In this section I shall rebut some possible objections which might be raised against the research which was reported in the previous sections.
6.1. The purpose of Hilbert's Program is to defend mathematics against skeptics. But why is mathematics in need of any defense? Doesn't everyone agree that mathematics is both valid and useful?
As to the usefulness of mathematics, opinion is divided. Some see mathematics as both a supreme achievement of human reason and, via science and industry, the benefactor of all mankind. (This is my own view.) Others believe that mathematics causes only alienation and war. Still others see mathematics as a useless but harmless pastime. The utility of mathematics can be argued only as part of a broad defense of reason, science, technology and Western civilization.
What chiefly concerns us here is not utility but scientific truth. Of course the two issues are related. Pragmatists might argue that mathematics is useful and therefore valid. But such an inference can cover only applied mathematics and is anyhow a non sequitur. It makes much more sense to argue that mathematics is true and therefore useful. In the last analysis, the only way to demonstrate that mathematics is valid is to show that it refers to reality.
And make no mistake about it -- the validity of mathematics is under siege. In a widely cited article , Wigner declares that there is no rational explanation for the usefulness of mathematics in the physical sciences. He goes on to assert that all but the most elementary parts of mathematics are nothing but a miraculous formal game. Kline, in his influential book Mathematics: The Loss of Certainty , deploys a wide assortment of mathematical arguments and historical references to show that ``there is no truth in mathematics.'' Kline's book was published by the Oxford University Press and reviewed favorably in the New York Times. (For a much more insightful review, see Corcoran .) Neither Wigner nor Kline is viewed as an enemy of mathematics. But with friends like these, who needs enemies? Arguments like those of Kline and Wigner turn up with alarming frequency in coffee-room discussions and in the popular press. Russell's famous characterization of mathematics, as ``the science in which we never know what we are talking about, nor whether what we say is true,'' is gleefully cited by every wisecracking sophist.
In the face of the attack on mathematics, what defense is offered by the existing schools of the philosophy of mathematics? Consider first the logicists. They say that mathematics is logic, logic consists of analytic truths, and analytic truths are those which are independent of subject matter. In short, mathematics is a science with no subject matter. What about the formalists? According to them, mathematics is a process of manipulating symbols which need not symbolize anything. Then there are the intuitionists, who say that mathematics consists of mental constructions which have no necessary relation to external reality, if indeed there is any such thing as external reality. Finally we come to the Platonists. They are better than the others because at least they allow mathematics to have some subject matter. But the subject matter which they postulate is a separate universe of objects and structures which bear no necessary relation to the real world of entities and processes. (They use the term ``real world'' referring not to the real real world but to their ideal universe of mathematical objects. The real real world is absent from their theory.) I submit that none of these schools is in a position to defend mathematics against the Russells and the Klines.
The four schools discussed in the previous paragraph are not very far apart. Each of them is based on some variant of Kantianism. Frequently they merge and blend. Most mathematicians and mathematical logicians lean toward an uneasy mixture of formalism and Platonism. Uneasiness flows from the implicit realization that neither formalism nor Platonism nor the mixture supports a comprehensive view of mathematics and its applications. There is urgent need for a philosophy of mathematics which would supply what Wigner lacks, viz. a rational explanation of the usefulness of mathematics in the physical sciences. Some form of finitistic reductionism may be relevant here.
I have argued elsewhere that the attack on mathematics is part of a general assault against reason. But this is not the burden of my remarks today. What is clear is that mathematicians and philosophers of mathematics ought to get on with the task of defending their discipline.
6.2. Hilbert's Program is exclusively concerned with the problem of validating infinitistic mathematics. But what's the big problem about the infinite? Isn't finitistic mathematics in equal need of validation?
There is a long history of doubts about the role of the infinite in mathematics. Aristotle's discussion of the infinite is more acute than modern ones but still inconclusive. Euclid achieved rigor in part by avoiding all reference to the infinite. Archimedes used infinite limit processes but never rigorously justified them. Later, infinitesimals in calculus were the occasion of intense philosophic controversy. Doubts about infinitesimals were exploited by Bishop Berkeley in his mystical assault on science and Enlightenment values. Weierstrass' arithmetization of calculus restored clarity and rigor, but the respite was only temporary. Controversy about the infinite was never more intense than in our own century.
The problem is that the infinite does not obviously correspond to anything in reality. The real world is made up of finite entities and processes. Everything that exists has a definite nature and is therefore in some sense limited. Aristotle argues for the real-world existence of the infinite, but only by recourse to a distinction between potential and actual infinity. Hilbert uses physical arguments to deny the existence of the infinite anywhere except in thought. Certainly any convincing account of the relationship between the infinite and the real world would have to be fairly subtle.
By contrast, the formulas of finitistic mathematics refer in a relatively unproblematic, common-sense way to various discrete or cyclical real-world processes. For this reason, finitistic mathematics has always been much less controversial than infinitistic mathematics. Only in our own time has there arisen an ultrafinitist school which posits bounds on the length of the natural number sequence. And the ultrafinitists have neither refuted finitistic mathematics nor shown us what an ultrafinitist textbook would look like. Finitistic mathematics is as firmly grounded as a science can be.
6.3. The essence of Hilbert's Program is to reduce infinitistic mathematics to finitistic mathematics. But what is the point of such a reduction? Does it really increase the reliability of infinitistic mathematics?
I grant that the reduction of infinitistic proofs to finitistic ones does not increase confidence in the formal correctness of infinitistic proofs. What such a reduction does accomplish is to show that finitistically meaningful end-formulas of infinitistic proofs are true in the real world. Hence formulas which occur in infinitistic proofs become more reliable in that they are seen to correspond with reality.
6.4. Why should we concern ourselves exclusively with finitistic reductionism? What about predicativistic or intuitionistic reductionism?
This objection has been partially answered in the digression at the end of §3. Finitism is much more restricted than either predicativism or intuitionism. Finitistic reasoning is unique because of its clear real-world meaning and its indispensability for all scientific thought. Nonfinitistic reasoning can be accused of referring not to anything in reality but only to arbitrary mental constructions. Hence nonfinitistic mathematics can be accused of being not science but merely a mental game played for the amusement of mathematicians. Proponents of predicativism and intuitionism have never tried to defend their respective doctrines against such accusations. Finitistic reductionism is an attempt to defend infinitistic mathematics by showing that at least some of it is more than a mental game and does correspond to something in reality. It is difficult to imagine how any such goal could be advanced by predicativistic or intuitionistic reductionism.
6.5. Are the possibilities of finitism really exhausted by PRA? Didn't Hilbert himself allow for an extended notion of finitism which would transcend the primitive recursive functions?
One might try to insist that certain multiply recursive functions such as the Ackermann function ought to be allowed as finitistic. However, Reverse Mathematics seems to indicate that such relatively minor changes would not significantly enlarge the class of finitistically reducible theorems. Hence the conclusions of §4 would remain essentially unaffected.
It is also true that Hilbert , p. 389, discussed a certain rather wide class of recursions of higher type. But Hilbert did not assert that such recursions are prima facie finitistic. Rather he presented them as part of his alleged proof of the Continuum Hypothesis, based on his incorrect belief that all of infinitistic mathematics is finitistically reducible. Certainly the recursions in question do not satisfy Hilbert's own criteria for finitism. (See also Tait , pp. 544-545.)
There are other possible objections to the identification of finitism with PRA. All such objections have been dealt with adequately by Tait .
6.6. The development of mathematics within Z2 or subsystems of Z2 involves a fairly heavy coding machinery. Doesn't this vitiate the claim of such subsystems to reflect mathematical practice?
It is true that the language of Z2 requires mathematical objects such as real numbers, continuous functions, complete separable metric spaces, etc. to be encoded as subsets of in a somewhat arbitrary way. (See [3,8,21,24].) However, this coding in subsystems of Z2 is not more arbitrary or burdensome than the coding which takes place when we develop mathematics within, say, ZFC. Besides, the coding machinery could be eliminated by passing to appropriate conservative extensions with special variables ranging over real numbers, etc. If this were done, the codes would appear only in the proofs of the conservation results. I do not believe that the coding issue has any important effect on the program of finitistic reductionism.
6.7. The systems and do not capture the full range of standard, infinitistic mathematics. Many well-known standard theorems cannot be proved at all in these systems. And even when a standard theorem is provable in or , the proof there is sometimes much more complicated than the standard proof. Doesn't this undercut the claim of and to embody a partial realization of Hilbert's Program?
Gödel's Theorem and Reverse Mathematics imply that many well-known standard mathematical theorems are not finitistically reducible at all. Therefore, the fact that these theorems are not provable in or does not disturb us in the least. It merely prevents our partial realization of Hilbert's Program from being a total one.
Somewhat more worrisome is the gap between standard proofs and proofs in or . However, this gap is certainly not wider than the one between Eulerian infinitesimal analysis and Weierstrassian - arguments. Moreover Hilbert explicitly embraced Weierstrass' reconstruction of analysis as a model for his own Program if not an integral part of it. ``Just as operations with the infinitely small were replaced by processes in the finite that have quite the same results and lead to quite the same elegant formal relations, so the modes of inference employing the infinite must be replaced generally by finite processes that have precisely the same results, that is, that permit us to carry out proofs along the same lines and to use the same methods of obtaining formulas and theorems.'' These words of Hilbert , p. 370, make me doubt that he would have been troubled by the above-mentioned gap.
6.8. The systems and do not seem to correspond to any coherent, sharply defined philosophical or mathematical doctrine. Aren't and mere ad hoc creations?
No, they are not mere ad hoc creations. The axioms of and embody compactness and Baire category respectively. These two principles are well known to be pervasive in infinitistic mathematics. (They are two different ways of affirming the existence of ``enough points'' in continuous media.) Moreover, the principal axiom of is known to be equivalent over to a number of key mathematical theorems. For instance, each of the Theorems 4.1 through 4.7, which were listed in §4 as being provable in , is in fact equivalent to over . These equivalences come from Reverse Mathematics and provide further evidence of the naturalness of .
Perhaps and do not correspond to any set of a priori ontological commitments such as might be proposed by philosophers unacquainted with the history and current state of mathematics. However, mathematics is entitled to define its own principles in accordance with its own needs, so long as these principles are compatible with the needs of the other sciences and with sound philosophy. This seems to leave room for systems such as and .
6.9. The claimed partial realization of Hilbert's Program is only patchwork. Infinitistic theorems are validated one at a time by laboriously reestablishing them within or or similar systems. Doesn't such a piecemeal procedure lack the ``once and for all'' grandeur of Hilbert's visionary proposal?
Let me say first that the work reported in §4 is much more systematic than it may appear from the outside. Many branches of infinitistic mathematics depend on a few key nonconstructive existence theorems. If these theorems or a reasonable substitute can be proved within , the rest follows routinely. Thus includes whole branches of mathematics and not only the theorems which were mentioned in §4 for illustrative purposes. It seems that most of the ``applicable'' or ``concrete'' branches of mathematics fall into this category. For example, the Artin-Schreier solution of Hilbert's 17th Problem can be carried out within . (See Friedman-Simpson-Smith .) I would estimate that at least 85% of existing mathematics can be formalized within or or stronger systems which are conservative over PRA with respect to sentences. Of course highly set-theoretical topics are excluded, but it is remarkable how many topics which at first may seem highly set-theoretical turn out not to be so. For instance, the Hahn-Banach Theorem for separable Banach spaces turns out to be provable in . (See Brown-Simpson .)
Having said this, I must admit that my plodding procedure lacks the grand sweep of Hilbert's plan. But to some extent this is inevitable in view of Gödel's Theorem. In any case, if there is a better procedure, I challenge the questioner to find it. Granted, it would be desirable to have a wholesale finitistic reduction of a large and easily identifiable part of infinitistic mathematics. But we do not know whether this is possible. In the meantime it seems desirable to establish finitistic reducibility for as much of infinitistic mathematics as we can. Moreover, the experience so gained may turn out to be useful in the larger task of validating infinitistic mathematics by methods not restricted to finitistic reductionism. It seems reasonable to hope that patience will pay off here.
6.10. The deduction of axioms from theorems seems like a very strange activity. Certainly such a wrong-headed enterprise would never have been tolerated by Hilbert. Can't we dismiss as mere propaganda the attempt to associate Reverse Mathematics with Hilbert's Program?
No, we cannot. It is true that the deduction of axioms from theorems is absent from Hilbert's formulation of his program. It is likewise absent from the final form of our results, discussed in §4 above, which constitute partial realizations of Hilbert's Program. However, Reverse Mathematics has played and will continue to play an important behind-the-scenes heuristic rôle in the discovery of such results. As explained and illustrated in §4, the interplay between ``forward mathematics'' and Reverse Mathematics leads to the discovery of formal systems such as and . That same interplay is essential to the ongoing process whereby we delimit the parts of mathematics that can be developed in such systems.
The fact that Hilbert's vision did not encompass Reverse Mathematics is of no consequence. Hilbert mistakenly thought that it would be possible to reduce all of infinitistic mathematics to finitism. Had he been right, there would have been no need to delimit the finitistically reducible parts of mathematics. Reverse Mathematics is instrumental in exploring the extent to which Hilbert's own Program can be carried out. For this reason I think that Hilbert would have recognized something of his own intention in the research which I have reported here.
6.11. What is the point of going on with Hilbert's Program once Gödel showed it to be impossible? Why not give up on finitistic reductionism and turn to some other method of validating infinitistic mathematics? For instance, why not appeal to Platonic intuition about the cumulative hierarchy? (This objection or one very much like it was raised at the symposium by Nick Goodman.)
The obituary for Hilbert's Program is premature to say the least. Gödel's Theorem rules out only the most thoroughgoing total realizations of Hilbert's Program. It does not rule out significant partial realizations. The results of §4 show that a substantial portion of the Program can in fact be carried out. (See also my answer to 6.9, above.) This is a remarkable vindication of Hilbert. It is also an embarrassing defeat for those who gleefully trumpeted Gödel's Theorem as the death knell of finitistic reductionism.
The need to defend the integrity of mathematics has not abated. On the contrary, Gödel's Theorem made this need more urgent than ever. Gödel supplied heavy artillery for all would-be assailants of mathematics. Authors such as Kline  cite Gödel with monotonous repetition and devastating effect. The assault rages as never before.
Platonic intuition is unsuitable as a weapon with which to defend the validity of mathematics. Only the first few levels of the cumulative hierarchy bear any resemblance to external reality. The rest are a huge extrapolation based on a crude model of abstract thought processes. Gödel himself comes close to admitting as much (, pp. 483-484). Arguing in favor of the cumulative hierarchy, Gödel (, pp. 477 and 485) proposes a validation in terms of testable number-theoretic consequences. Unfortunately such tests seem hard to carry out.
Finitistic reductionism is not the only plausible method by which to validate infinitistic mathematics. One might try to show that a substantial part of infinitistic mathematics is directly interpretable in the real world. Continuous real-world processes have not been sufficiently exploited. Aristotle's notion of potential infinity could be of value. Nevertheless, of all the possible approaches, the indirect one via finitism seems to be the most convincing.