Riemann on Deductive vs Creative Reasoning

[paalfis]

I understand that Riemann was very shy, so he didn't talk much. Something that he said was:

'If only I had the theorems! Then I should find the proofs easily enough' .

What do you think meant by that? I suspect he was comparing deductive reasoning (proofs) with imagination and the 'seeing over the walls'-kind of reasoning that is needed for creating theorems when facing new problems.

 

[Mark44]

I understand that Riemann was very shy, so he didn't talk much. Something that he said was:

'If only I had the theorems! Then I should find the proofs easily enough' .

What do you think meant by that? I suspect he was comparing deductive reasoning (proofs) with imagination and the 'seeing over the walls'-kind of reasoning that is needed for creating theorems when facing new problems.

Or maybe he was speaking literally and not figuratively -- that he didn't have the imagination to think up new theorems, but he did have the abilities to prove theorems that others posed.

 

[lavinia]

I understand that Riemann was very shy, so he didn't talk much. Something that he said was:

'If only I had the theorems! Then I should find the proofs easily enough' .

What do you think meant by that? I suspect he was comparing deductive reasoning (proofs) with imagination and the 'seeing over the walls'-kind of reasoning that is needed for creating theorems when facing new problems.

I think you are correct. Like all scientists, mathematicians form hypotheses - in their case these come from observation of mathematical objects and are inspired by observation of the so called "real word". They are looking for hypotheses that are in fact theorems. The proofs are secondary and verify the hypothesis. It seems that Riemann meant that you need to know what is true,the theorem, before you prove it.

Riemann, in fact, assumed certain hypotheses without any formal proof. The example I am thinking of is the Dirichlet principle which mathematically says that there is always a harmonic function in the interior of a region that has prescribed boundary values. He believed this because of examples from Physics. For instance, the Dirichlet principle would predict that if one heats up a metal plate all the time keeping the temperature on its edge constant, then a static equilibrium temperature throughout the entire plate will be reached. Finding the theorem in this case would be fomulating the temperature experiment as a statement about harmonic functions. Further evidence for the principle came from the experimental creation of static electric fields. There is a cool section in Feynmann's Lectures Book2 where he shows experimental setups that create harmonic functions with given boundary values.

As is turns out, the Dirichlet principle is false but it is true in enough cases to enrich mathematical theory.

Throughout the history of Mathematics, Physical laws have suggested theorems. Finding those laws then formulating them as mathematical theorems (and theories) is the key step. The proof is last.

Riemann was able to prove many of his hypotheses but at least one defies proof even today. That is the Riemann Hypothesis. Search for a proof has led to much new mathematics.

In mathematics itself, all Physics aside,hypotheses are formed based on observation of mathematical objects. These hypotheses are called conjectures and may or may not be theorems. The important conjectures, lead to new ideas and new research even when they are not known to be true.