The stuff Reality is made of – Part 2 - Induction

The stuff Reality is made of – Part 2 – Induction - by P.K. Odendaal - July 2019

We continue our exploration of the Universe with a second postulate.

 

POSTULATE 2 – GENERAL PROOF BY WAY OF INDUCTION IS A VALID UNIVERSAL PHENOMENON AND PRINCIPLE.

We are going to use Hilbert Space to prove this.

There is a principle in mathematics called induction which is a mathematical proof technique. It is essentially used to prove that a property holds for every natural number n, i.e. for n = 0, 1, 2, 3, and so on.

The problem arises from the … and so on. Is that up to ten, up to ten million or up to infinity? If we start at counting we know intuitively that it is valid for all integers up to infinity. The question which begs itself is: How far should we go until we accept it as being true. In the mentioned case ten is enough for me and a million is enough for a sceptic.

Let us say we settle for a hundred times and say that the probability of it being true for one hundred cases is about 99.9% or above. This can be calculated statistically more ‘accurately’ which I will not do now. Even if we calculate it to one hundred decimals, it will never be accurate enough for sceptics. What does accurate mean. For the Greeks PI was equal to 3 until they discovered irrational numbers which are numbers containing decimals. Today the record is held by Google who calculated PI to 3.14 trillion digits.

This is indeed a slippery slope. If we turn to social norms, we normally say that if it smells like fish it is most probably fish. We use only one instance and all of a sudden one is enough. If someone has lied to us once we say he or she is a liar even if he or she has only done it once.

With this as a background we will climb this slippery slope to dizzy heights using the last theory of Fermat.

In number theory, Fermat's Last Theorem or conjecture states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2.

The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, although most previous efforts used the method of induction.

We do not have 358 years to prove one conjecture true or false.

However, induction is a generally acceptable way of proving other things in other domains as well and that is what we will concentrate on.

In Part 1 of this series we have discovered that domains with four and five dimensions do exist. We can also prove further dimensional domains, but we will be happy to know that four and five dimensional domains exist. If that is so we can say that n-dimensional and even infinite-dimensional space or domains exist. Who is going to prove this?

That brings us to Hilbert space.

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

If vector algebra and calculus exist in an infinite-dimensional space, as Hilbert proved, then the phenomenon of induction is much wider than our terrestrial domain and then induction is a universal and valid phenomenon and principle.

We will use this postulate to expand our horizons to other worlds and other domains.

It follows that physics, as we know it on earth, is limited by three-dimensional space constrained by time and thus speed, and more specifically the speed of light, is a very, very limited way of looking at the Universe. There is so much out there! So limitless!

We as humankind, and more specifically scientists, limit our understanding of the Universe by what we experience on earth or see around us and in the sky and only in three dimensions. How provincial!