Issue #15, June, 2004 Streng geheim First in, Last out!
HOW DO WE THINK?
by von Ick
Have you ever asked yourself how your mind works?
Have you wondered what is the truth and how you know it?
Have you been thinking of the difference between the chaos and the order?
If yes, I would like to develop some little thoughts about these subjects.
It will be about the Gödel's theorem.
Some history first.
A famous German mathematician David Hilbert presented in 1900 the scientific community a list of 23 unsolved
problems that, once solved, should provide a definitive foundation for physics and mathematics.
One of the most important problems was to prove that the axioms of arithmetics are consistent or in other words that there is no possibility of contradiction in the logic we use.
Indeed as the whole huge mathematical apparatus begins only with the number 1 and the rules of logic that permit to derive new truths from already proven truths, it is paramount to prove that we will NEVER prove a theorem which is in contradiction with another already proven theorem.
Should it be possible even if only once, the whole construction of mathematics would break down like a castle of cards because the axioms being inconsistent, the system (mathematics) would be meaningless and there would be no Truth anymore.
At this stage many could say: "What do I care? I don't do any mathematics."
Nothing is more wrong - without knowing it you apply every day the axioms of logics and one could even
say that your life depends on it.
If a woman tells you that she will go to bed with you this evening, you will assume that her proposal is either true (she will do it and you'll enjoy every bit of it) or false (she will not do it and you will be extremely frustrated). If your employer tells you that he will increase your salary, you will similarly assume that it is either true or false.
By doing that you are applying the axiom of choice of formal logics that is saying that every correctly formed proposal (= sentence that makes sense within the rules of the language) is either true or false.
There could be said much about this Axiom but it is not the purpose here today.
In an argument about the merits of this or that political system you will use the rules of proving a theorem that say that given a set of valid axioms (valid = non contradictory) a proposal is true if it can be derived from these axioms while the negation of the proposal is automatically false.
So actually when you discuss politics (or football for that matter) you could ask yourself the same question as Hilbert did: "Are the axioms forming the base of my discussion consistent?" :)
Of a special interest is that it doesn't matter what these axioms EXACTLY are, what matters is that they are consistent.
In our human case we are using naturally only one set of axioms (even if we could imagine others) and these are the same regardless if we do high mathematics or talk football.
We have also all experienced this unpleasant sensation when the mind gets trapped in a logical loop where we suddenly don't know what to think.
Two examples of such loops:
1)
I say: "I am lying."
Is my sentence true or false?
If it is true, then I am lying but if I am lying, my sentence can't be true.
If it is false then I am not lying but if I am not lying, my sentence can't be false.
So my sentence is neither true nor false yet it is correctly formulated.
Hmmmm ... what happened to the axiom of choice?
2)
A prisoner is given the choice - "You must say a well formed sentence. If it is true you will be shot,
if it is false you will be hanged."
What do you say?
You think that you are already dead but then you remember the Axiom of choice and ... live :)
You say: "I will be hanged."
If it is true you should be shot but then it would be obviously false and you should be hanged.
If it is false you should be hanged but then it would be obviously true and you should be shot.
Again your well-formed sentence was neither true nor false and saved your life.
So it appears that Hilbert's concern has many connections to everyday's life and concerns some of the deepest aspects of the function of the logics and of the human mind.
As another side remark it is also to be noticed that man is not only a rational being and that in this whole matter the notion of "well formed sentence" which implies rules is extremely important.
Some could say "And what about poesy?"
"The green moon fell in my mouth and disquiet lions grovelled at my feet."
This is obviously a "well-formed sentence" in the sense that it respects grammar and vocabulary
(actually it is not really necessary) but you can't say if it is true or false.
It is because poesy is not a consistent formal system; it doesn't bother and doesn't need to do so if there are any consistent Axioms. There is no notion of truth in poesy - it talks to the irrational part of the human mind, to the emotions.
However everybody agrees that to build a computer that works, it is better to use logics than poesy :)
Thus everything at its place.
So now back to Hilbert's problem - is the arithmetics and following the whole mathematics consistent?
The genial Austrian mathematician Kurt Gödel born in Czechia brought in 1931 a very surprising answer to this question by proving his famous "Incompleteness theorem".
There are several formulations and Gödel has proven several theorems.
The one I want to mention here is that "It is IMPOSSIBLE to prove that Arithmetics IS consistent."
Even worse - a system, in which it is possible to prove that it is consistent, is actually inconsistent (=contradictory).
So if we had to stay at that, we would never know if one day the whole mathematics breaks down because we happen to prove a theorem that is in contradiction with another already proven theorem.
Luckily there is a way out because Gödel gives us the choice - completeness OR consistence.
If we want a complete system (e.g. a system in which each and every truth will be proven one day even if it is not today) than this system will be inconsistent.
Indeed, see above, if the system can "prove" that it is consistent (e.g. this statement is one of the truths achievable because of the completeness) then it is inconsistent and inconsistent systems are meaningless and uninteresting.
However if we choose the consistence, then the system is necessarily incomplete (e.g. there are truths that can't be proven within the rules of the system).
And that is lucky because we are in this case.
That is why the examples like the 2 perturbing above can exist - there are assertions that can't be decided within the system (btw every time I say "system" translate "language").
They are the trace of the incompleteness, which can only give us a chance (not a warranty) at consistence.
And to finish on a philosophical note, as this is true for every formal system that man could devise today and forever, it warns us that there will always be Truths that we won't be able to prove by any algorithm or any computer.
Does that mean that there is something in our mind that makes us different from intelligent machines (and I am not talking art here)?
Up to you to decide :)
von Ick