We should understand the need for certain types of self-restraint. However, intellectual asceticism (like all other forms of asceticism) cannot be the lot of many.
(a) Axiom of choice:
∀x(¬x=∅⇒ ∃y(“yis a function with domain of definitionx”
∧ ∀u(u∈x∧ ¬u=∅⇒ ∃w(w∈u∧“ u, w ∈y”)))).
That is, y chooses one element from each nonempty elementu∈x.
The belief that this axiom is true in V is at least as justified as the belief in the existence ofV itself. Over the past fifty years it has become customary for every working mathematician to accept this axiom, and the heated con- troversies about it at the beginning of the century are now all but forgotten.
The interested reader is referred to Chapter II ofFoundations of Set Theoryby Fraenkel and Bar-Hillel (North-Holland, Amsterdam, 1958).
4.10.General properties of axioms. Despite the wide variety of concepts reflected in these axioms, each of our sets of axioms for languages in L1 (tautologies;
AxL; special axioms of L1Ar and L1Set) have the following informal syntactic characteristics:
(a) An algorithm can be given that tells whether any given expression is an axiom (compare the syntactic analysis in §1 and the verification of the tautologies in Section 3.4).
(b) A finite number of rules can be given for generating the axioms.
It is clear that a priori, property (b) is less restrictive than (a). In fact, an algorithm as in (a) can be transformed into a rule for generating the axioms:
“Write out all possible expressions one by one in some order, and take those for which the algorithm gives a positive answer.”
It is actually natural to suppose that property (a) should characterize axioms, and property (b) should characterize deducible formulas, no matter how we explicitly describe the axioms and the deducible formulas in a given language. In Part III we make these intuitive ideas into precise definitions and show that (b) is strictly weaker than (a). See also the discussion in Section 11.6(c) of this chapter.
Thus, the method of deduction is a method of mathematics par excellence. (“Mathematical induction” clearly comes out of the same tradition.
Peano’s induction principle allows us to write only the first step and the general step of a proof, and is thereby in some sense the first metamathematical princi- ple. This point is observed by the tradition of listing Peano’s axiom among the special axioms (see 4.7(e)), but one way or another, it is one of the archetypes of mathematical thought.)
The longer the deductive argument, the more important it is for all its elementary components to be written in an explicit and normalized fashion. In the last analysis, the amount of initial data in formal mathematics is so small that failure to observe the rules of hygiene in long deductions would lead to the collapse of the system if we did not have external checks on the system.
In induction, on the other hand, relatively short deductions are based on a vast amount of initial information. Darwin’s theory of evolution is explained to school children, but life is not long enough to judge how persuasive the proofs are. We see a similar situation in comparative linguistics when the features of the so-called protolanguages are reconstructed. In such uses of induction, the
“rules of deduction” cannot be so very rigid, despite the critical viewpoint of the neo-grammarians.
2. The above observations concerning the method of deduction are supported by the fact that the notion of a formal deduction in languages of L1 is a close approximation to the concept of an ideal mathematical proof. It is therefore enlightening to examine the differences between deductions and the arguments we use in day-to-day practice.
(a)Reliability of the principles. Not only the mathematics implicit in the special axioms of L1Set and L1Ar, but even the logic of the languages ofL1 is not ac- cepted by everyone. In particular, Brouwer and others have called into question the law of the excluded middle. From their extremely critical perspective, our
“proofs” are at best harmless deductions of nonsense out of falsehood.
The mathematician cannot permit himself to be completely deaf to these criticisms. After thinking about them for a while, he should at least be willing to admit that proofs can have objectively different “degrees of proofness.”
(b) Levels of “proofness.” Every proof that is written must be approved and accepted by other mathematicians, sometimes by several generations of mathematicians. In the meantime, both the result and the proof itself are liable to be refined and improved. Usually the proof is more or less an outline of a formal deduction in a suitable language. But, as mentioned before, an assertion P is sometimes established by proving that a proof of P exists. This hierarchy of proofs of the existence of proofs can, in principle, be continued indefinitely. We can take down the hierarchy using sophisticated logical and set-theoretic principles; however, not everyone might agree with these principles. Papers on constructive mathematics abound with assertions of the type, “there cannot not exist an algorithm that computesx,”
whereas a classical mathematician would simply say “xexists,” or even “xexists and is effectively computable.”
(c)Errors. The peculiarities of the human mind make it impossible in practice to verify formal deductions, even if we agree that in principle, such a verification is the ideal form for a proof. Two circumstances act together with perilous effect: formal deductions are much longer than texts in argot, and humans are much slower at reading and comprehending such formal arguments than texts in natural languages.
A proof of a single theorem may take up five, fifteen, or even fifty pages. In the theory of finite groups, the proofs of the two Burnside conjectures occupy nearly five hundred pages apiece. Deligne has estimated that a complete proof of Ramanujan’s conjecture assuming only set theory and elementary analysis would take about two thousand pages. The length of the corresponding formal deductions staggers the imagination.
Hence, the absence of errors in a mathematical paper (assuming that none are discovered), as in other natural sciences, is often established indirectly: how well the results correspond to what was generally expected, the use of similar arguments in other papers, examination of small sections of the proof “under the microscope,” even the reputation of the author—in short, its reproducibility in the broadest sense of the word. “Incomprehensible” proofs can play a very useful role, since they stimulate the search for more accessible arguments.
The last two decades have seen the appearance of a very powerful method for performing long formal deductions, namely the use of computers. At first glance, it would seem that the status of formal deductions might greatly improve, so that the Leibnizian ideal of being able to verify truth mechani- cally would become attainable. But the state of affairs is actually much less trivial.
We first give two authoritative opinions on this question by C. L. Siegel and H. P. F. Swinnerton-Dyer. Both opinions relate to the solution by computer of concrete number-theoretic problems.
3. The present level of knowledge concerning Fermat’s last theorem is as follows. Letpbe a prime. It is called regular if it does not divide the numerator of any of the Bernoulli numbersB2= 16, B4= 301, . . . , Bp−3.Fermat’s theorem was proved for regular prime exponents by Kummer. For irregularpthere is a series of criteria for Fermat’s theorem to hold. These criteria reduce to checking that certain divisibility properties do not hold; if they hold, we must try cer- tain other divisibility properties, and so on. The verification for eachprequires extensive computer computations. As of 1955, this was successfully done for all p < 4002 (J. L. Selfridge, C. A. Nicol, H. S. Vandiver, Proc. Nat. Acad. Sci.
USA, 41, 970-973 (1955)).
Let v(x) denote the ratio of the number of irregular primes x to the number of regular primes x. Kummer conjectured that v(x) → 12 as x →
∞. Siegel (Nachrichten Ak. Wiss. G¨ottingen, Math. Phys. Klasse, 1964, No.
6, 51–57) suggests that √
e−1 is a more likely value for the limit, supports this opinion with probabilistic arguments, compares with the data of Selfridge–
Nicol–Vandiver, and concludes this discussion with the following unexpected sentence: “In addition, it must be taken into account that the above numerical
values forv(x) were obtained using computers, and therefore, strictly speaking, cannot be considered proved”!
4. Siegel’s point of view can be explained as a natural reaction to informa- tion received at second hand. But the excerpts below are from an article by a professional mathematician and experienced computer programmer (Acta Arithmetica, XVIII, 1971, 371–385). The article is devoted to the following problem:
LetL1, L2, L3 be three homogeneous linear forms inu, v, w with real coefficients and determinant ∆; and suppose that the lower bound of |L1L2L3|for integer values of u, v, w not all zero is 1. What can be said about the possible value for ∆?
The corresponding problem for the product of two linear forms is much easier, and was essentially completely solved by Markov. There are countably many possible values of
∆ less than 3, each of which has the form
∆ = (9−4n−2)1/2
for some integern; the first few values ofnare 1, 2, 5, 13, 29, and there is an algorithm for constructing all the permissible values ofn.
For three forms Davenport (1943) proved that ∆ = 7 or ∆ = 9 or
∆>9.1.In Swinnerton–Dyer’s paper, all values of ∆17 are computedunder the assumption that there are only finitely many such values and he gives a list of them: the third value is 148, and the last (the eighteenth) is
2597/9.
Discussing this result, he makes a very interesting comment:
When a theorem has been proved with the help of a computer, it is impossi- ble to give an exposition of the proof which meets the traditional test—that a sufficiently patient reader should be able to work through the proof and verify that it is correct. Even if one were to print all the programs and all the sets of data used (which in this case would occupy some forty very dull pages) there can be no assurance that a data tape has not been mispunched or misread. Moreover, every modern computer has obscure faults in its software and hardware—which so seldom cause errors that they go unde- tected for years—and every computer is liable to transient faults. Such errors are rare, but a few of them have probably occurred in the course of the cal- culations reported here.
The arguments on the positive side are also very curious:
However, the calculation consists in effect of looking for a rather small num- ber of needles in a six-dimensional haystack; almost all the calculation is concerned with parts of the haystack which in fact contain no needles, and an error in those parts of the calculation will have no effect on the final results.
Despite the possibilities of error, I therefore think it almost certain that the list of permissible ∆17 is complete; and it is inconceivable that an infinity of permissible ∆17 have been overlooked.
His conclusion:
Nevertheless, the only way to verify these results (if this were thought worth while) is for the problem to be attacked quite independently, by a differ- ent machine. This corresponds exactly to the situation in most experimental sciences.
We note that it is becoming more and more apparent that the processing, and also the storage, of large quantities of information outside the human brain leads to social problems that go far beyond questions of the reliability of math- ematical deductions.
5. In conclusion, we quote an impression concerning mechanical proofs, even ones done by hand, which is experienced by many.
After stating a proposition to the effect that “the functionTW,η0θ˜is correctly defined,” a gifted and active young mathematician writes (Inventiones Math., vol. 3, f.3 (1967), 230):
The proof of this Proposition is a ghastly but wholly straightforward set of computations. It took me several hours to do every bit and as I was no wiser at the end—except that I knew the definition was correct—I shall omit details here.
The moral: a good proof is one that makes us wiser.