Hilbert’s Progamme Early in C20, the mathematician Hilbert led a programme to formalise maths; to reduce it to a clear set of axioms and show that all true mathematical statements can be proved. It seemed like a natural idea, and the right way to use an axiomatic system.

Puzzle Think of a number. If it is even, half it. If it is odd, multiply by three and add one. Keep doing this. What do you find?

Puzzle Mathematicians have tested billions of cases, and the series always ends up at one, but no-one has yet proved it. Some people believe that the conjecture that ‘the series always ends at 1’ is true, but unproveable (not just that we haven’t done it yet, but that it is in principle not possible to prove it).

Puzzle Mathematicians have tested billions of cases, and the series always ends up at one, but no-one has yet proved it. Some people believe that the conjecture that ‘the series always ends at 1’ is true, but unprovable (not just that we haven’t done it yet, but that it is in principle not possible to prove it). What? Surely the whole point of Maths is to know that theorems are true by proving them!

Prize Competition I have 2 prizes (A and B) and it’s very simple to win one. If you make a false statement then you get no prize If you make a true statement then you get one of the prizes, but you don’t know which one. What should you say if you want to win prize A?

Prize Competition Statement: I will not get prize B TrueFalse Not only is the statement true, it ‘makes itself true’  False statement  no prize True statement  one prize (A or B but you don’t know which) You want prize A

The Island of Knights and Knaves A certain Island is populated only by Knights and Knaves who look the same, but All knights tell the truth all the time All knaves lie all the time All Knights are all members of either club A or club B but not both. Knaves are banned from both clubs.

The Island of Knights and Knaves You meet someone on a path and she says something. You do not know if she is a knave or a knight, but from what she says you can deduce that she is in club A. What did she say?

Knights and Knaves Statement: I am not in club B KnightKnave  Knaves  always lie and cannot be in either club Knights  always tell truth and are in one club You meet someone; she says something and you deduce she is in club A

The Island of Provable and Unprovable Statements An Island is populated by mathematical statements. Some statements are true (knights) Some statements are false (knaves) You cannot easily tell the difference. We are only interested in the true ones here. All true statements (knights) are provable (club A) or unprovable (club B) but not both.

The Island of Provable and Unprovable Statements Provable Unprovable TrueFalse eg Pythagoras’ theorem ? ? Godel showed that there are statements which cannot be proven either true or false. The example we did at the start may be one such example eg even + even = odd eg ? eg ?

I am not provable TrueFalse The Island of Provable and Unprovable statements Not only is the statement not false, the statement ‘makes itself’ impossible to be false Not only is the statement true, it ‘makes itself true’  This is the ‘translation’ of the mathematical theorem into English

Kurt Godel proved that there are mathematical equivalents of the statement “I am not provable”, which by their very nature must be true. So there are mathematical theorems which are true but for which no proof exists (n.b. not that we just haven’t yet found them; they really are unprovable). The World of Mathematical Theorems

Hilbert’s Progamme So (for this reason and others) Hilbert’s programme was shown to be impossible. This was a great surprise and shows that the notion of truth in maths is a stronger one that that of proof. Whereas it was once thought they were very closely linked we now realise that there is no necessary link. Where does this leave the notion of absolute truth in Maths?