I’ve been reading Rebecca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Gödel which, together with my finding David Mumford’s Why I am a Platonist, has kept me a bit more preoccupied, of late, with Platonism. This is not an entirely new preoccupation. I remember one of my early philosophy teacher’s periodically blurting out, “See, Plato was right!” And I hate to admit that I often wondered, “About what?” But Plato’s idea has become a more pressing issue for me now. It inevitably touches on epistemology, questions about the nature of the mind, as well as the nature of our physical reality. Gödel’s commitment to a Platonic view is particularly striking to me because of how determined he appears to have been about it. Plato, incompleteness, objectivity, they all came to mind again when I saw a recent article in Nature by Davide Castelvcchi on a new connection between Gödel’s incompleteness theorems and unsolvable calculations in quantum physics.
Gödel intended for his proof of mathematics’ incompleteness to serve the notion of objectivity, but it was an objectivity diametrically opposed to the objectivity of the positivists, whose philosophy was gaining considerable momentum at the time. Goldstein quotes from a letter Gödel wrote in 1971 criticizing the direction that positivist thought took.
Some reductionism is correct, [but one should] reduce to (other) concepts and truths, not to sense perceptions….Platonic ideas are what things are to be reduced to.
The objectivity preserved when we restrict a discussion to sense perceptions rests primarily on the fact that we believe that we are seeing things ‘out there,’ that we are seeing objects that we can identify. This in itself is often challenged by the observation that what we see ‘out there’ is completely determined by the body’s cognitive actions. Quantum physics, however, has challenged our claims to objectivity for other reasons, like the wave/particle behavior of light and our inability to measure a particles position and momentum simultaneously. While there is little disagreement about the fact that mathematics manages to straddle the worlds of both thought and material, uniquely and successfully, not much has advanced about why. I often consider that Plato’s view of mathematics has something to say about this.
What impresses me at the moment is that Gödel’s Platonist view and the implications of his Incompleteness Theorems, are important to an epistemological discussion of mathematics, . Gregory Chaitin shares Gödel’s optimistic view of incompleteness, recognizing it as a confirmation of mathematics’ infinite creativity. And in Chaitin’s work on metabiology, described in his book Proving Darwin, mathematical creativity parallels biological creativity – the persistent creative action of evolution. In an introduction to a course on metabiology, Chaitin writes the following:
in my opinion the ultimate historical perspective on the significance of incompleteness may be thatGödel opens the door from mathematics to biology.
Gödel’s work concerned formal systems – abstract, symbolic organizations of terms and the relationships among them. Such a system is complete if for, everything that can be stated in the language of the system, either the statement or its negation can be proved within the system. In 1931, Gödel established that, in mathematics, this is not possible. What he proved is that given any consistent axiomatic theory, developed enough to enable the proof of arithmetic propositions, it is possible to construct a proposition that can be neither proved nor disproved by the given axioms. He proved further that no such system can prove its own consistency (i.e. that it is free of contradiction). For Gödel, this strongly supports the idea that the mathematics thus far understood explores a mathematical reality that exceeds what we know of it. Again from Goldstein:
Gödel was able to twist the intelligence-mortifying material of paradox into a proof that leads us to deep insights into the nature of truth, and knowledge, and certainty. According to Gödel’s own Plantonist understanding of his proof, it shows us that our minds, in knowing mathematics, are escaping the limitations of man-made systems, grasping the independent truths of abstract reality.
In 1936, Alan Turing replaced Gödel’s arithmetic-based formal system with carefully described hypothetical devices. These devices would be able to perform any mathematical computation that could be represented as an algorithm. Turing then proved that there exist problems that cannot be effectively computed by such a device (now known as a ‘Turing machine’) and that it was not possible to devise a Turing machine program that could determine, within a finite time, if the machine would produce an output given some arbitrary input (the halting problem).
The paper that Castelvecchi discusses is one which brings Gödel’s theorems to quantum mechanics:
In 1931, Austrian-born mathematician Kurt Gödel shook the academic world when he announced that some statements are ‘undecidable’, meaning that it is impossible to prove them either true or false. Three researchers have now found that the same principle makes it impossible to calculate an important property of a material — the gaps between the lowest energy levels of its electrons — from an idealized model of its atoms.
Here, our ability to comprehend a quantum state of affairs and our ability to observe are tightly knit.
Again from Castelvecchi:
Since the 1990s, theoretical physicists have tried to embody Turing’s work in idealized models of physical phenomena. But “the undecidable questions that they spawned did not directly correspond to concrete problems that physicists are interested in”, says Markus Müller, a theoretical physicist at Western University in London, Canada, who published one such model with Gogolin and another collaborator in 2012.
The work described in the article concerns what’s called the spectral gap, which is the gap between the lowest energy level that electrons can occupy and the next one up. The presence or absence of the gap determines some of the material’s properties. The paper’s first author, Toby S. Cubitt, is a quantum information theorist and the paper is a direct application of Turing’s work. In other words, a Turing machine is constructed where the spectral gap depends on the outcome of a halting problem.
What I find striking is that the objectivity issues raised by the empiricists and positivists of the 1920s are not the same as the objectivity issues raised by quantum mechanics. But the notion of undecidability must be deep indeed. It persists and continues to be relevant.