In the middle of reading Melanie Mitchel’s book Complexity: A Guided Tour, I started thinking about all the ways mathematics has been losing the certainty it once had. Many of these thoughts have been put into my head by my friend Frank Kelly who is a real mathematician. Something I most definitely am not.
The Fifth Postulate
The loss of certainty in mathematics began in the early 19th century concerning the um-proveability of the fifth postulate.
1: Euclid’s geometry was once considered true absolutely. However Euclidean geometry has to satisfy all of Euclid postulates including the fifth postulate you. You have to be able to prove the fifth postulate in order for Euclidean geometry to work. If you can’t prove the fifth postulate then you don’t have Euclidean geometry you have non-euclidean geometry.
Unfortunately no one has ever been able to prove the 5th or parallel postulate.
Since the fifth postulate could not be proven Gaus decided in the early 1820 s to replace the fifth postulate with its negative. This was basically the idea that the sum of the three angles of a triangle can be less than 180 degrees. Evidently you still have a geometry that works and, I think, uses the rest of Euclids laws. But this is a strange non-euclidean geometry.
All of this really undercut the idea that mathematics always and absolutely if used correctly leads to the absolute truth.
Hilbert’s Three Questions
Around 1900 Hilbert asked three questions which turned out to very be very important.
1: given a set of axioms, is there a proof for every true statement, ie completeness. 2: Can only true statements be proved ie consistency 3: Is there a definite procedure that can tell us if the statement can be proven true or false. Ie, is it decidable.
As it turned out none of these things were true.
Godel’s Incompleteness theorem
Around 1900 the German mathematician Godel’s incompleteness theorem continued this deterioration of certainty in mathematics.
Goedel proved his theory mathematically but we can also understand it in a non-mathematical way.
Begin with this statement: “This statement is not provable.”
This statement can either be proven or not proven
If this statement can be proven then it is false. If this statement cannot be proven then it is true.
Thus a false statement can be proven . And it true statement cannot be proven.
Therefore arithmetic either has to be inconsistent or incomplete, one or the other. This replaced certainty in mathematics with a great deal of doubt.
Turing then went on to prove that there is no sure way to tell if any statement is decidable.
Einstein and relativity.
Time is not absolute
Heisenberg’s uncertainty principle.
Complex system uncertainty.
In complex systems, attempts to model the future very quickly turn to chaos