Timeline for What is... a grossone?
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 6, 2016 at 16:17 | comment | added | Mikhail Katz | As far as your comment that "they do indicate that Sergeyev's computational results can all be recovered via different techniques and are not inconsistent" I have to disagree with you. The grossone theory as presented by Sergeyev is clearly an inconsistent one. Moreover, Rosinger in various posts and even a published article seems to claim that there is nothing wrong with that. | |
| Jan 6, 2016 at 16:11 | comment | added | Mikhail Katz | Daniel, for a useful heuristic on infinite numbers and infinitesimals, I would recommend Keisler's book math.wisc.edu/~keisler/calc.html which happens to be both accessible and based on consistent mathematics. AND useful. Futhermore, it is a big hit with the students, as recent teaching experience shows. | |
| Jan 6, 2016 at 15:33 | comment | added | Daniel Moskovich | Sergeyev declared a list of properties of grossone and several mathematicians, Lolli, Kutateladze and Gutman, and Kauffman, defined objects which satisfy subsets of those properties. None of these is Sergeyev's grossone. But they do indicate that Sergeyev's computational results can all be recovered via different techniques and are not inconsistent. My conclusion would be that Sergeyev's grossone itself (as opposed to related objects) is not currently an well-defined object in mathematical logic, but, like naive infinitesimals, it is a useful and simple heuristic. | |
| Jan 6, 2016 at 12:29 | comment | added | Mikhail Katz | Daniel, only finite natural numbers can have both cardinal and ordinal properties. Postulating such things for an infinite number involves a potential inconsistency. As is clear from Kauffman's answer, his purely finite variables are decidedly not what Sergeyev is claiming for the grossone. | |
| Jan 6, 2016 at 12:04 | comment | added | Daniel Moskovich | The cardinal and ordinal properties of grossone are essentially those of a natural number. In this sense, Kauffman's "generic natural number" interpretation is the most faithful. It lacks one algebraic property (divisibility by all natural numbers), but as Kauffman's "transfer principle" shows, this algebraic property can be dropped without essential problem. Anyway, if it turns out that I care deeply about this property, can't I just take a "generic factorial"? | |
| Jan 5, 2016 at 7:54 | comment | added | Mikhail Katz | Daniel, it is true that Lolli continues this way; however, Sergeyev's claim that the grossone has these properties is merely a declamatory statement, rather than, as one might have expected in a mathematics paper, being justified by some clever definition. In research papers, whether in pure mathematics or applied mathematics, introducing an inconsistency is generally considered a shortcoming rather than an advantage. | |
| Jan 4, 2016 at 19:45 | comment | added | Daniel Moskovich | @katz The Lolli quote continues: "However such a factorial has none of the cardinal and ordinal properties of Grossone...". | |
| Jan 4, 2016 at 16:15 | comment | added | Mikhail Katz | ..."Gutman and Kutateladze have actually done [Sergeyev] a good service by pointing out that if you take a Robinson infinite natural number h, the factorial h! satisfies all the algebraic properties of Grossone (see below). This assures those who need reassurance that Sergeyev's system is as consistent as classical mathematics." (Lolli 2012, page 7987). In short, what Lolli is claiming here is that Gutman and Kutateladze justified Sergeyev via nonstandard analysis. | |
| Jan 4, 2016 at 16:13 | comment | added | Mikhail Katz | Daniel, you write that "even if Lolli's paper is indeed a reformulation of nonstandard arithmetic (and his answer indicates that he does not believe this to be the case), this has no bearing on the originality of Sergeyev's Grossone. Kutateladze's review also discusses a nonstandard arithmetic object, which I don't think is the same as Grossone." However, Lolli himself makes a connection between Sergeyev's theory and Kutateladze's approach. Thus, Lolli claims that... | |
| Dec 27, 2015 at 10:25 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
Comment on Lolli revised
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| Dec 26, 2015 at 17:35 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
updated
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| Dec 24, 2015 at 8:44 | history | answered | Daniel Moskovich | CC BY-SA 3.0 |