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596: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/25(日) 09:24:07.43 ID:eIdDsFH8

>>595
つづき

That is to say, it is precisely this sort of situation that is referred to by the term
“inter-universal”. Put another way,
a change of universe may be thought of [cf. the discussion of §2.7, (i)] as
a sort of abstract/combinatorial/arithmetic version of the classical notion
of a “change of coordinates”.
In this context, it is perhaps of interest to observe that, from a purely classical point of
view, the notion of a [physical] “universe” was typically visualized as a copy of Euclidean
three-space. Thus, from this classical point of view,
a “change of universe” literally corresponds to a “classical change of the coordinate system − i.e., the labelling apparatus − applied to label points in
Euclidean three-space”!
Indeed, from an even more elementary point of view, perhaps the simplest example of the
essential phenomenon under consideration here is the following purely combinatorial
phenomenon: Consider the string of symbols
010
− i.e., where “0” and “1” are to be understood as formal symbols. Then, from the
point of view of the length two substring 01 on the left, the digit “1” of this substring
may be specified by means of its “coordinate relative to this substring”, namely, as the
symbol to the far right of the substring 01. In a similar vein, from the point of view of
the length two substring 10 on the right, the digit “1” of this substring may be specified
by means of its “coordinate relative to this substring”, namely, as the symbol to the far
left of the substring 10. On the other hand,
neither of these specifications via “substring-based coordinate systems”
is meaningful to the opposite length two substring; that is to say, only the
solitary abstract symbol “1” is simultaneously meaningful, as a device
for specifying the digit of interest, relative to both of the “substring-based coordinate systems”.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1592654877/596

597: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/25(日) 09:24:26.06 ID:eIdDsFH8

>>596
つづき

Finally, in passing, we note that this discussion applies, albeit in perhaps a somewhat
trivial way, to the isomorphism of Galois groups ΨηX : GK〜→ GK induced by the
Frobenius morphism ΦηX in Example 2.6.1, (i): That is to say, from the point of view
of classical ring theory, this isomorphism of Galois groups is easily seen to coincide with
the identity automorphism of GK. On the other hand, if one takes the point of view
that elements of various subquotients of GK are equipped with labels that arise from
the isomorphisms ρ or κ of Example 2.6.1, (ii), (iii), i.e., from the reciprocity map of
class field theory or Kummer theory, then one must regard such labelling apparatuses
as being incompatible with the Frobenius morphism ΦηX . Thus, from this point
of view, the isomorphism ΦηX must be regarded as a “mysterious, indeterminate
isomorphism” [cf. the discussion of §2.7, (iii)].
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1592654877/597

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