the book
The Book is an invention of Paul Erdos, one of the most prolific mathematicians of the twentieth century. This book lists all the proofs of mathematics, written in the formal language of mathematical logic, according to length and in lexical order. With our knowledge of formal logic today, we know that a computer can be set up to start writing this book, and the computer will write as many pages as we desire, but we also know that the computer will never be able to complete the book. Fortunately for us, the author of The Book is not any computer, but the handiwork of The Almighty, and what we have is a totally finished book.
While the initial part of the book can be visualized without any difficulty, getting a mental picture of the last pages of the book does present a problem. The following is a visualization of the book for those who have no difficulty with the cardinals and ordinals of Cantor.
Anatomy of The Book. The salient features and the physical appearance of the book can be visualized as below.
• The front cover and the back cover are each one millimeter thick, and the entire book, including the covers, is three millimeters thick.
• The first sheet of paper is half-millimeter thick, the second sheet is half thick as the first, the third sheet is half thick as the second, and so on.
• On every odd page is written a full proof, and in the next even page is written the corresponding theorem.
• All the pages corresponding to the ordinals after \aleph_0 are stuck together so that not even the axiom of choice can be invoked to open them. For this reason, unfortunately, we will never know what His Last Theorem is, in fact, we will not even know whether there is a last theorem.
From the description of the book, we can infer that any formula which is a theorem can be found in the book, by sequentially going through the pages of the book. The only difficulty is that, if a formula is not a theorem, we will be eternally searching for it.
Note that the lexical order is with respect to proofs and not with respect to theorems, a cruel joke by The Almighty. David Hilbert, the high priest of formalism, once had high hopes of rewriting the book with theorems in the lexical order, but it did not take very long for logicians to find out that He is not in favor of the project.
A scientist can use The Book for taking his oath in the Ultimate Court of Nature. Years of cogitation on the limits of his knowledge makes the oath somewhat subdued: I solemnly swear and affirm that, if I am sane, I will tell the truth, nothing but the truth, however, not the whole truth.
Obviously, the following facts must have weighed heavily in the mind of the scientist:
• Mathematical logic is unable to decide the consistency of even the Elementary Arithmetic of Goedel. We believe our mathematics has no contradictions, just because it has worked well for us for the last two thousand years. Thus, the scientist is unable to categorically vouchsafe for the sanity of his logic.
• The exhilarating experience of the scientist is that whenever he writes his musico-logical composition strictly conforming to mathematical logic, he finds that nature is faithfully dancing to his tune. Earth dances around the sun, because Newton's laws forces it to do so, and electromagnetic wave radiates rhythmically through the universe because Maxwell's equation says, let there be light. Considering all this, the scientist feels that he is telling the truth and nothing but the truth, about all his observations of nature.
• The incompleteness theorems of Goedel and other developments in mathematical logic make it clear to the scientist that he will never be able to tell the whole truth regarding any subject matter. The talk of subquarks after the discovery of the final sixth quark reinforces this conviction of his.
Epilogue. Essentially, what The Book teaches us is humility, and consistent with this, it is not surprising that Bertrand Russell has the following advice for us:
United with his fellow-men by the strongest of all ties, the tie of a common doom,.... Be it ours to shed sunshine on their path, to lighten their sorrows by the balm of sympathy, to give them the pure joy of a never-tiring affection, to strengthen failing courage, to instil faith in hours of despair. Let us not weigh in grudging scales their merits and demerits, but let us think only of their need, of the sorrows, the difficulties, perhaps the blindnesses, that make the misery of their lives; let us remember that they are fellow-sufferers in the same darkness, actors in the same tragedy with ourselves.
The Book: Visualizing the Intellectual Universe
Tuesday, April 10, 2007
