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Infinitesimals and limits are the same thing


dasnulium

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18 hours ago, dasnulium said:

wtf: I did think carefully about word choice for the thesis, in particular I would never use the term 'infinitely small' to describe an infinitesimal - 'indefinitely small' does not mean smaller than 'every' positive number, it means smaller than any positive number to which you can assign a value (this is similar to Kant's ideas about the infinite as discussed by Bell). I don't use the word 'arbitrary' because in numerical analysis (unlike in regular calculus) it means something different to 'indefinite' - namely that the minimum value of the increment may be arbitrary (alternatively, depending on the functions concerned, it may have to meet certain criteria). Note that the term 'indefinitely small' is meaningless for numerical analysis for obvious reasons.

 

> it means smaller than any positive number to which you can assign a value

Can you give an example or an explanation of a positive number to which you can't assign a value? I can't imagine what that could possibly mean. Like 14. I can assign the value 14 to it. What does it mean to assign a value to a number? Isn't the value of a number the number itself? I cannot understand this remark at all.

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  • 2 years later...

This is Version 2 of Next to Nothing - a Single Paradigm, an essay and proof I first posted in January 2019 and linked from here. Here is that link:

Version 2 can be found here - https://drive.google.com/file/d/1k6gtN1k_WfRVyAUqwhjGXo10U3HWVpoA/view

Preface: Since posting Next to Nothing – a Single Paradigm over two and a half years ago I have been able to compose a more complete version of the paper. I here address the main criticism of the original, elaborate on one of its themes, and offer some thoughts on a related matter. Improvements have also been made throughout the original text (while retaining the general narrative) and many of the references are new. The proof itself remains unchanged. The additional Continuation section follows on from the main text.

Edited by dasnulium
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Swansont, against my wishes, decided to merge my previous comment with the 2019 post, rather than it being a new post. He then declined to respond to my complaint about this. Consequently, I won't be engaging any further with this site.

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!

Moderator Note

We've been doing this for a while, and know it's detrimental to the learning process to have more than one thread on the same subject. Learning through discussion is our focus, not on catering to individual wishes. We work for the owners, not you.

I recommend you start your own site and develop the rules you and your membership want. Thread closed for lack of engagement.

 
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