Gödel’s Proof and Einstein’s Dice: Undecidability in Mathematics and Physics – Part III

by Jochen Szangolies

There are countless virtual realities, albeit as of yet, not exactly a replacement for the real thing. Image credit: wikimedia commons.

The simulation argument, most notably associated with the philosopher Nick Bostrom, asserts that given reasonable premises, the world we see around us is very likely not, in fact, the real world, but a simulation run on unfathomably powerful supercomputers. In a nutshell, the argument is that if humanity lives long enough to acquire the powers to perform such simulations, and if there is any interest in doing so at all—both reasonably plausible, given the fact that we’re in effect doing such simulations on the small scale millions of times per day—then the simulated realities greatly outnumber the ‘real’ realities (of which there is only one, barring multiversal shenanigans), and hence, every sentient being should expect their word to be simulated rather than real with overwhelming likelihood.

On the face of it, this idea seems like so many skeptical hypotheses, from Cartesian demons to brains in vats. But these claims occupy a kind of epistemic no man’s land: there may be no way to disprove them, but there is also no particular reason to believe them. One can thus quite rationally remain indifferent regarding them.

But Bostrom’s claim has teeth: if the reasoning is sound, then in fact, we do have compelling reasons to believe it to be true; hence, we ought to either accept it, or find flaw with it. Luckily, I believe that there is indeed good reason to reject the argument. Read more »

Gödel’s Proof and Einstein’s Dice: Undecidability in Mathematics and Physics – Part II

by Jochen Szangolies

Commemorative plaque at Gödel’s former house in Vienna. Image credit: By Beckerhermann – Own work, CC BY-SA 4.0, via wikimedia commons

The previous column left us with the tantalizing possibility of connecting Gödelian undecidability to quantum mechanical indeterminacy. At this point, however, we need to step back a little.

Gödel’s result inhabits the rarefied realm of mathematical logic, with its crisply stated axioms and crystalline, immutable truths. It is not at all clear whether it should have any counterpart in the world of physics, where ultimately, experiment trumps pure reason.

However, there is a broad correspondence between physical and mathematical systems: in each case, we start with some information—the axioms or the initial state—apply a certain transformation—drawing inferences or evolving the system in time—and end up with new information—the theorem to be proved, or the system’s final state. An analogy to undecidability then would be an endpoint that can’t be reached—a theorem that can’t be proven, or a cat whose fate remains uncertain.

Perhaps this way of putting it looks familiar: there is another class of systems that obeys this general structure, and which were indeed the first point of contact of undecidability with the real world—namely, computers. A computer takes initial data (an input), performs a transformation (executes a program), and produces a result (the output). Moreover, computers are physical devices: concrete machines carrying out computations. And as it turns out, there exists questions about these devices that are undecidable. Read more »

Mind The Matter: Consciousness As Self-Representational Access

by Jochen Szangolies

Figure 1: Von Neumann’s replicator-design in its original cellular automaton guise. The tape stretches to the right, and the second-generation replicator is finishing up construction of the third.

There are two main problems that bedevil any purported theory of the mind. The first is the Problem of Intentionality: the question of how mental states can come to be about, or refer to, things in the world. The second is the Problem of Phenomenal Experience: the question of how come there is ‘something it is like’ to be in a certain mental state, how mental content is something that appears to us in a certain way (this is also often referred to as simply the ‘Hard Problem’).

These problems are often assumed to be separate issues. However, in a recent article published in the journal Erkenntnis (pre-print version), I propose that one can make progress on the Problem of Intentionality, but at the expense of leaving the Hard Problem unsolvable—indeed, making the task of ‘solving’ it a kind of conceptual confusion: an attempt of capturing the non-structural, non-relational in terms of structure and relation.

In a nutshell, I propose that states of mind are intentional because, through what I call the von Neumann-process, their own properties are represented to themselves; to the extent that these properties then reflect those of objects in the world, the properties of those objects are available to them. Hence, a mental state becomes ‘about’ the world by being, first and foremost, about itself. Read more »

Kurt Gödel’s Open World

by Ashutosh Jogalekar

Gödel and Einstein in Princeton (Source: Emilio Segre Visual Archives)

Two men walking in Princeton, New Jersey on a stuffy day. One shaggy-looking with unkempt hair, avuncular, wearing a hat and suspenders, looking like an old farmer. The other an elfin man, trim, owl-like, also wearing a fedora and a slim white suit, looking like a banker. The elfin man and the shaggy man used to make their way home from work every day. Passersby and motorists would strain their heads to look. Everyone knew who the shaggy man was; almost nobody knew who his elfin companion was. And yet when asked, the shaggy man would say that his own work no longer meant much to him, and the only reason he came to work was to have the privilege of walking home with the elfin man. The shaggy man was Albert Einstein. His walking companion was Kurt Gödel.

What made Gödel, a figure unknown to the public, so revered among his colleagues? The superlatives kept coming. Einstein called him the greatest logician since Aristotle. The legendary mathematician John von Neumann who was his colleague argued for his extraction from fascism-riddled Europe, writing a letter to the director of his institute saying that “Gödel is absolutely irreplaceable; he is the only mathematician about whom I dare make this assertion.” And when I made a pilgrimage to Gödel’s house during a trip to his native Vienna a few years ago, the plaque in front of the house made his claim to posterity clear: “In this house lived from 1930-1937, the great mathematician and logician Kurt Gödel. Here he discovered his famous incompleteness theorem, the most significant mathematical discovery of the twentieth century.”

The author in front of the house in Vienna where Gödel was living with his mother and brother when he proved his Incompleteness Theorems

The reason Gödel drew gasps of awe from colleagues as brilliant as Einstein and von Neumann was because he revealed a seismic fissure in the foundations of that most perfect, rational and crystal-clear of all creations – mathematics. Of all the fields of human inquiry, mathematics is considered the most exact. Unlike politics or economics, or even the more quantifiable disciplines of chemistry and physics, every question in mathematics has a definite yes or no answer. The answer to a question such as whether there is an infinitude of prime numbers leaves absolutely no room for ambiguity or error – it’s a simple yes or no (yes in this case). Not surprisingly, mathematicians around the beginning of the 20th century started thinking that every mathematical question that can be posed should have a definite yes or no answer. In addition, no mathematical question should have both answers. The first requirement was called completeness, the second one was called consistency. Read more »