坢（taira-ka） 60 Stars astrology A Summary of This Season?

60 Stars Astrology 

English　Version

By　TOKYOｰTANUKI

坢（taira-ka） 60-Stars astrology　

 A Summary of This Season?

1.　Well... before suddenly moving into Season ▲, I ended up packing quite a lot of ideas into the final posts.

There was my little thought experiment, "Between 0 and 1," about the birth order of numbers, and a short story inspired? by the Bolshevik Revolution.

One day my forehead suddenly became much larger, and that was enough reason to start writing.


...Now that my forehead has settled down quite a bit...
this is where I'll stop.




2.The main question I've been thinking about for many years is surprisingly simple:

"Why does 2 come after 0 and 1 ?"

I've read explanations of the Peano axioms, but...to be honest, as an amateur, I still don't really feel that I understand them.

Of course, I haven't even tried reading Principia Mathematica.
That would be far beyond poor Tanu-chan!


Personally, I don't think the next number absolutely has to be "2".


....Imagine Adam exists.

Let the state where Adam exists be 1, and the state where he doesn't exist be 0.

I'm not even sure whether the natural order should be 0 → 1 or 1 → 0.

If another Adam appears, then perhaps
0 → 1 → 2
makes perfect sense.

But if Eve appears instead...
perhaps it becomes
0 → 1 → Φ
Or maybe even
0 → 1 → Φ → 2


Well...that's simply one of my daydreams.

The real question is:
Should we assume that the order is fixed from the very beginning?



3. Suppose a world contains only 0 and 1.

Then perhaps the next number should also be describable using nothing but 0 and 1.

That idea became the basis for my short story.

To make the game more interesting, I invented a few completely arbitrary rules.
The smallest building block must be one of:
 (1 + 0)
 (1 + 1)
 (0 + 0)

and every new number must appear in the form
1 + X
where X is built under the following rules.

① Basic components
   * Only 0 and 1 may be used.
   * Fractions such as 1/(1+1) are completely forbidden.
② No three-layer exponents
   * Expressions like A^B^C are not allowed.
③ No three-term expressions
   * Three separate terms may not be combined in a single operation.
   * Only combinations of two blocks are permitted.
④ No triple parentheses
   * If three opening or closing parentheses appear consecutively, such as (((  or  ))) , the expression immediately fails.


For example,
1 + (1+1)^(1+1) = 5
may look perfectly fine.

However, if exponentiation is counted as an operator in the same way as addition or multiplication, then this expression effectively contains three elements, so I reject it.

Fortunately, the number 5 can also be written as

5 = 1 + ((1+1)^(1+1))
or
5 = 1 + ((1+1) + (1+1))
or
5 = 1 + ((1+1) × (1+1))


......These versions satisfy my imaginary rules because neither the inside nor the whole expression becomes a three-term construction.
......Yes, these are completely self-made rules.


At one point Chat-GPT even told me,"I'm not sure I understand what you mean."

Quite fair, actually.

The treatment of exponentiation, in particular, is rather arbitrary.



4. The idea behind "no three layers" is this.

For example,

257 = 1 + ((1+1)^(1+1)^(1+1))^(1+1)
is rejected because the part
((1+1)^(1+1)^(1+1))
already contains three exponent layers.

If exponentiation is treated as an independent operator, it also violates my "no three-term" rule.

Likewise,
257 = 1 + ((1+1)^(1+1))^(1+1)^(1+1)
is also rejected because it still creates a three-layer structure.

However, if exponentiation is not regarded as an independent operator, and only the three-layer structure itself is forbidden, then perhaps
257 = 1 + ((1+1)^(1+1))^((1+1)^(1+1))
might actually be acceptable.

In that case, perhaps even something like
65537 = 1 + (((1+1)^(1+1))^(1+1))^((1+1)^(1+1))
could also be allowed ?.


But then another question appears.

If this world truly contains only the concepts of 0 and 1, perhaps not only three-term expressions and three-layer exponents should be forbidden.
And maybe even three consecutive parentheses — ((( — should never be allowed.


By the way. another question also comes to mind.
Should something like
1/(1+1)
be allowed?

Since 1/(1+1) = 1 ÷ (1+1),
then
1 + (1/(1+1))
looks almost like two terms...
and almost like three.

.........Mmm......It's difficult to say.

Well...these are only daydreams after all.

5. Another of my daydreams is about geometry.

I imagine that shapes begin as

a point (a 1/Φ-gon),

then become

a straight line (a 1-gon),

and after that they split into two different paths:

curves (Φ-gons)
and
broken lines or polygons.


.....If that is true, perhaps numbers also branch in a similar way.

Strictly speaking, perhaps we first have
0, 1/Φ, and 1.

Adding another 1 produces 2.

Adding 1/Φ instead produces Φ.


....In other words,
perhaps there is more than one natural sequence in which numbers can emerge.

Curves eventually become complete when they close into a circle, provided they never grow into spirals.

Polygons, on the other hand, can continue forever by increasing their number of sides.

So,perhaps that is why
1, 2, 3, 4, 5,6,7,8,9,...
eventually became the dominant way of counting.


At least...
that's how my imagination likes to see it.




6. One last daydream.

Some people were taught by teachers, naturally, 
2^3^4
as something closely or related to repeated multiplication.

But to me,
it doesn't really feel that way.

After all,
2^3^4
and
4^3^2
produce completely different values.

....So perhaps exponentiation isn't really "multiplication-like."

People often say that in expressions like
a^b^c,
the numbers a and c are connected only through the middle exponent.

But sometimes I wonder...
what if a and c are somehow communicating directly?

.............Suppose we compare
a^b^c
with
a^d^c.

Could there be some hidden agreement between a and c that also appears in every expression of the form
a^e^c ?

At this point...
yes, I'm probably sounding like a crazy person.




Huh......Anyway,
my forehead has calmed down, so this season ends here.



Next time, the next season really begins.

That's all for today.





Tanu-chan 💓   TOKYO-TANUKI 💛

*** Note:** These are simply my thought experiments. I'm not trying to prove anything or replace mathematics. I'm just sharing the fun of thinking. Whether these ideas have any value is entirely up to each reader.(this note is from Chat-Gpt).