A professor of mine sent me a link once with the

class notes from his philosophy of mathematics course. Here's the section on Intuitionism, minus some graphic stuff that didn't copy/paste.

But I think its enough to give you the idea.

In his Philosophy of Mathematics class notes, Dr. Carl Posy wrote:

Part II: Intuitionism, A Third Direction

II.1 General Introduction to Intuitionism

The intuitionist's picture of mathematics was different. In represented a continuation of the unpopular Kantian approach to mathematics.

Intuitionism was founded in 1907 in the Ph.D. dissertation of L. E. J. Brouwer at the University of Amsterdam. After publishing his dissertation, he introduced several important topological theorems, thus giving him some mainstream mathematical prominence.

Intuitionism embraced two important philosophical theories:

Brouwer's Doctrine-- to be true is to be experienced; whatever exists has its origin in our conscious thought.

Mathematical objects are abstract, a priori, forms of our intuitions.

Two Differences from Kant

Brouwer was a solipsist. He believed the only mind was his own, and was less concerned with intersubjectivity than was Kant.

Brouwer rejected the claim of an a priori intuition of space. Instead, he thought mathematics was based entirely on an a priori intuition of time; Brouwer believed the structure of time guides all conscious activities. The presence of non-Euclidean geometry disallows a single a priori intuition of space.

Brouwer must reconstruct certain parts of mathematics given his own constraints. The positive program of intuitionism is a construction of mathematics as limited by Brouwer's Theory of Consciousness. The negative program of intuitionism argues that standard mathematics is actually wrong (or at least inconsistent).

Brouwer does argue that standard mathematics is inconsistent; his argument is based on his epistemological idealism. Brouwer makes little distinction between Hilbert and the Platonists.

Some of Brouwer's constructions depend on the assumption that if a proposition is true, we can come to know that it is true.

II.2 Intuitionist's Construction of the Natural Numbers

In constructing the natural numbers, there is one fundamental idea: We have the ability to distinguish one thing from another. Then, we must abstract from the difference between the two things, and get the concept of forming an entity with one part and another part. Through this activity, the natural numbers are constructed; the natural number one corresponds to the intuition before performing this mental activity, while two corresponds to the intuition after the activity. Repeating the mental abstraction gives the subsequent natural numbers. It is important to concentrate only on the act itself-- abstracted from the content; one must look at the empty form.

Through this method, the intuitionists also derive the basic numerical operators. An equation:

is the report of four activities: generating the numbers, looking at two of them together, and recognizing they're the same as a third. Arend Heyting(55) said the role of a math teacher is to make the student carry out the same mental activities as he did.

The point of abstract mathematical notation is to help us generate certain activities. Rules of mathematics abstract away from the actual mental activities. Ultraintuitionism concerns itself with physical limitations as well. Brouwer's standard intuitionism simply limits us to what is finitary.

According to intuitionist theory, reductio ad absurdum proofs are not allowable to prove that something exists (although they are acceptable for negative results.(56)

II.3 Intuitionist's Construction of the Real Numbers

Problem: The previous processes have finite limitations on the number of steps. The set of real numbers requires infinity in some way or another.

Brouwer, in his dissertation (1907), suggested a separate act of consciousness is needed for generating real numbers. He called it ur intuition-- a fundamental idea we all possess corresponding to the continuum. The ur is always growing. By 1918, however, Brouwer had given up the idea of the ur. His general idea was that it is the whole that is important when speaking of the real numbers. The elements of the whole are then abstracted through conditions or limitations. One of the main reasons the idea of ur intuition was discarded was that it lacked any mathematically useful structure.

Real numbers can be seen as a convergent sequence of rational numbers, but sequences are infinite. However, a constructable sequence, with a formula or rule given to generate the elements, is allowable. Thus, the convergent sequence must be capable of being generated by a rule. French mathematicians took the view that the only real numbers that exists are those that are convergent sequences of rationals which are calculable. However, Brouwer noticed the set of calculable algorithms is enumerable (has cardinal number 0). Thus, we can't restrict the real numbers to this set, because it would then not have properties that the uncountable reals have.

Brouwer's solution was a generalization of the notion of an algorithm or rule to give an uncountable number of algorithms to give what is needed for reals. This was the notion of a choice sequence.

II.4 Choice Sequences

Choice sequences represent Brouwer's main contribution to Kantian mathematics.

Ordinarily, an algorithm is a rule for calculating the elements in a sequence. Two things characterize an algorithm: 1) it is rule-like; and 2) it is deterministic (it gives exactly one value). Brouwer generalized algorithms by loosening the requirement that an algorithm be deterministic. The result is a sequence in which an element of a sequence is able to be chosen out of a set of candidates.

A choice sequence is given by a deterministic rule to give the first few elements, and a not-necessarily-deterministic rule for picking subsequent elements. Brouwer pointed out that this corresponds to an a priori intuition of time: the past is fixed, while the future depends on the past, but many possibilities remain.

Example

is a sequence of rationals.

(1) = ½, (2) = ½, (3) = ½, (4) = ½

(n+1) is some rational number such that:

Thus (5) is on , and (6) depends on what is picked for (5).

The canonical choice sequence is the sequence used to generate decimal fractions.

Consider defined exactly as is:

is a sequence of rationals.

(1) = ½, (2) = ½, (3) = ½, (4) = ½

(n+1) is some rational number such that:

Are and the same? Do they converge to the same real number? We don't and cannot know! Some important questions about choice sequences are not answerable in a finite amount of time. Thus, there is no truth concerning questions about the equality of and . We don't even know if we'll know the answer in a finite amount of time.

Brouwer had to rework set theory to coincide with his other constructions. Under his version of set theory, the distinction between an element of a set and the set itself is less well-defined.

The introduction of choice sequences result in contradictions with classical mathematical theorems. For example, there is a classical theorem stating a line has a total order;

This doesn't hold for numbers like ! Thus, order properties of the continuum are weaker in the intuitionist theory.

Brouwer proved a theorem stating that every real-valued function defined on a closed interval is uniformly continuous on that interval. Consider f(x) = 1 for x < ½, f(x) = 3 for x > ½. This function is clearly discontinuous at x = ½. It also appears to be defined over the interval [0,1]. However, in order for Brouwer's theorem to hold, he must show that the function is not defined at some point on the interval. One such point is . We can't tell what f() is equal to. Thus, this is not a counterexample to Brouwer's theorem.

From this, we can see that a function is defined if its value depends on only a finite amount of information about the input. This corresponds identically to continuity.

Two consequences of the property of discontinuity

Brouwer can easily prove the uncountability of the real numbers. Consider a function, f, mapping the reals into the natural numbers. If this is truly a continuous function, its value must be calculable based on a finite amount of information. However, since the natural numbers are discrete, such a function would have to be discontinuous. Let's say f(½) = n. Then, f() = n if = ½, or f() = k, if ½ (where k n). Thus, the function, f, must be undefined at x = . Therefore the function cannot be continuous (it isn't defined everywhere), and the real numbers must be uncountable.(57)

Suppose we want to divide the continuum into two sets, A and B (B = R - A). This activity of forming a subset of the continuum is perfectly natural. Using the method of characteristic functions, we can translate talking about sets into talking about functions. We define fA(x) = 1 if x A, fA(x) = 0 if x A. This method results in undetachable sets-- sets such that they cannot cleanly be picked out of the continuum. The real number may, or may not, be in the set, and the characteristic function for that set is discontinuous. In fact, for any subset of R, the characteristic function for R is discontinuous. In other words, there are no detachable subsets of the continuum. This view of the real number line is the same as Aristotle's. In a sense, we've come full circle, as the problems with the Aristotelian continuum re-appear.

Brouwer noticed that the properties of space thought to be purely geometric can be expressed temporally once we admit that what characterizes the structure of time is that the future is undecided.

The intuitionists and the Platonists agree on one important point: They both believe that the ideal parts of mathematics consist of actual objects created in the mind.

Brouwer, later in his career, admitted that there was a problem with choice sequences. The basic tenet that a real number is created by acts of choice seemed improper-- it required acts of humans, which Brouwer didn't feel should be introduced into mathematics. In the late 1940's, Brouwer introduced the method of the creating subject to generate real numbers. He said we should focus on an idealized mathematician, B, and divide his research into stages. At each stage we ask him the status of an unsolved mathematical problem. We then define the sequence :

(n) = ½ if at the nth stage, B hasn't yet proved or refuted the unsolved problem.

(n) = if at the nth stage, B has solved the problem.

This process forms a sequence which is a real number; there is no act of choice. Instead, there is an automatic procedure, capturing the same effect as choice sequences, without appealing to the non-mathematical act of choice.

Clearly, this method will not work if the unsolved problem is solved, so, in order for the method of the creating subjects to be an acceptable method, there must be an inexhaustible supply of unsolvable mathematical problems. Brouwer, as a matter of faith, believed this to be true. Hilbert, however, in a famous address to the congress of mathematicians in the late nineteenth century, remarked that there could be no problem which is unsolvable in principle. Brouwer obviously opposed this view.

II.5 General view of Brouwer's Intuitionism vs. Hilbert's Formalism

Both Hilbert and Brouwer were constructivists. Hilbert's Kantianism was very different from Brouwer's, though. Hilbert actually put a structure on the intuitive part of mathematics-- essentially that of finitary thought and formal systems. With Gödel's work, we can see that Hilbert's formal system fits the theory of recursive functions.

Brouwer was very much opposed to these ideas, especially that of formalizing systems. He even opposed the formalization of logic. Brouwer had a very radical view of mathematics and language's relationship. In language, we can communicate the output of mathematical construction, thus helping others recreate the mathematical experience. But, the proof itself--mathematical thought itself--construction itself--is a pre-linguistic, purely conscious activity which is much more flexible than language. Brouwer thought formal systems could never be adequate to cover all the flexible options available to the creative mathematician. Brouwer, in fact, thought formalism was absurd! In particular, Brouwer thought that it was crazy to think that codified logic could capture the rules for correct mathematical thought. He showed particular rules of logic are inadequate. The most famous of these was the law of the excluded middle: fails for . Another such rule, the rule of double negation () does not hold either. The inadequacy of the rule of double negation is another good reason for rejecting reductio ad absurdum proofs for positive results.

Brouwer hypothesized about the reason why philosophers and mathematicians included the law of the excluded middle. He supposed that logic was codified when the scientific community was concerned only with finite objects. Considering only finite objects, the law of the excluded middle holds. However, a mistake was made when mathematics moved into the infinitary: the rigid rules of logic were maintained without question. Brouwer suggested that no rigid codification should come before the development of mathematics.

A second major distinction between Brouwer and Hilbert was that they disagreed on the position of logic. While Hilbert thought logic was an autonomous, finished science that could be freely applied to other mathematics, Brouwer argued logic should only come after the mathematics is developed.

Brouwer's disciple, Arend Heyting, took on the challenge of explaining to the mathematical community what intutionism is all about. Contrary to Brouwer's wishes, Heyting formalized intuitionistic logic and intuitionistic number theory. Brouwer was furious, but in the end, Heyting's approach won; the intuitionism dealt with today is largely that which Heyting formalized.

Gödel, in the middle 1930s, proved the consistency of classical number theory relative to the consistency of intuitionistic number theory. Gödel, in 1958, gave an even more interesting proof to the effect that one can't proof the consistency of a formal system within a formal system with equivalent finitary limitations. Instead, one must use a less finitary formal system. Intuitionistic number theory is, in fact, less finitary than the formalist's number theory.

Intuitionism's drawbacks

Intuitionistic mathematics is much less familiar, and arguably more complicated than classical mathematical theory.

Many people were unhappy with Brouwer's ontological idealism.(58)

In summary, Brouwer's contribution was more philosophical than it was mathematical. The situation today remains that there is no single philosophy of mathematics that is entirely satisfactory.