The Three Crises in Mathematics

https://maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1980/0025570x.di021111.02p0048m.pdf


Although the formalization was not entirely complete, Russell and Whitehead thought that it was and planned to use it to show that mathematics can be reduced to logic. They showed that all classical mathematics, known in their time, can be derived from set theory and hence from the axioms of Principia. Consequently, what remained to be done, was to show that all the axioms of Principia belong to logic.

the formal set theory developed by Zermelo and Fraenkel (ZF) is so much better known than Principia

The formulation of the logicists' program now becomes: Show that all nine axioms of ZF belong to logic.

However, the above formulation of logicism is satisfactory for the purpose of showing that this school was not able to carry out its program.

what the logicists meant by "logic." The reason is that, whatever they meant, they certainly meant more than classical logic. Nowadays, one can define classical logic as consistingof all those theorems which can be proven in first order languages

They said: A logical proposition is a proposition which has complete generality and is true in virtue of its form rather than its content

the above law of the excluded middle "pV•P" is a logical proposition. Namely, this law does not hold because of any special content of the propositionp; it does not matter whether p is a proposition of mathematics or physics or what have you. On the contrary, this law holds with "complete generality," that is, for any proposition p whatsoever. Why then does it hold? The logicists answer: "Because of its form." Here they mean by form "syntactical form," the form of p V • P being given by the two connectives of everyday speech, the inclusive "or" and the negation "not" (denoted by V and ...,, respectively).

And now the logicists' task becomes clearer: It consists in showing that all nine axioms of ZF are logical propositions in the sense of logicism.

Hence, instead, we simply state that at least two of these axioms, namely, the axiom of infinity and the axiom of choice, cannot possibly be considered as logical propositions. For example, the axiom of infinity says that there exist infinite sets. Why do we accept this axiom as being true? The reason is that everyone is familiar with so many infinite sets, say, the set of the natural numbers or the set of points in Euclidean 3-space. Hence, we accept this axiom on grounds of our everyday experience with sets, and this clearly shows that we accept it in virtue of its content and not in virtue of its syntactical form. In general, when an axiom claims the existence of objects with which we are familiar on grounds of our common everyday experience, it is pretty certain that this axiom is not a logical proposition in the sense of logicism.

Since at least two out of the nine axioms of ZF are not logical propositions in the sense of logicism, it is fair to say that this school failed by about 20% in its effort to give mathematics a firm foundation.

The philosophy of logicism is sometimes said to be based on the philosophical school called "realism”

Russell was a realist and accepted the abstract entities which occur in classical mathematics without questioning whether our own minds can construct them. This is the fundamental difference between logicism and intuitionism, since in intuitionism abstract entities are admitted only if they are man made


intuitionists, on the contrary, felt that there was plenty wrong with classical mathematics

The logicists considered these paradoxes as common errors, caused by erring mathematicians and not by a faulty mathematics. The intuitionists, on 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.

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 I and, secondly, that the mental process which goes into the formation of the number I 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 I, 2, ... ,n for any natural number n. This mental con~truction 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

intuitionistic construction of natural numbers allows one to construct only arbitrarily long finite initial segments I, 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

 is inductive in the sense that, if one wants to construct, say, the number 3, one has to go through all the mental steps of first constructing the I, then the 2, and finally the 3; one cannot just grab the number 3 out of the sky.

According to intuitionistic philosophy, mathematics should be defined as a mental activity and not as a set of theorems (as was done above in 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 and effective. In- tuitionism maintains that human beings are able to recognize whether a given mental construc- tion 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: Mathemat- ics is the mental activity which consists 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

Another major consequence of the intuitionistic definition of mathematics is that mathemat- ics cannot be reduced to any other science such as, for instance, logic. This definition comprises too many mental processes for such a reduction. And here, then, we see a radical difference between logicism and intuitionism. In fact, the intuitionistic attitude toward logic is precisely the opposite from the logicists' attitude: According to the intuitionists, whatever valid logical processes there are, they are all constructs; hence, the valid part of classical logic is part of mathematics! Any law of classical logic which is not composed of constructs is for the intuitionist a meaningless combination of words. It was, of course, shocking that the classical law of the excluded middle turned out to be such a meaningless combination of words. This implies that this law cannot be used indiscriminately in intuitionistic mathematics; it can often be used, but not always.

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 classical mathematics. This does not bother the intuitionists since whatever parts of classical mathematics they cannot obtain are meaningless for them anyway. Intuitionism 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 logicism and intuitionism: The logicists wanted to justify all of classical mathematics

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 of 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 away 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 cannot 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 his work in (nonintuitionistic) topology.

there are the theorems which hold in intuitionism but are false in classical mathemat- ics. An example is the intuitionistic theorem which says that every real-valued function which is defined for all real numbers is continuous

A real-valued function f is defined in intuitionism for all real numbers only if, for every real number r whose intuitionistic construc-
tion has been completed, the real number f(r) can be constructed. Any obviously discontinuous function a classical mathematician may mention does not satisfy this constructive criterion.

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 sciences than is intuitionism.

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 occur in mathematics, whether sequences 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 between logicism and intuitionism is very similar to the contrast between realism and conceptualism

the modern concept of formalism, which includes finitary reasoning, must be credited to Hilbert.

 should guard against confusing axiomatization and formalization. Euclid axiomatized geometry in about 300 B.C., but formalization started only about 2200 years later with the logicists and formalists. Examples of axiomatized theories are Euclidean plane geometry with the usual Euclidean axioms, arithmetic with the Peano axioms, ZF with its nine axioms, etc. The next question is: "How do we formalize a given axiomatized theory?

Restricting ourselves to first order logic, "to formalize T" means to choose an appropriate first order language for T. The vocabulary of a first order language consists of five items, four of which are always the same and are not dependent on the given theory T. These four items are the following: (I) A list of denumerably many variables-who can talk about mathematics without using variables? (2) Symbols for the connectives of everyday speech, say -, for "not," A for "and," V for the inclusive "or," ~ for "if then," and~ for "if and only if"-who can talk about anything at all without using connectives? (3).The equality sign =; again, no one can talk about mathematics without using this sign. (4) The two quantifiers, the "for all" quantifier V and the "there exist" quantifier 3;

Since T is an axiomatized theory, it has so called "undefined terms." One has to choose an appropriate symbol for every undefined term of T and these symbols make up the fifth item.

the undefined terms of plane Euclidean geometry, occur "point," "line," and "incidence," and for each one of them an appropriate symbol must be entered into the vocabulary of the first order language. Among the undefined terms of arithmetic occur "zero," "addition," and "multiplication," and the symbols one chooses for them are of course 0, +,and X, respectively. The easiest theory of all to formalize is ZF since this theory has only one undefined term, namely, the membership relation. One chooses, of course, the usual symbol E for that relation. These symbols, one for each undefined term of the axiomatized theory T, are often called the "parameters" of the first order language and hence the parameters make up the fifth item

Since the parameters are the only symbols in the vocabulary of a first order language which depend on the given axiomatized theory T, one formalizes T simply by choosing these parameters. Once this choice has been made, the whole theory T has been completely for- malized.

For that kind of technical work, formalization is usually not only no help but a definite hindrance. If, however, one asks such foundational questions as, for instance, "Why is this branch of mathematics free of contradictions?", then formalization is not just a help but an absolute necessity

Suppose that Tis an axiomatized theory which has been formalized in terms of the first order language L. This language has such a precise syntax that it itself can be studied as a mathematical object. One can ask for instance: "Can one possibly run into contradictions if one proceeds entirely formally within L, using only the axioms of T and those of classical logic, all of which have been expressed in L ?" If one can prove mathematically that the answer to this question is "no," one has there a mathematical proof that the theory Tis free of contradictions!
This is basically what the famous "Hilbert program" was all about. The idea was to formalize the various branches of mathematics and then to prove mathematically that each one of them is free of contradictions. In fact if, by means of this technique, the formalists could have just shown that ZF is free of contradictions, they would thereby already have shown that all of classical mathematics is free of contradictions, since classical mathematics can be done axiomatically in terms of the nine axioms of ZF. In short, the formalists tried to create a mathematical technique by means of which one could prove that mathematics is free of contradictions. This was the original purpose of formalism.

The logicists wanted to use such a formaliza- tion to show that the branch of mathematics in question belongs to logic; the formalists wanted to use it to prove mathematically that that branch is free of contradictions. Since both schools "formalized," they are sometimes confused.

Suppose now that T is such a theory and that T has been formalized by means of the first order language L. Then Godel's theorem says, in nontechnical language, "No sentence of L which can be interpreted as asserting that T is free of contradictions can be provei}. formally within the language L."Although the interpretation of this theorem is somewhat controversial, most mathematicians have concluded from it that the Hilbert program cannot be carried out: Mathematics is not able to prove its own freedom of contradictions. Here, then, is the third crisis in mathematics.

Let again T be an axiomatized theory which has been formalized in terms of the first order language L. In carrying out Hilbert's program, one has to talk about the language L as one object, and while doing this, one is not talking within that safe language L itself. On the contrary, one is talking about Lin ordinary, everyday language, be it English or French or what have you. While using our natural language and not the formal language L, there is of course every danger that contradictions, in fact, any kind of error, may slip in. Hilbert said that the way to avoid this danger is by making absolutely certain that, while one is talking in one's natural language about L, one uses only reasonings which are absolutely safe and beyond any kind of suspicion. He called such reasonings "finitary reasonings," but had, of course, to give a definition of them.

“By a finitary argument we understand an argument satisfying the following conditions: In it we never consider anything but a given finite number of objects and of functions; these functions are well defined, their definition allowing the computation of their values in a univocal way; we never state that an object exists without giving the means of constructing it; we never consider the totality of all the objects x of an infinite collection; and when we say that an argument (or a theorem) is ·true for all these x, we mean that, for each x taken by itself, it is possible to repeat the general argument in question, which should be considered to be merely the prototype of these particular arguments.”

We have already compared logicism with realism, and intuitionism with conceptualism. The philosophy which is closest to formalism is "nominalism." This is the philosophy which claims that abstract entities have no existence of any kind, neither outside the human mind as maintained by realism, nor as mental constructions within the human mind as maintained by conceptualism. For nominalism, abstract entities are mere vocal utterances or written lines, mere names.

when formalists try to prove that a certain axiomatized theory T is free of contradictions, they do not study the abstract entities which occur in T but, instead, study that first order language L which was used to formalize T.

particular, they try to show that no sentence of L can be proven and disproven at the same time, since they would thereby have established that the original theory T is free of contradictions. The important point is that this whole study of L is a strictly syntactical study, since no meanings or abstract entities are associated with the sentences of L. This language is investigated by considering the sentences of L as meaningless expressions which are manipu- lated according to explicit, syntactical rules, just as the pieces of a chess game are meaningless figures which are pushed around according to the rules of the game. For the strict formalist "to do mathematics" is "to manipulate the meaningless symbols of a first order language according to explicit, syntactical rules." Hence, the strict formalist does not work with abstract entities, such as infinite series or cardinals, but only with their meaningless names which are the appropriate expressions in a·first order language.

Where do the three crises in mathematics leave us? They leave us without a firm foundation for mathematics. After Godel's paper [6] appeared in 1931, mathematicians on the whole threw up their hands in frustration and turned away from the philosophy of mathematics.

Modem mathematical logic, set theory, and intuitionism with its modifications are nowadays technical branches of mathematics, just as algebra or analysis, and unless we return directly to the philosophy of mathematics, we cannot expect to find a firm foundation for our science