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Known Unknowns
201

Rule 1: If I have no glyph in front of a symbol I can assume there is
an invisible Ψ there.
Rule 2: If I have a positive letter (or a letter with no symbol in front
of it) I can put a ♮ in front of it and put it on the other side
of the >
Rule 3: I can swap the Ψ and ♮ symbols of all the symbols in my
equation if I do it to all of them.

The proof in symbols

γ > β is the same as Ψ γ > Ψ β. (rule 1)
Ψ γ > Ψ β is the same as ♮β > ♮γ (rule 2 twice)
♮β > ♮γ is the same as β > γ (rule 3)

Any collection of symbols will do. The symbols have no meaning
in themselves other than the meaning we have given them. A tribe in
the Amazon jungle could demonstrate a proof without knowing any
mathematics. All I need say is, "Hey, I want to play a game with you. Can
anyone make this into that, in the fewest possible steps, while obeying
these rules?"

But, is it true we can ignore the meaning behind the symbols. Does
it matter that we were talking of numbers rather than spears, counters, or
crocodiles? If we look at the marathon winning analogy again, we know
the nature of a game is important. In a running race we can interpret
holding hands to mean the two athletes are treated as one, the existing
rules can then be applied as normal and the pair become a single winner.
But, in tennis, there would be a problem. I wouldn't want to come on court
and find I'm playing against two opponents! On consideration though
I'd be happy if they had to hold hands while they played so that they
constituted a single player. When we examine the actual circumstances,
we can add a rule and show the rule works, but we have to see something
about the specific sport that makes the rule fair and workable.

Hilbert was convinced mathematical truth is not like this and
that proofs follow from the rulebook without any knowledge of the
circumstances, i.e., the sport being played or any other analogous thing.
He was to be proven wrong by Kurt Gödel.

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