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

the whole of no limitation

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 1,2,4,8, .... , and the center word means infinity. The numbers 1,2,4,8, .... show the numbers of parts generated by cutting the wholeness shown in the blank page. If you cut a sheet of paper in two, put one on the top of the other, and cut again, it becomes four sheets. If you stack 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. In general, if you cut the wholeness n times, it becomes 2^n ( two to the nth power) parts, where n=0,1,2,3, ... . Note that two to the 0th power is one.

Here, if the first sheet of paper has an infinite area, the smallest sheet' area has the same size as the first one after the infinite cuttings. In the first place, the first paper itself maybe a piece of paper which was generated by cutting endlessly. Baien writes in GENGO that no matter how small it is, it has the same size as the original.

Baien might have understood the countable infinity. We can match the natural numbers to concentric circles endlessly. But I don't think Baien distinguished between endlessness and mathematical infinity in the strict sense.

I think that the blank sheet, which means wholeness, includes all the natural numbers, all the rational or the irrational numbers, and all the transcendental numbers, and infinity, and that it has everything from elementary particles to the entire universe.

Because nothing exists outside of totality. Even if the totality is cut in half, and then repeatedly cut in half, it keeps the entirety.

Baien thought that half of infinity is infinity. However he believed that the two halves get the antithetical character each other according to the principle of duality. It is similar to the one-to-one correspondence between positive and negative integer sequences.


The relationship between the white circle and the black circle in the figure above is the same as the relation between the white page and the black page in the figure below, where the left page is obtained by inverting black and white of the right page written "一不上図" (wholeness cannot be expressed) .In both figures, the center line indicates the book binding.

When you close the book, you will not see anything. Opening a book has a logical meaning. I think it can be called expansion or cutting. When you open the book, you can see the left and right white and black figures and the binding of the book.

Let us take an example of a sequence. Suppose all integers are arranged in a row from the smallest to the largest.
-∞,... , -3, -2, -1, 0, 1, 2, 3, ... , ∞

The right side of 0 is
1,2,3, ... , ∞,
and the left side of 0 is
-1, -2, -3, ... , -∞.

In this example 0 corresponds the book binding, and the operation of closing a book corresponds to addition. If you adjust the left and right numbers based on 0 and add the two overlapped numbers you will get 0s. And 0s disappear.

When you close the book, you can not see anything. Opening the facing pages of Gengo which Baien put as one-pair-diagrams, has a logical meaning. Some of them are arranged right and left page, and others former page and next. Of course, the arrangement has the reason.


As I mentioned in other commentaries, his idea is almost the same as Sheffer stroke. A complete collection of Baien was published in 1912, and some copies of the book were 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 universities where copies of "A complete collection of Baien" were donated. However, even if it is not the case, Baien was the first person who discovered the stroke function in history, and there is no doubt the function used throughout his thinkings. But, 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 was used as the basic formula of his idea. Though not so often be used in his sentences, it was so powerful an engine for his thinking. And that can be rewritten as "一 反(一反一) " by his descriptions. This "反" is an operator that has the same functionality as the stroke function. It is not just a kanji but works as a logical operator.

That is the basic logic of GENGO and Baien did not often use in the book, but it is the most important concept as a logical principle. And he used a thin straight line to express the logic in the figure. He used a seam line of the open book as the centerline of the pair figure and took advantage of the fact that readers can't see the front and back pictures at the same time.

Opening and closing a book, or turning and returning the page with figures has nearly the same function as the Rubin vase. In the sentences, it is expressed by a pair of lines, where the left line ends with a black comma and the right one with a white comma. The lines look like this.

Baien intuitively recognized that the logical operation that humans can do on the unrecognizable totality is a cutting. And then he intuited that the most straightforward case is the case when the two logical objects obtained by a cutting get opposite and complementary properties each other like the seqence example above.

Baien visited Ise Shrine precisely the year he got the inspiration above, but whether it was before or after the visit could only be determined by carefully examining his diary and letters.

Therefore the 一 反(一反一) can be rewritten as 一 |(一|一). This formula is the same as t| (t | t) , which is a peculiar application of the stroke function. I think Baien showed the stroke function in his figure as follows.

I remember that Bertrand Russell explained a taughtology "t implies itself" in his book "Introduction to Mathematical Philosophy".

Baien started writing the first version GENGO when he was 30 years old. And his work continued to the 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.

Therefore, we can regard the figure below as a graphical representation of the Schaeffer function, which is a tautology. However, "t" is replaced with "一" in the picture, and they are entirely different. While t is a set that does not include itself as an element, "一" is a collection that includes itself as an element.

Bertrand Russell found the Russell paradox from the existence of a set that contains itself as an element while Baien reached the self-reference paradox from "一". And Baien saw the paradox not as a contradiction to be solved logically, but as an essential form of existence that encompasses everything.

The upper figure has Holon, binary tree, and Sheffer stroke.
Click back the map above for more detail if you want.

Logically, t | (t | t)t, and t → tt, so naturally, t | (t | t)t → t holds. Using the truth tables, you can find that the Schaeffer function is a complete set of functions. About the completeness of the stroke function see here. Note that the stroke function is logically the same as nand, that is p|q≡¬(p∧q).

t | (t | t) means "if t then t", or "t implies t". The following diagram in which t is defined as all the ""s in small circles in the figure above. And the center small circle is logically congruent to the large circle on the outermost circumference.

What is essential when you look at the figure is that the "一" in the center and the large circle on the outermost circumference are congruent. I once called this logical relationship, "the principle of equivalence." I think you can understand it intuitively by clicking the following figure with the mouse.

binary tree structure of GENGO

The figure above shows binary tree structure of GENGO. Baien drew that as the one pair picture. "一合" means "one pair". The pair chart above which illustrated by drawing software, but the original figure drawn by Baien. You can see the layout by the thumbnail below.

A pair of diagrams in which the left one is a black and white inverted version of the right one.
The idea of the figure is not wrong, but Baien did not draw the left diagram.

Baien drew all "一"s in the right diagram of the figure above. Naturally, there is no logical problem even if Baien had used the left diagram which can be obtained by inverting black and white of the right diagram. But over 200 years ago, there was no easy way to invert tones. There were many diagrams Baien could not draw. The reason was limitation of the time when there were only ink, brushes, paper, and a traditional compasses. The figure above is therefore an interpretation of the original figure by the author of this site.

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.

And the "一" in the small circle is similar to the Holon proposed by Arthur Koestler and may be similar to Totality and Infinity, too. The Holon can be split into two parts by cutting, but the two don't lose the characters of both whole and part.

At the quantum level, matter has both wave and particle properties, and elementary particles can be classified into bosons, fermions, and others. In other words, every substance in the universe has a dual characteristic from the time of its birth.

The kanji "一" in the small circle means one also in Japan today, but, in GENGO, it does not mean the natural number one. Because if you divide it into two, each part becomes the original "一", and if you unify them you get "一" again. So, the number one, which is written "一" in this figure represents totality.

That is not an arihemetical culiculation, but a logical operation on the existence. "一" may be a little similar to infinity. Infinity keeps infinity whether it is divided into two or two of them are combined into one. The natural sequence can be divided into even and odd sequences. These three sequences have the same infinite size aleph-zero.

The subject of Baien's ideas transcended the subject of natural science, and he constructed it not as philosophical speculation but as general system theory. Sueki Takehiro (1921 -2007) pointed out that the Baien's world system is very similar to Kitaro Nishida's "self-identity of absolute contradiction".
The subject of Baien's ideas transcended the subject of natural science, and he constructed it not as philosophical speculation but as general system theory. Sueki Takehiro (1921 -2007) pointed out 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 a systematic philosophical speculation. The two thinkers, Baien and NISHIDA, were similar in that they started thinking from when they got the 'Pure Experience'.

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