As answered by senor
daniil
Cisfinitum
Letter to Leonid Savelyevich Lipavsky,
The Fall of the Stem.
Dear Leonid Savelyevich:
I
We shall regard a branch of knowledge as creative if it does not rely on postulates of category E.
Any branch of knowledge relying on postulates of category E will be considered non-creative.
The science of logic ("formal logic", "laws of thinking") relies on postulates of category E, therefore it is non-creative.
Art cannot rely on postulates of category E, therefore it is a creative branch of knowledge.
I am considering a creative science that cannot rely on postulates of category E.
II
A stem stands on a postulate E. Create a postulate E as the primary stem. Then the stem E will stand on a new postulate P1. Create the postulate P1 as the most-primary stem. Then the stem E will rely on the postulate P2. Create the postulate P2 as the foremost-primary stem etc.
I must remark that the stem of formal logic (Bool, Peirce et al.) is famous precisely for not having to consider the origin of the postulate E.
Postulates E, E', E''... can at any time be replaced by P1, P2, P3... and the new postulates can be regarded as postulates of category E.
But only with an infinite shifting of P into the subsequent P1, P2, P3... the stem grows or, more exactly, falls into the uncut field of the postulation, and the postulation becomes
P1P2P3 ... Pw .
Depict a new stem Sw. Notice that to create a field (P1 ... Pw) one must explore each P in turn. We shall, by convention, mark an explored field by the letter a.
Then we write that the new stem Sw relies on an already explored postulation a (P1 ... Pw) or
Sw
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a (P1 ... Pw)
III
Define an exploration of a postulatory field. For this, we shall define a postulate as the limit of the fall of a stem relying on it. Also notice that if a stem is some continuum K, then the postulate will be expressed by the formula
K – H = 1 (for all H).
Taking (K – H) for some stem and exploring it, we find the postulate P1.
P1 = K – H – H1 = 1.
Following further, we obtain:
P2 = K – H – H1 – H2 = 1.
Opening the fall, we see that
Pw = K – H – H1 – H2 – ... – Hy – Hw = 1.
And finally, the exploration of the postulatory field will be expressed as*:
a (P1...Pw) = (K – H) ~~(K – H – H1)~~ ... (K – H – H1 – ... – Hy) ~~(K – H – H1 – ... – Hy – Hw) .
* The sign ~~ reads: "deals with..." (Author's remark).
IV
Let us look at what happens to the stem. First of all, we shall call this process a "fall of the stem".
Consequtive moments of the fall can be expressed as follows:
K postulated by E, or
K
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.
E
K – H
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.
P1
K – H – H1
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.
P2
K – H – H1 – ... – Hw
mw –1
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Pw
Following further, we obtain:
Sw K +(K – H) +( K – H – H1) +...( K – H – H1 – ... – Hy)
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=
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a (P1 ... Pw) (K – H) ~~ (K – H – H1) ~~ ... ~~ (K – H – H1 – ... – Hy) ~~(K – H – H1 – ... – Hy –Hw)
V
I am very interested in Your opinion about non-postulable science. The thing is that by postulating Sw by an infinitely decreasing field (P1 ... Pw) we cannot refer to this as our previous unit of reliance. The new unit of reliance will be O (null).
a (P1 ... Pw) = O.
This is the first and the only statement that can be a new postulate not of category E.
According to the first condition of §1, a stem relying on a (P1 ... Pw) shall be considered creative.
VI
If we assume that a creative science Sw can exist, then we can anticipate that by the 4th condition of §1 it will be similar to art.
If a creative science will have to deal with quantitative notions, then one can anticipate that the number system must be different from our solaric corpus. I modestly note that the new number system will be null and its domain of exploration will be Cisfinitum.
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Thus, my dear Leonid Savelyevich, allow me to finish the letter and wish you a good night.
Meanwhile I, thinking of the cisfinite void, am ready to stand and wait, while people hurry to sleep by counting to a hundred, whereas the insidious Moock with his dogs is going hunting.
October 16, 1930.[/url]
