## A of mathematics.In this description, Brouwer uses the “move

A clear description of intuitionism by Brouwer can be found in his Cambridge lectures on intuitionism. Here he splits the philosophy into two acts. Firstly: Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognizing that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics.In this description, Brouwer uses the “move of time” in the sense of Kant whereby this is one of the most basic intuitive concepts shared by everyone and which forms the basis of all human experience. From this, we have the idea of a mental construction, the taking of constructive steps one after the other, each time forming a new mental image (see cite{snap}).We see clearly here the desire of Brouwer to detach the practice of mathematics from everything but the mind, even doing away with any mathematical language. It is then not immediately clear how we would proceed in the communication of mathematical ideas. Brouwer admits that some language is necessary, but that there will always be inconsistencies between the concepts created by the mind and the description created using any mathematical language cite{cambr}. However, according to Brouwer, it is possible to use the concepts of contradiction and deductive reasoning in an intuitionistic way. It is when we try to use the law of excluded middle that we encounter a problem.We say that mathematical assertion can be emph{judged} if a proof can be found or if we can reduce it to absurdity (which can be seen as a proof of the negation). For assertions about finite structures we can try all constructions which may yield an example which affirms the assertion, thereby giving a proof or giving an absurd statement which proves the negation. In this way, all such finite assertions can be judged in finite time and are therefore true or false in this sense. Clearly we can then use LEM. If we instead consider conjectures, such as the Riemann hypothesis, that can not be tested in this way, we may not yet say anything about the truth or falsity of such a statement in an intuitionistic way. This does not mean that such a statement will never have any value; with time a proof or contradiction may be found. However, at the moment, for an intuitionist there is no value to this conjecture and we can not base any work on its statement.A natural question which arises from this discussion is whether intuitionism yields any useful form of mathematics. We can, for instance, look at the real numbers and see that these can not be constructed by taking steps in time as suggested in Brouwer’s first act; we will not even take enough steps in our mental construction to obtain all numbers between 0 and 1. This means that, for example, limits or other analytic concepts are not yet recoverable from this first act.The main problem here is clearly infinity and infinite sequences. Brouwer accepted that it was possible to have a so called potential infinity, in which it is not possible to hold in the mind an infinite set in its totality, but one may accept that a sequence can be extended to infinity. In the words of Hermann Weyl describing the infinite sequence of natural numbers: “The sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation” cite{weyl}. To get around this problem, an important aim being to allow the use of the continuum, Brouwer introduces the second act of intuitionism:egin{displayquote} Admitting two ways of creating new mathematical entities: firstly in the shape of more or less freely proceeding infinite sequences of mathematical entities previously acquired (so that, for decimal fractions having neither exact values, not any guarantee of ever getting exact values admitted); secondly in the shape of mathematical species, i.e. properties supposable for mathematical entities previously acquired, satisfying the condition that if they hold for a certain mathematical entity, they also hold for all mathematical entities which have been defined to be `equal’ to it, definitions of equality having to satisfy the conditions of symmetry, reflexivity and transitivity.~ldots citep.~8{cambr}end{displayquote}Together with the first act this allowed Brouwer and later his student Arend Heyting to develop a much larger theory of mathematics (see, for example cite{matann} and cite{heytunt}).It is of course natural to question how successful intuitionism is as a foundation for mathematics, especially compared to other such systems. If we were to ask an intuitionist for an answer to this question, they would undoubtedly say that it is very successful. An intuitionist believes that mathematics is precisely that which takes place in the mind and so the mathematics brought forth by intuitionistic thought is the only work which is of interest. They do not take into account mathematics which is yet to be judged. Of course, mathematicians who do not adhere to these principles, will give a different answer. The mere existence of mathematical conjecture should serve to indicate a willingness of some to work outside that which is purely constructive. We shall touch more on the success of intuitionism later.We shall now look at an important element of intuitionistic work, namely intuitionistic logic.