A. Heyting

Intuitionistic views on the nature of mathematics

One of the questions which philosophers ask about mathematics is: Why are mathematical theorems so certain? Whence does mathematics take its evidence, its indubitable truth? The answer of intuitionists to these questions is: The basic notions of mathematics are so extremely simple, even trivial, that doubts about their properties do not rise at all. Intuitionism is not a philosophical system on the same level with realism, idealism, or existentialism. The only philosophical thesis of mathematical intuitionism is that no philosophy is needed to understand mathematics. On the contrary, every philosophy is conceptually much more complicated than mathematics. Logic in the usual sense does depend upon philosophical questions. One of its basic notions is that of a proposition being true. But what is a proposition? Does it coincide with the sentence by which it is expressed or is it something behind the sentence, some meaning? If so, what is the relation between the proposition and the sentence? And what does it mean that the proposition is true? Does this notion presuppose the existence of an external world in which it is true? If the proposition is the same as the sentence analogous questions can be asked. I am not going to answer them; they have been solved in a hundred different ways, none of them quite convincing, and all of them showing that logic is complicated and therefore unsuitable as a basis for mathematics. I shall come back to the relations of logic to mathematics later in this talk. We look for a basis of mathematics which is directly given and which we can immediately understand without philosophical subtleties. The first that presents itself is the process of counting. However, counting establishes a correspondence between material or non-material objects and the natural numbers, so it can only be understood if both an external world (or at least some sort of objects) and abstract numbers are given. It is still too complicated to serve as a basis for mathematics. An analysis

After Thirty Years

Publisher Summary None of the conceptions of mathematics is today as clear-cut as it was in 1930. The chapter discusses about formalism and logicism. Formalism is the least vulnerable, but for metamathematical work, it needs some form of intuitive mathematics. As to logicism, many axiomatic systems of logic and of set theory compete. There are several definitions of logicism that are proved by several authors to be equivalent, though Brouwer had proved in 1920 from the constructive point of view that they are not. A new form of mathematics is born in which it can be defined that which part of the work is purely formal, and which platonistic assumptions are made. In this connection, the chapter reviews a remark made by Hahn that intuitionism and formalism are important investigations inside mathematics but are not the foundations of mathematics. Kreisel made a similar remark in 1953 stating that some people develop preferences for certain methods in mathematics and dislikes of others. Hahn and Kreisel both seem to consider mathematics as something given beforehand and the research of foundations as a special subject inside it.

Axioms for Intuitionistic Plane Affine Geometry

Publisher Summary The axiomatic methods are used in intuitionistic mathematics. Every theorem can be expressed in the form of an axiomatic theory. Conversely, every axiomatic theory can be read as a general theorem. However, the content of the theory is often influenced by the intuitionistic point of view. The chapter essentially describes a problem that states a system of axioms for intuitionistic plane projective geometry and which is satisfied by the intuitionistic analytical geometry. It also describes the axiom system for projective geometry, the axiom system for affine geometry, elementary theorems, projective points, projective lines, and the proof of the projective axioms.