Do you know the famous number pi=3.1415...?
It is the ratio of the circumference and the diameter of a circle.
By now, millions of decimals of pi have been calculated. So there are some more decimals
known except the five mentioned above.
To find the decimals of pi is not as easy as to find those of (for instance) 1/7 = 0.142857 142857 142857 ... In the decimal representation of 1/7 we meet an infinite number of blocks 142857, and an infinite number of blocks 285 as well. Some people have been wondering whether there are an infinite number of blocks consisting of ninety-nine consecutive 9's in the decimal representation of pi or not. This question seems to be fundamentally unsolvable. Foolish question, you say? We are going to see things that are even more foolish.
There is a current in maths, the so-called intuitionism, that wants to start from no
other things than those that are intuitively clear. The intuitionists write down
decimals of sqrt(2) = 1.4142... (the square root of 2) and continue to do this (only in their
minds, of course) until they have written down as many decimals of sqrt(2) as are known of pi:
millions. And as soon as we know more decimals of pi, they write down as many additional
decimals of sqrt(2). And now comes the point: whenever a block of ninety-nine consecutive 9's
shall be found, they count how many decimals have come before that block. If that number is
even, they write for "sqrt(2)" only 9's for the time being, until a new block of ninety-nine
9's for pi is found. If the number is odd, they write for "sqrt(2)" as many provisional 0's
. When a new block of ninety-nine 9's is found, then they substitute with retro-active
effect for the provisional 9's or 0's in "sqrt(2)" the right decimals of sqrt(2) in these
positions, and thereafter they repeat with the new block of ninety-nine 9's the same procedure
that had been done with the old one. Etcetera ad infinitum.
You may naturally say: then this decimal representation doesn't necessarily represent sqrt(2). It represents a number that is somewhat greater than sqrt(2) or somewhat smaller than sqrt(2), or (only if there are no blocks of ninety-nine 9's in the decimal representation of sqrt(2) or if there are an infinite number of such blocks) sqrt(2) itself. You are right. But the intuitionists say that they construct in this way a number A that "hovers around sqrt(2)". This is nonsense, of course. We are not able to judge the number before all decimals are known. We shall never know whether A is smaller than sqrt(2), or greater than sqrt(2), or equal to sqrt(2). But exactly one of these three possibilities is true (God knows which one), and A does not hover. Because, in reality, to each point on the line of real numbers belongs one and only one real number, and vice versa.
The intuitionists do believe that 'hovering numbers' exist, but they don't believe in
'discontinuous functions'. Because, they say, take for example the function f with f(x) = 0 for
x smaller than sqrt(2) , and f(x) = 2 for x greater than or equal to sqrt(2). Then we will never
be able to approximately calculate the value f(A) of the function f in the number A, mentioned above, that "hovers around
sqrt(2)". So, they say, f is not really a function according to our standards.
Hurray for the intuitionists! All functions are continuous, and some numbers are hovering. Then maths is only a little game for fools.