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opencog-learn commits / 88a7ab5

Start work on this mornings thoughts.
88a7ab52269782c0fceffb2fb10b0115a7f5dd68
by Linas Vepštas <linasvepstas@gmail.com> · 2026-01-14T19:11:45.000Z

Changed files (1) +356 -1

learn-lang-diary/here-and-now/here-and-now.lyx +356 -1
……@@ -731,7 +731,362 @@ Well,
731731\end_layout
732732
733733\begin_layout Standard
734Part
734The rest of this text is the diary proper.
735 I'll try to avoid making apologies for how demented some of it may sound.
736 I'm not interested in fighting with cranks.
737 The Dunning–Kruger effect is not limited to stupid people:
738 some brilliant people suffer from it as well.
739 Don't be one of them.
740\end_layout
741
742\begin_layout Subsection*
74314 January 2026
744\end_layout
745
746\begin_layout Standard
747So as I am tossing and turning in bed this morning,
748 planning this diary entry,
749 I develop the following chain of ideas.
750 The culmination of these ideas is an attempt to understand non–locality and wave function collapse in QM.
751 Working on this,
752 I kept making steps that go backwards,
753 getting simpler and more basic at each step,
754 until I arrive at a starting point that is absurdly trite.
755 Hah.
756 Rather than walking backwards for this text,
757 let me start at the start.
758\end_layout
759
760\begin_layout Standard
761Consider a function
762\begin_inset Formula
763\[
764f:I\to\mathbb{R}
765\]
766
767\end_inset
768
769where
770\begin_inset Formula $I=\left[0,1\right]$
771\end_inset
772
773 the closed unit interval of the reals endowed with the standard topology.
774 Likewise,
775
776\begin_inset Formula $\mathbb{R}$
777\end_inset
778
779 is the unbounded reals,
780 with the standard topology.
781 Other topologies are possible,
782 e.g.
783 the
784\begin_inset Formula $p$
785\end_inset
786
787–adic topology,
788 so this is already a strong statement.
789 The natural topology allows a metric to be defined,
790 so these are metric spaces.
791\end_layout
792
793\begin_layout Standard
794What is
795\begin_inset Formula $f$
796\end_inset
797
798?
799 Well,
800 I already said that its a function;
801 thus,
802 it is not a relation.
803 It is a section through
804\begin_inset Formula $I\times\mathbb{R}$
805\end_inset
806
807 viewed as a line bundle;
808 the base space is
809\begin_inset Formula $I$
810\end_inset
811
812 and the fiber is
813\begin_inset Formula $\mathbb{R}$
814\end_inset
815
816.
817 (N.B.
818
819\begin_inset Formula $f$
820\end_inset
821
822 is a pre–prototype for a wave–function
823\begin_inset Formula $\psi$
824\end_inset
825
826 and
827\begin_inset Formula $I$
828\end_inset
829
830 is a stand–in for space–time.
831 It will take a few steps to get there,
832 though.)
833\end_layout
834
835\begin_layout Standard
836There is a spot of philosophical confusion here.
837 When I write
838\begin_inset Formula $f:I\to\mathbb{R},$
839\end_inset
840
841this notation suggests,
842 or implies,
843 or has the natural interpretation that there is
844\begin_inset Quotes eld
845\end_inset
846
847one copy
848\begin_inset Quotes erd
849\end_inset
850
851 of
852\begin_inset Formula $\mathbb{R}$
853\end_inset
854
855 and that
856\begin_inset Formula $f$
857\end_inset
858
859 is a request to choose
860\begin_inset Quotes eld
861\end_inset
862
863one value
864\begin_inset Quotes erd
865\end_inset
866
867 out of this
868\begin_inset Formula $\mathbb{R}$
869\end_inset
870
871 for each
872\begin_inset Quotes eld
873\end_inset
874
875point
876\begin_inset Quotes erd
877\end_inset
878
879 of
880\begin_inset Formula $I$
881\end_inset
882
883.
884 However,
885 when I say that
886\begin_inset Formula $f$
887\end_inset
888
889 is a section through the line bundle
890\begin_inset Formula $I\times\mathbb{R}$
891\end_inset
892
893,
894 the natural interpretation is that each fiber is a
895\begin_inset Quotes eld
896\end_inset
897
898distinct copy
899\begin_inset Quotes erd
900\end_inset
901
902 of
903\begin_inset Formula $\mathbb{R}$
904\end_inset
905
906,
907 and that each copy can be labeled by a point
908\begin_inset Formula $x\in I$
909\end_inset
910
911.
912 So which is it?
913 Are there many copies,
914 or is there just one?
915\end_layout
916
917\begin_layout Standard
918Since I have not asked that
919\begin_inset Formula $f$
920\end_inset
921
922 be continuous,
923 it would seem that the following rather hallucinatory statement is possible:
924\begin_inset Formula
925\[
926I\times\mathbb{R}\simeq\prod_{x\in I}\mathbb{R}_{x}=\mathbb{R}\times\mathbb{R}\times\cdots
927\]
928
929\end_inset
930
931where
932\begin_inset Formula $\mathbb{R}_{x}$
933\end_inset
934
935 is the fiber above the point
936\begin_inset Formula $x\in I$
937\end_inset
938
939 of the base space.
940 The product
941\begin_inset Formula $\prod$
942\end_inset
943
944 is the Cartesian product.
945 This product
946\begin_inset Formula $\prod_{x\in I}$
947\end_inset
948
949 is a product of uncountably many copies of
950\begin_inset Formula $\mathbb{R}$
951\end_inset
952
953.
954 When written in this way,
955 it's quite the nasty little space,
956 because this notation suggests that the appropriate topology is the so–called
957\begin_inset Quotes eld
958\end_inset
959
960box topology
961\begin_inset Quotes erd
962\end_inset
963
964 or
965\begin_inset Quotes eld
966\end_inset
967
968fine topology
969\begin_inset Quotes erd
970\end_inset
971
972.
973 This topology is very very different from what one might naturally think of,
974 when writing down the Cartesian product
975\begin_inset Formula $I\times\mathbb{R}$
976\end_inset
977
978.
979
980\end_layout
981
982\begin_layout Standard
983Somewhere in high–school algebra,
984 before the word
985\begin_inset Quotes eld
986\end_inset
987
988Cartesian
989\begin_inset Quotes erd
990\end_inset
991
992 is used,
993 and long long before the idea of a topology is presented,
994 students are already encouraged to think of the natural topology on
995\begin_inset Formula $I\times\mathbb{R}$
996\end_inset
997
998.
999 It is like a strip of paper,
1000 one inch wide,
1001 and
1002\begin_inset Quotes eld
1003\end_inset
1004
1005infinitely long
1006\begin_inset Quotes erd
1007\end_inset
1008
1009.
1010 Why is this
1011\begin_inset Quotes eld
1012\end_inset
1013
1014natural
1015\begin_inset Quotes erd
1016\end_inset
1017
1018?
1019 Because we live in 3D space,
1020 and a piece of paper is naturally 2D,
1021 and Euclid could articulate the notion of a flat plain,
1022 because in the physical reality in which we live,
1023 there are lots of flat planes,
1024 and the topology,
1025 the natural topology,
1026 as defined by Hausdorff and the Poles of the 1930's,
1027 and the
1028\begin_inset Quotes eld
1029\end_inset
1030
1031Polish spaces
1032\begin_inset Quotes erd
1033\end_inset
1034
1035 that they give us,
1036 are naturally metric spaces that can be intuited by high–school students.
1037 Metric spaces are axiomatized in mathematics,
1038 but ordinary common every–day experience with a ruler provide a natural intuition for metricity.
1039 This
1040\begin_inset Quotes eld
1041\end_inset
1042
1043ordinary experience
1044\begin_inset Quotes erd
1045\end_inset
1046
1047 is also why mathematicians accept the axioms of metric spaces as being somehow obvious or reasonable or expected or normal,
1048 and not some space alien from outer space.
1049\end_layout
1050
1051\begin_layout Standard
1052This
1053\begin_inset Quotes eld
1054\end_inset
1055
1056ordinary experience
1057\begin_inset Quotes erd
1058\end_inset
1059
1060 is grounded in the physical world,
1061 as we live in it,
1062 having an
1063\begin_inset Quotes eld
1064\end_inset
1065
1066obvious
1067\begin_inset Quotes erd
1068\end_inset
1069
1070 3D structure,
1071 perceived by our brains,
1072 or rather,
1073 the neurons in our brains.
1074 Of course,
1075 the brains in mammals and birds perceive this as well;
1076 and we can split hairs as to whether spiders and flies perceive space as being 3D as well.
1077 I think they do,
1078 but there is always someone looking for an argument.
1079 Why am I talking about brains in a text ostensibly about QM?
1080 Because these same brains that have no issue with metric spaces and natural topologies go hog–wild when confronted with Hilbert spaces.
1081\end_layout
1082
1083\begin_layout Standard
1084which suggests or implies the natrual topol
1085\end_layout
1086
1087\begin_layout Standard
1088fine top box paradoxes e.g.
1089
7351090\end_layout
7361091
7371092\begin_layout Section*