In classical physics, probabilistic or statistical knowledge has been always related to ignorance or inaccurate subjective knowledge about an actual state of affairs. This idea has been extended to quantum mechanics through a completely incoherent interpretation of the FermiDirac and Bose-Einstein statistics in terms of “strange” quantum particles. This interpretation, naturalized through a widespread “way of speaking” in the physics community, contradicts Born’s physical account of Ψ as a “probability wave” which provides statistical information about outcomes that, in fact, cannot be interpreted in terms of ‘ignorance about an actual state of affairs’. In the present paper we discuss how the metaphysics of actuality has played an essential role in limiting the possibilities of understating things differently. We propose instead a metaphysical scheme in terms of powers with definite potentia which allows us to consider quantum probability in a new light, namely, as providing objective knowledge about a potential state of affairs.

Quantum mechanics arguably provides the best evidence we have for strong emergence. Entangled pairs of particles apparently have properties that fail to supervene on the properties of the particles taken individually. But at the same time, quantum mechanics is a terrible place to look for evidence of strong emergence: the interpretation of the theory is so contested that drawing any metaphysical conclusions from it is risky at best. I run through the standard argument for strong emergence based on entanglement, and show how it rests on shaky assumptions concerning the ontology of the quantum world.

During its history of more than 40 years, AIT knew a significant variation in terminology. In particular, the main measures of complexity studied in AIT were called Solomonoff-Kolmogorov-Chaitin complexity, Kolmogorov-Chaitin complexity, Kolmogorov complexity, Chaitin complexity, algorithmic complexity, program-size complexity, etc. Solovay’s handwritten notes [22]3 , introduced and used the terms Chaitin complexity and Chaitin machine.4 The book [21] promoted the name Kolmogorov complexity for both AIT and its main complexity.5 The main contribution shared by AIT founding fathers in the mid 1960s was the new type of complexity—which is invariant up to an additive constant—and, with it, a new way to reason about computation.

¿Qué son las matemáticas? Una sencilla respuesta: lo que hacen los matemáticos. (También, por supuesto, lo que hacemos los no matemáticos cuando tratamos de encontrar procedimientos para contar, medir, etc.) Ahora bien: ¿Qué hacen los matemáticos? Una primera manera de ver su actividad sería como sigue: los matemáticos descubren entidades matemáticas (por ejemplo, los números), y las propiedades de esas entidades, de igual manera que un científico natural encuentra especies de seres vivos, o genes que corresponden a características de seres humanos, o compuestos químicos, o estrellas lejanas. Los seres descubiertos, y sus propiedades, no dependen del matemático: el número 3 es primo e impar aunque uno no quiera. Hasta aquí todo bien, y muchos sostienen esta posición sin ver problemas en ella. Pero, ¿qué ocurriría si un matemático nos dijera que acaba de descubrir el último número que existe, más allá del cual no hay ninguno? ¿Le darían un premio por su esfuerzo? Otro ejemplo: ¿saldría un matemático a la prensa a declarar que el número 3, contra todas las expectativas, ahora se está comportando como un número par? ¿O que se ha descubierto que a temperaturas muy bajas 2+2=5? Obviamente no. A pesar de que en cierto modo hacer matemáticas es descubrir cosas cuyas propiedades no dependen de la voluntad de uno, parece evidente que términos como “descubrir”, “entidad”, “propiedad” y “existir” no tienen en las matemáticas el mismo significado que en biología o química. Cual sea la diferencia es una de las tareas inconclusas en filosofía de las matemáticas

We investigate the strength of a randomness notion R as a set-existence principle in second-order arithmetic: for each Z there is X that is R-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in RCA0. We verify that RCA0proves the basic implications among randomness notions: 2-random ⇒ weakly 2-random ⇒ Martin-Lof random ⇒ computably random ⇒ Schnorr random. Also, over RCA0 the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Lof randoms, and we describe a sense in which this result is nearly optimal.

I outline Bell’s vision of the “great enterprise” of science, and his view that conventional teachings about quantum mechanics constituted a betrayal of this enterprise. I describe a proposal of his to put the theory on a more satisfactory footing, and review the subsequent uses that have been made of one element of this proposal, namely Bell’s transition probabilities regarded as fundamental physical processes.

This is a brief introduction to the world of Noncommutative Algebra aimed for advanced undergraduate and beginning graduate students.

The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

This dissertation pursues two tightly interwoven goals: to bring out the relevance of questions for the field of logic, and to establish a solid theory of the logic of questions within a classical logical setting. These enterprises feed into each other: on the one hand, the development of our formal systems is motivated by our considerations concerning the role to be played by questions; on the other hand, it is via the development of concrete, workable logical systems that the potential of questions in logic is made clear and tangible.

In this paper I put forward an approach to the problem of describing a particle in a field without assuming the space-time continuum. I deduce as much as possible from very simple assumptions concerning interactions between the elements of a « bootstrap >) type assemblage in which each particle in the assemblage is built out of the interactions of all the others. Interaction either exists (a situation denoted by the digit (1) or else it does not exist (denoted by the digit (0) and there is no other possibility. No dynamical properties are assumed for the particles beyond the discrete, all-or-none interactions, and dynamics, therefore, including the momentum concept has to be built later. The theory being proposed differs vitally in this respect from the bootstrap theories that are based on the S-matrix technique.