Timeline for Does it make sense to take an infinitesimal volume of shape other than a cube?
Current License: CC BY-SA 4.0
31 events
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| S Sep 27, 2022 at 18:27 | history | suggested | Glorfindel | CC BY-SA 4.0 |
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| Sep 27, 2022 at 18:27 | review | Suggested edits | |||
| S Sep 27, 2022 at 18:27 | |||||
| S Sep 27, 2022 at 18:27 | history | suggested | Glorfindel | CC BY-SA 4.0 |
broken link fixed
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| Sep 27, 2022 at 18:18 | review | Suggested edits | |||
| S Sep 27, 2022 at 18:27 | |||||
| Sep 23, 2020 at 17:09 | comment | added | Dale | @Sidarth you said "I think what I am doing is questioning whether then such a definition is "correct" (physicist point of view)". You already know the answer to that. In your own words: "I know the answer to this is of course yes and I know it's usefulness." From the physicist point of view it works, as you already know. There is nothing more to it from the physics point of view. FYI, the hyperreal infinities are also ordered, just like the infinitesimals. | |
| Sep 23, 2020 at 16:28 | comment | added | Sidarth | "That is exactly....to be ordered." - I agree. I think what I am doing is questioning whether then such a definition is "correct" (physicist point of view) but then again, mathematical definitions don't care about physical reality or daily parlance ( since this argument on the other end of the number line would be like comparing different infinities and saying one infinity is greater than the other, even though infinity means something that's bigger than the biggest number I can think of)." I don't know how.... ordered." - you sound like an exasperated professor. Love it :) Thank you ,sir! | |
| Sep 23, 2020 at 16:16 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 23, 2020 at 15:01 | comment | added | Dale | @Sidarth "I would not want to take one term (which is boundlessly small) in the product as "lesser" than another boundlessly small quantity , right?" I don't know how many ways I can devise to tell you that yes you can and often do take one boundlessly small quantity as smaller or larger than another boundlessly small quantity. That is exactly what it means for the infinitesimals to be ordered. Here is a question for you: Given the definitions of hyperreals, in the paragraph I added, which infinitesimal is larger $dx$ or $\epsilon$? They are not equal, one is definitely smaller than the other | |
| Sep 23, 2020 at 14:53 | comment | added | Sidarth | I understand that there might be no "absolute smallest" @Dale, but in a given problem, when I am solving, in a given line, when I write a product of infinitesimals, keeping in touch with the word infinitesimal as boundlessly small, I would not want to take one term (which is boundlessly small) in the product as "lesser" than another boundlessly small quantity , right? (Hope this is not irking you!). Also, its very acceptable that there is no one absolute smallest thing. | |
| Sep 23, 2020 at 11:54 | comment | added | Dale | @Sidarth “Just a definition and lost it’s meaning”. The definition of a word IS the meaning of that word. It didn’t lose any meaning, it just never meant “absolute smallest” to begin with. I added a paragraph above about the ordering of the hyperreals that should help see why there is no absolute smallest | |
| Sep 23, 2020 at 11:53 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 23, 2020 at 7:21 | comment | added | Sidarth | @Dale Hey Dale. Your recent addendum seemed spot on then I recollected my own doubts (see OP : .."I have trouble accepting ...") but I am not denying this. It is perfectly acceptable that I can go to smaller volumes given that the infinitesimals are "ordered". This seems to be like the playing around the concept of "boundlessly small" - no matter where I am, I can go smaller. One thing I have trouble :in dXdydz, dX is smaller than the other two "infinitesimals". Now, "infinitesimals" is just a definition and lost it's meaning.Any comments? | |
| Sep 23, 2020 at 0:57 | history | rollback | Dale |
Rollback to Revision 9
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| Sep 23, 2020 at 0:28 | history | edited | Deschele Schilder | CC BY-SA 4.0 |
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| Sep 22, 2020 at 21:41 | comment | added | Dale | @Sidarth I added a section addressing your current question. | |
| Sep 22, 2020 at 21:39 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 21, 2020 at 20:25 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 14, 2020 at 13:35 | history | edited | Brian | CC BY-SA 4.0 |
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| Sep 14, 2020 at 13:30 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 14, 2020 at 13:25 | comment | added | Dale | @Buraian excellent suggestion! I have added several links to the answer. By no means is that a complete list, but it is roughly organized in order of increasing rigor and decreasing accessibility. | |
| Sep 14, 2020 at 13:23 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 14, 2020 at 12:58 | comment | added | Brian | Could you give some further readings/ citations on this answer? I found it quite interesting | |
| Sep 14, 2020 at 2:16 | comment | added | Sidarth | Right. I think it's safe to say that I am completely in rigorous mathematics now? | |
| Sep 14, 2020 at 1:56 | comment | added | Dale | @Sidarth yes, I agree. But structures that make initially non-intuitive things seem plausible are my kind of structures! Particularly if the new structures are consistent with the established but non-intuitive approach, and are easier to use. The extra definitions are then worth learning. Of course, that is pure personal opinion and you are under no obligation to agree! | |
| Sep 14, 2020 at 0:52 | comment | added | Sidarth | Yes. Definitely it has helped. But I also feel that an extra structure has been imposed in order to make something that was initially non-intuitive seem more plausible. I feel that it did not address the problem straightforwardly but rather justified somethings by using extra definitions. (I am not a mathematician. So this approach is a little different to me!) | |
| Sep 14, 2020 at 0:48 | comment | added | Dale | @Sidarth I am glad it helped. I have only recently learned of hyperreal numbers and now wish that calculus were taught using them | |
| Sep 14, 2020 at 0:46 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 14, 2020 at 0:28 | comment | added | Sidarth | "...An infinitesimal volume merely needs to be smaller than any positive real volume, not smaller than other infinitesimal volumes...." I did not know this! Very non -intuitive since I was initially thinking that the infinitesimal volume has to be vanishingly small, meaning there is only one possible infinitesimal volume, one made of infinitesimal lengths. | |
| Sep 13, 2020 at 19:40 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 13, 2020 at 17:34 | history | edited | Dale | CC BY-SA 4.0 |
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| Sep 13, 2020 at 17:01 | history | answered | Dale | CC BY-SA 4.0 |