1- Complementary Mathematics

2- Hilbert’s 24th Challenge
At the end of his celebrated ICM lecture in 1900, Prof. David Hilbert was concerned that mathematics would split into separate, non-communicating branches. He stated:

“I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts”

3- Complementary Mathematics
This work is motivated by Hilbert’s organic paradigm. Our solution is based on the relationship between the local and the non-local, as observed among 5-year-olds, who described the relationship between a point and a line.

4- 5-year-olds’ Observation
(- Our work is based on insight about the relations between a point and a line.

- We start by researching the Membership concept.

5- Membership: Two Options

An element is either a set or an urelement.

A urelement is a member of a set, but not a set by itself.

A sub-element is an element that defines another element.

- In Option 1 each member is also a sub-element of a given element, and it is notated by an Epsilon-like symbol.

- Option 2 is an extension of option 1, where each member is not necessarily a sub-element of another element, and this kind of membership is called bridging, and notated by a Euro sign.

- In the next slide we define the logical basis of a point and a line, but first we wish to use them in order to demonstrate an option 2 membership.
- If a line-segment is a non-composed object (it is an urelement) just like a point, then a point can be a member of a line-segment, but it is not its sub-element, since a non-composed line-segment is not a collection of sub-elements.

From this point of view, a line and a point are mutually independent just like two axioms.

6- Locality and Non-locality
- In this slide we have the logical definition of Locality and Non-locality, which is based on the bridging between x and A where x and A are place holders of an element.

- Definition 1 is the logical basis of Locality (_._ XOR __ .) and the Point concept is a particular case of Locality.

- In general, any set's member is a particular case of Locality.

- Definition 2 is the logical basis of Non-locality (_._ AND _._) and the Line concept is a particular case of Non-locality.

7- Bridging and Symmetry

8- Bridging and Modern Math

9- Organic Natural Numbers
The standard Natural Numbers are both cardinals and ordinals because Peanos' or ZF axioms are based on induction.

10- Uncertainty and Redundancy

11- An example of Uncertainty and Redundancy
- Let us use an analogy that is based on the relations between a light ray and a prism.

- On the one side of the prism we can see a white light and on the other side we can see a colorful light.

- Peano or ZF axioms that stand in the basis of the Natural Numbers, produce only a broken-symmetry products, where each N member is clearly distinguished (as can be seen in the colorful-light analogy).

- We can also see that by using the Set concept in the standard way, we must ignore redundancy and uncertainty as first-order properties.

- By using symmetry as a first-order property of the Organic Natural Numbers, we expand our research in order to include also symmetrical and partial-symmetrical products.

-It allows us to use redundancy and uncertainty as first-order properties of the Organic Natural Numbers.

- In this analogy, symmetrical products are represented by the white light, and the partial-symmetrical products are represented by the different symmetrical states that can be found within the prism.

- Let us see some examples of these first-order symmetrical states.

12- Organic Natural Numbers 1 to 5
- By using symmetry as a first-order property of the Organic Natural Numbers, we discovered that N members are the particular case of the broken symmetry (marked by the orange rectangles).

- Since each bridging between the local and the non-local is a first-order state, then each Organic Natural Number is both a particular case of the number system, and a general state of the number system.

- In this case, the number system has a fractal-like shape, in which each part has a self similarity over scales.

13- Symmetry and arithmetic
-By using symmetry as a first order property of the Organic Natural Numbers system, we can distinguish between multiplication and addition.

-Multiplication can be used only in the case of a symmetrical bridging.

-Addition can be used only in the case of a broken-symmetry bridging.

- According to this distinction 1*3 is not equal to 1+1+1 , for example.

14- Further Research
Finally, research done by Dr. Linda Kreger Silverman over the last two decades demonstrates that there are two kinds of learners: auditory-sequential learners (ASL) and visual-spatial learners (VSL). Complementary mathematics is a model that bridges between ASL and VSL.

We believe that further research into various degrees of the number's distinction (measured by symmetry and based on bridging between locality and non locality) is the right way to fulfill Hilbert's organic paradigm of the mathematical language.

There are certain common Privileges of a Writer,
the Benefit whereof, I hope, there will be no Reason to doubt;
Particularly, that where I am not understood, it shall be concluded,
that something very useful and profound is coucht underneath.

Jonathan Swift, Tale of a Tub, Preface (1704)
(as quoted by D. E. Knuth in "Sorting and Searching")

What was Wednsday's achievement?

A system is any framework which at least enables to research the logical connectives between "in" , "out"