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Reality is made of – Part 2 – Induction - by P.K. Odendaal - July 2019
We continue our exploration of the Universe with a second postulate.
– 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
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.
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.
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.
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.
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.
not have 358 years to prove one conjecture true or false.
induction is a generally acceptable way of proving other things in other
domains as well and that is what we will concentrate on.
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?
brings us to Hilbert space.
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.
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.
use this postulate to expand our horizons to other worlds and other domains.
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
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!