The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in hisSubstanzbegriff und Funktionsbegriff (1910). The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice.

El objetivo del presente artículo es someter a escrutinio la afirmación de Hintikka según la cual la verdadera lógica elemental no es la clásica sino la lógica IF y, en consecuencia, el marco en que ordinariamente son pensadas las relaciones entre lógica y matemáticas es por completo inadecuado. Para ello, primero se exponen las funciones o características que una lógica debe poseer y, en segundo lugar, se presentan las ideas constitutivas de la lógica IF. Más adelante se demuestran algunas de las propiedades matemáticas de la lógica IF y se analizan las complejidades a que da lugar la negación en este sistema. Por último, se ofrecen algunas razones para matizar o poner en duda las conclusiones que Hintikka extrae de su propuesta para la filosofía de las matemáticas.

In this paper we discuss the relation of quantum theory to the problem of metaphysics. Based on metaphysical and anti-metaphysical stances, we put forward an ‘interpretational map’ of quantum mechanics in general and of the modal interpretation in particular. Thus, within the modal interpretation, we distinguish between: Modal Interpretations (which start from) the Mathematical Formalism (MIMF) and Modal Interpretations (which start from) Metaphysical Principles (MIMP). Finally, we argue for a middle path in between metaphysical principles and the formal conditions imposed on quantum mechanics

April 25, 2003, marked the 100th anniversary of the birth of Andrei Nikolaevich Kolmogorov, the twentieth century’s foremost contributor to the mathematical and philosophical foundations of probability. The year 2003 was also the 70th anniversary of the publication of Kolmogorov’s Grundbegriffe der Wahrscheinlichkeitsrechnung. Kolmogorov’s Grundbegriffe put probability’s modern mathematical formalism in place. It also provided a philosophy of probability – an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe – the work of the earlier scholars whose ideas Kolmogorov synthesized.

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.