.

The intuitionists have a different notion of truth. To them, a proposition can be said to be true only if someone has rigorously demonstrated its truth. Undecided questions, like the Goldbach conjecture, are neither true nor false. Intuitionists do not accept the law of the excluded middle in logic which says that every proposition is either true or false. Thus a standard technique in mathematics, proof by contradiction, is not allowed by them. One version of proof by contradiction works like this: in order to prove a formula first assume that is false. From this assumption deduce a contradiction (like 1 = 2). If this can be done then infer that the assumption, of being false, does not hold and therefore is true. The intuitionists, however, will not accept this type of argument.

Intuitionism differs from standard mathematics in that an object may be said to exist only if an actual construction of it can be demonstrated. Thus irrational numbers like and are fine because there are algorithms which calculate their decimal expansions to any required accuracy. But consider the following argument, due to Cantor, about the cardinality of the set of irrational numbers. Cantor showed that it is possible to compare the cardinalities even of infinite sets. The counting numbers, for example, have the same cardinality as the set of fractions because there is a way of matching one counting number with one fraction so that none are counted twice and there are none left over. However Cantor showed, using proof by contradiction, that the set of real numbers has a strictly larger cardinality because there is no way of matching the counting numbers with the reals. Sets like the reals or the irrationals are called uncountable. However, this argument does not satisfy the intuitionist. They do not accept that such sets exist, they want to see them actually constructed.

Intuitionism is a form of subjectivism because mathematics is seen as a construction of individual humans. This is certainly true of Brouwer's exposition, though other intuitionists do present more sophisticated versions. Here mathematics is not a process of discovery (as in Platonism) but of invention. The complex number was not discovered but constructed, as indeed were all other numbers. Pythagoras' theorem only became true after Pythagoras gave the proof.

Frege (himself a Platonist) criticised this subjectivism saying:

If the number two were a [subjective -- RH] idea, then it would have straight away to be private to me only. Another man's idea is, ex vi termini [from the power of the boundary line], another idea. We should have to speak of my two and your two, of one two and all twos. ...and in the course of millennia these might evolve, for all we could tell, to such a pitch that two of them would make five. [#!Gil90!#]