Did Poincare Anticipate Gödel?

Two of the most profound ideas of the 20th century were Einstein’s theory of relativity and Gödel’s incompleteness theorem. Much has been written about how Poincare anticipated many of the foundational concepts that formed the basis of special relativity, but his anticipation of some of the basic aspects of Gödel’s theorem seems to be less well known. Many of Poincare’s ideas on these subjects were presented in three popular books, Science and Hypothesis (1903), The Value of Science (1905), and Science and Method (1908). We know that the first of these was read avidly by Einstein and his friends Solovine and Habicht at meetings of their self-styled Olympia Academy. The book contains many of Poincare’s thoughts on what he called “relativity”, both as it relates to pure geometry and as it relates to the laws of mechanics. In addition to explaining why he believed that “some day the ether will be thrown aside as useless”, he also asserted that there is no absolute time, that our measures of time can acquire meaning only by convention, and that “we have no direct intuition of the simultaneity of two events occurring in two different places”. He referred back to his 1898 paper “The Measure of Time” (reprinted in The Value of Science in 1905), in which he explicitly described the operational meaning of simultaneity based on light signals together with the convention that the speed of light is taken to be invariant. This of course is reminiscent of Einstein’s moment of epiphany in the summer of 1905, as he recalled it in his Kyoto address:

My solution was really for the very concept of time, that is, that time is not absolutely defined but there is an inseparable connection between time and the signal velocity. With this conception the foregoing extraordinary difficulty could be thoroughly solved. Five weeks after my recognition of this, the present theory of special relativity was completed.

In a striking illustration of simultaneity, both Poincare and Einstein wrote papers in the summer of 1905 describing their respective views of the principle of relativity and its consequences and implications. All this has been thoroughly discussed in the scholarly literature on the origins of special relativity.

Less often do we find discussions of Poincare’s role as a precursor to Gödel in the critique of Hilbert’s attempt to give a finitistic proof of the consistency of arithmetic. It is usually said that Hilbert’s program was shown to be impossible by Gödel in 1931, but already in the book Science and Method we find that Poincare had argued that Hilbert’s program was inherently impossible. Of course, it’s well known that Poincare was an early champion of what came to be called intuitionism, and that he was opposed to both the logicism of Russell and the formalism of Hilbert, but the extent to which Poincare anticipated Gödel’s results (if not his methods) does not seem to have received much attention.

Recall that Gödel proved the impossibility of demonstrating the consistency of arithmetic within the formal system of arithmetic itself. He did this by first showing that all possible statements about arithmetic can be mapped to the integers (as Gödel numbers), which are the objects of arithmetic. Letting N_k(n) denote the natural number to which is mapped the kth propositional statement about the arbitrary natural number n, Gödel considered the proposition N_k(k), which is the kth proposition applied to the number k itself. Now let N_m(k) denote the proposition that “the proposition corresponding to N_k(k) cannot be proven within arithmetic”. Then we set k equal to m, so N_m(m) maps to the proposition that “the proposition N_m(m) cannot be proven”. Thus if N_m(m) cannot be proven, then N_m(m) is true (by definition), whereas if it can be proven then it is false. In the latter case, arithmetic is inconsistent, so we’re forced to conclude that if arithmetic is consistent then the proposition N_m(m) is both true and unprovable within arithmetic. This is called Gödel’s “incompleteness theorem”, because it implies that any consistent formal system (sufficient to encompass arithmetic) entails statements that are true but unprovable within that system. Gödel then went on to show that the proposition “If the consistency of arithmetic is provable then N_m(m) is provable” is a provable statement within arithmetic, from which it follows that the consistency of arithmetic is not provable within the formal system of arithmetic.

On the surface, Poincare’s critique of Hilbert’s program may not appear to resemble Gödel’s theorem, but it’s undeniable that Poincare’s conclusion was the same, i.e., it is not possible to prove, within the formal system of arithmetic, the consistency of arithmetic. This is to be found in the book Science and Method, wherein Poincare considered the prospects for proving the consistency of arithmetic – as formalized by Peano’s axioms – using finitistic reasoning. Not surprisingly he focused on the axiom of induction:

If a set S contains the natural number 1, and if for any number x in S the number x+1 is also in S, then S contains all the natural numbers.

How do we know this axiom (together with the others) never leads to a contradiction? Poincare argues that any purported proof of the consistency of arithmetic must ultimately rely on a complete induction, the consistency of which is the very thing we are trying to prove. Clearly we can’t simply assume the consistency of a formal system containing this axiom to prove that a formal system containing this axiom is consistent.

Hilbert’s initial efforts to formalize arithmetic grew out of his work on the foundations of geometry, in which he believed he had reduced the question of the consistency of geometry to the question of the consistency of arithmetic. Hence the natural next step was to prove the consistency of arithmetic itself. He outlined his ideas in this direction around 1905, and this work was the subject of Poincare’s critique. Hilbert later said that Poincare had been unable to accept his theory due to a strong prejudice against Cantor’s ideas, and that Poincare’s great authority had a large influence on many young mathematicians. Hilbert may have been referring to himself, because he seems to have given up on his attempts to formalize arithmetic at the time, and did not return to the project until about 1914 (coincidentally, just a year or two after Poincare’s death). By about 1922 he had developed the final version of his ideas on the subject. In a review lecture given in 1927 Hilbert explained how he believed Poincare’s criticism could now be answered:

Poincaré denied from the outset the possibility of a consistency proof for the arithmetic axioms, maintaining that the consistency of the method of mathematical induction could never be proved except through the inductive method itself. But, as my theory shows, two distinct methods that proceed recursively come into play when the foundations of arithmetic are established, namely, on the one hand, the intuitive construction of the integer as numeral, that, is, contentual induction, and, on the other hand, formal induction proper, which is based on the induction axiom and through which alone the mathematical variable can begin to play its role in the formal system. Poincaré arrives at his mistaken conviction by not distinguishing between these two methods of induction, which are of entirely different kinds.

Since Poincare was no longer living when this response to his criticism was formulated, we can’t know for sure how he would have reacted, but I’ll venture to say that he probably would not have concurred with this diagnosis of his “mistaken conviction”. Indeed  he seems to have addressed this very argument in his original critique. He wrote that even the recursive definition of the natural numbers (which is evidently what Hilbert now calls contentual induction) cannot be accepted as a formally established proposition (even as a postulate), because

To legitimately lay down a system of postulates, we must be assured that they are not contradictory… Does this mean that we must be sure of not meeting with contradiction after a finite number of propositions, the finite number being, by definition, that which possesses all the properties of a recurrent nature in such a way that if one of these properties were found wanting – if, for instance, we came upon a contradiction – we should agree to say that the number in question was not finite? In other words, do we mean that we must be sure of not meeting a contradiction, with this stipulation that we agree to stop just at the moment when we are on the point of meeting one? The mere statement of such a proposition is its sufficient condemnation. Thus not only does Mr. Hilbert's reasoning assume the principle of induction, but he assumes that this principle is given us, not as a simple definition, but as an a priori synthetic judgment.

Clearly Poincare did not accept even what Hilbert later named “contentual induction”, because in order to accept that the “content” has been legitimately established we must be assured that the existence of this “content” (e.g., the natural numbers) does not imply a contradiction, and this can only be established by the application of what Hilbert called “formal induction proper”. Poincare says the full concept of the natural numbers can’t be established by introducing them one at a time, building up the content of the set recursively; it requires the concept of complete induction. Perhaps Hilbert was trying to address this by drawing a distinction between the construction of a natural number and some subsequent step that enables it to “begin to play its role in the formal system”. This is reminiscent of Aristotle’s idea of a two-phase process by which things come into being, first as a generic existent thing with no properties, and then with the assignment of properties. Whether or not there is any validity in this idea, it’s unclear that it answers Poincare’s original critique.

Of course, following the appearance of Gödel’s theorem in 1930, it’s generally conceded that Hilbert was overly optimistic about the possibility of a finitistic proof of the consistency of arithmetic. Gödel’s incompleteness theorem was immediately seen to imply the corollary that no proof of the consistency of arithmetic was possible within the formal system of arithmetic itself. This doesn’t necessarily imply that a finitistic proof is impossible, partly because of the ambiguity as to the precise meaning of “finitistic”, which unfortunately Hilbert never seems to have explicitly defined. Some of his followers, notably Gentzen, continued to pursue Hilbert’s program, but only by availing themselves of certain elements, such as transfinite induction, that most people would regard as being outside the original scope of “finitistic methods”.

It’s notable that these later efforts relied on the assumption of something like transfinite induction, considering that it was precisely the inability to establish (in a finitistic and non-circular way) the axiom of induction that Poincare had identified as the obstacle to any finitistic proof of the consistency of arithmetic. From this point of view, one could argue that, contrary to how the story is usually told, Hilbert’s dream of a formalistic proof of the consistency of arithmetic had already been shown to be unattainable by Poincare soon after the idea was first proposed in 1905. Nothing that Hilbert did in the 1920’s invalidated Poincare’s fundamental critique. According to this view, the appearance of Gödel’s theorem in 1931 simply provided additional confirmation of what Poincare had already proven (by much simpler considerations) a quarter century earlier.

Admittedly, Gödel’s incompleteness theorem explicitly entails more than just the corollary about the impossibility of establishing the consistency of a formal system within the system itself. As it’s name suggests, the incompleteness theorem shows that any formal system is inherently incomplete in the sense that there exist propositions that are both  true and unprovable within the system. Poincare did not explicitly assert (let alone prove) this fact. On the other hand, it seems safe to say that Poincare accepted the consistency of arithmetic and first-order logic, and he certainly would have agreed that it could be consistent, and yet he asserted that the consistency could not be proven within the system itself. This is a particular example of incompleteness, and it isn’t difficult map the reasoning that led to this conclusion to any other formal system (with sufficient complexity to entail arithmetic). Hence, similar to how Poincare may be said to have anticipated many of the results of special relativity, despite having used methods that seem weaker than those associated with canonical special relativity, we might says he also anticipated many of the results of Gödel’s work on the foundations of mathematics, albeit by methods that seem less robust than Gödel’s.

An interesting irony here is the fact that Gödel’s argument  (which Poincare more or less anticipated) can be seen as an application of Cantor’s diagonal argument, in the sense that even if the undecidable proposition is added to the formal system as a new axiom, the augmented system is subject to the same reasoning, leading to yet another undecidable proposition. The irony is that Hilbert attributed Poincare’s rejection of formalism to his (Poincare’s) antipathy to Cantor’s style of reasoning, and yet the ultimate realization of Poincare’s critique in the form of Gödel’s theorem can be regarded as an application of one of Cantor’s central arguments.

On the other hand, to actually achieve a rigorous proof of the incompleteness theorem, Gödel had to delve more deeply into Richard’s paradox than Poincare had done. According to Poincare, the “true solution” to the antinomies was to be found in a letter of Jules Richard himself appearing in the Revue generale des sciences of June 30, 1905 (which is, coincidentally, also the date of Einstein’s submittal of his paper on the electrodynamics of moving bodies to the Annalen der Physik, about three weeks after Poincare had submitted his paper on the same subject to the French Academy of Science). Recall that Richard had constructed a paradox using Cantor’s own diagonal argument as the mechanism. He first asks us to consider the natural numbers that are definable in a finite number of words. The set consisting of these numbers he calls E, and he notes that this set is denumerable. He then assigns a number (from one to infinity) to each of the numbers in this set. Then, following Cantors diagonal construction, he points out that we can construct a natural number N whose nth decimal digit is one greater than the corresponding digit of the nth number in the set E (except we map both the digits 8 and 9 to 1, to avoid duplication of infinitely repeating 9s). Hence N differs from every element of E, and yet we have defined N in a finite number of words. Richard’s explanation, endorsed by Poincare, is that the set E is actually defined as the set of all natural numbers definable in a finite number of words without referring to the set E. We cannot legitimately refer to the set E in the definition of the set E, since this would be circular. Therefore, although N has been defined in a finite number of words, it is not properly an element of E because its definition refers to E.

This remains the accepted resolution of Richard’s paradox, and of the other antinomies, which are blocked by forbidding non-predicative definitions. (Poincare says “the definitions that are to be regarded as non-predicative are those which contain a vicious circle”.) However, as can be seen by reviewing the outline of Gödel’s proof of the incompleteness theorem presented above, Gödel’s argument is patterned very closely on Richard’s paradox, but it avoids any reliance on non-predicative definitions. This is done by establishing a mapping between statements about arithmetic to statements of arithmetic. (This mapping is reminiscent of the axiom of reduction in Russell’s theory of types, by which numbers of many different “types” are mapped to entities of the same type, but the axiom of reduction is seemingly more ambitious, as it asserts that this reduction is accomplished for infinitely many types.)

Had Poincare not died in 1912 following an operation for appendicitis at the age of 58, he would have been only 62 when Einstein’s general theory of relativity appeared in 1916, and 72 when quantum mechanics was founded in 1926, and had he lived to the age of 77 he could have seen Gödel’s incompleteness theorem. It would have been fascinating to hear his views of these developments. He was undeniably quite conservative in his outlook, and this conservatism seems to have been facilitated by his conventionalism, which allowed him to acknowledge the possibility of different ways of looking at things, while at the same time denying that the new ways were necessarily better. In Science and Method he included sections on both “The New Logics” and “The New Mechanics”, each containing lucid accounts of many of the (then) new ideas in those fields. It’s tempting to interpret these accounts, taken out of context, as endorsements of the new conceptions, but Poincare almost always concludes with an expression of his conservative instincts, suggesting that it’s all really just a matter of convention, and we may as well continue with our old views. For example, he ends his discussion of the “new mechanics” by saying

Suppose that in a few years these theories are subjected to new tests and come out triumphant. Our secondary education will then run a great risk. Some teachers will no doubt wish to make room for the new theories. Novelties are so attractive, and it is so hard not to appear sufficiently advanced! The children will be warned that ordinary Mechanics has had its day, and that at most it was only good for such an old fogey as Laplace… Is it good to warn them that [the old mechanics] is only approximate? Certainly, but not till later on; when they are steeped to the marrow in the old laws and are no longer in danger of unlearning them. Then they may safely be shown their limitations. It is with the ordinary mechanics that they have to live; it is the only kind they will ever have to apply. Whatever be the progress of motoring, our cars will never attain the velocities at which its laws cease to be true. The other is only a luxury, and we must not think of luxury until there is no longer any risk of its being detrimental to what is necessary.

Likewise at the end of another essay on “the new mechanics” published posthumously in 1913, he wrote:

What shall be our position in view of these new conceptions? Shall we be obliged to modify our conclusions? Certainly not; we had adopted a convention because it seemed convenient and we had said that nothing could constrain us to abandon it. Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one in order not to disturb their old habits, I believe, just between us, that this is what they shall do for a long time to come.

Poincare’s last collection of essays gives us some hint as to how he would have reacted to things like quantum mechanics. It includes an essay on the quantum theory at the time of the 1911 Solvay conference, which he attended along with Lorentz, Curie, Einstein, and many of the other notable physicists of the day. The essay seems to corroborate Einstein’s appraisal of Poincare’s contribution to the conference (which Einstein referred to as a witch’s sabbath). In a letter to his friend Henrich Zanngar, Einstein wrote that “Poincare was simply negative in general, and, all his acumen notwithstanding, he showed little grasp of the situation”. Indeed we find that Poincare’s essay is focused on what he saw as difficulties for the quantum theory, although, to be fair, this was his way of approaching most new subjects, and his criticisms were often instrumental in advancing a subject (as shown by his immensely fruitful interaction with Lorentz). It’s also interesting the Poincare already in 1912 seems to have identified the fact that photons (quanta of energy) must not necessarily be conserved. He wrote

Electrons maintain their individuality throughout the most diverse vicissitudes. Is this equally true of the so-called atoms of energy? There are, for instance, 3 quanta of energy on a resonator whose wave length is 3; this energy passes to a second resonator whose wave length is 5. Therefore, it no longer represents 3 but 5 quanta, since the quantum of the new resonator is smaller; and since in the transformation the number of atoms and the size of each of them have changed. This is why the theory does not yet satisfy the mind.

In the same collection of posthumous essays there appears one entitled “The Relations Between Matter and Ether”. Aside from the title, the essay contains virtually no mention of the ether. (The word appears only twice, in passing.) Instead it consists of a survey of the latest theories. At the end he wrote

I shall leave this quandary, and I shall conclude with the following reflection. As science progresses, it becomes more and more difficult to make room for a new fact which does not fit in naturally. The older theories rest on a large number of numerical coincidences which cannot be attributed to chance. We cannot therefore put asunder that which they have joined together; we can no longer destroy the framework, we must try to "bend" it. And it does not always lend itself to this. The principle of equipartition of energy explained so many facts that it must contain some truth; on the other hand, it is not entirely true since it does not explain all the facts. We can neither discard it nor retain it without modification, and the modifications which seem imperative are so strange that we hesitate to accept them. In the present state of science, we can only admit these difficulties without resolving them.

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