1619:
you said. For me, in this case, it's not very complicated. I'm not concerned exactly with some general definition of "random sequence", I'm interested in seeing how our world (or something essentially similar, like perhaps the cellular automota) produces something that looks for all the world as though it were random. By random, I mean something along the lines of "all instances are independent and all possibilities are equally likely". You know, random. I know the answer should be simple and straightforward, it's just damnably hard to get across to other people what I'm asking. So, please be patient.
29:! Thank you for your contributions. I hope you like the place and decide to stay. I was delighted to see your responses at the (mathematics) reference desk; We do not have too many regulars there with enough knowledge to tackle the harder questions. I do hope you will create a user page, and more importantly, to see you contributing to articles as well as the RefDesk. When you do, you may find the following pages useful:
1551:
has introduced additional solutions, which must be detected and deleted in the context of the original problem. There's no way around it: when cancelling or multiplying by an expression that may or may not have an inverse, one must verify that the solution set is unchanged, not merely contained. --
873:
way to prove his inference false is by stating values (and a model, although he doesn't mention this) which makes the statement false. Of course, there are other methods, including derivation of a contradiction from the proposed statement (the one I started with) and the somewhat more esoteric proof
1618:
I'm not familiar with the word "ontological", but I looked it up and it apparently means "An explicit formal specification of how to represent the objects, concepts and other entities that are assumed to exist in some area of interest and the relationships that hold among them." That fits with what
790:
which is quite clearly a division and is so described by him. The action that was used was left cancellation by an element without an inverse. (The algebra of polynomials does not have inverses for the "variable".) I attempted to point this out in-discussion and then on his Talk page using the
420:
You don't give quickly up, do you? In the discussion of 597 days ago, we were dealing with the field of real numbers, and not some arbitrary algebraic structure. The original issue you raised was that of division by a quantity that was potentially zero. If you allow nilpotent elements, you cannot
874:
of dependence of the statement on the set of axioms with a proof of consistency of the negated statement. Regardless of the discursively incorrect assertion that only one form of argument of falsehood was valid, the previous question was still adequately dismissed by me.
1086:... And on the off-chance you care at all why I took ~600 days to reply, I have lately been constructing a representation of a monoid ring with nilpotent elements and I was reminded of the challenge to find a value that proves the statement false. --
1249:
1100:
I admit that
Lambiam's original choice of words was poor. The "Divide both sides", in retrospect, should not have been taken literally, and Lambiam clarified his intention in later comments. He sought to produce a sufficient condition
237:
of the form P(x) ⇒ Q(x) is invalid, you have to be able to give a value x in the domain of the implied quantification such that P(x) is true, while Q(x) is false. I defy you to produce a value x such that x − x = 0 but x − x ≠ 0.
1346:
1065:
966:
699:
432:
1611:
If you don't mind, I'd like to continue the discussion from the
Reference Desk. You sound like you know what you're talking about. I would have continued it then, but I've been offline for three days.
1431:
860:
581:
507:
780:
1488:
1545:
1072:... and the additional solution ignored by Lambiam in his original demonstration should be explicitly checked and/or rejected by the student in the context of the modeled problem. --
1583:=0), but which still has solutions and all of those solutions are also solutions to the original problem. This is what Lambiam aspires to have found, and that much is correct. --
388:
368:
1108:
407:
Perhaps you do not understand because you wish not to understand. But in any case, please stop posting incoherent strings of statements on my talk page. I give up. --
1348:
ought to be decidable, thus if it is not true, it must be false and a counterexample must exist. Considering the finer points of logic is an unneeded digression. --
1362:
This I understood, but where's the proof that the set of solutions in the original problem is not reduced by one when the degree is reduced by one. Because while
583:, Lambiam has correctly mentioned that this implication also happens to hold (for real numbers). The proof requires more than one step, but is valid nonetheless.
393:
Try to avoid insulting terms like "gibberish", please. Maybe you should first look closer to home for the cause of your lack of ability to understand. -- --
73:
53:
1260:
972:
1254:
To find all solutions, you need a two-sided implication - the one you gave doesn't really tell you that what you have found is a solution.
68:
882:
615:
38:
22:
1490:
will appear equally valid to the student and is of course wrong. Using
Lambiam's inference, but one degree lower gives
1365:
794:
515:
58:
441:
866:
can be used to derive a contradiction even in the field of the reals, as I demonstrated using the simpler statement.
714:
131:
Sounds like a really, really bad idea. Msin(i) is just the mass times the sine of i. What's next, an article about
1575:
But
Lambiam has said very clearly that he had no interest in keeping the solution set unchanged. By removing an
33:
1588:
1436:
1353:
593:
231:
140:
99:
88:
1493:
80:
262:
Case B: x ≠ 0. Then we may divide both sides of an equation (even according to
Fuzzyeric) by x, giving us:
512:
Since you (FuzzyEric) failed to understand this, and diverted the discussion to the unrelated implication
84:
1620:
48:
1244:{\displaystyle \pi r(r+{\sqrt {r^{2}+h^{2}}})=\pi r^{2}h/3\Leftarrow 3(r+{\sqrt {r^{2}+h^{2}}})=rh,}
373:
230:
I don't understand what you mean by "Using the proposed inference form". In general, to show that a
1584:
1349:
589:
234:
136:
95:
1615:
I like the idea of using cellular automata to figure this out. I'll see what I can do with that.
421:
even in general divide by non-zero numbers, so the "counterexample" fails to make the point. --
289:
I really don't get what you are trying to say. You write something on my talk page of the form:
422:
411:
397:
349:
280:
132:
118:
186:
212:
1433:
is valid in a field without zero divisors for a perhaps inobvious reason, the statement
196:
1561:
1553:
1087:
1073:
435:). I will say now what I should have said back then - Lambiam is right, on all counts:
216:
203:
199:
152:
1579:, he obtained an equation which lacks some solutions of the original (those involving
165:
122:
43:
272:
Conclusion: for all possible values of x, it is the case that x − x = 0 ⇒ x − x = 0.
1105:
solution, not necessarily all solutions), and for this the relevant implication is
603:
408:
394:
277:
324:
only because I understood from what you wrote that you challenged the validity of
26:
250:
182:
121:
as a redlink, if you'd like to create an article (your first maybe?). Cheers,
1341:{\displaystyle \forall x\in \mathbb {R} (x^{3}-x^{2}=0\Rightarrow x^{2}-x=0)}
509:. This is true for any ring, as it involves multiplication, not division.
1623:
1592:
1569:
1357:
1095:
1081:
597:
425:
415:
401:
284:
219:
206:
189:
168:
155:
144:
125:
103:
1060:{\displaystyle \Rightarrow (3(r+{\sqrt {r^{2}+h^{2}}})=rh)\lor (r=0).}
869:
On his talk, he makes two incorrect claims. The first is that the
181:
Would you happen to know of any interesting applications of the
877:
The correct inference in the original context has always been:
606:
explicitly writes in the initial context (which you reference):
431:
For some reason I remember this correspondnce (which emerged
961:{\displaystyle \pi r(r+{\sqrt {r^{2}+h^{2}}})=\pi r^{2}h/3.}
694:{\displaystyle \pi r(r+{\sqrt {r^{2}+h^{2}}})=\pi r^{2}h/3.}
588:
Please re-read the discussion before commenting further. --
64:
And let's not forget the hub of the mathematics community:
346:
Don't you really see that there is a difference between
241:
Here is a proof of the implication you are challenging:
114:
1496:
1439:
1368:
1263:
1111:
975:
885:
797:
717:
618:
518:
444:
376:
352:
1560:) 23:09, 17 May 2008 (UTC) (edited very slightly by
1539:
1482:
1426:{\displaystyle x^{3}-x^{2}=0\Rightarrow x^{2}-x=0}
1425:
1340:
1243:
1059:
960:
855:{\displaystyle x^{3}-x^{2}=0\Rightarrow x^{2}-x=0}
854:
774:
693:
576:{\displaystyle x^{3}-x^{2}=0\Rightarrow x^{2}-x=0}
575:
501:
382:
362:
335:It is easy enough to give an independent proof of
502:{\displaystyle x^{3}-x^{2}=0\Leftarrow x^{2}-x=0}
775:{\displaystyle 3(r+{\sqrt {r^{2}+h^{2}}})=rh.}
1257:It was assumed that a statement as simple as
8:
74:Knowledge (XXG) talk:WikiProject Mathematics
1483:{\displaystyle x^{2}-x=0\Rightarrow x-1=0}
195:I can't think of anything better than the
79:I hope you enjoy editing here and being a
1540:{\displaystyle x^{2}-x=0\Leftarrow x-1=0}
1501:
1495:
1444:
1438:
1405:
1386:
1373:
1367:
1317:
1298:
1285:
1274:
1273:
1262:
1218:
1205:
1199:
1176:
1167:
1146:
1133:
1127:
1110:
1013:
1000:
994:
974:
950:
941:
920:
907:
901:
884:
834:
815:
802:
796:
749:
736:
730:
716:
683:
674:
653:
640:
634:
617:
555:
536:
523:
517:
481:
462:
449:
443:
375:
351:
358:
353:
69:Knowledge (XXG):WikiProject Mathematics
438:Lambiam's original statement was that
7:
248:Case A: x = 0. Then x − x = 0, so,
34:The five pillars of Knowledge (XXG)
1264:
339:; you do not need the validity of
14:
862:. Use of left cancellation by
383:{\displaystyle \Longleftarrow }
306:I did not claim any such thing.
1519:
1462:
1398:
1335:
1310:
1278:
1226:
1190:
1184:
1154:
1118:
1051:
1039:
1033:
1021:
985:
979:
976:
928:
892:
827:
757:
721:
661:
625:
548:
474:
416:01:27, 29 September 2006 (UTC)
402:00:41, 29 September 2006 (UTC)
377:
355:
285:15:39, 28 September 2006 (UTC)
220:05:56, 23 September 2006 (UTC)
207:05:03, 23 September 2006 (UTC)
190:04:31, 23 September 2006 (UTC)
169:20:57, 18 September 2006 (UTC)
156:20:46, 17 September 2006 (UTC)
145:07:37, 16 September 2006 (UTC)
126:04:13, 16 September 2006 (UTC)
104:17:11, 12 September 2006 (UTC)
83:! If you need help, check out
1:
1624:19:50, 16 December 2006 (UTC)
226:Invalid division proposition
54:How to write a great article
1639:
363:{\displaystyle \implies }
85:Knowledge (XXG):Questions
1593:10:03, 18 May 2008 (UTC)
1570:23:26, 17 May 2008 (UTC)
1565:
1557:
1358:23:02, 17 May 2008 (UTC)
1251:which is trivially true.
1096:22:38, 17 May 2008 (UTC)
1091:
1082:22:17, 17 May 2008 (UTC)
1077:
598:17:55, 17 May 2008 (UTC)
426:10:36, 17 May 2008 (UTC)
309:In what you wrote, both
245:Either x = 0 or x ≠ 0.
1541:
1484:
1427:
1342:
1245:
1061:
962:
856:
776:
705:Divide both sides by π
695:
577:
503:
384:
364:
265:x − x = 0 ⇒ x − x = 0.
257:x − x = 0 ⇒ x − x = 0.
21:Hello, Fuzzyeric, and
1542:
1485:
1428:
1343:
1246:
1062:
963:
857:
777:
696:
578:
504:
385:
365:
1494:
1437:
1366:
1261:
1109:
973:
883:
795:
715:
616:
516:
442:
374:
350:
317:are true statements.
94:Again, welcome! --
1537:
1480:
1423:
1338:
1241:
1057:
958:
852:
772:
691:
573:
499:
380:
360:
359:
354:
320:I gave a proof of
117:response. I offer
39:How to edit a page
1224:
1152:
1019:
926:
791:simpler sentence
755:
659:
413:
399:
328:, not to justify
293:"your claim that
282:
1630:
1546:
1544:
1543:
1538:
1506:
1505:
1489:
1487:
1486:
1481:
1449:
1448:
1432:
1430:
1429:
1424:
1410:
1409:
1391:
1390:
1378:
1377:
1347:
1345:
1344:
1339:
1322:
1321:
1303:
1302:
1290:
1289:
1277:
1250:
1248:
1247:
1242:
1225:
1223:
1222:
1210:
1209:
1200:
1180:
1172:
1171:
1153:
1151:
1150:
1138:
1137:
1128:
1066:
1064:
1063:
1058:
1020:
1018:
1017:
1005:
1004:
995:
967:
965:
964:
959:
954:
946:
945:
927:
925:
924:
912:
911:
902:
861:
859:
858:
853:
839:
838:
820:
819:
807:
806:
781:
779:
778:
773:
756:
754:
753:
741:
740:
731:
700:
698:
697:
692:
687:
679:
678:
660:
658:
657:
645:
644:
635:
582:
580:
579:
574:
560:
559:
541:
540:
528:
527:
508:
506:
505:
500:
486:
485:
467:
466:
454:
453:
412:
398:
389:
387:
386:
381:
369:
367:
366:
361:
297:is justified by
281:
1638:
1637:
1633:
1632:
1631:
1629:
1628:
1627:
1609:
1497:
1492:
1491:
1440:
1435:
1434:
1401:
1382:
1369:
1364:
1363:
1313:
1294:
1281:
1259:
1258:
1214:
1201:
1163:
1142:
1129:
1107:
1106:
1009:
996:
971:
970:
937:
916:
903:
881:
880:
830:
811:
798:
793:
792:
745:
732:
713:
712:
670:
649:
636:
614:
613:
551:
532:
519:
514:
513:
477:
458:
445:
440:
439:
372:
371:
348:
347:
228:
213:Talk:Nine lemma
179:
111:
59:Manual of Style
27:Knowledge (XXG)
19:
12:
11:
5:
1636:
1634:
1608:
1605:
1604:
1603:
1602:
1601:
1600:
1599:
1598:
1597:
1596:
1595:
1585:Meni Rosenfeld
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1504:
1500:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1447:
1443:
1422:
1419:
1416:
1413:
1408:
1404:
1400:
1397:
1394:
1389:
1385:
1381:
1376:
1372:
1350:Meni Rosenfeld
1337:
1334:
1331:
1328:
1325:
1320:
1316:
1312:
1309:
1306:
1301:
1297:
1293:
1288:
1284:
1280:
1276:
1272:
1269:
1266:
1255:
1252:
1240:
1237:
1234:
1231:
1228:
1221:
1217:
1213:
1208:
1204:
1198:
1195:
1192:
1189:
1186:
1183:
1179:
1175:
1170:
1166:
1162:
1159:
1156:
1149:
1145:
1141:
1136:
1132:
1126:
1123:
1120:
1117:
1114:
1084:
1070:
1069:
1068:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1016:
1012:
1008:
1003:
999:
993:
990:
987:
984:
981:
978:
968:
957:
953:
949:
944:
940:
936:
933:
930:
923:
919:
915:
910:
906:
900:
897:
894:
891:
888:
875:
867:
851:
848:
845:
842:
837:
833:
829:
826:
823:
818:
814:
810:
805:
801:
788:
786:
785:
784:
783:
782:
771:
768:
765:
762:
759:
752:
748:
744:
739:
735:
729:
726:
723:
720:
703:
702:
701:
690:
686:
682:
677:
673:
669:
666:
663:
656:
652:
648:
643:
639:
633:
630:
627:
624:
621:
607:
590:Meni Rosenfeld
586:
585:
584:
572:
569:
566:
563:
558:
554:
550:
547:
544:
539:
535:
531:
526:
522:
510:
498:
495:
492:
489:
484:
480:
476:
473:
470:
465:
461:
457:
452:
448:
405:
404:
391:
379:
357:
344:
333:
318:
307:
303:
302:
274:
273:
270:
269:
268:
267:
266:
260:
259:
258:
227:
224:
223:
222:
209:
200:User:John Baez
178:
175:
174:
173:
172:
171:
159:
158:
148:
147:
137:Meni Rosenfeld
113:Thank you for
110:
107:
96:Meni Rosenfeld
77:
76:
71:
62:
61:
56:
51:
46:
41:
36:
18:
15:
13:
10:
9:
6:
4:
3:
2:
1635:
1626:
1625:
1622:
1616:
1613:
1606:
1594:
1590:
1586:
1582:
1578:
1574:
1573:
1571:
1567:
1563:
1559:
1555:
1550:
1534:
1531:
1528:
1525:
1522:
1516:
1513:
1510:
1507:
1502:
1498:
1477:
1474:
1471:
1468:
1465:
1459:
1456:
1453:
1450:
1445:
1441:
1420:
1417:
1414:
1411:
1406:
1402:
1395:
1392:
1387:
1383:
1379:
1374:
1370:
1361:
1360:
1359:
1355:
1351:
1332:
1329:
1326:
1323:
1318:
1314:
1307:
1304:
1299:
1295:
1291:
1286:
1282:
1270:
1267:
1256:
1253:
1238:
1235:
1232:
1229:
1219:
1215:
1211:
1206:
1202:
1196:
1193:
1187:
1181:
1177:
1173:
1168:
1164:
1160:
1157:
1147:
1143:
1139:
1134:
1130:
1124:
1121:
1115:
1112:
1104:
1099:
1098:
1097:
1093:
1089:
1085:
1083:
1079:
1075:
1071:
1054:
1048:
1045:
1042:
1036:
1030:
1027:
1024:
1014:
1010:
1006:
1001:
997:
991:
988:
982:
969:
955:
951:
947:
942:
938:
934:
931:
921:
917:
913:
908:
904:
898:
895:
889:
886:
879:
878:
876:
872:
868:
865:
849:
846:
843:
840:
835:
831:
824:
821:
816:
812:
808:
803:
799:
789:
787:
769:
766:
763:
760:
750:
746:
742:
737:
733:
727:
724:
718:
711:
710:
708:
704:
688:
684:
680:
675:
671:
667:
664:
654:
650:
646:
641:
637:
631:
628:
622:
619:
612:
611:
610:
609:
608:
605:
601:
600:
599:
595:
591:
587:
570:
567:
564:
561:
556:
552:
545:
542:
537:
533:
529:
524:
520:
511:
496:
493:
490:
487:
482:
478:
471:
468:
463:
459:
455:
450:
446:
437:
436:
434:
430:
429:
428:
427:
424:
418:
417:
414:
410:
403:
400:
396:
392:
345:
342:
338:
334:
331:
327:
323:
319:
316:
312:
308:
305:
304:
300:
296:
292:
291:
290:
287:
286:
283:
279:
271:
264:
263:
261:
256:
255:
253:
252:
247:
246:
244:
243:
242:
239:
236:
233:
225:
221:
218:
214:
210:
208:
205:
201:
198:
194:
193:
192:
191:
188:
184:
176:
170:
167:
163:
162:
161:
160:
157:
154:
150:
149:
146:
142:
138:
134:
130:
129:
128:
127:
124:
120:
116:
108:
106:
105:
101:
97:
92:
90:
87:or ask me on
86:
82:
75:
72:
70:
67:
66:
65:
60:
57:
55:
52:
50:
47:
45:
42:
40:
37:
35:
32:
31:
30:
28:
24:
16:
1621:Black Carrot
1617:
1614:
1610:
1580:
1576:
1548:
1102:
870:
863:
706:
602:Incorrect.
419:
406:
340:
336:
329:
325:
321:
314:
310:
298:
294:
288:
275:
249:
240:
229:
180:
164:Hey, sorry.
112:
93:
89:my talk page
78:
63:
20:
1607:Probability
235:implication
211:(copied to
187:HappyCamper
151:Agreed. --
1547:is valid,
251:a fortiori
202:found. --
183:nine lemma
177:Nine lemma
81:Wikipedian
44:Help pages
1562:Fuzzyeric
1554:Fuzzyeric
1088:Fuzzyeric
1074:Fuzzyeric
343:for that.
232:universal
217:Fuzzyeric
204:Fuzzyeric
153:Fuzzyeric
166:Marskell
123:Marskell
49:Tutorial
17:Welcome!
604:Lambiam
423:Lambiam
409:Lambiam
395:Lambiam
278:Lambiam
133:gcos(θ)
119:Msin(i)
109:Msin(i)
23:welcome
215:) --
135:? --
1589:talk
1566:talk
1558:talk
1354:talk
1092:talk
1078:talk
871:only
709:/3:
594:talk
433:here
370:and
313:and
197:link
185:? --
141:talk
115:this
100:talk
1549:but
25:to
1591:)
1572:)
1568:)
1526:−
1520:⇐
1508:−
1469:−
1463:⇒
1451:−
1412:−
1399:⇒
1380:−
1356:)
1324:−
1311:⇒
1292:−
1271:∈
1265:∀
1185:⇐
1161:π
1113:π
1094:)
1080:)
1037:∨
977:⇒
956:3.
935:π
887:π
841:−
828:⇒
809:−
689:3.
668:π
620:π
596:)
562:−
549:⇒
530:−
488:−
475:⇐
456:−
378:⟸
356:⟹
276:--
254::
143:)
102:)
91:.
1587:(
1581:r
1577:r
1564:(
1556:(
1535:0
1532:=
1529:1
1523:x
1517:0
1514:=
1511:x
1503:2
1499:x
1478:0
1475:=
1472:1
1466:x
1460:0
1457:=
1454:x
1446:2
1442:x
1421:0
1418:=
1415:x
1407:2
1403:x
1396:0
1393:=
1388:2
1384:x
1375:3
1371:x
1352:(
1336:)
1333:0
1330:=
1327:x
1319:2
1315:x
1308:0
1305:=
1300:2
1296:x
1287:3
1283:x
1279:(
1275:R
1268:x
1239:,
1236:h
1233:r
1230:=
1227:)
1220:2
1216:h
1212:+
1207:2
1203:r
1197:+
1194:r
1191:(
1188:3
1182:3
1178:/
1174:h
1169:2
1165:r
1158:=
1155:)
1148:2
1144:h
1140:+
1135:2
1131:r
1125:+
1122:r
1119:(
1116:r
1103:a
1101:(
1090:(
1076:(
1067:"
1055:.
1052:)
1049:0
1046:=
1043:r
1040:(
1034:)
1031:h
1028:r
1025:=
1022:)
1015:2
1011:h
1007:+
1002:2
998:r
992:+
989:r
986:(
983:3
980:(
952:/
948:h
943:2
939:r
932:=
929:)
922:2
918:h
914:+
909:2
905:r
899:+
896:r
893:(
890:r
864:x
850:0
847:=
844:x
836:2
832:x
825:0
822:=
817:2
813:x
804:3
800:x
770:.
767:h
764:r
761:=
758:)
751:2
747:h
743:+
738:2
734:r
728:+
725:r
722:(
719:3
707:r
685:/
681:h
676:2
672:r
665:=
662:)
655:2
651:h
647:+
642:2
638:r
632:+
629:r
626:(
623:r
592:(
571:0
568:=
565:x
557:2
553:x
546:0
543:=
538:2
534:x
525:3
521:x
497:0
494:=
491:x
483:2
479:x
472:0
469:=
464:2
460:x
451:3
447:x
390:?
341:Q
337:P
332:.
330:P
326:Q
322:Q
315:Q
311:P
301:.
299:Q
295:P
139:(
98:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.