it provides a more intuitive and direct approach to calculus, making it easier to understand and apply. SIA also allows for the use of non-standard analysis, which can provide new insights and solutions to problems. Additionally, SIA can be used to solve problems that are difficult or impossible to solve using traditional calculus.

4. What are the limitations of SIA?

While SIA has many advantages, it also has some limitations. One limitation is that it is not as widely accepted or studied as traditional calculus, so there may be less resources and support available for learning and using SIA. Additionally, SIA can be more complex and abstract, which may make it more difficult for some people to understand and apply.

5. How is SIA used in real-world applications?

SIA has been used in various real-world applications, such as physics, economics, and computer science. In physics, SIA has been used to study quantum mechanics and general relativity. In economics, SIA has been used to model economic systems and analyze market behavior. In computer science, SIA has been used to develop algorithms and analyze data. However, SIA is still a relatively new and niche approach, so its applications may not be as widespread as traditional calculus.

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