What Are Infinitesimals – Simple Version

[bhobba]

Introduction
When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when required, you can throw infinitesimals away as being negligible.
This is fine for applied mathematicians, physicists, actuaries etc., who want it as a tool to use in their work. But mathematicians, while conceding it is OK to start that way, eventually will need to rectify using handwavey arguments and be logically sound. The usual way of doing it is using limits.
Instead, I will justify the idea of infinitesimals as legitimate.  Not with full rigour; I leave that to specialist texts, but enough to satisfy those interested in the fundamental ideas. About 1960, mathematicians (notably Abraham Robinson) did something nifty. They created hyperreal numbers, which...

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[fresh_42]

I'm still no friend of the hyper (att: <-- link is a joke) concept, but I'm happy that someone explained it. I am in principle very interested in the history of mathematics and physics, and in that respect, the article is very helpful. People tend to forget that the epsilontic is actually a rather new concept. Infinitesimals as independent quantities were in use from Leibniz and Newton until Lie and Noether at the beginning of the 20th century.

Here is Noether's article from 1918:
https://de.wikisource.org/wiki/Invariante_Variationsprobleme (German transcription)
https://arxiv.org/pdf/physics/0503066.pdf (English transcription)
and I'm sorry that I didn't find the facsimile on the server of the University of Göttingen right now. Anyway, it shows - and the facsimile shows it even more - that the entire Lie theory was developed along the concept of infinitesimals.

Btw., I didn't know that there were also hyperrationals (TIL).

 

[bhobba]

Btw., I didn't know that there were also hyperrationals (TIL).

Hi Fresh

I can see why it looks like a joke. The idea is to use the concept of infinitesimals; the reader can make it less of a joke.

I am a bit concerned about having two articles - a simple version and an advanced version. Also, the advanced version has a link to how natural numbers, integers, rationals and reals are constructed. It is a bit advanced for the audience I had in mind, so am working on an article at a more basic level. You may be interested in that.

It also goes into a bit of the history of why these more formal approaches were devised, and ZFC set theory (or the ZFCA version I use with Urelements). As you would know the axiom of infinity is basically Von Neumann's construction of the naturals.

You may find it interesting. For me, it may allow the combining of the more advanced article and simplified version by separating out the more advanced material.

Thanks
Bill

 

[fresh_42]

Hi Fresh

I can see why it looks like a joke.

I think this is a serious misunderstanding. My link to the youtube techno song titled "Hyper, hyper" was the joke, not your article. The article is fine. Maybe I should stop to make fun of everything.

I would be interested in the transition process. Infinitesimals were so common that all physicists and mathematicians used them as we use ordinary numbers today. However, modern textbooks switched to epsilontic. Did it come before, with, or because of Bourbaki? Or was it parallel to the rise of topology? What triggered the transition?

 

[bhobba]

´What triggered the transition?

Sorry for the confusion.

This all grew out of studies in logical model theory (see the section on Ultraproducts):
https://en.wikipedia.org/wiki/Model_theory

Thanks
Bill

 

[fresh_42]

Sorry for the confusion.

This all grew out of studies in logical model theory (see the section on Ultraproducts):
https://en.wikipedia.org/wiki/Model_theory

Thanks
Bill

One would expect that the emphasis on semantics over syntax favored the classical model with infinitesimals instead of the rather syntactic epsilontic. Infinitesimals were common, and epsilontic is an obstacle till today. Even I have to ensure myself each time I use it that the order of quantifiers is correct: $\forall\;\exists\;\forall$ - not very intuitive.