This is an excerpt in from the book , Quantum PhilosophyAlthough this chapter is to say that formalism is required in mathematics .This post is in awe of what a display of Intellectual Integrity somebody can showcase THE CRISIS IN THE FOUNDATIONS OF SET THEORYWhat is known as the crisis in set theory is a striking event that deserves to be replayed in the spirit of what it was, that is, a drama lacking neither heartbreaking nor noble undertones. Here it is then, more or less as it might be seen on the stage.
Two characters are present as the curtain rises, Gottlob Frege and Bertrand Russell. The action is set in a temple, that of the goddesses Mathematics and Logic. At the back are full-size portraits of the great priests of the time: David Hilbert and Henri Poincaré. Other pictures, in subdued tones, depict Dedekind, Peano, and Cantor. A portrait of Frege himself appears on an easel in the foreground; it has just been retrieved from the storage room after
a long stay there.
The actor playing Frege appears to be in his fifties. He is unassuming but betrays a unique passion that can only be inspired by truth. Almost twenty-five years have elapsed since the publication of his short book on logic which had initially gone unnoticed. Bertrand Russell is thirty years old. He has the unmistakably sharp traits of an aristocrat and speaks with a slight Cambridge accent.
FREGE: Yes, my final book on the theory of sets is due to appear soon. It took me twenty long years of hard work, but it was perhaps worth the effort.
RUSSELL: You know very well what I think of it. Nothing as important as your first book had been written in logic since Aristotle; and your latest one, I believe, should definitely establish mathematics on a solid base. What an achievement for the honor of the human mind!
FREGE: Let us not exaggerate. It is true, at any rate, that the logic is sufficiently clear. As for the mathematics, I think one should begin with set theory and build everything on it. In fact, there is nothing simpler or more transparent than a set. When you speak of a collection of objects, everybody knows what you are talking about.
RUSSELL: Yes, it appears to be quite obvious, and yet, I have one nagging reservation.
FREGE: Which one?
RUSSELL: Something in your Begriffschrift that puzzles me. You say there, essentially, that an arbitrary set, and I insist on the term “arbitrary,” may always be taken as an element of another set. Do you still think so?
FREGE: More than ever. A major part of my new book is based on that fact, and the idea is repeatedly exploited. Do you have an objection? I thought it to be obvious. What’s wrong with the idea that any object can always be included in a set along with other objects?
RUSSELL: That is certainly what our intuition tells us. But I wonder if we can always trust it, and if it is not possible that intuition may deceive us when left unchecked even for an instant.
FREGE: All right, I can see that you have found a skeleton in the closet. Better take it out. What is it? This expression is due to Hilbert.
RUSSELL: Do you agree that, in principle, certain sets may contain themselves as elements?
FREGE: It is at any rate a direct consequence of what we said earlier. If you asked me for an example, I would propose the catalogue of a library, which can be one of the books placed on a shelf of the same library; or the word “dictionary” in a dictionary; or God, who says “I am who I am”; or the table of contents of a book, which contains the table of contents, or even . . .
RUSSELL: I see. But let us consider all the others instead, and designate by A the set of all those sets that are not elements of themselves. Now let me ask you a question: Does this set A belong to itself?
FREGE: Let’s see, this should not be difficult. Suppose it does, that is, that A belongs to A. Now, by definition, the elements of A are those sets that do not belong to themselves. Thus, assuming that the answer to your question is “yes,” we have a contradiction. Therefore, the answer must be “no.”
RUSSELL: Are you sure?
FREGE: If I answer “no,” this implies that A does not belong to A. But then, by the very definition of A, it follows that A does belong to A. Good Lord, you are absolutely right! No matter which path we choose, it leads to a contradiction.
This is a paradox, what am I saying? An aporia, a catastrophe! It is the principle of the excluded middle that you have just called into question. But this is impossible, we cannot reject this principle, for there would be no logic left, all thought would collapse.
RUSSELL: I can see only one way out: to repeal what you have said
in the past and start all over from the beginning.
FREGE, after a moment’s reflection: There is no other solution. Naturally, my great project of rebuilding mathematics is shattered to pieces. Just when I thought I had succeeded! But, you know, what you have found is truly amazing, extraordinary. Congratulations!
It is a while since I came across something so interesting! (He leaves walking unsteadily, smiling and talking to himself.)
RUSSELL, watching Frege leave:
What a demonstration of intellectual integrity! Such grace! I have never seen anyone pursue truth as honestly as he does. He was about to culminate at last a life-long endeavour, he who had been so often passed over in favour of others who did not deserve it. . . . He did not care, and when told that one of his most fundamental hypotheses is wrong, how does he react? His intellectual pleasure overwhelms his personal dis-appointment. It’s almost superhuman. What an interior strength a man can summon if he devotes himself entirely to knowledge and creation, rather than to a vain search for honours and celebrity! What a lesson! (He also leaves.)
THE CHOIR: The temple has been shaken and it is cracking. Is it an earthquake? Paradoxes are piling up. The Cretan liar has been resuscitated. There are also Richard’s and Burali-Forti’s paradoxes, besides Russell’s. Are we to become everybody’s laughing- stock when it has been pointed out that an eleven-word sentence suffices to define “the smallest number impossible to name with less than twelve words”? Is logic only an illusion?
HILBERT, entering the room: Calm down, please, and do not panic. Look those fearsome paradoxes straight in the eye. They are all alike. They all carry the same sign, that of the whole considered as a part. The library’s catalog is a list of all books. Epimenides, the
Cretan, says that all Cretans are liars. Your eleven-word sentence refers to all possible definitions of a number.
This story shows only one thing: that Frege had not gone far enough in his efforts to formalize mathematics. He thought he could trust his intuition, if only a little, regarding sets, which appear to be so limpid. It was his sole mistake, and it is our duty to correct it. From now on, logic and mathematics will be entirely formal. (He leaves, followed by a thoughtful Zermelo, who would take up the task proclaimed by Hilbert.)
