Limits To Science: God, Godel, Gravity

Gödel's friendship with Princeton Institute colleague Einstein lasted till Einsteins death. They both admired each other, and late in his life Einstein commented that "his own work no longer meant much, that he came to the Institute merely [..] to have the privilege of walking home with Gödel".

Gödel's brilliance is beyond any doubt, but his mind also played tricks on him. Gödel's perfectionism created obsessions that in his later years grew far beyond his control. He stopped publishing his work out of fear for imperfections. Gödel's paranoia thereby deprived the world of any further contributions of this great mind. Other obsessions, however, affected Gödel himself. One of these was a fear of getting poisoned. Gödel trusted only the food prepared by his wife. When in 1977 she got hospitalized for six months, Gödel refused to eat. Late December 1977 the giant of mathematics was admitted to Princeton Hospital, weighing no more than 65 pounds. He died two weeks later, with the diagnosis "malnutrition and inanition due to personality disturbance".

Since Laplace replied "Sire, I had no need of that hypothesis" to Napoleon's questioning why 'the author of the universe' was missing from his theories, no church and no God seem capable to stop physicists making progress.

Enter Gödel. Is he God's revenge? Is Gödel putting limits to what physicist can achieve?

Physicists build theories that are based on mathematics. They use mathematics to derive conclusions from these theories. They make predictions about physical reality based solely on mathematics and logic. Surely Gödel's results put limitations on what can be achieved in theoretical physics!

The above picture is misleading as it focusses on one single system of axioms. Mathematicians are not bound to work in a single such system, and are free to construct novel and more powerful systems. It is a freedom that is almost a defining characteristics of mathematics. A more accurate picture is therefore the following:

Since Gödel's 1931 publication, further work has elucidated the meaning of his incompleteness theorems. In particular, in 1936 Alan Turing*** gave a specific example of uncomputability: a problem that can not be answered by any computer no matter how powerful or how long one is prepared to wait. This uncomputability implies Gödel's incompleteness: the grey areas in above animations.

Turing's uncomputability refers to the generic problem of deciding whether any given computer program with given inputs will stop. Turing demonstrated that there exist computer program's with given inputs that can not be predicted to ever halt. You simply have to start the program and wait. As long as it keeps running you have no clue if it ever will halt.

So it is gravity, and not Gödel, that prevents us predicting the future. That actually makes into a nice bumper sticker: "Gravity is to physics as Gödel is to math". This is the point Hawking is referring to in his Dirac lecture where he states "quantum gravity is essential to the argument".

As far as the question "is a TOE possible?" is concerned, all of the above is absolutely irrelevant. Uncompuability does not imply that a Theory Of Everything can not be constructed. Gödel does not prevent physicists from doing so, and neither does gravity. Stating that Godel (or Turing, or gravity) implies the logical impossibility of a TOE, is the same as stating that because of the incompleteness theorem an axiomatic logic can not be constructed.