Mathematical Problems

Answers

• Q10JUDGMENT

  Can a fixed procedure for reasoning settle a disagreement when the participants accept the same premises?

  Yes. every problem is decidable. States the conviction that every definite mathematical problem can be settled, either by solution or proof of impossibility: 'in mathematics there is no ignorabimus'.

  It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. Take any definite unsolved problem, such as the question as to the irrationality of the Euler-Mascheroni constant , or the existence of an infinite number of prime numbers of the form . However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the [Pg 9] firm conviction that their solution must follow by a finite number of purely logical processes. … This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus .

  Paris International Congress lecture 1900; full German text on de.wikisource with the 'kein Ignorabimus' sentence (search results).