Intuitionism

Brouwer's Intuitionism
• Founded by Luitzen Brouwer (1881-1966), intuitionism is the mathematical philosophy that the proper objects of mathematics are given in intuitions.
• Given as such, the objects of mathematics require no justification or foundations. Thus, intuitionists reject logicism and formalism.
• Brouwer and most other intuitionists believe that only some mathematical objects are directly given, however. All others must be constructed clearly out of given objects, or they must be rejected.
• Intuitionists deny, for example, the principle of the excluded middle (the principle that P or not-P is true for any mathematical or logical sentence P). Their reasoning is that if I show that I cannot show P, I have not shown not-P; this requires a separate proof.
• For this reason, intuitionists are often very similar in approach to the more recent demands that some make for "constructive proofs" -- that is, a constructive proof is typically one that does not use a reductio ad absurdum, but "constructs" the thing to be shown with a direct proof of some kind. But it is important to note still that one can be inclined to favor or even require constructive proofs without being an intuitionist.

Comparing views on Infinity
• Logicists: infinite collections are acceptable, although sometimes they are proposed as hypotheticals.
• Formalists: infinite collections are acceptable if we can prove our reasoning about them is consistent.
• Intuitionists: reference to infinite collections is to be avoided, but we can refer to unbounded collections when we show some property is true of any arbitrary element (for example). Consider the construction of the natural numbers: we show that for any n there is an n+1 element.

Next: Russell.