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Groups > sci.math > #359184 > unrolled thread
| Started by | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| First post | 2017-01-03 07:24 -0800 |
| Last post | 2017-01-05 08:53 -0800 |
| Articles | 20 on this page of 137 — 15 participants |
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Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-03 07:24 -0800
Re: Déjà vu Jürgen R. <jurgenr@web.de> - 2017-01-03 18:47 +0100
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-03 11:27 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-04 03:34 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-04 06:34 -0800
Re: Déjà vu "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2017-01-05 00:17 +0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-04 11:14 -0800
Re: Déjà vu "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2017-01-05 20:39 +0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 08:53 -0800
Re: Déjà vu "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2017-01-07 17:55 +0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 03:07 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 03:13 -0800
Re: Déjà vu Simon Roberts <retenshun@gmail.com> - 2017-01-07 04:03 -0800
Re: Déjà vu "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2017-01-08 00:54 +0800
Re: 4-figure tables and geometry "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2017-01-08 00:58 +0800
Re: 4-figure tables and geometry Apollyon <adravirgo@gmail.com> - 2017-01-07 09:01 -0800
Re: 4-figure tables and geometry "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-07 12:25 -0800
Re: 4-figure tables and geometry Peter Percival <peterxpercival@hotmail.com> - 2017-01-07 20:36 +0000
Re: 4-figure tables and geometry "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-07 12:41 -0800
Re: 4-figure tables and geometry bursejan@gmail.com - 2017-01-07 16:04 -0800
Re: 4-figure tables and geometry Robert Wessel <robertwessel2@yahoo.com> - 2017-01-07 23:30 -0600
Re: 4-figure tables and geometry "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2017-01-08 21:52 +0800
Re: 4-figure tables and geometry "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2017-01-10 22:06 +0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-07 04:44 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-04 14:12 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-04 14:50 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 02:18 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-04 15:01 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 02:19 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 02:21 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 02:41 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-05 02:47 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 03:00 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 04:08 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 08:26 -0800
My GeoPosition is a point at infinity Vinicius Claudino Ferraz <boralaemcasa@gmail.com> - 2017-01-05 16:24 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-05 02:36 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 02:50 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 03:02 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 04:09 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 08:28 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 09:00 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 09:15 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-05 11:20 -0800
Re: Déjà vu Jürgen R. <jurgenr@web.de> - 2017-01-05 21:53 +0100
vu uv datum abu.kuanysh05@gmail.com - 2017-01-05 13:06 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 16:33 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 18:41 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-06 02:20 -0800
Re: Déjà vu Jürgen R. <jurgenr@web.de> - 2017-01-06 12:45 +0100
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-06 05:30 -0800
Re: Déjà vu Jürgen R. <jurgenr@web.de> - 2017-01-06 15:59 +0100
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-06 09:11 -0800
Re: Déjà vu Jürgen R. <jurgenr@web.de> - 2017-01-06 18:54 +0100
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-06 13:25 -0800
Re: Déjà vu Vinicius Claudino Ferraz <boralaemcasa@gmail.com> - 2017-01-06 13:40 -0800
Re: Déjà vu Jürgen R. <jurgenr@web.de> - 2017-01-06 23:19 +0100
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-06 14:37 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 03:34 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 03:36 -0800
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-07 12:21 -0800
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-07 18:41 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-06 10:09 -0500
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-06 09:10 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-06 09:14 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-06 18:19 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-06 16:53 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-07 01:02 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 03:01 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-07 10:20 -0500
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-07 09:22 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-07 09:51 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-07 09:52 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-07 10:12 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 10:21 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-07 10:24 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-07 10:57 -0800
Re: Déjà vu genmailus@gmail.com - 2017-01-07 11:05 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 11:32 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-07 11:47 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-07 11:15 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-07 17:21 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-07 20:48 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-08 01:36 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-08 01:38 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-08 03:04 -0800
Re: Déjà vu Jürgen R. <jurgenr@web.de> - 2017-01-08 13:27 +0100
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-07 10:01 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 12:34 -0500
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-09 10:08 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 17:18 -0500
Re: Déjà vu genmailus@gmail.com - 2017-01-09 14:28 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 17:55 -0500
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-09 17:12 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 20:55 -0500
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-09 18:08 -0800
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-09 23:08 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 10:55 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-10 05:13 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 09:49 -0500
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-10 07:44 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 11:45 -0500
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-10 09:51 -0800
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 10:33 -0800
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 10:35 -0800
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 13:00 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-10 11:05 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 18:08 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-10 17:25 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 01:48 -0500
Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 23:24 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 00:34 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 11:26 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 08:55 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 12:13 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 09:20 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 09:22 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 09:23 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 13:11 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 10:26 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 13:42 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 10:50 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 10:58 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 11:01 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 11:03 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-11 11:11 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-10 00:03 -0800
Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 10:21 -0500
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-06 07:29 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-06 07:27 -0800
Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-06 09:17 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-06 09:35 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-06 13:54 -0800
Re: Déjà vu bursejan@gmail.com - 2017-01-05 03:13 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-05 08:46 -0800
Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-05 08:51 -0800
Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-05 08:53 -0800
Page 4 of 7 — ← Prev page 1 2 3 [4] 5 6 7 Next page →
| From | "Ross A. Finlayson" <ross.finlayson@gmail.com> |
|---|---|
| Date | 2017-01-07 12:21 -0800 |
| Message-ID | <0c4f6cb7-7d41-48df-aefb-49ad109c599b@googlegroups.com> |
| In reply to | #359878 |
On Saturday, January 7, 2017 at 3:34:28 AM UTC-8, burs...@gmail.com wrote: > Am Freitag, 6. Januar 2017 23:38:04 UTC+1 schrieb John Gabriel: > > Actually, WM has some blind spots, ... > > Not only WM, also you bird brain Janus Garbriel > brainbrain have are full of blind spots. > > And the PDF that WM calls a source book in transfinite > is truely not a neutral source, its rather a pile of WMs > strange world and his crappy "proof" attempts. > > But anyway here is something from another source book: > > Galilei never wrote a book on the purely mathematical > aspects of his work, but we find them discussed among > other questions. His pupils Cavalieri and Torricelli > later gave full elaboration to his ideas on problems > dealing with infinitesimaLs (see Selections IV.5, 8, 9). > In the Discorsi of 1638, in which Galilei laid the > foundations of modern mechanics, we see him (as Salviati) > discussing with Simplicio some of the fundamental > difficulties one meets in the concepts of infinite > and infinitesimal. Such difficulties were already > widely discussed in antiquity, and equally among > the scholastic writers. Galilei made some points > clear, such as the difference between actual and > potential infinity and the "equality" in number of > the natural numbers and their squares, which play > an important role in the modern theory of aggregates > as developed by Georg Cantor. The text shows that > Galilei was not afraid of the scholastic "indivisible," > of letting a line segment be an aggregate of infinitely > many points, and of accepting the line continuum as > an actual infinite. > > A Source Book in Mathematics, 1200-1800 > von D. J. Struik > http://press.princeton.edu/titles/2415.html Thanks again. Struik the historian helps remind that the infinitesimal analysis and limits and series were equipped with bumper rails and safety gauges (and rightly) in the formalization of Cauchy/Weierstrass as otherwise about the infinitesimal analysis, the real analysis. Calculus (and correspondingly, modern mathematics) isn't the pinnacle of mathematics, concept, or thought: it's the manicured lane. Please don't feed the trolls, and ignore them as they defecate under the bridge, they choose to live there. (Euh..., that would better reflect a usual sensitivity to the plight of the destitute, but these trolls are petty monsters tolerated only in the interests of free speech.) Please ignore them and don't reward their cropojectile habits. About the learning curve there yet rise above the learning mountains, and it's not the way everyone chooses, but there is a high road, and there's only one way up: up. In continuing metaphor from the foundations, involves all the way from the center of the Earth to the Sun. "The text shows that Galilei was not afraid of the scholastic "indivisible," of letting a line segment be an aggregate of infinitely many points, and of accepting the line continuum as an actual infinite." Neither was Berkeley.
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| From | "Ross A. Finlayson" <ross.finlayson@gmail.com> |
|---|---|
| Date | 2017-01-07 18:41 -0800 |
| Message-ID | <1662453e-dfd4-41dd-b706-8bb2e4e91b18@googlegroups.com> |
| In reply to | #359969 |
On Saturday, January 7, 2017 at 12:21:50 PM UTC-8, Ross A. Finlayson wrote: > On Saturday, January 7, 2017 at 3:34:28 AM UTC-8, burs...@gmail.com wrote: > > Am Freitag, 6. Januar 2017 23:38:04 UTC+1 schrieb John Gabriel: > > > Actually, WM has some blind spots, ... > > > > Not only WM, also you bird brain Janus Garbriel > > brainbrain have are full of blind spots. > > > > And the PDF that WM calls a source book in transfinite > > is truely not a neutral source, its rather a pile of WMs > > strange world and his crappy "proof" attempts. > > > > But anyway here is something from another source book: > > > > Galilei never wrote a book on the purely mathematical > > aspects of his work, but we find them discussed among > > other questions. His pupils Cavalieri and Torricelli > > later gave full elaboration to his ideas on problems > > dealing with infinitesimaLs (see Selections IV.5, 8, 9). > > In the Discorsi of 1638, in which Galilei laid the > > foundations of modern mechanics, we see him (as Salviati) > > discussing with Simplicio some of the fundamental > > difficulties one meets in the concepts of infinite > > and infinitesimal. Such difficulties were already > > widely discussed in antiquity, and equally among > > the scholastic writers. Galilei made some points > > clear, such as the difference between actual and > > potential infinity and the "equality" in number of > > the natural numbers and their squares, which play > > an important role in the modern theory of aggregates > > as developed by Georg Cantor. The text shows that > > Galilei was not afraid of the scholastic "indivisible," > > of letting a line segment be an aggregate of infinitely > > many points, and of accepting the line continuum as > > an actual infinite. > > > > A Source Book in Mathematics, 1200-1800 > > von D. J. Struik > > http://press.princeton.edu/titles/2415.html > > Thanks again. Struik the historian helps remind > that the infinitesimal analysis and limits and series > were equipped with bumper rails and safety gauges > (and rightly) in the formalization of Cauchy/Weierstrass > as otherwise about the infinitesimal analysis, the > real analysis. > > Calculus (and correspondingly, modern mathematics) > isn't the pinnacle of mathematics, concept, or thought: > it's the manicured lane. > > Please don't feed the trolls, and ignore them as > they defecate under the bridge, they choose to > live there. (Euh..., that would better reflect > a usual sensitivity to the plight of the destitute, > but these trolls are petty monsters tolerated only > in the interests of free speech.) Please ignore > them and don't reward their cropojectile habits. > > About the learning curve there yet rise above the > learning mountains, and it's not the way everyone > chooses, but there is a high road, and there's only > one way up: up. > > In continuing metaphor from the foundations, involves > all the way from the center of the Earth to the Sun. > > > > "The text shows that Galilei was not afraid of the > scholastic "indivisible," of letting a line segment > be an aggregate of infinitely many points, and of > accepting the line continuum as an actual infinite." > > Neither was Berkeley. No, don't reply, don't copy what's already been written, a proper point only needs to be made once. He doesn't deserve your attention, and we don't deserve your spam. We already know your opinion. It's only necessary to (as you see fit) remark on relevant updates or aspects of your opinion, otherwise this pasting useless replies to each other is wasteful garbage and insult to the medium. That's wasteful garbage and insult to the medium. It's like a cry for help, wrapped in shit, dumped in a paper bag, lit on fire on the porch, then shitbag there yelling "bring it in, bring it in". The "cusp of reason" Ancient Greece is a bit behind the times of our roots of reason of Ancient Greece. What a maroon. And the Pythagoreans would've drowned him. https://www.youtube.com/watch?v=Jd_41tM6H2Y https://www.youtube.com/watch?v=w6KS6ofEAjs
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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Date | 2017-01-06 10:09 -0500 |
| Message-ID | <8c50e54e-7308-9132-5d91-93ce9208e4b7@att.net> |
| In reply to | #359676 |
On 1/6/2017 8:30 AM, WM wrote:
> Am Freitag, 6. Januar 2017 12:45:19 UTC+1 schrieb Jürgen R.:
>> The name of the number is not the number.
>
> The number is the set of its representations, nothing else.
>
>> How many H-atoms to you need to represent the state
>> in phase space of a mole of H_2 atoms?
>
> In simplest form 6*10^23
0 pts.
No partial credit possible, as you did not show your work.
While it is possible for someone to be so situated that
they feel that they can blindly guess with an air of great
authority (for example, as an instructor of a captive
audience of students) without fear of contradiction
_at the time_ , that is in general a poor strategy for
avoiding looking like a fool.
Students do not remain your students forever. There is
a definite risk, unavoidable once they leave your care,
of their having your errors pointed out to them. Imagine
their laughter.
On the positive side of the balance sheet, if you were
frank and open with them about what you _don't_ know
(about phase spaces and infinity, for starters), that
lends authority to what you say you _do_ know. There is
a message sent, implicitly or explicitly, that if
you _didn't_ know about XYZ you would say so. ("See?"
is the message. "I told you I didn't know about TUV.
So, if I didn't know about XYZ, I would say so, if I
needed to. But I'm not saying so, so I don't need to.")
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2017-01-06 09:10 -0800 |
| Message-ID | <4ab89379-19be-417e-bf39-346783eae530@googlegroups.com> |
| In reply to | #359681 |
Am Freitag, 6. Januar 2017 16:09:59 UTC+1 schrieb Jim Burns: > On 1/6/2017 8:30 AM, WM wrote: > > Am Freitag, 6. Januar 2017 12:45:19 UTC+1 schrieb Jürgen R.: > > >> The name of the number is not the number. > > > > The number is the set of its representations, nothing else. > > > >> How many H-atoms to you need to represent the state > >> in phase space of a mole of H_2 atoms? > > > > In simplest form 6*10^23 > > 0 pts. Rennekampff did ask for one state, not for the number of states. One state of a mole, positions and velocities of all particles, can be represented by a mole. But since you have reappeared after my explanation to your question why I use [0, 1] as the limit of [0, 1 - 1/n]: Have you understood it? Regards, WM
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| From | Me <franz.fritschee.ff@gmail.com> |
|---|---|
| Date | 2017-01-06 09:14 -0800 |
| Message-ID | <9d118d7d-c2c0-43b9-ba6b-0c44797b0d76@googlegroups.com> |
| In reply to | #359704 |
On Friday, January 6, 2017 at 6:10:21 PM UTC+1, WM wrote: > I use [0, 1] as the limit of [0, 1 - 1/n] Maybe you should use the correct value [0, 1) instead? Hint: [0, 1) =/= [0, 1]. See: https://en.wikipedia.org/wiki/Set-theoretic_limit
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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Date | 2017-01-06 18:19 -0500 |
| Message-ID | <c2f888c3-82fe-1be1-23e4-744e5fc3367d@att.net> |
| In reply to | #359704 |
On 1/6/2017 12:10 PM, WM wrote:
> Am Freitag, 6. Januar 2017 16:09:59 UTC+1 schrieb Jim Burns:
>> On 1/6/2017 8:30 AM, WM wrote:
>>> Am Freitag, 6. Januar 2017 12:45:19 UTC+1 schrieb Jürgen R.:
>>>> The name of the number is not the number.
>>>
>>> The number is the set of its representations, nothing else.
>>>
>>>> How many H-atoms to you need to represent the state
>>>> in phase space of a mole of H_2 atoms?
>>>
>>> In simplest form 6*10^23
>>
>> 0 pts.
>> No partial credit possible, as you did not show your work.
>>
>> While it is possible for someone to be so situated that
>> they feel that they can blindly guess with an air of great
>> authority (for example, as an instructor of a captive
>> audience of students) without fear of contradiction
>> _at the time_ , that is in general a poor strategy for
>> avoiding looking like a fool.
>>
>> Students do not remain your students forever. There is
>> a definite risk, unavoidable once they leave your care,
>> of their having your errors pointed out to them. Imagine
>> their laughter.
>>
>> On the positive side of the balance sheet, if you were
>> frank and open with them about what you _don't_ know
>> (about phase spaces and infinity, for starters), that
>> lends authority to what you say you _do_ know. There is
>> a message sent, implicitly or explicitly, that if
>> you _didn't_ know about XYZ you would say so. ("See?"
>> is the message. "I told you I didn't know about TUV.
>> So, if I didn't know about XYZ, I would say so, if I
>> needed to. But I'm not saying so, so I don't need to.")
>
> Rennekampff did ask for one state, not for the number of
> states. One state of a mole, positions and velocities of
> all particles, can be represented by a mole.
It is to be expected that you have your own idea of
what it means to represent something. The, uh,
unusual nature of your idea is to be expected, too.
Do you have civil engineering students?
Suppose one asks you, "How do I _represent_ this
bridge that I am asked to design, so that I can use
what I learned here to know it will not fall down?"
How do you answer? "Your bridge can be _represented_
by a bridge. Build it, and if it doesn't fall down,
it won't fall down."
> But since you have reappeared after my explanation to
> your question why I use [0, 1] as the limit of
> [0, 1 - 1/n]: Have you understood it?
You pretend that the [0, 1 - 1/n] are not sets.
_Why_ you want to pretend the [0, 1 - 1/n] are not sets
is a question for your psychologist. I don't claim
to understand that. On the other hand, I don't claim
to care, either.
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| From | John Gabriel <thenewcalculus@gmail.com> |
|---|---|
| Date | 2017-01-06 16:53 -0800 |
| Message-ID | <91137ab6-a8f6-44ea-983e-726c3f207d25@googlegroups.com> |
| In reply to | #359790 |
On Friday, 6 January 2017 15:19:54 UTC-8, Jim Burns wrote:
> On 1/6/2017 12:10 PM, WM wrote:
> > Am Freitag, 6. Januar 2017 16:09:59 UTC+1 schrieb Jim Burns:
> >> On 1/6/2017 8:30 AM, WM wrote:
> >>> Am Freitag, 6. Januar 2017 12:45:19 UTC+1 schrieb Jürgen R.:
>
> >>>> The name of the number is not the number.
> >>>
> >>> The number is the set of its representations, nothing else.
> >>>
> >>>> How many H-atoms to you need to represent the state
> >>>> in phase space of a mole of H_2 atoms?
> >>>
> >>> In simplest form 6*10^23
> >>
> >> 0 pts.
> >> No partial credit possible, as you did not show your work.
> >>
> >> While it is possible for someone to be so situated that
> >> they feel that they can blindly guess with an air of great
> >> authority (for example, as an instructor of a captive
> >> audience of students) without fear of contradiction
> >> _at the time_ , that is in general a poor strategy for
> >> avoiding looking like a fool.
> >>
> >> Students do not remain your students forever. There is
> >> a definite risk, unavoidable once they leave your care,
> >> of their having your errors pointed out to them. Imagine
> >> their laughter.
> >>
> >> On the positive side of the balance sheet, if you were
> >> frank and open with them about what you _don't_ know
> >> (about phase spaces and infinity, for starters), that
> >> lends authority to what you say you _do_ know. There is
> >> a message sent, implicitly or explicitly, that if
> >> you _didn't_ know about XYZ you would say so. ("See?"
> >> is the message. "I told you I didn't know about TUV.
> >> So, if I didn't know about XYZ, I would say so, if I
> >> needed to. But I'm not saying so, so I don't need to.")
> >
> > Rennekampff did ask for one state, not for the number of
> > states. One state of a mole, positions and velocities of
> > all particles, can be represented by a mole.
>
> It is to be expected that you have your own idea of
> what it means to represent something. The, uh,
> unusual nature of your idea is to be expected, too.
>
> Do you have civil engineering students?
> Suppose one asks you, "How do I _represent_ this
> bridge that I am asked to design, so that I can use
> what I learned here to know it will not fall down?"
>
> How do you answer? "Your bridge can be _represented_
> by a bridge. Build it, and if it doesn't fall down,
> it won't fall down."
Your analogies are even more incoherent than your attempts at logic. Chuckle.
>
> > But since you have reappeared after my explanation to
> > your question why I use [0, 1] as the limit of
> > [0, 1 - 1/n]: Have you understood it?
>
> You pretend that the [0, 1 - 1/n] are not sets.
>
> _Why_ you want to pretend the [0, 1 - 1/n] are not sets
> is a question for your psychologist. I don't claim
> to understand that. On the other hand, I don't claim
> to care, either.
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2017-01-07 01:02 -0800 |
| Message-ID | <75ee80ae-6449-4a1b-ab8f-6e199fb50e05@googlegroups.com> |
| In reply to | #359790 |
Am Samstag, 7. Januar 2017 00:19:54 UTC+1 schrieb Jim Burns: > On 1/6/2017 12:10 PM, WM wrote: > > Rennekampff did ask for one state, not for the number of > > states. One state of a mole, positions and velocities of > > all particles, can be represented by a mole. > > It is to be expected that you have your own idea of > what it means to represent something. The, uh, > unusual nature of your idea is to be expected, too. > > Do you have civil engineering students? Yes. > Suppose one asks you, "How do I _represent_ this > bridge that I am asked to design, so that I can use > what I learned here to know it will not fall down?" > > How do you answer? "Your bridge can be _represented_ > by a bridge. Build it, and if it doesn't fall down, > it won't fall down." In fact this is an important tool. The colleagues of the faculty of civil engineering perform a competition every year: http://www.brueckenbau-wettbewerb.de/ > > > But since you have reappeared after my explanation to > > your question why I use [0, 1] as the limit of > > [0, 1 - 1/n]: Have you understood it? > > You pretend that the [0, 1 - 1/n] are not sets. No, it is irrelevant how you call the intervals. Of course they are sets of points or real numbers. > > _Why_ you want to pretend the [0, 1 - 1/n] are not sets > is a question for your psychologist. I don't pretend that. I proved that the set limit or union differs from the analytical limit. > I don't claim to understand that. It seems that is what happens to most of your environment. > On the other hand, I don't claim to care, either. Why then did you ask for my proof? Can't understand anymore either? Or is it only inconvenient because of the result? Regards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2017-01-07 03:01 -0800 |
| Message-ID | <df9cf677-e63e-4637-a02b-9d83f4c352e0@googlegroups.com> |
| In reply to | #359860 |
Am Samstag, 7. Januar 2017 10:02:37 UTC+1 schrieb WM: > In fact this is an important tool. > http://www.brueckenbau-wettbewerb.de/ These are still model bridges, not real size bridges. There would be also a transfer to reality problem left.
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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Date | 2017-01-07 10:20 -0500 |
| Message-ID | <1eb23cf1-3e29-c29c-e650-862d50b8fb83@att.net> |
| In reply to | #359860 |
On 1/7/2017 4:02 AM, WM wrote:
> Am Samstag, 7. Januar 2017 00:19:54 UTC+1 schrieb Jim Burns:
>> On 1/6/2017 12:10 PM, WM wrote:
>>> Rennekampff did ask for one state, not for the number of
>>> states. One state of a mole, positions and velocities of
>>> all particles, can be represented by a mole.
>>
>> It is to be expected that you have your own idea of
>> what it means to represent something. The, uh,
>> unusual nature of your idea is to be expected, too.
>>
>> Do you have civil engineering students?
>
> Yes.
>
>> Suppose one asks you, "How do I _represent_ this
>> bridge that I am asked to design, so that I can use
>> what I learned here to know it will not fall down?"
>>
>> How do you answer? "Your bridge can be _represented_
>> by a bridge. Build it, and if it doesn't fall down,
>> it won't fall down."
>
> In fact this is an important tool.
> The colleagues of the faculty of civil engineering
> perform a competition every year:
> http://www.brueckenbau-wettbewerb.de/
Whatever value there may be in building models, it's
not what you were talking about, nor what I was talking
about. The civil-engineering equivalent of "representing"
a mole of hydrogen by a mole of hydrogen is "representing"
a bridge (or bridge-to-be) by _that same bridge_ .
This is in line with your idea of using sets as names
of themselves. I assumed that you still held on to that.
Am I mistaken? Have you recanted this particular part of
Mueckemania while I wasn't looking? Or are you merely
uncomfortable talking about your idea and thus wish to
change the subject?
----
Your ideas about hydrogen atoms are at least as ridiculous
as your ideas about infinity and about names, but they're not
so important, since you aren't expected to know anything
about physics.
One notion of yours, in particular, assaults me like
a piece of chalk squeaking on a blackboard. Spin is
not classical, _especially_ spin 1/2 (your "bit" because
of its two z-spin states +1/2 and -1/2) which has no
classical analog. Talking about the spin of a hydrogen atom
electron (which I generously interpret your mumblings as
referring to) as though it were a classical bit (as opposed
to a qubit, a quantum bit) is ignorant.
>>
>>> But since you have reappeared after my explanation to
>>> your question why I use [0, 1] as the limit of
>>> [0, 1 - 1/n]: Have you understood it?
>>
>> You pretend that the [0, 1 - 1/n] are not sets.
>
> No, it is irrelevant how you call the intervals.
> Of course they are sets of points or real numbers.
You replace the limit set of a sequence of sets with
with a set with endpoints equal to the limits of
of the sequences of endpoints of the set sequence.
Having changed the definition, you find that you disagree
with everyone else about the limit of that sequence of sets.
The conclusion you draw is that _everyone else_ is wrong.
As I say, a question for your psychologist.
----
Do you invent a new definition of limit for each question?
I wonder how you deal with an increasing sequence of sets of
rationals with endpoints that converge to irrationals.
Do you say it converges? If so, to what?
Let's be specific. Consider the sequence of sets
{x| x e Q & 0 =< x =< 3 },
{x| x e Q & 0 =< x =< 3.1 },
{x| x e Q & 0 =< x =< 3.14 },
{x| x e Q & 0 =< x =< 3.141 },
{x| x e Q & 0 =< x =< 3.1415 },
...
Does it have what you call a limit? If so, what?
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2017-01-07 09:22 -0800 |
| Message-ID | <1ac56b66-fea1-4d27-b60c-a9baa9e2dcec@googlegroups.com> |
| In reply to | #359896 |
Am Samstag, 7. Januar 2017 16:20:22 UTC+1 schrieb Jim Burns: > You replace the limit set of a sequence of sets with > with a set with endpoints equal to the limits of > of the sequences of endpoints of the set sequence. Yes. lim [a_, b_n] = [lim a_, lim b_n]. > > Having changed the definition, you find that you disagree > with everyone else about the limit of that sequence of sets. The reason is that set theory puts union = limit, which is wrong. But most have not comprehended that, in particular for the sequence of FISONs or finite rows 1 1, 2 1, 2, 3 ... The union is not a transfinite set, i.e., a set larger than all FISONs. > The conclusion you draw is that _everyone else_ is wrong. > As I say, a question for your psychologist. No it is a simple proof that matheologians require psychological care, because everybody else understands this example: An infinite but not transfinite (= larger than every finite) set is the above set of finite rows which does not contain a transfinite row, neither when written as above nor when written in one single row. Proof: If every row is shifted into the first row and extinguished before the next row is shifted, then no transfinite set appears in the first row. If the extinguishing is omitted however, nothing is changed, because every row covers its predecessor. > Do you invent a new definition of limit for each question? No, I invented this limit only for the very special case of inclusion monotonic seqeuences of real intervals, in order to have another, simpler example where union and limit differ in set theory. By the way, the psychological defects of matheology seem to be more grave than the mathematical aspects. Therefore you can find a section Psychological aspects of set theory on p. 323 of https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf Regards, WM
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| From | Me <franz.fritschee.ff@gmail.com> |
|---|---|
| Date | 2017-01-07 09:51 -0800 |
| Message-ID | <c40a4e36-589e-4345-ac0a-87e4e616c18c@googlegroups.com> |
| In reply to | #359919 |
On Saturday, January 7, 2017 at 6:22:16 PM UTC+1, WM wrote:
> Am Samstag, 7. Januar 2017 16:20:22 UTC+1 schrieb Jim Burns:
> >
> > You replace the limit set of a sequence of sets
> > with a set with endpoints equal to the limits of
> > of the sequences of endpoints of the set sequence.
> >
> Yes. lim [a_, b_n] = [lim a_, lim b_n].
What you mean, idiot, is
WM_lim [a_n, b_n] = [lim a_n, lim b_n] .
this "Mückenheim-limit" differs from the usual set-theoretic limit of the a sequence of sets, which usually is denoted by "lim ...".
Note, that your "formula" does not even work for all sequences of closed intervals (even if they are "inclusion monotonic").
Consider the sequence (A_n) with A_n = [n, n] (i.e. a_n = n and b_n = n) or the sequence (B_n) = [-n, n] (i.e. a_n = -n and b_n = n).
Hence it seems that your "Mückenheim-limit" is restricted to a) sequences of closed intervals that b) are "inclusion monotonic" and where c) the limits of the sequences of the endpoints [...] exists in each case.
Great. So I fail to see any benefit of your Mückenheim-limit in mathematics.
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| From | Me <franz.fritschee.ff@gmail.com> |
|---|---|
| Date | 2017-01-07 09:52 -0800 |
| Message-ID | <e37e2a9b-43ad-4021-8e8b-a1e7bf10b9b9@googlegroups.com> |
| In reply to | #359919 |
On Saturday, January 7, 2017 at 6:22:16 PM UTC+1, WM wrote:
> Am Samstag, 7. Januar 2017 16:20:22 UTC+1 schrieb Jim Burns:
> >
> > You replace the limit set of a sequence of sets
> > with a set with endpoints equal to the limits of
> > of the sequences of endpoints of the set sequence.
> >
> Yes. lim [a_, b_n] = [lim a_, lim b_n].
What you mean, idiot, is
WM_lim [a_n, b_n] = [lim a_n, lim b_n] .
this "Mückenheim-limit" differs from the usual set-theoretic limit of the a sequence of sets, which usually is denoted by "lim ...".
Note, that your "formula" does not even work for all sequences of closed intervals (even if they are "inclusion monotonic").
Consider the sequence (A_n) with A_n = [n, n] (i.e. a_n = n and b_n = n) or the sequence (B_n) with B_n = [-n, n] (i.e. a_n = -n and b_n = n).
Hence it seems that your "Mückenheim-limit" is restricted to a) sequences of closed intervals that b) are "inclusion monotonic" and where c) the limits of the sequences of the endpoints [...] exists in each case.
Great. So I fail to see any benefit of your Mückenheim-limit in mathematics.
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2017-01-07 10:12 -0800 |
| Message-ID | <b2b6ff47-dabf-495e-8159-d636b1c83198@googlegroups.com> |
| In reply to | #359941 |
Am Samstag, 7. Januar 2017 18:53:01 UTC+1 schrieb Me: > Hence it seems that your "Mückenheim-limit" is restricted to a) sequences of closed intervals that b) are "inclusion monotonic" and where c) the limits of the sequences of the endpoints [...] exists in each case. > > Great. So I fail to see any benefit of your Mückenheim-limit in mathematics. It would easen the understanding of the following difficult fact: There are infinit sets which are not transfinite. An infinite set is for instance the set of finite rows 1 1, 2 1, 2, 3 ... which does not contain a transfinite row, neither when written as above nor when written in one single row. Proof: If every row is shifted into the first row and extinguished before the next row is shifted, then not even you will claim that a transfinite set appears in the first row. If the extinguishing is omitted however, nothing is changed. Even you should understand that there is nothing transfinite in this little example. Nevertheless the set of rows and of numbers is infinite. IT is the union of all FISONs which differs from their limit, a transfinite set larger than all FISONs. Regards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2017-01-07 10:21 -0800 |
| Message-ID | <72c98623-4c19-4e24-bb9c-5ccd3dad6444@googlegroups.com> |
| In reply to | #359948 |
Muckefunk, did you try a fire extinguisher? Am Samstag, 7. Januar 2017 19:13:03 UTC+1 schrieb WM: > ... If the extinguishing is omitted however, nothing is changed. ... > Regards, WM
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| From | Me <franz.fritschee.ff@gmail.com> |
|---|---|
| Date | 2017-01-07 10:24 -0800 |
| Message-ID | <31a5ef49-bf1e-48ea-ad93-38f92683ef5f@googlegroups.com> |
| In reply to | #359948 |
On Saturday, January 7, 2017 at 7:13:03 PM UTC+1, WM wrote: > Am Samstag, 7. Januar 2017 18:53:01 UTC+1 schrieb Me: > > > > Hence it seems that your "Mückenheim-limit" is restricted to a) sequences of > > closed intervals that b) are "inclusion monotonic" and where c) the limits > > of the sequences of the endpoints [...] exists in each case. > > > > Great. So I fail to see any benefit of your Mückenheim-limit in mathematics. > > > It would easen the understanding of the following difficult fact [...] If you say so. > [...] And where did you apply your Mückenheim-limit here?
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2017-01-07 10:57 -0800 |
| Message-ID | <6b9e0848-7f1a-48eb-ba51-5fc238b1233d@googlegroups.com> |
| In reply to | #359952 |
Am Samstag, 7. Januar 2017 19:24:22 UTC+1 schrieb Me: > On Saturday, January 7, 2017 at 7:13:03 PM UTC+1, WM wrote: > > Am Samstag, 7. Januar 2017 18:53:01 UTC+1 schrieb Me: > > > > > > Hence it seems that your "Mückenheim-limit" is restricted to a) sequences of > > > closed intervals that b) are "inclusion monotonic" and where c) the limits > > > of the sequences of the endpoints [...] exists in each case. > > > > > > Great. So I fail to see any benefit of your Mückenheim-limit in mathematics. > > > > > It would easen the understanding of the following difficult fact [...] > > If you say so. > > > [...] > > And where did you apply your Mückenheim-limit here? The infinite but not transfinite union of the rows is comparable with (0, 1]. In this case however set theory claims a limit that would be analogous to [0, 1], namely the limit of the rows, a set with aleph_0 elements. Regards, WM
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| From | genmailus@gmail.com |
|---|---|
| Date | 2017-01-07 11:05 -0800 |
| Message-ID | <97a22a95-1d04-44ab-b39a-b22251b4d701@googlegroups.com> |
| In reply to | #359961 |
On Saturday, 7 January 2017 10:58:01 UTC-8, WM wrote: > Am Samstag, 7. Januar 2017 19:24:22 UTC+1 schrieb Me: > > On Saturday, January 7, 2017 at 7:13:03 PM UTC+1, WM wrote: > > > Am Samstag, 7. Januar 2017 18:53:01 UTC+1 schrieb Me: > > > > > > > > Hence it seems that your "Mückenheim-limit" is restricted to a) sequences of > > > > closed intervals that b) are "inclusion monotonic" and where c) the limits > > > > of the sequences of the endpoints [...] exists in each case. > > > > > > > > Great. So I fail to see any benefit of your Mückenheim-limit in mathematics. > > > > > > > It would easen the understanding of the following difficult fact [...] > > > > If you say so. > > > > > [...] > > > > And where did you apply your Mückenheim-limit here? > > The infinite but not transfinite union of the rows is comparable with (0, 1]. In this case however set theory claims a limit that would be analogous to [0, 1], namely the limit of the rows, a set with aleph_0 elements. What can you expect from a theory which equates indistinguishable points on the number line with "real" numbers? A point is reifiable on the number line if and only if: 1. The magnitude can be produced geometrically. 2. There is a number describing the magnitude, that is, it has a measure. Otherwise a point on the number line is meaningless nonsense. Points do not appear on lines. Point are to a number line as road signs are to a road. Lines do NOT consist of points. They are defined by the distance between two points. There is no such thing as an infinite line. To say that a line can be extended indefinitely in either direction does NOT imply anything about infinity. > > Regards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2017-01-07 11:32 -0800 |
| Message-ID | <f164b0a5-cd2a-4d5a-bf91-a98e36d550d9@googlegroups.com> |
| In reply to | #359962 |
WM and JG, assume you live on flat earth (possibly WM and JG anyway believe this). And this flat earth is infinitely wide and it is inhabited by little people, with little villages and littel roads. These road have road sign. You yourself will never travel all the villages and pass all the road signs. Neverthells the little people are always mending their roads, and there are proper road signs. Since this flat earth is infinitely wide and has a constant road density, there will infinitely many road signs, even if you yourself, will never see them all. Am Samstag, 7. Januar 2017 20:05:56 UTC+1 schrieb genm...@gmail.com: > Otherwise a point on the number line is meaningless nonsense. > Points do not appear on lines. Point are to a number line as > road signs are to a road. > To say that a line can be extended indefinitely in either > direction does NOT imply anything about infinity.
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| From | bursejan@gmail.com |
|---|---|
| Date | 2017-01-07 11:47 -0800 |
| Message-ID | <53d1579b-9548-41e6-8102-1173e263ce12@googlegroups.com> |
| In reply to | #359967 |
Warning JG: If you tell us, only those signs exist that I can see myself on this flat earth, I will tell you your new calculus method doesn't exist either. Assume each villiage is a function f, I might want to apply your new calculus to. I also will never be able to travel all villages. Nevertheless you tell me in advance, your new calculus will always work, and you have a proof JG. How can this be possible if there is no infinite? Am Samstag, 7. Januar 2017 20:32:19 UTC+1 schrieb burs...@gmail.com: > WM and JG, assume you live on flat earth (possibly WM and JG > anyway believe this). And this flat earth is infinitely wide > and it is inhabited by little people, with little villages > > and littel roads. These road have road sign. You yourself > will never travel all the villages and pass all the road > signs. Neverthells the little people > > are always mending their roads, and there are proper road > signs. Since this flat earth is infinitely wide and has a > constant road density, there > > will infinitely many road signs, even if you yourself, will > never see them all. > > > Am Samstag, 7. Januar 2017 20:05:56 UTC+1 schrieb genm...@gmail.com: > > Otherwise a point on the number line is meaningless nonsense. > > Points do not appear on lines. Point are to a number line as > > road signs are to a road. > > > To say that a line can be extended indefinitely in either > > direction does NOT imply anything about infinity.
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