Klaas Landsman

The aim of this paper is to argue that the (alleged) indeterminism of quantum mechanics, claimed by adherents of the Copenhagen interpretation since Born (1926), can be proved from Chaitin's follow-up to Goedel's (first) incompleteness theorem. In comparison, Bell's (1964) theorem as well as the so-called free will theorem-originally due to Heywood and Redhead (1983)-left two loopholes for deterministic hidden variable theories, namely giving up either locality (more precisely: local contextuality, as in Bohmian mechanics) or free choice (i.e. uncorrelated measurement settings, as in 't Hooft's cellular automaton interpretation of quantum mechanics). The main point is that Bell and others did not exploit the full empirical content of quantum mechanics, which consists of long series of outcomes of repeated measurements (idealized as infinite binary sequences): their arguments only used the long-run relative frequencies derived from such series, and hence merely asked hidden variable theories to reproduce single-case Born probabilities defined by certain entangled bipartite states. If we idealize binary outcome strings of a fair quantum coin flip as infinite sequences, quantum mechanics predicts that these typically (i.e. almost surely) have a property called 1-randomness in logic, which is much stronger than uncomputability. This is the key to my claim, which is admittedly based on a stronger (yet compelling) notion of determinism than what is common in the literature on hidden variable theories.

We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining these ideas is topos theory, a framework earlier proposed for physics by Isham and collaborators. Our "Principle of general tovariance" states that any mathematical structure appearing in the laws of physics must be definable in an arbitrary topos (with natural numbers object) and must be preserved under so-called geometric morphisms. This principle identifies geometric logic as the mathematical language of physics and restricts the constructions and theorems to those valid in intuitionism: neither Aristotle's principle of the excluded third nor Zermelo's Axiom of Choice may be invoked. Subsequently, our "Equivalence principle" states that any algebra of observables (initially defined in the topos Sets) is empirically equivalent to a commutative one in some other topos.

Bell’s Theorem from Physics 36:1–28 (1964) and the (Strong) Free Will Theorem of Conway and Kochen from Notices AMS 56:226–232 (2009) both exclude deterministic hidden variable theories (or, in modern parlance, ‘ontological models’) that are compatible with some small fragment of quantum mechanics, admit ‘free’ settings of the archetypal Alice and Bob experiment, and satisfy a locality condition akin to parameter independence. We clarify the relationship between these theorems by giving reformulations of both that exactly pinpoint their resemblance and their differences. Our reformulation imposes determinism in what we see as the only consistent way, in which the ‘ontological state’ initially determines both the settings and the outcome of the experiment. The usual status of the settings as ‘free’ parameters is subsequently recovered from independence assumptions on the pertinent (random) variables. Our reformulation also clarifies the role of the settings in Bell’s later generalization of his theorem to stochastic hidden variable theories

Our laws of nature and our cosmos appear to be delicately fine-tuned for life to emerge. First, if the initial conditions prevailing immediately after the Big Bang had been ever so slightly different, then the universe would either have recollapsed immediately, or would have expanded far too quickly into a chilling, eternal void. Second, if any one of the fundamental forces of nature had been a tiny bit different in strength, or if the masses of some elementary particles had been a little unlike they are, there would have been no stars and no recognizable chemistry, and hence no Sun, no Earth, no carbon, et cetera. These highly special conditions and numbers suggest some sort of a conspiracy of literally cosmic proportions, which seems rather hard to attribute to chance. Some theologians and religiously oriented scientists and philosophers have taken advantage of this apparent fine-tuning of the clockwork of the universe to revive the scholastic Argument from Design, but also secular thinkers have felt the need to explain fine-tuning, typically by proposing the existence of a `Multiverse', i.e., a vast family of universes existing in parallel, of which ours is merely one, singled out by the `Anthropic Principle' to the effect that it is the only one that can possibly contain observers and hence can be observed. We analyze this issue from a sober perspective. Having reviewed the literature and having added several observations of our own, we conclude that cosmic fine-tuning turns out to support neither Design nor a Multiverse, both of which fail to explain fine-tuning or even to increase its likelihood. In fact, fine-tuning and Design rather seem to be at odds with each other, except when one makes an additional assumption that is practically indistinguishable from Design and hence renders the whole argument circular. Likewise, the inference from fine-tuning to a Multiverse only works if the latter is underwritten by a startling metaphysical hypothesis we consider unwarranted. Instead, we conclude that fine-tuning requires no special explanation at all, since it is not the Universe that is fine-tuned for life, but life that has been fine-tuned to the Universe.

The Free Will Theorem of Conway \& Kochen on the one hand follows from uncontroversial parts of modern physics and elementary mathematical and logical reasoning, but on the other hand seems predicated on an undefined notion of free will. Although Conway and Kochen informally claim that their theorem supports indeterminism and, in its wake, a libertarian agenda for free will, inferring the former from the Free Will Theorem is a \emph{petitio principii}. Of course, this also considerably weakens the case for the latter. A recent reformulation of the Free Will Theorem due to Cator and the author uses a mathematically precise formulation of the idea that experimentalists freely choose their settings even in a deterministic world, and thence construes the theorem as providing constraints on determinism. In the present paper we argue that the concept of free will used in this version of the theorem is essentially the one proposed by Lewis, also known as `local miracle compatibilism'. To do so, we give a mathematical reformulation of this idea, which we believe to be well within its spirit and which may be of independent interest. On this double reformulation the Free Will Theorem shows that local miracle compatibilism contradicts the parts of modern physics used in its assumptions. Therefore, although the intention of Conway and Kochen was to support free will through their theorem, what they actually achieved is the opposite: a well-known, philosophically viable version of free will now turns out to be incompatible with physics!

This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low probability, their very existence implies that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements should be reinterpreted statistically, like the second law of thermodynamics. In three appendices I discuss the Born rule and its status in both single and repeated experiments, review the notion of 1-randomness that in various guises was investigated by Kolmogorov and others and treat Bell’s Theorem and the Free Will Theorem with their implications for randomness.

I argue that Bohmian mechanics cannot reasonably be claimed to be a deterministic theory. If one assumes the “quantum equilibrium distribution” provided by the wave function of the universe, Bohmian mechanics requires an external random oracle in order to describe the algorithmic randomness properties of typical outcome sequences of long runs of repeated identical experiments. This oracle lies beyond the scope of Bohmian mechanics, including the impossibility of explaining the randomness property in question from “random” initial conditions. Thus the advantages of Bohmian mechanics over other interpretations of quantum mechanics, if any, must lie at an ontological level, and in its potential to derive the quantum equilibrium distribution and hence the Born rule.

In the light of his recent (and fully deserved) Nobel Prize, this pedagogical paper draws attention to a fundamental tension that drove Penrose’s work on general relativity. His 1965 singularity theorem (for which he got the prize) does not in fact imply the existence of black holes (even if its assumptions are met). Similarly, his versatile definition of a singular space–time does not match the generally accepted definition of a black hole (derived from his concept of null infinity). To overcome this, Penrose launched his cosmic censorship conjecture(s), whose evolution we discuss. In particular, we review both his own (mature) formulation and its later, inequivalent reformulation in the pde literature. As a compromise, one might say that in “generic” or “physically reasonable” space–times, weak cosmic censorship postulates the appearance and stability of event horizons, whereas strong cosmic censorship asks for the instability and ensuing disappearance of Cauchy horizons. As an encore, an “Appendix” by Erik Curiel reviews the early history of the definition of a black hole.

This expository paper relates the Hole Argument in general relativity (GR) to the well-known theorem of Choquet-Bruhat and Geroch (1969) on the existence and uniqueness of globally hyperbolic solutions to the Einstein field equations. Like the Earman–Norton (1987) version of the Hole Argument (which is originally due to Einstein), this theorem exposes the tension between determinism and some version of spacetime substantivalism. But it seems less vulnerable to the campaign by Weatherall (2018) and followers to close the Hole Argument on the basis of “mathematical practice,” since the theorem only talks about isometries and hence does not make the pointwise identifications via diffeomorphisms that Weatherall objects to. Among other implications of the theorem for the philosophy of GR, we reconsider Butterfield’s (1987) influential definition of determinism. This should be amended if its goal is to express the idea that GR is deterministic in the absence of Cauchy horizons, although its original form does capture the way GR is indeterministic in their presence! Furthermore, in GR isometries come out as gauge symmetries, as do Poincaré transformations in special relativity. Finally, I discuss some implications of the theorem for the philosophy of science: Accepting the determinism horn still requires a choice between Frege-style abstractionism and Hilbert-style structuralism; and, within the latter, between structural realism and empiricist structuralism (which I favor).

This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory. This book is Open Access under a CC BY licence.

This book presents a multidisciplinary perspective on chance, with contributions from distinguished researchers in the areas of biology, cognitive neuroscience, economics, genetics, general history, law, linguistics, logic, mathematical physics, statistics, theology and philosophy. The individual chapters are bound together by a general introduction followed by an opening chapter that surveys 2500 years of linguistic, philosophical, and scientific reflections on chance, coincidence, fortune, randomness, luck and related concepts. A main conclusion that can be drawn is that, even after all this time, we still cannot be sure whether chance is a truly fundamental and irreducible phenomenon, in that certain events are simply uncaused and could have been otherwise, or whether it is always simply a reflection of our ignorance. Other challenges that emerge from this book include a better understanding of the contextuality and perspectival character of chance (including its scale-dependence), and the curious fact that, throughout history (including contemporary science), chance has been used both as an explanation and as a hallmark of the absence of explanation. As such, this book challenges the reader to think about chance in a new way and to come to grips with this endlessly fascinating phenomenon.

Wat is toeval? Bestaat het? Hoe moeten we toeval interpreteren? Is alles logisch te verklaren, of is er meer tussen hemel en aarde? In dit boek kijken we naar een vakantieganger die op Bali onverwacht zijn buurman tegenkomt, de Shakespeare-onderzoeker die per toeval de sleutel tot de ware identiteit van de bard in diens teksten denkt te ontwaren, de man die altijd gezond leefde maar plotseling een ongeneeslijke vorm van kanker krijgt, gelovigen die de hand van God denken te zien in het perfecte samenspel van de natuurconstanten dat leven mogelijk maakt, fysici die de grondslagen van de kwantummechanica of de richting van de tijd onderzoeken, biologen die Darwins evolutietheorie proberen te doorgronden, de filosofie van de vrije wil, en ten slotte Jezus als een toevallig of een noodzakelijk fenomeen. In zijn nieuwste boek geeft mathematisch fysicus Klaas Landsman zowel een inleiding als een visie op al deze onderwerpen, die iedereen met belangstelling voor de combinatie van wetenschap en levensbeschouwing zal raken en op nieuwe gedachten zal brengen.