Abstract
Science constructs tools for knowledge and, occasionally, this bold enterprise may let a few believe in the “completeness” of a given theoretical frames: as for these phenomena, we can predict, derive, compute ... everything. Yet, negative results, often based on the very tools proposed by the scientific approaches, set the limits to knowledge construction and opened the way to new science. Scientism instead assume to increasingly and completely occupy reality by pre-given scientific tools. Early positivism, with Laplace, expected to obtain the predictability of classical dynamics, of the Solar system in particular, from their explicit determination by suitable sets of equations. Poincaré’s negative answer set the limits of this hypothesis of complete deducibility of “all astronomical facts” by the equational approach to classical mechanics. Less than one century later, Hilbert, by his novel meta-mathematical foundation of mathematics, hoped to completely and consistently derive all mathematical properties by formal deduction. Gödel disproved this conjecture by tools that are internal to the formalist approach, similarly as Poincaré had disproved Laplace’s dream by a formal analysis of the equations. Also Einstein worked at the possible incompleteness of Quantum Mechanics, from a relativistic perspective. Finally, we will then address the supposed completeness of molecular descriptions in Biology, that is, of DNA seen as the locus of hereditary information and as the complete program, the “blue print”, of ontogenesis.
A preliminary version of this paper appeared as “Incompletezza”, in C. Bartocci and P. Odifreddi (Eds.), La Matematica, vol. 4, (pp. 219–262), Einaudi, Torino, 2010.
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- 1.
Poincaré “sees” the geometry and the complexity of chaos, without even drawing it:
To represent the figure formed by these two curves [the stable and unstable periodic “trajectories”] and their infinitely numerous intersections [homocline points], these intersections form a sort of lattice, of fabric, a sort of network made up of infinitely tight meshes; each of these curves must never intersect itself, but must fold upon itself in a very complex manner so as to intersect an infinite number of times with all other meshes in the network. One is struck by the complexity of this figure, which I will not even attempt to draw. Nothing is more apt for giving an idea of the complexity of the three-body problem and in general of all problems of dynamics where there is no uniform integral (Poincaré 1892).
- 2.
A pendulum can be conceived as a bar connected to a pivot. If we attach another bar to the bottom of this first bar, one that is also free to rotate, what we have is a double pendulum. A recent and amusing theorem (Béguin 2006) demonstrated the following: if we choose a sequence of integer numbers a 1, a 2, a 3, ... , we can put the double pendulum in an initial position so that the second limb will make at least a 1 clockwise turns, and then change direction to make at least a 2 counterclockwise turns, and then at least a 3 clockwise turns, etc. If we choose a random sequence a 1, a 2, a 3, ... (see Sect. 6), this purely mathematical result makes chaos and unpredictability “understandable” (but does not demonstrate it) in one of the simplest deterministic systems possible. We can also see this by observing an actual physical double pendulum or a computer simulation (such simulations can be found on the Web, we will return to this).
- 3.
It is interesting to compare explicit theorization and geometric analysis of the unpredictability of a Newtonian system in Poincaré (1892) and the references in Lighthill (1986), (as well as the title: “The recently recognized failure of predictability in Newtonian dynamics” ... “recently”?). Indeed, two schools particularly distinguished themselves in the twentieth century regarding the theory of dynamic systems: the French school (Hadamard, Leray, Lévy, Ruelle, Yoccoz ... ) and the Russian school (Lyapunov, Pontryagin, Landau, Kolmogorov, Arnold ... ). To these, we must add, namely, the Americans Birkhoff and Lorentz. But works on the subject by Hadamard, Lyapunov and Birkhoff have long remained isolated and seldom quoted. Up until the results by Kolmogorov and Lorentz in the 1950s and 1960s, and well beyond, Classical Rational Mechanics—of which Lighthill presided the international association in 1986—has dominated the mathematical analysis of physical dynamics (as well as the way the subject matter was taught to the author of this article, prior to such apologies, alas).
- 4.
Ruelle and Takens (1971a,b) faced many difficulties for getting published. As has been said above and as will be said again as for genocentric approaches in Biology, the Laplacian mentality (but Laplace, two centuries ago, was a great mathematician) is still present in many minds of today, although outside of the sphere of Mathematical Physics. And, in general, “negative results” are the most difficult to accept. Thus, they are the most difficult ones to finance, even if they are most likely to open up new horizons. And this is exactly what the institutional administrators of research steered towards positive projects and towards patents will succeed in hindering even more definitively, bolstered by their bibliometric indices: the role of critical thinking and of intrinsically “anti-orthodox” innovation that are characteristic of scientific research, see MSCS Ed. Board (2009) and Longo (2014, 2018b).
- 5.
Besides those already mentioned, there are numerous other highly important “negative results”, particularly in Physics (not to mention that in Mathematics, by a skillful use of double negations, any result can be presented as “negative”). The results of which it is question here are among those which contradicted major projects of knowledge, or theories that marked the history of science and which sometimes continue to guide common sense. They are also results that are linked to the negation of an assumed completeness (in its various forms) of these theoretical propositions.
- 6.
We can, for example, recall the first formalization of one of these fundamental systems of computability, Church’s untyped lambda-calculus (1932). The ensuing “Curry paradox”, an antinomy similar to that of the barber’s, will first entail a refinement of the calculus (1936) and the invention of another one, with “types” (1940). The first formal system of types by Martin-Löf will also be contradictory (1970). The formalizations, which loose “meaning” along the way (we will return to this), easily tend to produce contradictions: “Logic is not sterile”, said Poincaré (1906), “it has created contradictions”. Given the innovations they brought and the responses they were quick to receive, it must be noted that these formal theories, in spite of these errors of syntax due to the lack of a mathematical interpretation, were at the origin of very interesting ideas and systems and not of a major crisis, except among logicists (Longo 1996).
- 7.
The properties of 0 and of the successor symbol (0 is not a successor and the successor operation is bijective) and, especially, of induction: suppose A(0) and that from A(n) one is able to deduce A(n + 1), then deduce A for all integers, i.e. A(m) for all m, see (Gödel 1986–2003) for details.
- 8.
In Gödel’s proof, point (3) requires a hypothesis only slightly stronger than consistency: ω-consistency. This is a technical, yet very reasonable—and natural—hypothesis: the natural numbers are a model of PA—it would be “unnatural” to assume less. This hypothesis was later weakened to consistency, see (Smorynski 1977).
- 9.
There has been some debate on the actual meaning of Cons: does “Cons” really expresses consistency? Piazza and Pulcini (2016) rigorously confirm the soundness of the approach we informally followed here and clarify the issue by a close proof-theoretic analysis.
- 10.
Piazza and Pulcini (2016) prove the truth of Cons in the natural or standard model of PA, by applying Herbrand’s notion of “prototype proof”—a proof of a “for all” statement, by using a “generic” element of the intended domain, instead of induction. This is a key notion also for the analysis of true and interesting (non-diagonal, like G) but unprovable propositions of PA, see below and Longo (2011). Formal induction is not the bottom line of the foundation of mathematics, even not for the (meta-)theory of Arithmetic.
- 11.
In the ontological search for an unprovable mathematical truth, sometimes the “fact” that G must either be true or false is used. Or—this amounting to the same thing—that either G or ¬G must be true, without saying which because it would be necessary to prove it. This “weak ontology of truth” comes from a classical and legitimate hypothesis (the excluded middle) but one which is very strong and semantically unsatisfactory (or highly distasteful—and this is important in Mathematics) when it is question of discussing the truth of assertion G: this is a formal rewriting of the liar paradox which is precisely neither true nor false. Gödel also uses the excluded middle (G is independent from classical PA) but precisely to give us, in Proof Theory, the “middle”: the undecidable.
- 12.
Let’s recall for the reader who may be somewhat numbed by this wonderful pun that the question resides in the difference between the meta-theoretical hypothesis of consistency and Cons, the theoretical hypothesis of consistency which encodes consistency in PA. When this extra assumption is added to PA, then, we insist, Gödel could formally derive G, within PA, thus its truth in the standard model, which realizes Cons, if PA is consistent (see the work in Piazza and Pulcini (2016) and the previous notes as for the meaning of Cons).
- 13.
We can mention Gentzen’s ordinal analysis (1935). Larger infinities, as orders beyond integers or as cardinals beyond the countable, provide tools for the analysis of proof in order to fill the incompleteness of PA—or to postpone it to stronger theories. Set Theory, with an axiom of infinity, in its formal version (ZF or NBG) extends and proves consistency of PA, but it does not prove its own consistency—it is incomplete, of course, nor is able to answer the questions for which it was created: the validity of the axiom of choice and of the continuum hypothesis. The respective independence results cast additional light on the expressivity and on the limits of formal systems (Kunnen 1980).
- 14.
Gödel moved there permanently in 1939, after a spectacular escape from Nazi-occupied Austria.
- 15.
Deduction in EPR may remind of another, from Aristotle:
the void is impossible, because in it, all objects would fall at the same speed (La Physique, vol. 4, chap. 8).
Great theoretical minds, even when they are mistaken, propose very interesting ideas indeed.
- 16.
Husserl (1933):
Original certainty can not be confused with the certainty of axioms, because axioms are already the result of the formation of meaning and always have such formation of meaning as a backdrop.
- 17.
For technical details regarding order and symmetries in the demonstrations we refer to, see Longo (2011).
- 18.
The date at which Husserl’s manuscript was written reminds us that almost all of the story we have told took place during the first and dramatic half of the twentieth century, 1933 being a pivotal year, with the rise of Nazism and the flight from Germany of so many people we have met in these pages. During that year, Husserl, who was 74 years old at the time, was prohibited from publishing and even from accessing the University Library. And this frequent appearance of some illustrious names reminds of another important/small academic/political story. In 1923, Einstein, having recently been awarded the Nobel prize, thought about returning to Italy, maybe for a long period, after a short stay in Bologna. He had a very good knowledge of the results by Levi-Civita and was in contact with several colleagues, among whom Volterra and Enriques. The latter, in the previous years had become familiar in dealing with the governments, managed to obtain a meeting with the new prime minister, formally not yet dictator, Benito Mussolini: he hoped to obtain exceptional financing for the guest. This was in early 1924. The Duce’s response was: “Italy has no need for foreign geniuses” — this reminds by contraposition of the great Princes of the Renaissance or of Princeton in the 1930s (and afterwards). And so Einstein did not return to Italy. In 1929, Marconi added to a list of his colleagues drafted for Mussolini a little e. (for Jew — “ebreo” in Italian) in front of the names of the three aforementioned Italian mathematicians, the greatest of their time. The Duce, nine years prior to his racial laws, excluded them from the Academy of Italy (Faracovi et al. 1998).
- 19.
In 1999, Collins was pleased to stress the difference between humans and Caenorhabditis Elegans, a one millimeter worm with less than 1000 cells, whose DNA had just been decoded: it has only 19,000 genes!
- 20.
Jacob (1965):
The surprise is that genetic specificity is written not with ideograms, like in Chinese, but with an alphabet.
In this perspective, see Jacob (1974) for more, also the philosophy of biology, reduced to Molecular Biology, is transformed in an annex of a philosophy of (alphabetic) language, cf. Sect. 5 above. As a matter of fact, Molecular Biology deals with information, programs, expressions, signals ... since “life is fully coded” in chromosomes, following Schrödinger (1944), thus in discrete sequences of meaningless signs, as theorized also by Maynard-Smith (1999), Gouyon et al. (2002) and many others.
- 21.
The empirical evidence on the incompleteness of the genocentric approach as for a dramatic phenotype, cancer, and some general consequences on the understanding of causality, in particular on the etiology of that disease, are discussed in Longo (2018a).
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Appendix: On Gödelitis in Biology
Appendix: On Gödelitis in Biology
In an attempt to bypass the mechanistic-formal approach, enriched by some noise, Danchin (2003, 2009) tried to bring Gödel’s theorem into the genocentric view of biology. Within the formal-linguistic approach to biology, Gödel’s incompleteness would prove the “creativity” of biological dynamics by recursion and diagonalizing on the programs for life: in short, the DNA would generate unpredictable novelty by a creative encoding of phenotypes, a la Gödel. A remarkable attempt for a leading biologist, as these issues in Logic are far from common sense, as we hinted above.
Indeed, (Rogers 1967), a classic in Computability Theory, calls “creative” the set of (encoded) theorems of arithmetic, i.e. the formal-mechanical consequences of its axioms. As we know, by Gödel’s first theorem, this set is not computable ( not decidable)—and, to the biologist, its evocative name may recall Bergson’s Creative Evolution. However, this set is semi-computable (semi-decidable), meaning that it may be effectively generated and, as such, is far from “unpredictable”, since an algorithm produces all and exactly all its infinite elements—the set of encoded theorems. Moreover, the generation of Gödel’s undecidable formula is effective as well: it is an incredibly smart recursive and “diagonal” construction (it recursively uses the encoding of logical negation), as we have seen, which allows to construct a formula not derivable from the axioms. This procedure may be indefinitely and effectively iterated.
In short, Gödel’s undecidabile sentence is effectively produced by an effective encoding of the metatheory into the formal theory and it does not finitely “create” any “unpredictable” information: the diagonal formula may be constructed, even though it is not derivable from the axioms. In summary, on one side, formal derivability is not decidability (Gödel’s first theorem), as the “information” in the axioms does not allow to decide all formulae, typically Gödel’s diagonal formula. Yet it still yields semi-computability or semi-decidability: the theorems can be effectively generated, by passing through the encoded metatheory (what would be the metatheory in evolutionary biology?). On the other side, the construction of the sentence that escapes the given axioms is also effective (semi-computable), as we have seen.
Theoretical unpredictability, instead, that is the least property one expects for “creativity” in nature, is at least (algorithmic) randomness, for infinite sequences (Sect. 6). This yields a very strong form of incomputability, far from semi-computa-bility. As observed in Sect. 6, a random set of numbers and its complement cannot even contain an infinite semi-computable subset. This form of randomness may be soundly compared, asymptotically, to unpredictability in physics, as we observed (note that biological unpredictability includes both classical and quantum randomness (Buiatti and Longo 2013; Calude and Longo 2016a)).
We also observed that finite incompressibility does not soundly relate to randomness in nature: an incompressible sequence may be programmable—by a program of its length or just one bit longer; moreover, there are no sufficiently long incompressible sequences, Calude and Longo (2016b)—except by a restriction on the allowed machines, a la Chaitin (Calude 2002).
In summary, physical/biological randomness is unpredictability relative to the intended theory (Calude and Longo 2016a), and a time related issue: it concerns the future and is associated to time irreversibility (Longo and Montévil 2014: chap. 7). It relates only asymptotically to algorithmic randomness; it is necessary, but insufficient, for analysing evolutionary changes. It goes well beyond Gödel’s constructive diagonal craftiness.
The merit of Danchin’s remarks, though, is that they are based on precise mathematical notions, thus they may be proved to be wrong. This is in contrast to the commonsensical abuses of vague notions of information and program, as mostly used in Molecular Biology, from which strong consequences have been too often derived, see Longo (2018a).
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Longo, G. (2019). Interfaces of Incompleteness. In: Minati, G., Abram, M., Pessa, E. (eds) Systemics of Incompleteness and Quasi-Systems. Contemporary Systems Thinking. Springer, Cham. https://doi.org/10.1007/978-3-030-15277-2_1
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