Content uploaded by Egger Mielberg

Author content

All content in this area was uploaded by Egger Mielberg on Mar 04, 2021

Content may be subject to copyright.

Content uploaded by Egger Mielberg

Author content

All content in this area was uploaded by Egger Mielberg on Mar 04, 2021

Content may be subject to copyright.

Sense Theory

(part 2)

Sense Function

[P-S Standard]

Egger Mielberg

egger.mielberg@gmail.com

25.07.2019

Abstract.

The Sense Theory is not a part of traditional mathematics. It is a new

paradigm of how we can formalize complex cognitive processes of

the human brain. The basis of the theory is a sense function which

determines sense conformity between a set of objects or/and events

and a single subject (described object/event).

The sense function has a series of unique properties that can help

find associative connections between trillions different-type

objects/events.

By the function, we can investigate a whole process of forming a

single sense of big data set of different business or scientific events.

1. Introduction

It has been far away in the past when a first mathematical function was

formulated. Since antiquity, people used quantitative analysis in their

everyday life. After hundreds of years, a great number of different

mathematical sets and spaces were postulated. But none of them was not

focused on such qualitative characteristic as a sense. In this article, we

describe a new sense-centered function which is the main element of

Sense Space (see next articles).

2. Problem

In the traditional mathematical theory, a function is a relation that

associates each element “x” of a set “X”, the domain of the function, to a

single element “y” of another set “Y”, the codomain of the function. In many

cases, set X, as well as set Y, are numeric. Such function was originally

focused on how the elements of one numeric set depends on varying

elements of another set.

In the advent of AI concept, many complicated and unsolvable tasks were

formulated. However, neither differential calculus nor other mathematical

methods helped to even approximate to a solution of the tasks. The reason

for that is the codomain of the function operates elements of numeric

nature whereas AI operates senses.

Below we introduce a new-paradigm function which is sense-centered.

3. Solution

As in classical mathematics, we need to define a "coordinate" system for

operating with such an abstract object like a sense unit. Unlike rules of

setting a point on the line, in Sense Theory, we need to set three

conditions:

1. sense units (SU) -

units

The table shows how many variants of the sense units can be chosen for a

procedure of sense approximation.

S

O(N)

S

C

2. initial set (IS) is a set of Non-Sense Sets, Object Non-

Sense Sets, Complete Sense Sets and Incomplete Sense Sets which is

used for first iteration of the sense approximation.

3. sense direction (SD): subject or event.

The subject of the sense approximation is any element (sub-element) of

Text Set which is a noun with qualitative properties.

The event of the sense approximation is a thing that happens during some

time.

As well as the coordinate method of classical mathematics, in Sense

Theory, there is own method, the method of sense units.

The method of sense units is the algorithm for determining the meaning

(sense direction) of a text set by a chosen element or group of sense units.

For clarity's sake, we need to introduce such notation as levels of the

sense approximation.

Definition 1: The first level of the sense approximation is a result of the first

iteration of the calculating algorithm where the Text Set is the Initial Set.

Given:

SU – chosen variant of the sense units.

IS – chosen Text Set for the sense approximation.

SD – chosen “subject” or “event”.

Definition 2: The second or N level of the sense approximation is a result of

the second (N’s) iteration of the calculating algorithm where the Text Set is

the first (N-1) level result.

Given:

SU – chosen variant of the sense units.

IS – first (N-1) level result.

SD – chosen “subject” or “event”.

Definition 3: Sense function is a sense conformity between No-Sense

Set (Object No-Sense Set) and zero object .

Notation: or or , where .

Like in traditional mathematics, in Sense Theory we introduce such

notations as to the domain and codomain of the function. In our case, the

domain of is a sense sequence. The codomain of is always one of the

set of . For example, in case of we have

the following functional expression:

or

Specifying a function ():

1. Analytical method [1]:

or

2. Graphical method.

This method shows the depth of the object description as well as the level

of correlation with other objects with the same set of No-Sense Set

elements.

Fig.1

Definition 4: The function that has N No-Sense Set (Object No-Sense

Set) elements and determines only one zero object is a -complete

function.

Definition 5: The function that has N No-Sense Set (Object No-Sense

Set) elements and determines more than one zero object is a -partial

function.

It is obvious that in many practical cases the function graph will be looking

like this:

Fig.2

3. Table method.

This method allows for defining all possible combinations of No-Sense Set

elements that relates to one of Object Set elements.

Sense determinant and limit.

Function

is called sense determinant. It determines a rule of sense

conformity between No-Sense Set (Object No-Sense Set) elements and a

single zero object of Object Set.

Zero object in the following expression:

is called sense limit. It is a maximum achieved level of the sense

approximation.

Unlike traditional mathematics, in Sense Theory, there is no one-to-one

inverse function. Otherwise, from the following formulas:

or

we would get the following result:

or

and

or

The result is a direct contradiction to Definition 1 [1] and the main definition

of the sense function.

Theorem (Equivalence of functions).

“Two functions

and

are equivalent if and only if the following

expression is met:

“.

Proof.

The theorem is proofed by using two Axioms, “The Axiom of Object Limit”

and “The Axiom of Object Equality”.

Theorem (Nonexistence of composition functions).

“There is no composition function

as the following expression does

not make a sense:

”.

Proof.

The proof of the theorem deduces from the main definition of and

Definition 3 [1].

Properties of :

is strictly surjective.

Theorem (Surjection of function).

“Each element of has semantic connection to at least one sense

unit of “.

Proof.

The proof of the theorem deduces from Definition 3 and Definition 11 [1].

Definition 6: In the case when the number of elements tends to N,

the number of semantically connected zero objects tends to 1 or 0. In this

case, is called an semantically increasing function.

Definition 7: In the case when the number of elements tends to 1,

the number of semantically connected zero objects tends to N. In this case,

is called an semantically decreasing function.

Definition 8: is defined if and only if there is a sense limit:

or

Definition 9: is not defined if and only if there is no sense limit:

or

4. Conclusion

In this article, we presented the new “sense-focused” function. Unlike

classical mathematical function, the sense-based codomain of our function

is purely qualitative. It helps design different semantical models for

purposes of any kind.

We hope that our decent work will help other AI researchers in their life

endeavors.

To be continued.

References

[1] E. Mielberg, “Sense Theory”, Part 1, 2019,

http://vixra.org/abs/1905.0105

[2] E. Titchmarsh, “The Theory of Functions”, 1939,

https://archive.org/details/TheTheoryOfFunctions/page/n5

[3] S. Sternberg, “Theory of functions of a real variable”, 2005,

http://www.math.harvard.edu/~shlomo/docs/Real_Variables.pdf

[4] J. Harkness, F. Morley, “The Theory of Functions”, 1898,

https://www.ams.org/journals/bull/1899-06-02/S0002-9904-1899-00671-

2/S0002-9904-1899-00671-2.pdf

[5] R. Remmert, “Theory of Complex Function”, 1991,

https://www.matem.unam.mx/~hector/Remmert-TheoryCpxFtns.pdf

[6] I. Natanson, “Theory of Functions of a Real Variable”,

https://epdf.pub/theory-of-functions-of-a-real-

variable345a969e7942ecf12b11e3b8055dd80725702.html

[7] C. Pinter, “A Book of Set Theory”, 2014,

http://matematicas.uis.edu.co/adrialba/sites/default/files/SetTheoryDover-

%20Charles%20C%20Pinter.pdf

[8] K. Kunen, “Set Theory”, 1980,

https://pdfs.semanticscholar.org/8929/ab7afdb220d582e9880b098c23082d

a8bc0c.pdf

[9] R. Andr´e, “Axioms and Set Theory”, 2014,

http://www.math.uwaterloo.ca/~randre/1aaset_theory_140613.pdf

[10] H. Schwichtenberg, “Mathematical Logic”, 2004,

http://www.mathematik.uni-

muenchen.de/~schwicht/lectures/logic/ws03/ml.pdf

[11] J. Aspnes, “Notes on mathematical logic”, 2010,

http://www.cs.yale.edu/homes/aspnes/pinewiki/attachments/MathematicalL

ogic/mathematical-logic.pdf

[12] M. Abramowitz, I. Stegun, “Handbook of Mathematical Functions with

Formulas, Graphs and Mathematical Tables”, 1972,

http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf

[13] J. Nicholas, J. Hunter, J. Hargreaves, “Functions and Their Graphs”,