In the first part of this post, I described how science put Auguste Comte to shame when he postulated the impossibility of knowing the chemical composition of the stars. To compensate for this optimistic view that science can do everything, I will here consign another elegant ridicule of a scientific prediction, in this case the fruit of blind optimism. Its protagonist is the great mathematician David Hilbert (1862-1943).
Unlike Comte, Hilbert had an almost blind faith in the human capacity to know through rational analysis. In 1930, Könisberg, the town where Hilbert was born 68 years earlier, made him an honorary citizen on the occasion of his retirement. Hilbert gave a moving speech in which he defended the idea that there are no unsolvable problems in science: “In an effort to give an example of an unsolvable problem,” Hilbert asserted, “the philosopher Comte once said that science would never discover the secret of the chemical composition of the bodies of the universe. A few years later the problem was solved. In my view, the real reason why Comte could not find an unsolvable problem is that such a thing does not exist”. And he added: “Wir müssen wissen, wir werden wissen”, that is, “We must know, we will know”. Then Hilbert went to the local radio station, where he was to repeat the final part of his lecture for the whole town. In front of the microphone in the solitude of the studio, but with the same intensity as he had used a few moments earlier in the warmth of those gathered in the conference room, Hilbert again recited the last sentence of his speech: “We must know, we will know”. And then he smiled. According to Constance Reid, Hilbert’s biographer: “If you listen very carefully you can hear Hilbert’s laughter at the end”. A recording of his words is still preserved and can be listened to here (I must admit that, however much I have listened to it, I do not hear Hilbert’s laughter at the end; perhaps the reader will be luckier).
Faced with the paradoxes that allegedly affected the foundations of mathematics at the turn of the 19th to the 20th century, Hilbert had proposed an ambitious programme that sought to establish systems of axioms for each area of mathematics, from which any true property in that area could be deduced by rigorous proof. This project was called the formalist programme. With his statement, “We must know, we shall know”, Hilbert showed absolute confidence in the success of his formalist programme; few sentences in mathematics have been so inopportune.
Hilbert uttered his sentence on 8 September 1930 in Königsberg. During the three previous days, a conference on the epistemology of the exact sciences had been held precisely in Königsberg; on the 6th – at three o’clock in the afternoon, to be precise – the young Austrian logician Kurt Gödel – 24 years old – had spoken for twenty minutes to communicate a couple of results he had obtained in the previous months, and the following day at a round table he stated: “I can give examples of true but unprovable arithmetical propositions in the formal system of classical mathematics”. Despite its importance, the statement went unnoticed by almost everyone present, with the exception of John von Neumann. Von Neumann was very impressed with what the twenty-four-year-old said he could do; from then on he was one of his most ardent admirers. In 1931, Gödel published his incompleteness results under the title: On Formally Undecidable Sentences of Principia Mathematica and Related Systems. It is undoubtedly the most famous paper ever published on mathematical logic.
In the first place, Gödel showed that axiomatised formal systems, provided they are sufficiently interesting, such as Russell and Whitehead’s Principia Mathematica, or those of set theory, are necessarily incomplete, that is, there will be formulae which, being true properties in the domains that those axioms formalise, will not be theorems of that formal system. In other words, starting from a set of axioms, syntactic correctness will not allow us to reach all the possible truths. And without the possibility of a fix: the problem cannot be solved by adding more axioms, because, no matter how many are added, there will still be properties of the formal system that are undemonstrable truths. In the second place, if the system is consistent – it does not give rise to contradictions – that will be precisely one of those indemonstrable truths.
What Gödel went to Königsberg to explain meant the impossibility of completing Hilbert’s formalist programme and struck a death blow to that beautiful sentence, “We must know, we shall know”, which Hilbert uttered in the same city and on almost the same day.
References
A.J. Durán, Pasiones, piojos, dioses… y matemáticas, Destino, Barcelona, 2009.
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