“底部”，对直觉主义而言就是数学的开端，是对自然数1,2,3…的解释（请注意，在自然数中不包括“零”这个数）。根据直觉主义哲学，人类对自然数有一种内在的原始直觉，这首先意味着，我们对数字1的意义有一种直接的肯定，其次，形成数字1的心理过程可以重复。当我们重复它时，我们得到了数字2的概念；当我们再次重复它时，得到了数字3的概念；以这种方式，人类可以为任何自然数n构造任何有限的初始段1，2，…，n。 如果我们内在没有时间意识，这种一个接一个的自然数的心智构造是永远不可能的。“之后（After）”指的是时间，布劳威尔同意哲学家伊曼纽尔-康德（1724-1804）的观点，即人类对时间有一种直接的认识，康德用“直觉”（intuition）一词来表示“即时意识”（immediate awareness），这就是“直觉主义”这一名称的由来。
这个定义的一个主要后果是，所有的直觉主义数学都是能行的，或者像人们通常所说的那样是“构造的”。从现在开始，我们将使用 “构造的”这个形容词作为 “能行的”同义词。也就是说，每一个构造都是“构造性”的，而直觉主义数学不过是在不断地进行构造。例如，如果一个实数r出现在一个直觉主义的证明或定理中，它绝不会仅仅因为存在证明而出现在那里，它出现在那里是因为它被从上到下地构造起来了。例如，这意味着r的每个小数位在原则上是能计算出来的。简言之，所有直觉主义的证明、定理、定义等等，都是完全构造性的。
This school was begun about 1908 by the Dutch mathematician, L.E.J. Brouwer (1881-1966). The intuitionists went about the foundations of mathematics in a radically different way from the logiciels. The logiciels never thought that there was anything wring with classical mathematica; then silly wanted to show that classical mathematics is part of logic. The intuitionists, on the contrary, felt that there was plenty wring with classical mathematics.
By 1908，several paradoxes had arisen in Cantor’s set theory. Here, the word « paradoxe » is used as synonymous with « contradiction ». George Cantor created set theory, starting around 1870, and he did his work « naively », meaning non axiomatically. Consequently he formed sets with such abandon that he himself, Russell and others found several paradoxes within his theory. The logicists considered these paradoxes as common errors, caused by erring mathematicians and not by a faulty mathematics. The intuitionists, one the other hand, considered these paradoxes as clear indications that classical mathematics itself is far from perfect. They felt that mathematics had to be rebuilt from the bottom on up.
The « bottom », that is, the beginning of mathematics for the intuitionists, is their explanation of what the natural numbers 1,2,3,… are. (Observe that we do not include the number zero among the natural numbers). According to intuitionistic philosophy all human beings have a primordial intuition for the natural numbers within them. This means in the first place that we have an immediate certainty as to what is meant by the number 1 and, secondly, that the mental process which goes into the formation of the number 1 can be repeated. When we do repeat it, we obtain the concept of the number 2; when we repeat it again, the concept of the number 3; in this way, human beings can construct any finite initial segment 1,2, .., n for any natural number n. This mental construction of one natural number after the other would never have been possible if we did not have an awareness of time within us. « After » refers to time and Brouwer agrees with the philosopher Immanuel Kant (1724-1804) that human beings have an immediate awareness of time. Kant used the word « intuition » for « immediate awareness » and this is where the name « intuitionism » comes from.
It is important to observe that the intuitionistic construction of natural numbers allows one to construct only arbitrarily long finite initial segments 1,2, …n. It does not allow us to construct that whole closed set of all the natural numbers which is so familiar from classical mathematics. It is equally important to observe that this construction is both « inductive and « effective ». It is inductive in the sense that, if one wants to construct says the number 3, on has to go through all the mental steps of first constructing the 1, then the 2, and finally the 3; one cannot just grab the number 3 out of the sky. It is effective in the sense that, once the construction of a natural number has been finished, that natural number has been constructed in its entirely. It stands before us as a completely finished mental construct, ready for our study of it. When someone says, « I have finished the mental construction of the number 3 », it is like a bricklayer saying, « I have finished that wall », which he can say only after he has laid every stone in place.
We now turn to the intuitionistic definition of mathematics. According to intuitionistic philosophy, mathematics should be defined as a mental activity and not as a set of theorems (as was done above on the section on logicism). It is the activity which consists in carrying out, one after the other; those mental constructions which are inductive and effective in the sense in which the intuitionistic construction of the natural numbers is inductive effective. Intuitionism maintains that human beings are able to recognize whether a given mental construction has these two properties. We shall refer to a mental construction which has these two properties as a construct and hence the intuitionistic definition of mathematics says : Mathematics is the mental activity which consist in carrying out constructs one after the other.
A major consequence of this definition is that all of intuitionistic mathematics is effective or « constructive » as one usually says. We shall use the adjective « constructive » as synonymous with « effective » from now on. Namely, every construct is constructive, and intuitionistic mathematics is nothing but carrying out constructs over and over. For instance, if a real number r occurs in an intuitionistic proof or theorem, it never occurs there merely on grounds of an existence proof. It occurs there because it has been constructed from top to bottom. This implies for example that each decimal place in the decimal expansion of r can in principle be computed. In short, all intuitionistic proofs, theorems, definitions, etc?, are entirely constructive.
Once the intuitionistic definition of mathematics has been understood and accepted, all there remains to be done is to do mathematics the intuitionistic way. Indeed, the intuitionists have developed intuitionistic arithmetic, algebra, analysis, set theory, etc. However, in each of these branches of mathematics, there occur classical theorems which are not composed of constructs and, hence, are meaningless combinations of words for the intuitionists. Consequently, one cannot say that the intuitionists have reconstructed all of the classical mathematics. This does not bother the intuitionists since whatever parts of classical mathematics they cannot the cannot obtain are meaningless for them anyway. Intuitionsm does not have as its purpose the justification of classical mathematics. Its purpose is to give a valid definition of mathematics and then to « wait and see » what mathematics comes out of it. Whatever classical mathematics cannot be done intuitionistically simply is not mathematics for the intuitionist. We observe here another fundamental difference between logician and intuitionist : The logicists wanted to justify all of classical mathematics.
Let us now ask how successful the intuitionistic schools has been in giving us a good foundation for mathematics, acceptable to the majority of mathematicians. Again, there is a sharp difference between the way this question has to be answered in the present case and in the case go logicism. Even hard-nosed logicists have to admit that their school so far has failed to give mathematics a firm foundation by about 20%. However, a hard-nosed intuitionist has every right in the world to claim that intuitionism has given mathematics an entirely satisfactory foundation. There is the meaningful definition of intuitionistic mathematics, discussed above; there is the intuitionistic philosophy which tells us why constructs can never give rise to contradictions and, hence, that intuitionistic mathematics is free of contradictions. In fact, not only this problem (of freedom from contradiction) but all other problems of a foundational nature as well receive perfectly satisfactory solutions in intuitionism.
Yet if one looks at intuitionism from the outside, namely, from the viewpoint o the classical mathematician, one has to say that intuitionism has failed to give mathematics an adequate foundation. In fact, the mathematical community has almost universally rejected intuitionism. Why has the mathematical community done this, in spite of the many very attractive features of intuitionism, some of which have just been mentioned ?
One reason is that classical mathematicians flatly refuse to do aways with the many beautiful theorems that are meaningless combinations of words for the intuitionists. An example is the Brouwer fixed point theorem of topology which the intuitionists reject because the fixed point confit be constructed, but can only be shown to exist on grounds of an existence proof. This, by the way, is the same Brouwer who created intuitionism; he is equally famous for this work in (nonintuitionistic) topology.
A second reason comes from theorems which can ne proven both classically and intuitionistically. It often happens that the classical proof of such a theorem is short, elegant, and devilishly clever, but not constructive. The intuitionists will of course reject such a proof and replace it by their own constructive proof go the the same theorem. However, this constructive proof frequently turns out to be about ten times as long as the classical proof and often seems, at least to the classical mathematician, to have lost all of its elegance. An example is the fundamental theorem of algebra which in classical mathematics is proved in about half -a page, but takes about ten pages of proof in intuitionistic mathematics. Again, classical mathematicians refuse to believe that their clever proofs are meaningless whenever such proofs are not constructive.
Finally, there are the theorems which hold in intuitionism but are false in classical mathematics. An example is the intuitionistic theorem which says that every real-valued function which is defined for all real numbers is continuous. This theorem is not as strange as it sounds since it depends on the intuitionistic concept of a function : A real-valued function f is defined in intuitionism for all real numbers only if, for every real number r whose intuitionistic construction has been completed, the real number f(r) canoe constructed. Any obviously discontinuous function a classical mathematician may mention does not satisfy this constructive criterion. Even so, theorems such as this one seem so far out to classical mathematicians that they reject any mathematics which accepts them.
These three reasons for the rejection of intuitionism by classical mathematicians are neither rational nor scientific. Nor are they pragmatic reasons, based on a conviction that classical mathematics is better for applications to physics or other science than is intuitions. They are all emotional reasons, grounded in a deep sense as to what mathematics is all about. (If one of the readers knows of a truly scientific rejection of intuitionism, the author would be grateful to hear about it). We now have the second crisis in mathematics in front of us : It consists in the failure of the intuitionistic school to make intuitionism acceptable to at least the majority of mathematicians.
It is important to realize that, like logicism, intuitionism is rooted in philosophy. When, for instance, the intuitionists state their definition of mathematics, given earlier, they use strictly philosophical and not mathematical language. It would, in fact, be quite impossible for them to use mathematics for such a definition. The mental activity which is mathematics can be defined in philosophical terms but this definition must, by necessity, use some terms which do not belong to the activity it is trying to define.
Just as logicism is related to realism, intuitionism is related to the philosophy called « conceptualism ». This is the philosophy which maintains that abstract entities exist only insofar as they are constructed by the human mind. -This is very much the attitude of intuitionism which holds that the abstract entities which occurs in mathematics, whether sequence or order-relations or what have you, are all mental constructions. This is precisely why one does not find in intuitionism the staggering collection of abstract entities which occur in classical mathematics and hence in logicism. The contrast v-between logicism and intuitionism is very similar to the contrast between realism and conceptualism.