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They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor through

by ianai 3y ago

They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor through https://mathworld.wolfram.com/NonstandardAnalysis.html https://mathworld.wolfram.com/NonstandardAnalysis.html I have long thought such would be easier to work with on modern computers.

monista 3y ago
Note though, that nonstandard analysis isn't compatible with more "intuitionistic" https://en.wikipedia.org/wiki/Axiom_of_determinacy https://en.wikipedia.org/wiki/Axiom_of_determinacy (in place of axiom on choice), which free you from Banach–Tarski paradox and have some other appealing properties.
- zozbot234 3y ago
  On the contrary, in an intuitionistic setting you can use https://en.wikipedia.org/wiki/Synthetic_differential_geometry https://en.wikipedia.org/wiki/Synthetic_differential_geometr... and https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis which aren't nearly as easy to phrase classically. (Sure you can phrase things in terms of toposes, but that's just a roundabout way of saying that you're operating in a constructive setting.)
- yuppiemephisto 3y ago
  Can do it in pure ZF: https://www.youtube.com/watch?v=YE68yptMMeQ https://www.youtube.com/watch?v=YE68yptMMeQ https://u.cs.biu.ac.il/~katzmik/spot.html https://u.cs.biu.ac.il/~katzmik/spot.html
scythe 3y ago
Nonstandard analysis is one of those things that sounds nice in the abstract, but even though I first learned about it more than ten years ago, I have yet to see a worked didactic-level example of how you would, for example, obtain the derivative of sine using nonstandard analysis, or derive the natural logarithm.
- ducttapecrown 3y ago
  I literally saw the sin one yesterday, so I'll rewrite it!! Let e be an infinitesimal. We write st(x) for the function dropping the infinitesimal part of a number. Then f'(x) = st(1/e (f(x+e)-f(x))) Now use the angle sum identity and cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos... So st(1/e (sin(x+e)-sin(x))) = st(1/e (sin(x)cos(e)+cos(x)sin(e)-sin(x)) = st(1/e (sin(x)(1-e^2)+cos(x)e - sin(x))) = st(1/e (sin(x) - sin(x) - e^2 sin(x) + cos(x)e)) = st(e sin(x) + cos(x)) = cos(x)
  - ianai 3y ago
    "cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos..." Isn't that from the definition of cos and sin, even geometrically?
    - karatinversion 3y ago
      The version of rigorous infinitesimals I've seen didn't let you have these identities, because e^2 != 0 (because it still maintains the field axiom xy = 0 => x=0 V y=0); but the argument still goes through by carrying the whole power series in place of these simplifications – you can write the higher order terms as e*f(x, e) and drop them when you evaluate st(...).
  - scythe 3y ago
    You can get the relationship: 1 - x^2 < cos(x)^2 < 1 / (1 + x^2) from ordinary trigonometry, although you can't just do this with nilsquare infinitesimals; you need a more sophisticated setup.
- yuppiemephisto 3y ago
  https://alok.github.io/2022/09/12/y539043/ https://alok.github.io/2022/09/12/y539043/
adgjlsfhk1 3y ago
Infinitesimals are commonly used in scientific computing (see forward mode automatic differentiation).

by ianai 3y ago

monista 3y ago

Note though, that nonstandard analysis isn't compatible with more "intuitionistic" https://en.wikipedia.org/wiki/Axiom_of_determinacy https://en.wikipedia.org/wiki/Axiom_of_determinacy (in place of axiom on choice), which free you from Banach–Tarski paradox and have some other appealing properties.

zozbot234 3y ago
On the contrary, in an intuitionistic setting you can use https://en.wikipedia.org/wiki/Synthetic_differential_geometry https://en.wikipedia.org/wiki/Synthetic_differential_geometr... and https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis which aren't nearly as easy to phrase classically. (Sure you can phrase things in terms of toposes, but that's just a roundabout way of saying that you're operating in a constructive setting.)
yuppiemephisto 3y ago
Can do it in pure ZF: https://www.youtube.com/watch?v=YE68yptMMeQ https://www.youtube.com/watch?v=YE68yptMMeQ https://u.cs.biu.ac.il/~katzmik/spot.html https://u.cs.biu.ac.il/~katzmik/spot.html

scythe 3y ago

Nonstandard analysis is one of those things that sounds nice in the abstract, but even though I first learned about it more than ten years ago, I have yet to see a worked didactic-level example of how you would, for example, obtain the derivative of sine using nonstandard analysis, or derive the natural logarithm.

ducttapecrown 3y ago
I literally saw the sin one yesterday, so I'll rewrite it!! Let e be an infinitesimal. We write st(x) for the function dropping the infinitesimal part of a number. Then f'(x) = st(1/e (f(x+e)-f(x))) Now use the angle sum identity and cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos... So st(1/e (sin(x+e)-sin(x))) = st(1/e (sin(x)cos(e)+cos(x)sin(e)-sin(x)) = st(1/e (sin(x)(1-e^2)+cos(x)e - sin(x))) = st(1/e (sin(x) - sin(x) - e^2 sin(x) + cos(x)e)) = st(e sin(x) + cos(x)) = cos(x)
- ianai 3y ago
  "cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos..." Isn't that from the definition of cos and sin, even geometrically?
  - karatinversion 3y ago
    The version of rigorous infinitesimals I've seen didn't let you have these identities, because e^2 != 0 (because it still maintains the field axiom xy = 0 => x=0 V y=0); but the argument still goes through by carrying the whole power series in place of these simplifications – you can write the higher order terms as e*f(x, e) and drop them when you evaluate st(...).
- scythe 3y ago
  You can get the relationship: 1 - x^2 < cos(x)^2 < 1 / (1 + x^2) from ordinary trigonometry, although you can't just do this with nilsquare infinitesimals; you need a more sophisticated setup.
yuppiemephisto 3y ago
https://alok.github.io/2022/09/12/y539043/ https://alok.github.io/2022/09/12/y539043/

adgjlsfhk1 3y ago

Infinitesimals are commonly used in scientific computing (see forward mode automatic differentiation).