Bismillahi rahmani rahim

In the name of God, the most gracious, the most merciful

For Elena, my love

Whitepaper

by Anton Spass

The better we understand the world around us,

the better we know how to deal with information

The idea is to remove the gap between

human-developed abstractions and our feelings

Preface

We all used to refer to digits as a convenient way to represent quantity. Actually, probably, the only way. But what if there are alternatives?

Ok. Let’s start counting. With apples, just like in school.

Let’s say we got some apples -

How would we communicate how many apples we got? Assuming we don’t know numbers, we can just draw the apples. This way -

Or this -

Or even draw smth else -

- which is probably too long :)

Anyway, it gives us an interesting observation - it actually does not matter how we arrange our apples or elephants - the quantity meaning will be preserved.

Let’s take something more generic & easy - for instance circles -

Let’s go on.  Let’s outline the circles in a circle so that our notation makes feeling it is complete - there’s nothing to add to this quantity

Now if we add more circles to the heap -

- we can continue arrange them the same way -

Oops. Seems like at the end we got less circles than necessary to have all the big circles. No worries. Let’s outline them the same way except that the last big circle will be incomplete. Let’s keep empty circle placeholders as well -

Adding more circles, eventually we’ll have enough to construct a yet bigger one -

OK. Each complete unity we can notate with a big circle -

Great. Seems like that approach can be extended so that we can introduce bigger and bigger circles -

These circles are our abstractions. So idea is upper level abstractions are constructed by concatenation of a lower level abstractions -

When an abstraction gets completed, it becomes a building block for upper level abstractions -

Let’s call them Vector Abstractions

1. Vector abstractions are based on concatenation. Larger circles constructed from smaller
2. Vector abstractions are non-positional. No matter how we arrange our circles - small and large - the quantity meaning will not suffer

We can use size/color encoding to distinguish abstracts of different levels. We can think of vector abstraction as smth that appears as a result of outlining basic circles, level by level.

Light upon light

Division

OK, let’s say we have seven apples -

In terms of our vector abstractions it is

Or just this -

Now let’s say we want to express this quantity via three -

We just rearrange the circles and get -

Or just this -

This leads us to other observations -

1. Our notation always works with discrete abstractions

2. Vector abstractions are constructed from the ground up 

Some examples of using color encoding are below

Modeling quantity

We live in a world where we don't have a reliable model for quantity.

We need to have a model of quantity that will consider both original qualities of the phenomenon AND have semantics allowing for unity and extension.

Abstractions used for valid quantity model should be

 - Non-positional. From any perspective it should maintain the same meaning

 - Discrete

 - Based on concatenation: L3 = concatenation (L2) = concatenation (L1),

        where L1, L2 and L3 are abstractions of levels 1, 2 and 3 respectively.

This is the Vector Abstractions Manifesto.

Overall, unity is the core idea for abstraction. Abstractions used for a model are nothing but a set of unities that are constructed from unities. Again, we can think of vector abstraction as smth that appears as a result of outlining circles, level by level.

X/Y/Z axis in a coordinate grid cannot serve as a model for quantity - they are

- continuous, where quantity is discrete

- they don’t suggest semantics for quantity but focus on position. Quantity here serves more as a measurement unit and position ID rather than object being researched.

The shape of abstractions we use actually does matter, because it affects our understanding of the phenomenon. Ultimately, it defines the way we think...

Decimal notation from quantity model perspective

Decimal notation is to be re-considered.

If we think about numbers, both technically and theoretically we don't have to be bound to the base 10 system anymore. We can express quantity via a numeric system that gives us an accurate representation for any given quantity.

Other observations -

Repeating decimals - are the result of using inappropriate numeral systems.

Irrational numbers - do not refer to quantity. From a quantity perspective, the square root of 2 is 1. Just like for any primitive number.

                                                                  * * *

Let’s take the decimal result of 10/3 as an example.

3.33 - 3 where 10 is a whole + 33 where 100 is a whole -

It is a lie.

If we take 3.333 as a result, i.e. 3 where 10 is a whole + 333 where 1000 is a whole, it will still be a lie. And now it is obvious. It is just an endless lie, no matter how deep we go after the decimal point.

If we write 3 ⅓, we are using 2 numeric systems to express the quantity: 3 where 10 is a whole and 1 where 3 is a whole.

We are missing the level 1 where 3 is constructed. How is level 1 different from upper levels? If we look at vector abstraction model, it is obvious:

Or just this -

So we can write 101,3 instead, where 3 is a numeric system. It will be true. Think of 3 in this notation as a template for constructing abstractions of different levels.

We can also add dots to the notation to distinguish different levels - 1.0.1,3. This would allow for things like 7.11.3,16

Also, to express any quantity, we do not need non-primitive numbers, they are redundant as such. If we take programming, non-primitive numbers fail to comply with DRY principle.

Other examples of vector abstractions

Google maps satellite images -

The example view below is constructed from 4 source geo images.  

On the upper zoom level, those 4 images will become one common image to create upper level abstraction. And so on -