Computation, Information, and Description

Department of Computer Science
The University of Iowa

Lecture Notes

Last modified: 12 March 1997

Hilbert's Program

David Hilbert was arguably the most ingenious mathematician of this century. He solved many difficult problems in particular branches of mathematics, and he also was concerned with the foundations of mathematics as a whole. The attempt to give all of mathematics a secure foundation in set theory had foundered on the ``paradoxes'' of set theory, which were actually presentations of an inherent self-contradiction in the assumptions of set theory. Those assumptions were so intuitively appealing that many great mathematicians, particularly Gottlob Frege, accepted them as a secure starting point from which to develop geometry, algebra, number theory, real analysis, and all other branches of mathematics. The discovery of a contradiction was rather scary to those who cared for the certainty of mathematical reasoning.

Hilbert proposed a program to fix this problem: ``I should like to eliminate once and for all the questions reagarding the foundations of mathematics, in the form in which they are now posed, by turning every mathematical proposition into a formula that can be concretely exhibited and strictly derived, thus recasting mathematical definitions and inferences in such a way that they are unshakable and yet provide an adequate picture of the whole science.'' Specifically, Hilbert's program has two parts:

  1. Provide a single formal system of computation capable of generating all of the true assertions of mathematics.
  2. Prove mathematically that this system is consistent, that is, that it contains no contradiction. This is essentially a proof of correctness.
Strictly speaking, conistency does not guarantee that a system is correct, just that it doesn't contradict itself. But, the detail of Hilbert's program and his argument in favor of it shows that he really intends to show correctness. Hilbert argues that only certain simple assertions are judged to be true or false based on their content. These simple assertions are essentially the equations between integer expressions with no variables, which can be checked by the arithmetic operations that we all learned in grade school. More complex assertions, particularly those expressed with variables, and asserting the existence of numbers with certain properties, or asserting that some property holds of all numbers, are ``ideal'' propositions, and there is no fixed standard of truth for them. Rather, a system for deriving ideal propositions is deemed correct if the numerical equations that it produces are true, and it is deemed useful if we find it practically helpful for organizing our understanding and manipulation of numerical equations. In the standard sorts of logical systems that Hilbert and others use for mathematics, consistency implies this sort of correctness, which is more properly called conservative extension.

Hilbert leaves the question of usefullness out of his theory, since it is a pragmatic question having to do with the applications of mathematics, rather than a mathematical question per se. But, he clearly believes that the most useful systems of mathematics may be incorporated on his foundation.

Hilbert's program founded a loosely defined school in the philosophy of mathematics, called formalism. Mathematicians of today seem to acknowledge formalism as the basis for their work, but they mostly seem not to understand it. Kurt Gödel showed that Hilbert's program is impossible. But, the clear statement of the program was an immense contribution to our understanding. As you read Hilbert's key papers on this topic, try to separate out the conceptual essentials from the particulars that were determined by mathematical practice and convention at the time. For example, Hilbert's ideas have nothing specifically to do with integer numerical mathematics vs. non-numerical mathematics. At the time, and to this day, the mathematics of numbers attracts a large share of attention. And, it is well known that integer numbers can represent all strictly finite mathematical objects. I will separate out the crucial from the accidental as much as I can in the notes.

Aside from a general understanding of Hilbert's ideas, please try to judge whether his papers support the following
Thesis: The first half of Hilbert's program, to provide a single absolutely rigorous formal computational foundation for all of mathematics, is essentially Descarte's program restricted to mathematics, and expressed more precisely. Hilbert goes further, and insists on a proof that the computational foundation is consistent.

Über das Unendliche
David Hilbert

This is the text of a lecture by Hilbert in Münster on 4 June 1925 at a meeting of the Westphalian Mathematical Society in honor of the memory of Weierstrass. I am using an English translation, ``On the infinite,'' by Stefan Bauer-Mengelberg in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, edited by Jean van Heijenoort, Harvard University Press, 1967. I refer to page number in the van Heijenoort book. I number paragraphs starting with the first paragraph to begin on a given page. A partial paragraph carried over from the previous page, I number as 0.

  1. Title. Hilbert wished to place all of mathematics on a completely reliable computational foundation. He realized that all finite objects are trivially computable. Computation is essentially a way of presenting the infinite number of values of a function in a precise way through a finite program. So, Hilbert expressed his problem as one of rigorously explaining the infinite.
  2. 369, par 1-2. The word ``arithmetic'' in this paper means the theory of real analysis. Today, it is often used to mean the theory of the integers, or even the calculation of sums, differences, products, and quotients on the integers. Real analysis was particularly important because it supported the differential and integral calculus and other key mathematical techniques used in the physical sciences. It was well known that rationals may be represented by pairs of integers, and that reals may be represented by infinite sequences of rational approximations. While Turing was concerned with the computability of individual real numbers, Hilbert was more concerned with the computable verification of correctness of proofs regarding the real numbers.
  3. 370 par 1. ``Modes of inference employing the infinite must be replaced generally by finite processes that have precisely the same results.'' That is, intuitive references to the infinite must be converted into objective computations.
  4. 370 par 2. Hilbert's (translated) words, but not quite in order: ``The aim of my theory is to endow mathematical method with definitive reliability.''
  5. 370-371. ``The definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences, but rather for the honor of the human understanding itself.'' This has nothing to do with my thesis, but it's a marvellous example of an emotional attitude toward mathematics.
  6. 371, par 2. Notice the word ``contentual.'' Hilbert uses it a lot in this paper. I am pretty sure that the German word is ``inhaltlich.'' This word is mentioned in Combinatory Logic by Haskell B. Curry, Robert Feys, William Craig, North-Holland (1958), but they don't specifically say that it comes from Hilbert. I am not sure whether this is a commonly used German word, or whether Hilbert cooked it up for his study. There is certainly no direct translation into commonly used English. Curry translated it as ``contensive.'' It is sometimes rendered as ``intuitive,'' but that seems to me very misleading. ``Contentual'' here means referring to the content of assertions or formulae. It is used as the opposite of ``formal,'' which means referring to the form of assertions. Most people, even many mathematicians, misunderstand the use of the word ``formal'' in logic, and think that its opposite is ``relaxed,'' or ``unrigorous,'' rather than ``contentual.'' The proper understanding of formal vs. contentual reasoning is crucial to our study, and to the understanding of mathematical logic. Formal reasoning is essentially computation. Contentual reasoning is essentially inspiration. Sometimes they can be translated into one another, but only formal judgements can be thoroughly objective and reliable by themselves.
  7. 371 par 3, through 372 par 0. This is off of my thesis again, but Hilbert states the incorrectness of Euclidean geometry as a description of physical space (at least according to today's physics) very nicely here. These paragraphs could be the end of a long trace of the impact of geometry on European thought, starting with Euclid and the ancient Greeks.
  8. 372 par 1. Hilbert contrasts ``specific numerical equations,'' which might as well have been 2+2=4 instead of his more complicated examples, with ``the solution of an arithmetic problem'' given by a formula with variables, which represents the ``infinitely many propositions'' resulting from substitutions for the variables. The language may mislead you here. He disparages the importance of ``specific numerical equations,'' since they ``can be verified by computation.'' The point is that each such equation can be verified by a single computation. Once we all look at that computation, there is no real problem agreeing objectively that the equation is true. On the other hand, formulas with variables stand for global properties of the computations of certain programs on all possible inputs. The crucial distinction is between verifying the correctness of a single computation, and discovering a property of all the infinitely many computations by a particular program.
  9. 372 par 2. This is the first use of the word ``ideal,'' but the concept was used without the name in 370 par 4, with the more familiar example of the imaginary numbers. Hilbert claims that mathematics consists of a very small core of contentual concepts, whose meanings are fixed in advance. Assertions about the core must be absolutely true. Other concepts are introduced to help us generate the true core assertions in a systematic and understandable way. These additional concepts are the ``ideal'' elements. Assertions about ideal elements do not have fixed meanings. There is no question of correctness regarding the ideal elements, only regarding the core assertions that they help generate. I think that essentially all mathematicians accept this sort of use of ideal elements. But Hilbert classifies only the specific numerical equations as core assertions, and leaves all other assertions in the world of ideals. In particular, he treats all assertions of existence, or of universal properties, as ideals. Many mathematicians, particularly intuitionists, insist on a larger class of core assertions.
  10. 374 bottom of page. The thingumajigs shown here with all the omegas are called the ordinals. They extend the positive integers into the infinite, using them for ordering rather than expressing size. A different system of infinites, called the cardinals, is used for size. The ordinals are not directly relevant to my thesis of the day. But, we will discuss them briefly later. In an interesting and paradoxical way, they represent the simplest known way to present complexity. So, take a look at them as you go by.
  11. 375 par 4-5. Hilbert's program could have been an entirely positive application of Descarte's longing for certainty to the whole scope of mathematics, where it seemed likely that certainty could be achieved once and for all. But, earlier attempts at a fixed foundation for all of mathematics led to the ``paradoxes,'' that is iherent contradictions, of naive set theory. This shook some of mathematicians' confidence that they knew precisely what they were talking about. Hilbert is trying to repair the damage, and create a security that will never again suffer such an attack. There were also inconsistencies in the early development of the differential calculus, associated with the treatment of infinitesimals as if they were finite quantities. I am less familiar with these problems, but Hilbert seems to regard them more or less on a par with the inconsistencies of naive set theory.
  12. 375 par 5. ``And where else would reliability and truth be found if even mathematical thinking fails?'' This sentence can be read as an emotional justification for pulling Descarte's program back to mathematics, as a sort of intellectual strategic retreat.
  13. 376 par 1. ``It is necessary to make inferences everywhere as reliable as they are in ordinary elementary number theory.'' Notice the rough analog between Descarte's desire to extend the certainty of geometry to all knowledge, and Hilbert's more modest desire to extend the certainty of number theory to all mathematics. I think that the substitution of number theory for geometry is not particularly significant, but just represents a different emphasis of mathematical interest in different centuries. It is ironic, though, that while the number theory of Hilbert's day seems as secure today as he thought it, we now know that it is incomplete, and has no secure completion.
  14. 376 par 3. Here Hilbert admits contentual judgements into his secure mathematics, but only ``when applied to real objects or events.'' He has already argued that real objects are all finite. I am a bit puzzled by the exact scope of ``real objects or events.'' I think that he does not mean all physically real objects, but just the ones that constitute mathematical assertions. Physical uncertainty appears to preclude objective certainty regarding quantum events. Toward the end of the paragraph, he requires that ``it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction.'' This sound a lot like Turing's justification of the primitive steps of the Turing machine, and it applies only to discrete symbols, not to arbitrary physical objects. The requirement that basic objects and their basic relations cannot ``be reduced to anything else'' makes no sense to me. Every object can be described in terms of smaller components. The crucial part is that we do not need such a reduction to make objective judgements. The last two sentences support my restriction to mathematical formulae, and not all physical objects: ``This is the basic philosophical position that I consider requisite of mathematics and, in general, for all scientific thinking, understanding, and communication. And in mathematics, in particular, what we consider is the concrete signs themselves, whose shape, according to the conception we have adopted, is immediately clear and recognizable.'' Notice that Hilbert refers to ``thinking, understanding, and communication,'' and in particular to ``concrete signs,'' but not to other sorts of physical objects, such as electrons. Hilbert doesn't use the word ``formal'' here, but the ``shape'' of signs, and the relations in which they are placed (``they follow each other, or are concatenated''), while they are contentual properties of the signs, they are formal properties of the assertions expressed with those signs.
  15. 377 par 0. ``It is possible to obtain in a purely intuitive and finitary way, just like the truths of number theory, those insights that guarantee the reliability of the mathematical apparatus.'' I think that the ``truths of number theory'' here are just the ``numerical equations'' mentioned earlier in the paragraph. In the first sentence of the paragraph, he referred to ``ordinary finitary number theory,'' and I believe that ``number theory'' in the later sentence is an abbreviation for ``ordinary finite number theory,'' that is numerical equations, rather than a reference to the theory of universal and existential assertions about integers. First, Hilbert is asking for other forms of mathematical assertion to be placed on a computational basis like numerical equations. Second, he is particularly asking for a proof of the consistency of mathematics to be done in this way.
  16. 377 par 3 through 378 par 3. A ``disjunction'' here is a finite sequence of alternatives, joined by ``or.'' Assuming that the individual alternatives are testable by computation (for example, they can be numerical equations), a disjunction can be tested for truth or falsehood by checking each of the alternatives. But, the obvious direct way to test an assertion of existence leads to an infinite computation. Because one may stop the computation as soon as a true alternative arises, but one never gets an answer when the existential assertion is false, Hilbert says that existentials are ``from the finitist point of view incapable of being negated.''
  17. 379 par 2. ``Let us remember that we are mathematicians and as such have already often found ourselves in a similar predicament.'' Another marvellous appeal to emotion. I find it moving. Hilbert doesn't make it explicit here, but he is touching on his faith that he and his colleagues can meet all possible mathematical challenges. Is this hubris? Will someone write a term paper analyzing Hilbert's program as a Greek tragedy, according to Aristotle's definition?
  18. 380 par 1. The repetition of ``a+b=b+a'' in two different fonts may puzzle you. In the first version, the germanic a and b are actually being used as variables, to indicate the contentual assertion that the infinitely many equations with numbers substituted for a and b are all true. Hilbert allows us to judge the truth of each individual numerical equation contentually, but not the universal truth of infinitely many of them. The second version, with the roman a and b, is understood formally instead of contentually, and an ideal assertion rather than a basic one from the core of mathematics.
  19. 381 par 0. Hilbert finally uses the f word: ``It is necessary to formalize the logical operations and also the mathematical proofs themselves.'' That is, he will express mathematical proofs as computations, that can be checked for correctness objectively and reliably by considering their formal symbolic structure, rather than the content expressed by each assertion in the proof. If we find an error in a proof, we do not detect the falsehood of an assertion, we merely detect a manipulation of symbols that does not follow the rules.
  20. 381 par 2 through 383 par 1. The details aren't important to us. Hilbert is giving some of the particular computation rules that he proposes for a universal foundation of mathematics. The important thing to us is that they are computation rules.
  21. 383 par 3-5. Now, Hilbert notes that an acceptable formal system of mathematics, although it may use ideal assertions freely and without a contentual standard of correctness, must generate only correct numerical equations. He calls this property ``consistency.'' There is some confusion here, due to accepting certain conventions as if they were incontrovertible. Strictly speaking, a logical system is consistent as long as it does not prove an assertion and also the negation of the same assertion. Because Hilbert's system of numerical equations is complete (that is, every true equation is verified), and because he admits inequalities along with equalities, an erroneous equation must provide an inconsistency. And, because the particular logical rules that everybody uses allow every formula to be proved in an inconsistent system, all possible errors can be converted to the form 1!=1. If he were more careful here, Hilbert would mention that in general a system of ideals need only be a conservative extension of the core contentual theory. That is, it must not provide proofs of core assertions unless they are contentually true. In the popular systems of logic today, theories are consistent precisely when they are conservative extensions of the numerical equations. But Hilbert, after arguing that we may define the ideals any way we like, should not have relied so implicitly on particular properties of his particular ideals when he expressed the requirements for a formal system.
  22. 383 par 4. ``A formalized proof, like a numeral, is a concrete and surveyable object. It can be communicated from beginning to end.'' That is, a proof can be judged objectively as a computation, just like a numerical equation.
  23. 384 par 1. ``Mathematics in a certain sense develops into a tribunal of arbitration, a supreme court that will decide questions of principle---and on such a concrete basis that universal agreement must be attainable and all assertions can be verified.'' This pretty well nails my thesis, I think.
  24. 384 par 3. ``Every mathematical problem can be solved.'' Here's the hubris again. To paraphrase, ``are we not mathematicians?'' I couldn't resist this recasting of Hilbert's phrase analogous to a quote from a piece of sensational literature, even though the comparison is grossly unfair to even the grossest of mathematicians. 1 week bragging rights to anyone who knows the quote that I refer to.
    Jason Boer pointed out a problematic sentence at the end of the same paragraph. I think that I need to quote the whole paragraph.
    As an example of the way in which fundamental questions can be treated I would like to choose the thesis that every mathematical problem can be solved. We are all convinced of that. After all, one of the things that attract us most when we apply ourselves to a mathematical problem is precisely that within us we always hear the call: here is the problem, search for the solution; you can find it by pur thought, for in mathematics there is no ignorabimus. Now, to be sure, my proof theory cannot specify a general method for solving every mathematical problem; that does not exist. But the demonstration that the assumption of the solvability of every mathematical problem is consistent falls entirely within the scope of our theory.
    In what precise sense can we not ``specify a general method for solving every mathematical problem,'' although ``every mathematical problem can be solved''? I am pretty sure that Hilbert intended a single formal system sufficient for all of mathematics; in 383 par 2 he says, ``... we are able ... to construct the system of provable formulas, that is, the science of mathematics.'' The definite article seems to indicate a single system. The first note for the ``Grundlagen'' paper also supports the requirement of a single formal system. What then?
  25. 384 par 3. ``The demonstration that the assumption of the solvability of every mathematical problem is consistent falls entirely within the scope of our theory.'' Ironically, Hilbert's method were used to show the contrary.
  26. 384 par 4 through 392 par 1. This is just mathematics, not related to our topic and you can skip it. It is Hilbert's attempt to prove an important assertion in set theory, called the ``continuum hypothesis.'' Today, the continuum hypothesis is known to be independent of the widely accepted facts about sets. One may consistently assume it to be true, or to be false. There seems to be no solid intuitive consensus that can settle it one way or the other.
  27. 392 par 2. Hilbert probably overemphasizes his reliance on contentual judgements regarding numerical equations, versus Frege's and Dedekind's attempts to derive mathematics from logic without content. I am not convinced that there is a real distinction in the work, as opposed to a distinction merely in the way that it is described. Hilbert's contentual judgements about numerical equations work precisely because they can also be treated as formal judgements: in some sense the content of numerical equations is a sort of form. In terms of foundational accomplishement, Frege's work led to a detected contradiction in the mathematical content. Hilbert's appears to have been all self consistent in the mathematics, but his metamathematical proposals were incorrect.

I think that you need to read this paper a few times, accumulating questions each time and trying to answer them on subsequent passes. There are some subtle shifts in the types of concepts being discussed. Perhaps you can avoid being rattled by these shifts if you consider that they are a lot like the shifts between different levels of abstraction in computing systems: machine, program, data, etc.

Pay particular attention to the content vs. form axis. The subtlety here is that the form of a mathematical assertion becomes part of the content of a metamathematical assertion. A mathematical assertion with variables, such as a+b=b+a, represents an infinite number of equations, and cannot be judged contentually as it stands. Instead, the form of the assertion, which is just a certain pattern of the signs ``a'', ``b'', ``+'', and ``='', is embedded into a larger formal structure, called a proof. The assertion that the proof is correct according to a set of computational rules can be judged contentually, just as a specific numerical equation (e.g. 2+2=4). The content of the assertion of correctness of the proof has to do with the form of the assertion a+b=b+a.

Recall that the problem of the infinite arose for Hilbert when mathematics went from assertions that can be judged each by a single computation, to assertions about a program P that carries out infinitely many computations on infinitely many arguments. Hilbert's method involves taking P as data to a verifier program. The verifier program tries to generate a proof of some universal property of all the computations of P. When a verifier succeeds, we now have a single computation, whose correctness can be judged objectively. We take this single computation to be a demonstration of some property of the infinitely many computations of good old P. But, the usefulness of this verification now depends on verifying the verifier. Hilbert thinks we can do that, too.

Hilbert did a marvellous job of advancing the precision with which we understand what is at stake in mathematical logic. His conclusions were essentially all wrong. Probably his immense success in other areas of mathematics made him extrapolate the success of his program in logic.

Die Grundlagen der Mathematik
David Hilbert

This is the text of another lecture by Hilbert in Hamburg in July 1927. I am using an English translation, ``The foundations of mathematics,'' by Stefan Bauer-Mengelberg and Dagfinn Follesdal from From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, edited by Jean van Heijenoort, Harvard University Press, 1967. I refer to page number in the van Heijenoort book. I number paragraphs starting with the first paragraph to begin on a given page. A partial paragraph carried over from the previous page, I number as 0.

  1. 464 par 1 of the paper. ``I should like to eliminate once and for all the questions reagarding the foundations of mathematics, in the form in which they are now posed, by turning every mathematical proposition into a formula that can be concretely exhibited and strictly derived, thus recasting mathematical definitions and inferences in such a way that they are unshakable and yet provide an adequate picture of the whole science.'' That's the first half of Hilbert's program, quite explicitly. I think that the second half is implicit in the word ``unshakable.''
  2. 464-465. This is verbatim repetition from ``On the infinite.''
  3. 465 par 2 through 467 par 2. These are just the particular computation rules for Hilbert's particular system. We care only that they are computation rules.
  4. 467 par 3. ``In my theory contentual inference is replaced by manipulation of signs according to rules; in this way the axiomatic method attains that reliability and perfection that it can and must reach if it is to become the basic instrument of all theoretical research.''
  5. 467 par 4 through 469 par 6. More detailed computation rules. Not important to us, but you might find the explicit use of recursion soothing, since it looks more like programming than the proof rules of the earlier segment.
  6. 469 par 7-8. Here is the contrast between contentual judgements regarding numerical equations (``elementary number theory'') treated as core assertions, and the ideal propositions of ``number theory'' using variables.
  7. 470 par 2. Hilbert calls the two aforementioned sorts of assertions ``real propositions'' vs. ``ideal objects.''
  8. 470 par 4 through 471 par 1. The focus on the law of the excluded middle is misleading here. This is the ``law'' that every proposition is either true, or its negation is true. This rule allows indirect proof. For example, if the assumption that every number has the property P leads to a contradiction, excluded middle allows the conclusion that there exists a number without the property P. Some mathematicians, called ``constructivists'' or ``intuitionists'' refuse to accept such indirect evidence for existence, and demand a direct construction of the number in question. The controversy regarding excluded middle is fascinating, but it has nothing to do with the viability of Hilbert's program. Precisely the same logical problems arise with or without excluded middle. This still hasn't sunk in to everybody's thinking.
  9. 471 par 3. Hilbert requires that the ideals form a conservative extension of the theory of the real propositions: ``the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain.'' He equates this with ``consistency'': ``no contradiction is thereby brought about.'' Conservative extension is the real requirement. For the conventional sorts of logic, including Hilbert's, it is equivalent to consistency. But, it doesn't have to be.
  10. 471 par 4-5. Here you see some of the properties of conventional logic that tie conservative extension to consistency. The only point of these paragraphs is to make it obvious to people who never programmed in Perl that consistency can be expressed in terms of a simple sort of symbolic pattern matching.
  11. 479 par 2. ``Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.'' Well, the tone sounds like hubris again, but this is actually a relatively modest claim, and largely substantiated by events. Mathematics may arguably be understood as the science of forms, which have an objective quality independent of individual beliefs. What is missing here is the claim that a single formal system may embrace all of mathematics at once, and that we may prove that it contains no error. Notice that Hilbert objects to the ``actual, contentual assumptions'' of Russell and Whitehead. But, recall that Hilbert claims to treat numerical equations contentually, and even seems to regard that as a virtue. It is not the mere contentual quality of Russell's and Whitehead's assumptions that Hilbert objects to, but the fact that each particular assumptions is not verifiable by a single computation or finite observation, and furthermore that there is not even a proof that they are consistent with basic numerical equations. Of course, Hilbert's proposed system never gets its consistency proof either. Oh well.