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Counting in a different way? What does it mean? That 2 and 2 would not make 4? Yes they do, but 2 + .1 will not make 2.1 every time. For we will have to reconsider the relation between whole numbers and their decimal digits. And this will enable us to see fractional dimensions of Mandelbrot in a more pertinent way, for precisely these dimensions are dimensions with decimal values.

But before seeing these dimensions, we have to see first, why 0.5 for example, has nothing to do with the path which goes from 0 to 1. Why most of the time, 0.5 cannot possibly be between 0 and 1.
Strangely enough, the simplest way to understand how to travel from 0 to 1 without ever reaching 0.5, is to begin with a detour through infinity.

Then we go.
 
 
 

Cantor's absurd infinities

Ask a particles physicist about the infinitesimal, he will tell you how very strange things happen there, which are unthinkable at our scale. Such as that a particle can be a well-located corpuscle, and a wave of infinite size endlessly expanding in space, both at the same time.
Ask a mathematician about the infinitely great, he will tell you the same thing: when they reach infinity, numbers gain properties which are unthinkable just before infinity.
 
 

For example, count the even whole-numbers : 2, 4, 6, 8, etc.  You can easily calculate that whatever the moment you stop counting, their number will be half the total number of the whole numbers you have seen. Because, for every even number, you have to add one uneven number to have the total number of whole numbers. And you can see clearly, that this will continue, even if you count very far toward infinity: an even number, an uneven number, etc. That's a perfect and very simple alternation you can follow toward infinity. But when you are right at infinity, and count the infinite number of whole numbers you have encountered, and the number of even numbers you have picked up one time every two whole numbers, then you find that these numbers are not double and half one of the other: they are perfectly and strictly equal.

 

Thus, strange things happen at infinity, since a property true ... until infinity, suddenly might be totally untrue when arriving right at infinity. Of course, mathematicians are reasonable people. If they have to acknowledge such an insane fact, that's because it has been proved. It's the logical conclusion of a demonstration, and a mathematician cannot elude a logical conclusion. This demonstration was made by Cantor, the father of set theory. We shall now summarize his method.

It begins with a trick to compare all the sets, without needing to count all their elements. This enables us to compare even the infinite sets we cannot count.
The trick is to pair the two sets, by associating each element of one of the sets with one element of the other set. And we see whether anything remains in the second when the first is exhausted. Of course, in the case of infinite sets, we do not really make the pairing for all the elements. But we find a method used as principle, and we wonder if anything would remain by applying this method.
For example, to compare the infinite number of whole numbers, and the infinite number or irrational numbers, the method is as follows:
      1/ we begin by supposing that we draw up an exhaustive list with all irrational numbers, and we put them all in a vertical column.

[We recall that irrational numbers are numbers which have an infinite number of decimal digits. In that infinite series, no periodicity can be found to imagine next digits by studying previous digits. 'Pi' for instance, is an irrational number]

      2/ then, we pair every irrational number with a whole number, beginning with 1.
In short, after having made a column of irrational numbers, we count how many lines that column holds, from 1 to infinity.
      3/ then these 2 columns of whole numbers and irrational numbers are put side by side, we ask the following question: the number of whole numbers being infinite, are there even more irrational numbers than whole numbers? That is, does the column of irrational numbers continue after that of whole numbers has reached infinity?

The answer is yes.
To find an irrational number which is not paired with a whole number, we only have to pick up the 1st irrational number in the list and change its 1st digit, then to change its 2nd digit in order to make it different from the 2nd digit of the 2nd irrational number of the list, then to change its 3rd digit to make it different, etc.
When we have reached the last irrational number paired with a whole number just at infinity, we have then formed a new irrational number. That new one is different from all those which are paired for it is different from every of them by one digit at least.
Thus, the column of irrationals, holds at least one number which is not paired with a whole number. Therefore, irrational numbers are more numerous than the infinite number of whole numbers.
 

Probably, the comparison between whole numbers and irrational numbers, does not have many practical consequences for itself. But with the same processes, Cantor demonstrated 'more serious' things. Serious, that is if we try to use numbers to represent physical phenomena, and in particular to define space properties. Thus, we can demonstrate that there are more points in a tiny segment of a straight line than there are numbers in the infinite set of whole numbers. 'Worse', we can demonstrate that every segment of a straight line, whatever its size, holds the same number of points. And that there are as many points in a straight line 1 D, than in a 2 D plane, and even than in a 3 D volume. In short, all the commonsense properties about the relation of space dimensions, which are likely to prolong themselves continuously and regularly until infinity, would be invalidated all in a sudden, precisely after infinity. Properties usually incompatible with each other, would turn to be so, when at infinity.

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