Introduction to Another Mathematics

In the years that I worked with Frans Coppelmans a whole new Mathematics was developed.
Up till now I hadn't the time to write about this new theme, but now I have so….
Frans Coppelmans studied drawings of children and discovered 'laws of composition'.

The beginning.

Where to start?  Lets take one single point to start with . 
And the other points?

                    .                            ……………….


                   one                                        many

One opposed to many. How many? As we all know this many is a tricky thing. We don't know whether the one many is one more or less than the other many.  In nature we also see this movement. We take one seamen put this seamen in the ground and then a plant will grow. This plant will flourish and many seeds are produced.

The other movement

                     ……………….                         .


                      from many             to            one     

we also know as we for instance assemble many parts to one car.

Our Mathematical challenge will be to show how we come from one point to many points (law of parsimony).
The most simple many is two points so:

                             .                              .  .


                             1                               ?

So we have one point and how do we come to two points?

How can we describe this transition? We can describe it by saying that a division takes place and the result of this division is two points.

                       .                                 .        .


                       1               :              two points

By dividing the one point it disappears:

                                                         .        .


                         t1                                    t2

How will we look at the result?
When we look at the separate points than we can repeat the division and with each division we will end up with one point extra. And so we can generate as many points as we like.

But we also can look differently to the first result. When we look at the result as a whole; as a relation between two points than we have something different something new; a distance. How can we describe this distance? We can call it 2( ). The two brackets indicate that this two is in fact a variable, because we don't know yet whether this distance is 2 yards or 3 yards or whatever. The difference with our starting-point 1 is that this 2 ( ) has a value and we entered the first dimension.





                          1                                     2()                                   ?

What next?
We lost something and we won something what will be the next step? We know that the next result will consist of three points and when we say three points than we think immediately of a triangle, don't we? But we must remember that we just entered the first dimension and a triangle supposes the existence of two dimensions so we must find something within the line; how will this look like?

             .                  .        .   
like this   

             .   .                     .     
or this

          .                     .                              .
       or like this?

Now we must remember a second thing namely that we have a distance of a certain length say 2 yards and so we must preserve this distance and this means that the third point will lie between the two 'value'-points of the distance. Where will it lie?

                 .         .     .    
2(2 yards)

                 .  .            .     2(2 yards)

What will it be? Or is it indecisive?  The most simple assumption is that this third point will lie in the middle because we cannot assume different masses, so the dividing-movement will in either direction be the same. So the third result will be:

                                      .                          .

            .                                                    .

                                      .                          .


            1                        2(2 yards)         3(2yards)

             t1                       t2                       t3

And how will we call this transition from 2( ) to 3( )?
Because this transition is in a sense the reverse operation of 1 to 2(); it is in fact a reconstruction of that movement; we call this operation a multiplication. And this means that we can reproduce this value over and over again. So this is not a static three but a dynamic three as well.

Is this the end of the story? No its only the beginning. We have to develop this value in the second and third dimension and we have to restore the value free nature of the first structure one.

For a quick view on how things develop, you can click here. For an elaboration on the basis of this new mathematics into a perceptual system: The reconstruction of the Contour



The Reconstruction
of the Contour