﻿To prove

# To prove or ergotism

To prove or ergotism searching for and finding evidence is intended to find a truth by accuracy.
In many sciences is proven something by gathering relevant data and by trying to reason according to
the rules of logic. This happens in mathematics and all science.
An important scientific application is found in the forensic investigation the criminal law.

To court not only found 'silent witnesses' are used but also the human witnesses.
In some countries one witness is enough for serious crimes, in other countries at least two witnesses are required to decisive evidence.
The witness may fantasize, can tell a tale, exaggerate, he can make a mistake or tell lies flatly.
How many victims have not been convicted of conspiracy?
So there is a distinction between scientific and legal evidence.

Inductive evidences
This example shows the following physical law: all lenses conform to the lens formula.
Sometimes you look through a lens and sometimes you project an image.
In general a lens makes a resized image of the object.
Sometimes the image is seen upside down.
The lens formula gives a mathematical relationship between the object-to-lens distance,
the lens-to-screen distance and the focal length of the lens
The mathematical form of the lens formula for this argument irrelevant.

A non deformable lens has a fixed focal length.
With a not rigid lens as our eye-lens, the len-to-image distance remains the same.

Let's see how such a law is created.

Step 1: the experiment.
• We take a lighted candle, and we project the flame in a darkened room with a lens on a white cloth.
• Every time we choose another object-to-lens distance and
• after focusing we measure the lens-to-image distance.
certainly we always ensure to have the strongest/sharpest possible impression.
• We record all observations in a table with two columns. In the first column comes the object-to-lens
distance and in the second column the lens-to-image distance.
Step 2 The processing of the measurement is performed at the desk.
Processing takes almost always a number of mathematical steps.
The first processing step is to create a graph.

Step 3 From the graph follows a conclusion.
In this case that is the provisional lens formula.

Step 4 The experiment is repeated with many different lenses. The final conclusion of these
experiments is that there is always found the same mathematical relation between the object-to-lens
distance, the lens-to-screen distance and the focal length of the lens.

Step 5 Global cooperation.
Simply it is impossible to perform to measure all lenses in the world. After comparing many
publications everyone found the same relationship between the object-to-lens distance,
the lens-to-screen distance and the focal length of the lens being the lens formula.

Step 6 The inductive proof of the lens formula.
Finally the physicists say: globally we never found a lens not meeting the lens formula,
so we believe that all lenses meet the lens formula,
not only ever investigated lenses but as well all lenses of the future.
This method is called inductive argument.

Many laws of sciences are formed thereby. Thus they are based on belief.
Here believing means a rock solid confidence.

It only needs one lens found to be not complying with the lens formula,
and the world of opticians collapses!

Danger
Inductive proofs can be dangerous if not enough research has been done carefully.
In practice it is impossible to sufficiently careful examination.
A sample is just more reliable if more measured. In this respect absolute certainty does not exist.

Example, if I know ten people in a given population and all ten behave criminally,
it is a horrible blunder to determine that the entire population would be criminal.

Mathematical induction
We add the first five natural numbers: 1 + 2 + 3 + 4 + 5.
We calculate: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 and finally 10 + 5 = 15.
So 1 + 2 + 3 + 4 + 5 = 15. That's great, isn't it! With five numbers this is a piece of cake
but for example to add in that way 9999 consecutive numbers is a long time consuming task.

Take again 1 + 2 + 3 + 4 + 5 as an example.
There are 5 numbers to be added, starting at 1.
Multiply 5 with its successor, so with 6, and divide by 2.
We obtain 5 × 6/2 = 15. Fortunately we find again 15.

Can this calculation rule also be used with 9999 consecutive numbers?
The sum of all those numbers would be 9,999 × 10,000/2 = 49,995,000.

The mathematician proves it very elegant.

Step 1 He assumes that the calculation rule is true for n consecutive counting numbers
from 1.Herein n is any number.

Step 2 Then he shows from this assumption follows the calculation rule also applies to n + 1 numbers.
Therefore the sum of n + 1 numbers must equal to the sum of n numbers plus the last number n+1.
So (n + 1) × (n + 2) / 2 = {n × (n + 1)/2} + (n + 1).
And that is a mathematical truth.

Stap 3 The math person shows the rule is true for n = 1 so for the first two
consecutive numbers. Indeed 1 + 2 equals 2 × 3 / 2 = 3.

Stap 4
According to step 2: if it is true for 2 numbers, it is true for 3 numbers.
According to step 2: if it is true for 3 numbers, it is true for 4 numbers.
According to step 2: if it is true for 4 numbers, it is true for 5 numbers.
. . .
This process never stops. Then the calculation rule is true for each set of numbers
always beginning with 1.