A pudding of proofs

'Can you do Addition?' the White Queen said. 'What's one and
one and one and one and one and one and one and one and
one and one?'
'I don't know', said Alice. 'I lost count'.
'She can't do Addition', the Red Queen interrupted.

- Lewis Carroll

What is two and two what is one plus one and what is half of that plus that?

How many is more than all and less than none?

Calculate one plus two times two plus three times three!
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As an observation on the stock market I sketch a simple example.

Our share price closed at R100, it loses 50% tomorrow, closing on R50. The next day it gains 50% again and a very easy calculation gives R75 a share. Whereas simply adding gives  - 50% + 50% = 0 which is very mistaken, it would mean the answer is R100 again but in fact it is R75.

To generalise suppose P is the beginning share price. It increases by A% and subsequently decreases by A% as in our example. The formula for the resultant share price is found as
P.[1- (A/100)^2] and by inspection we may deduce amongst more that this price is always strictly less than P.

Suppose now with a given initial P, we calculate the price after an increase of A% followed by a decrease of B% this may then be compared to first a decrease of B% and following that, an increase of A% in effect then the other way round.

The formula for the second problem can be obtained in the same fashion. The question is, are the two answers always the same, or not? These manipulations are elementary they are on beginning secondary school level. Or should we resort to putting the matter to the vote?

You cannot rush blindfolded into this and just start adding percentages of gains and losses you'll go completely off-track and quickly.
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SOS !

Does a round peg go into a square hole?

Our hero Hans in the Netherlands discovered  a hole in a dike with no-one around to call for help. The hole being 5cm square with water spouting out and he had a rod of 5cm diameter. Could he manage to block the hole? Could you save our country?
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How many mathematicians can change a light-bulb?
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If a strict subset B of A has the same number of elements as A then the sets must be infinite. It is similar to saying that if for an integer, n = n + 1 then n is infinite, n = +∞
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Crucial fundamental issues arise in "ordinary understanding", as common ideas and beliefs on sets, or collections, for instance Russell's paradox:

Let (define) N as the set of all non-self membered sets, i.e. the collection of every set not belonging / being an element to itself, or every set not having itself as a member.

Is N a member of itself? If it is a member of itself, then it must meet the condition of it’s not being a member of itself. At the same time, if it is not a member of itself, then it precisely meets the condition of being a member of itself.

A contradiction.

This reasoning then also eliminates a possibility of the existence of a "set of all sets". In turn Cantor's Theorem and also dated around 1900 demonstrates, and in effect it was proved there is no largest set and no largest cardinality so that the order of infinities is unbounded. Thus if there exists an infinity, there must exist an infinity that is even larger, as with ordinary numbers.

Gödel later on took these ideas further and showed that there always is a theorem that is true but that cannot be proved. His reasoning by Cantor's diagonal method is simple and elegant but the logic now is more sophisticated. These concepts are quite advanced. When working with infinities there is always trouble even today. And it might be always!
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Notes on some ideas

The first part is arithmetic and actually is very easy you just have to read it for exactly what it says, and keep to the rules! Any given number of mathematicians can change a lightbulb, for amusement one may prove a result by mathematical induction but it really boils down to simple logic. These facts are all meant as being concrete and litteral. As for the SOS problem the answer is reasonably obvious.

Postscript

Modern mathematics as eternal truths consists of structures and patterns, as axioms and layers of abstraction and proof-derived theorems. It often can be applied very successfully to models with the purpose of studying, and to solve real-world problems.

And as an after-thought . . .

Where and what in the precise meaning of the four given phrases might there be ambiguity, or different meanings amongst themselves, "for each" "for every" "for all" "for any" if any?

Similar and related consider "there is", "there exists" "let there be" and even "let it be"? Also "therefore", "thus" and "that is"
QED