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The name of the figure is 'Number of the Whole'.
'天' directly means 'heaven' and '數' means 'number'.

I don't know what kind of translation word is right. I think the 'number of totality' is also good. The literal translation may be "number of heavens". The Chinese numerals in the figure are the same as ...., and the center word means infinity. The numbers show the number of cutting times of the wholeness, which shows in the blank figure bellow.

the whole of no limitation

If you cut a sheet of paper in two, and put on each other, and cut again, it becomes four sheets. If you stacked them up and cut them, you get eight sheets, and it will increase to 16.32.64, and finally, it will be an infinite number of sheets.

However, if the first sheet of paper has an infinite area, the smallest sheet' area has the same size after the infinite cutting. In the first place, the first paper itself maybe a piece of paper cut endlessly. BAIEN writes in GENGO that no matter how small it is, it is the same size as the original.

BAIEN might have understood the countably infinite. We can match the natural numbers to concentric circles endless. But I don't think BAIEN distinguished between endless and mathematical infinity in the strict sense.

This blank sheet has a natural number and has any rational or irrational number and transcendental numbers. It includes everything from elementary particles to the entire universe. Why because nothing can exist out of the wholeness. Even if the totality cut in half, and then cut the half many times, that keeps the entirety. BAIEN thought that half of infinity is infinity. And he believed that the two halves get the antithetical character each other.


The upper dual circle figure is the same as follows, which I made. Because the opposite two, both infinity fields, and no one can recognize whether the whole being of no limitation, which is before every logical thought is white or black.The first logical judgment is to determine that it is unrecognizable, and the first logical operation is to divide that into two opposite parts. The thin strokes in those figures and the seam of the dual figure are logical symbols of that.


As I mentioned in other commentaries, his idea is almost the same as Sheffer stroke. A complete collection of BAIEN published in 1912, and it donated to university libraries around the world.

Sheffer published the stroke function in 1913, so he might have seen the GENGO diagrams. That may determined by examining the records left at the university where the "A complete collection of BAIEN" was donated. However, even if it is not so, BAIEN was the first person who discovered the stroke function in history, and there is no doubt the function used throughout his thinkings. However, no one, including myself, has been able to verify the details.

Also, the 一即一一 and 一一即一, which is a famous thinking pattern of BAIEN, used as the basic formula of his idea. That does not so often use in his sentences but was the so powerful engine for his thinking. And that can be rewritten as 一 反(一反一) by his descriptions. This is an operator that has the same functionality as Stroke. It is not just kanji but works as a logical operator.

That is the basic logic of GENGO and does not often use in his books, but it is the most important concept as a logical expression. And he uses a thin straight line in the figure to represent this formulation. And he had used a seam line of the opened book as the centerline of the dual figure and took advantage of the fact that people can't see the front or back picture at the same time. It has nearly the same function as the Rubin vase. In the sentence, it expressed by a two-line, one-pair description of white and black dots like this.

BAIEN intuited that the logical operation that human reason could perform on an unrecognizable totality was cutting. And then he intuited the most straightforward case was when the two cutting logical subjects get the antithetical characters each other.

Therefore the 一 反(一反一) can rewrite 一 |(一|一). This formula is the same
t| (t | t) which is the peculiar case of Stroke function. I think Baien showed that the stroke function in his figure as follows.

I remember that Bertrand Russell explained the logic as "t implies itself" in his book "Introduction to Mathematical Philosophy".

BAIEN had started writing the first version GENGO when he was 30 years old. And his work had continued to last day of his life. Though he could not finish the last version of GENGO, his logical system had completed already. I interpret that the blank page marked "impossible to express" wraps all the small circles in the following figure. They are all both a part and a whole.
The upper figure has Holon, binary tree, and Sheffer stroke.

In this figure, "一" is written on the right side of the following picture, so naturally, there is no logical problem even if a figure inverted to white "一" drawn on a black background. But more than 200 years ago, there wasn't a tool that allowed a mouse click to reverse a negative and positive. There are many diagrams that BAIEN does not draw. It was a limitation of the time when there were only ink, brushes, paper and a traditional compasse.

a half size picture
binary tree structure of GENGO

And the "一" in the small ring is similar to the Holon proposed by Arthur Koestler and maybe nearly with Totality and Infinity too. The Holon can split into two parts by cutting, but the two don't lose both characters with a whole and apart.

In the quantum level, that divides into waves and particles and particles separate bosons and fermions. That means the original materials of the universe have a dual character from the time of occurrence.

The "一" in the small circle, which written in kanji, means one also in usual in Japan, but it is not 1 in natural number in GENGO. Because if you divide it into two, the two become the same "一", and if you combine the divided two into, it's one. So, the number one, which is written "一" in this figure represents totality.

That is not a quantity calculation, but a logical operation of the value of existence. Maybe, the number of infinity is a little similar. Infinity keeps infinity, whether divided into two or combined into one. Natural numbers consist of an even sequence and an odd sequence. These three sequences have the same infinite size aleph-zero.

The following figure shows the natural numbers described by modulo five residues. There is no zero. In BAIEN's mathematics, zero means no division. I interpret the natural numbers sequence as an exponent of the cutting algebra '2'.
The subject of BAIEN's ideas was transcendental from the subject of natural science, and he constructed it not as philosophical speculation but as general system theory. It is pointed out by Sueki Takehiro (1921 -2007) that the BAIEN's world system is very similar to Kitaro Nishida's "self-identity of absolute contradiction".

However, there is a decisive difference between BAIEN's and Nishida's ideas. While BAIEN developed a general system theory, Nishida developed systematic philosophical speculation. The two thinkers, BAIEN and NISHIDA, are similar in that they started thinking from when they got the 'Pure Experience'.

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